John Christian William McKinley LaTeX Archive

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\lhead{Cosmogenesis Requires an Initiating Increment}
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\title{\textbf{Cosmogenesis Requires an Initiating Increment}}
\author{John C. W. McKinley\,\orcidlink{0009-0005-7097-5035}}
\date{May 27, 2026}

\begin{document}
\maketitle

\blfootnote{\scriptsize This version published at \url{https://doi.org/10.5281/zenodo.20405016}.}

\begin{abstract}
Two prior results are taken as proven \cite{mckinley_9A, mckinley_1AC}: that no closed physical description internally fixes the onset of its own new causal chain; and that if a new causal chain begins, then its beginning requires an actual added increment of energy, denoted $\Delta E_{\mathrm{init}}$. The present paper applies those results to the limit case. If the closed physical description of the universe has a beginning, then its beginning required an actual added initiating increment. The argument is structural, not cosmological. It does not depend on any particular cosmological model and does not identify the ontology of the contributor. It states only that wherever the closed physical description has a beginning, the beginning required $\Delta E_{\mathrm{init}}$.
\end{abstract}

\section{Introduction}

Two prior results are taken as proven here. First, no closed physical description internally fixes the onset of its own new causal chain \cite{mckinley_9A}. Second, if a new causal chain begins, then its beginning requires an actual added increment of energy $\Delta E_{\mathrm{init}}$ \cite{mckinley_1AC}.

The present paper does not re-argue either claim. It applies them to a single case: the onset of the universe.

The claim defended here is narrow. If the closed physical description of the universe has a beginning---under any cosmological model that admits one---then by the prior results, that beginning required an actual added initiating increment. The argument does not require a particular interpretation of cosmogenesis. It applies to any onset at which the closed physical description does not yet contain anything earlier.

\section{Definitions}

\begin{definition}[Closed physical description]
A \emph{closed physical description} is a description containing only the laws, state-terms, and causal resources internal to the causal chain under discussion.
\end{definition}

\begin{definition}[Onset of the universe]
The \emph{onset of the universe} is the definite point at which the closed physical description of the universe begins, rather than not yet beginning. The present paper takes no position on which cosmological model identifies that onset; the claim applies to any model in which one exists.
\end{definition}

\begin{definition}[Initiating increment]
The \emph{initiating increment}, denoted $\Delta E_{\mathrm{init}}$, is the minimal actual added increment of energy required for the beginning of a new causal chain.
\end{definition}

\section{The Argument}

\begin{proposition}[The onset of the universe is not internally fixed]
\label{prop:cosmo-not-fixed}
The closed physical description of the universe does not internally fix the onset of its own beginning.
\end{proposition}

\begin{proof}
By prior result \cite{mckinley_9A}, no closed physical description internally fixes the onset of its own new causal chain. The closed physical description of the universe is a closed physical description. Its beginning is the onset of a new causal chain. Therefore the closed physical description of the universe does not internally fix the onset of its own beginning.
\end{proof}

\begin{proposition}[Cosmogenesis requires an initiating increment]
\label{prop:cosmo-increment}
If the universe has an onset, then that onset required an actual added initiating increment $\Delta E_{\mathrm{init}}$.
\end{proposition}

\begin{proof}
By \cref{prop:cosmo-not-fixed}, the closed physical description of the universe does not internally fix the onset of its own beginning. By prior result \cite{mckinley_1AC}, if a new causal chain begins, then its beginning requires an actual added increment of energy $\Delta E_{\mathrm{init}}$ not contained in the closed physical description of the chain. The onset of the universe is the beginning of such a chain. Therefore the onset of the universe required an actual added initiating increment $\Delta E_{\mathrm{init}}$.
\end{proof}

\section{Scope and Clarifications}

\begin{remark}[Model-independence]
The argument does not depend on any particular cosmological model. It applies to any model in which the closed physical description of the universe has an onset, including standard Big Bang cosmology, inflationary cosmology with a beginning, and any other model in which the closed physical description does not extend indefinitely backward. The argument does not apply to models in which the closed physical description has no onset at all.
\end{remark}

\begin{remark}[No ontology of the contributor is identified]
The present paper does not identify the ontology of whatever supplies $\Delta E_{\mathrm{init}}$ at cosmogenesis. It does not characterize the contributor, its persistence, its further properties, or any interpretation of its nature. Those are downstream questions. The result here is structural: wherever the closed physical description has a beginning, the beginning required an actual added initiating increment.
\end{remark}

\begin{remark}[Boundary of applicability]
The claim here is not that ordinary physics fails within already-running cosmological evolution. The claim is narrower. The closed physical description of the universe does not account for its own onset, and if that onset occurred, an actual added increment was required.
\end{remark}

\section{Falsifier}

The claim of this paper fails only if the universe has an onset and that onset occurred without any actual added increment of energy.

\section{Conclusion}

The prior results stand. Closed physical descriptions do not internally fix the onset of new causal chains; and if such a chain begins, then its beginning requires an actual added increment of energy $\Delta E_{\mathrm{init}}$. The present paper applies those results to the limit case. The closed physical description of the universe does not internally fix the onset of its own beginning. If the universe has an onset, then that beginning required an actual added initiating increment $\Delta E_{\mathrm{init}}$.

\begin{thebibliography}{9}

\bibitem{mckinley_9A}
J. C. W. McKinley, \emph{No Closed Physical System Internally Fixes the Onset and Direction of a New Causal Chain}, Zenodo, \href{https://doi.org/10.5281/zenodo.19464780}{10.5281/zenodo.19464780} (2026).

\bibitem{mckinley_1AC}
J. C. W. McKinley, \emph{The Initiating Increment: Why the Beginning of a New Causal Chain Requires an Actual Added Increment of Energy}, Zenodo, \href{https://doi.org/10.5281/zenodo.20369221}{10.5281/zenodo.20369221} (2026).

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\title{\textbf{The Initiating Increment: Why the Beginning of a New Causal Chain Requires an Actual Added Increment of Energy}}
\author{John C. W. McKinley\,\orcidlink{0009-0005-7097-5035}}
\date{May 25, 2026}

\begin{document}
\maketitle

\blfootnote{\scriptsize This version published at \url{https://doi.org/10.5281/zenodo.20369221}.}

\begin{abstract}
Two prior results are taken as proven \cite{mckinley_19A, mckinley_1AB}: that if a new causal chain begins, then a non-internal contribution to its onset exists; and that no inert candidate can satisfy the role of that contribution, which must be efficacious toward the onset. The present paper states: A contribution that is efficacious toward the onset and is not contained in the closed physical description of the chain is, relative to that description, an actual added increment of energy. The minimal such increment is denoted $\Delta E_{\mathrm{init}}$.
\end{abstract}

\section{Introduction}
Two prior results are taken as proven here. First, if a new causal chain begins, then a non-internal contribution to its onset exists \cite{mckinley_19A}. Second, no inert candidate can satisfy the role of the required non-internal contribution; the contribution must be efficacious toward the onset \cite{mckinley_1AB}.

Those results leave one further step. The present paper draws the consequence: relative to the closed physical description, such a contribution enters as an actual added increment of energy, denoted $\Delta E_{\mathrm{init}}$.

\section{Definitions}

\begin{definition}[Closed physical description]
A \emph{closed physical description} is a description containing only the laws, state-terms, and causal resources internal to the causal chain under discussion.
\end{definition}

\begin{definition}[New causal chain]
A \emph{new causal chain} is a chain whose onset is the issue under discussion.
\end{definition}

\begin{definition}[Efficacious toward the onset]
A contribution is \emph{efficacious toward the onset} if it is among what brings the onset about: among what makes the transition from not-yet-beginning to beginning occur. A contribution that is not among what brings the onset about is \emph{inert toward the onset}.
\end{definition}

\begin{definition}[Actual added increment of energy]
A contribution is an \emph{actual added increment of energy} relative to a closed physical description if it is an increment of energy that is efficacious toward the onset and is not contained within that description.
\end{definition}

\begin{definition}[Initiating increment]
The \emph{initiating increment}, denoted $\Delta E_{\mathrm{init}}$, is the minimal actual added increment of energy required for the beginning of a new causal chain.
\end{definition}

\section{The Required Contribution}

By the prior results, if a new causal chain begins, a non-internal contribution exists \cite{mckinley_19A} and is efficacious toward the onset \cite{mckinley_1AB}. The present paper draws the immediate consequence.

\begin{axiom}[Beginning is a threshold-crossing requiring energy]
\label{axiom:threshold-energy}
The beginning of a new causal chain is a physical threshold-crossing, and a physical threshold-crossing is brought about only by an increment of energy.
\end{axiom}

\begin{proposition}[Initiation requires an actual added increment of energy]
\label{prop:added-increment}
If a new causal chain begins, then its beginning requires an actual added increment of energy.
\end{proposition}

\begin{proof}
By \cite{mckinley_19A, mckinley_1AB}, if a new causal chain begins, a non-internal contribution exists and is efficacious toward the onset. By \cref{axiom:threshold-energy}, that beginning is a physical threshold-crossing brought about only by an increment of energy; the efficacious contribution required for the onset is therefore an increment of energy. Not being contained in the closed physical description of the chain, it is, relative to that description, an actual added increment of energy. 
\end{proof}

\section{The Initiating Increment}

The notation $\Delta E_{\mathrm{init}}$ marks the claim that the required contribution is an actual, added increment of energy, rather than merely formal, descriptive, or contained within the closed physical description. The argument does not require that $\Delta E_{\mathrm{init}}$ be large. It requires only that the beginning of a new causal chain introduces a real added increment of energy.

The claim is relative to the closed physical description: the required contribution is added with respect to that description.

\section{Falsifier}

The claim of this paper fails only if a new causal chain begins without any actual added increment of energy.

\section{Conclusion}


The prior results stand. If a new causal chain begins, then a non-internal contribution is required for its beginning, and that contribution must be efficacious toward the onset.

This paper adds: A contribution that is efficacious toward the onset and is not contained in the closed physical description of the chain is, relative to that description, an actual added increment of energy. That minimal increment is denoted $\Delta E_{\mathrm{init}}$.

\begin{thebibliography}{9}

\bibitem{mckinley_19A}
J. C. W. McKinley, \emph{If a New Causal Chain Begins, a Non-Internal Contribution Exists}, Zenodo, \href{https://doi.org/10.5281/zenodo.19752797}{10.5281/zenodo.19752797} (2026).

\bibitem{mckinley_1AB}
J. C. W. McKinley, \emph{An Inert Contribution Does Not Begin a Causal Chain: A Structural No-Go Result}, Zenodo, \href{https://doi.org/10.5281/zenodo.20351786}{10.5281/zenodo.20351786} (2026).

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\title{\textbf{An Inert Contribution Does Not Begin a Causal Chain:\\
A Structural No-Go Result}}
\author{John C. W. McKinley\,\orcidlink{0009-0005-7097-5035}}
\date{May 23, 2026}

\begin{document}
\maketitle

\blfootnote{\scriptsize This version published at \url{https://doi.org/10.5281/zenodo.20351786}.}

\begin{abstract}
Two prior results are taken as proven \cite{mckinley_9A, mckinley_19A}: no closed physical system internally fixes the onset and direction of its own new causal chain among multiple lawful possibilities, and if such a chain nonetheless begins, then a non-internal contribution exists. The present paper states a structural no-go on candidates for that contribution. No inert candidate can satisfy the role of the non-internal contribution required for the beginning of a new causal chain. A candidate that is merely present, or otherwise established to exist, but is not among what brings the onset about, is not the contribution required by the prior result. The prior result establishes that a non-internal contribution exists; the present result excludes inert candidates from filling that role.
\end{abstract}

\section{Introduction}

Two prior results are taken as proven here. First, no closed physical system internally fixes the onset and direction of its own new causal chain among multiple lawful possibilities \cite{mckinley_9A}. Second, if such a chain nonetheless begins, then a non-internal contribution exists \cite{mckinley_19A}.

Those results establish that a contribution exists but do not settle the character of that contribution. The present paper states one constraint, as a denial: no inert candidate can satisfy the role of the non-internal contribution required for the beginning of the chain. The argument rests on two cited prior results and the definitions below; it is structural and interpretive, and alters no equation of standard physics.

\section{Definitions}

\begin{definition}[Closed physical description]
A \emph{closed physical description} is a description containing only the laws, state-terms, and causal resources internal to the causal chain under discussion.
\end{definition}

\begin{definition}[New causal chain]
A \emph{new causal chain} is a chain whose onset is the issue under discussion, rather than a process already underway and simply continuing under known laws.
\end{definition}

\begin{definition}[Non-internal contribution]
A \emph{non-internal contribution} is a contribution that is not contained within the closed physical description. This is the \emph{contribution in question}: the contribution whose existence at the onset of a new causal chain is established by the prior result \cite{mckinley_19A}.
\end{definition}

\begin{definition}[Efficacious toward the onset]
A contribution is \emph{efficacious toward the onset} if it is among what brings the onset about: among what makes the transition from not-yet-beginning to beginning occur.
\end{definition}

\begin{definition}[Inert toward the onset]
A contribution is \emph{inert toward the onset} if it is present, or is otherwise established to exist, but is not among what brings the onset about.
\end{definition}

\section{The No-Go}

\begin{proposition}[No-go on inert candidates]
\label{prop:nogo-inert-candidates}
No inert candidate can satisfy the role of the non-internal contribution required for the beginning of a new causal chain.
\end{proposition}

\begin{proof}
By \cite{mckinley_19A}, if a new causal chain begins where the closed physical description does not internally fix its onset and direction, then a non-internal contribution exists. The role of that contribution is to account for the chain's beginning. A candidate contribution that is inert toward the onset is not among what brings the onset about. It therefore cannot satisfy the role of the contribution required for the chain's beginning. Hence no inert candidate can satisfy the role of the required non-internal contribution.
\end{proof}

\section{Falsifier}

The claim fails if a new causal chain begins where the non-internal contribution required to account for its beginning is established to be inert toward the onset.

\section{Conclusion}

No inert candidate can satisfy the role of the non-internal contribution required for the beginning of a new causal chain. The prior results establish that such a contribution exists. The present result excludes a merely present or inert candidate from filling that role. The required contribution must be efficacious toward the onset.

\begin{thebibliography}{9}

\bibitem{mckinley_9A}
J. C. W. McKinley, \emph{No Closed Physical System Internally Fixes the Onset and Direction of a New Causal Chain}, Zenodo, \href{https://doi.org/10.5281/zenodo.19464780}{10.5281/zenodo.19464780} (2026).

\bibitem{mckinley_19A}
J. C. W. McKinley, \emph{If a New Causal Chain Begins, a Non-Internal Contribution Exists}, Zenodo, \href{https://doi.org/10.5281/zenodo.19752798}{10.5281/zenodo.19752798} (2026).

\end{thebibliography}

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\lhead{General Timelessness and the Atom}
\rhead{John C. W. McKinley}
\cfoot{\thepage}

\title{\textbf{General Timelessness and the Atom}\\
\large Stable and Radioactive Bound Configurations as Two Presentations of Timeless Lawful Structure}
\author{John C. W. McKinley\,\orcidlink{0009-0005-7097-5035}}
\date{May 20, 2026}

\begin{document}

\maketitle

\blfootnote{\scriptsize This version published at \href{https://doi.org/10.5281/zenodo.20279574}{10.5281/zenodo.20279574}.}

\begin{abstract}
General timelessness is the architectural condition under which lawful structure is timeless and spacetime registration is the special case in which the structure is invoked~\cite{Inversion}. The matter-side instance of general timelessness was given a formal grounding by the time-independence of stationary-state structural content in standard quantum mechanics~\cite{Atom}: a bound material system is the lawful availability of a stationary structure, in force at the region of spacetime supporting the bound state, registered when called on. The present paper supplies the empirical companion to that formal account. Stable bound configurations exhibit indefinite non-decay; they accumulate no effects from exposure to time across cosmological intervals. Radioactive bound configurations exhibit lawful-rate channel realization; their decay events register along the worldline with rate fixed by structural content, independent of accumulated proper time. The two cases are presented here as a joint empirical pattern supporting the matter-side reading, not as central case plus counterexample. They differ in whether the lawful content includes a decay channel; they do not differ in ontological category. In both cases the system is timeless lawful structure, registered when called on. Stable configurations have no admissible spontaneous decay channel under the operative description; nothing registers; nothing accumulates. Radioactive configurations have at least one decay channel in their content; channel realizations register along the worldline at rate $\lambda$ fixed by content. Both are presentations of the same structural reading. Neither is a substance aging through time. The claim is interpretive only; the result alters no equations of atomic physics, nuclear physics, or quantum mechanics, and modifies no predictive content.
\end{abstract}

\section{Introduction}

General timelessness has been established in the corpus as the architectural condition under which lawful structure is timeless and spacetime registration is the special case~\cite{Inversion}. The Atom paper~\cite{Atom} supplied the matter-side instance on formal terrain: a bound material system is the lawful availability of a stationary structure, in force at the region of spacetime supporting the bound state, registered when called on. Persistence is the non-lapse of the lawful structure across timelike ordering. Between detector invocations no internal process occurs; no traveler-substance, no granule, and no component process is required to maintain the state.

The Atom paper grounded the matter-side reading on formal time-independence of stationary-state structural content. The present paper supplies the empirical companion. Two empirical observations from standard atomic and nuclear physics jointly support the matter-side timelessness reading: (i) stable bound configurations exhibit indefinite non-decay, accumulating no effects from exposure to time across cosmological intervals; (ii) radioactive bound configurations exhibit lawful-rate channel realization, with decay events registering at rate $\lambda$ fixed by structural content and independent of accumulated proper time.

The two observations are presented here as a joint empirical pattern supporting the matter-side reading, not as central case plus counterexample. Earlier presentations of the matter-side reading treated radioactivity defensively: stable atoms were the central case for timelessness, and radioactive decay was the apparent counterexample that the reading had to neutralize. The present paper drops the defensive framing. Stable and radioactive nuclei are both timeless lawful structure in force; they differ in whether the lawful content includes a decay channel; they do not differ in ontological category. Stability and radioactivity are content-level distinctions, not ontological categories. The reading covers both equally; one is not protected by treating the other as the opposing case.

The unified presentation matters because the defensive framing concedes too much. Treating radioactivity as ``the strongest apparent objection to timelessness'' allows the substance-aging picture to set the agenda: it suggests that radioactivity is where substance-aging is most naturally located, and the timelessness reading must work hard to defuse the obvious case. Once the unified frame is adopted, that asymmetry disappears. Radioactivity is not where substance-aging is most tempting; it is where lawful-content-with-a-channel is most clearly visible. The memoryless property of exponential decay is the empirical signature of this: the lawful structure of an unrealized radioactive nucleus is the same at every event on its worldline prior to channel realization; the channel realizes with rate fixed by content; no accumulated age modifies the rate. Memoryless decay is timeless lawful structure with a decay channel. Indefinite non-decay is timeless lawful structure without one. Two presentations, one structural reading.

The argument is conservative and standard-physics-only. It introduces no new equations, predicts no new outcomes, and modifies no predictive content of atomic or nuclear physics. The contribution is interpretive: stable and radioactive bound matter jointly exhibit the matter-side instance of general timelessness on standard-physics territory.

\section{Background and Prior Results}

The registration-discipline assumed here is fixed by the Bedrock statement of the Timeless Light Model~\cite{Bedrock}. The present paper presupposes the following published results and does not re-argue them.

\begin{itemize}
\item Timelessness is the general condition of lawful structure; spacetime registration is the special case in which the structure is invoked~\cite{Inversion}.
\item The bound material system is the lawful availability of a stationary structure, in force at the region of spacetime supporting the bound state, registered when called on; persistence is the non-lapse of the lawful structure across timelike ordering, with no traveler-substance, no granule, and no component process between invocations~\cite{Atom}.
\item Decay outcomes for unstable systems are worldline-internal: whether an unstable particle has decayed at a given event is determined by the proper time accrued along its worldline relative to the proper time required for decay~\cite{Accrual}.
\end{itemize}

The Atom paper established the matter-side instance of general timelessness on formal terrain: the time-independence of stationary-state structural content in standard quantum mechanics licenses the reading of bound matter as the registered availability of a timeless lawful structure. The present paper supplies an empirical companion: indefinite non-decay of stable configurations and memoryless-rate channel realization of radioactive configurations are joint empirical signatures of the same structural reading.

\section{Definitions}

\begin{definition}[Bound configuration]
\label{def:bound-config}
A bound atomic or nuclear configuration is a system describable by a stationary state of a time-independent Hamiltonian under the applicable atomic, electroweak, and strong-interaction selection rules.
\end{definition}

\begin{definition}[Lawful content]
\label{def:content}
The lawful content of a bound configuration is the structural specification of the configuration under the specified physical conditions: the time-independent Hamiltonian, the applicable selection rules, the admissible transitions, the conservation principles, and the decay channels (if any). The lawful content is a structural feature of the configuration, not a dynamical variable; it does not evolve in proper time.
\end{definition}

\begin{definition}[Decay channel]
\label{def:decay-channel}
A decay channel is an admissible spontaneous state-change pathway present in a bound configuration's lawful content, by which the configuration transitions to a configuration of lower total energy under the applicable selection rules. The channel's existence is structural: it is a feature of the lawful content, not an internal process. The channel's lawful rate $\lambda$ is fixed by the structural content.
\end{definition}

\begin{definition}[Stable bound configuration]
\label{def:stable}
A bound configuration is stable if its lawful content contains no admissible spontaneous decay channel: no lawful state-change pathway to a lower-energy configuration under the applicable selection rules.
\end{definition}

\begin{definition}[Radioactive bound configuration]
\label{def:radioactive}
A bound configuration is radioactive if its lawful content contains at least one admissible spontaneous decay channel.
\end{definition}

\begin{definition}[Channel realization]
\label{def:realization}
A channel realization is a spacetime-side registration event in which a decay channel present in a configuration's lawful content statistically transitions the configuration to a lower-energy state. The realization's statistics are governed by the channel's lawful rate $\lambda$: the probability of realization in proper-time interval $d\tau$ is $\lambda\, d\tau$, independent of accumulated proper time along the worldline.
\end{definition}

\begin{definition}[Indefinite non-decay]
\label{def:non-decay}
A stable bound configuration exhibits indefinite non-decay if, absent external disruption, it persists across open-ended timelike intervals without registering any state change traceable to exposure to time.
\end{definition}

\begin{definition}[Substance-aging]
\label{def:substance-aging}
Substance-aging is the cumulative modification of a system's properties as a function of the proper time over which it has existed, attributable to the system being a primitive material persisting through time and accumulating exposure-effects from that persistence. Substance-aging is distinct from channel realization (\cref{def:realization}), which is the statistical registration of an admissible lawful transition rather than the cumulative wear of an enduring substance.
\end{definition}

\section{Joint Empirical Claim}

\begin{proposition}[Stable presentation: indefinite non-decay]
\label{prop:stable-empirical}
Stable bound configurations exhibit indefinite non-decay: across cosmological timescales, they register no state change traceable to exposure to time.
\end{proposition}

\begin{proof}
The hydrogen ground state, the closed-shell electron configurations of noble gases, the ground-state electron configurations of stable atoms, and paradigmatic stable nuclear ground states such as \(^1\)H, \(^2\)H, \(^3\)He, \(^4\)He, \(^{12}\)C, \(^{16}\)O, and \(^{56}\)Fe are described in standard atomic and nuclear physics as stable bound configurations governed by time-independent Hamiltonians under their applicable selection rules. The mathematical content of this description is that the time-evolution operator $e^{-i\hat{H}t/\hbar}$ acts on an energy eigenstate $\psi$ satisfying $\hat{H}\psi = E\psi$ as the global phase $e^{-iEt/\hbar}$, so that the probability density $|\psi(t)|^2 = |\psi(0)|^2$ is independent of $t$ across all $t$. No parameter, no expectation value, and no structural property of the stationary state evolves with $t$ within this description. By \cref{def:stable}, none has an admissible spontaneous decay channel. Empirical bounds on proton stability place a lower limit on its mean lifetime in excess of \(10^{34}\) years~\cite{ParticleDataGroup}; this is many orders of magnitude longer than the age of the universe and is consistent with the proton being absolutely stable under standard-model selection rules. Cosmological matter abundances confirm that stable nuclide configurations persist across the age of the universe without diminution traceable to internal degradation. A hydrogen atom formed billions of years ago and a hydrogen atom newly formed by recombination have no intrinsic age-marker distinguishing them: if both are in the same state, they are the same lawful configuration, with identical structural content and identical predictions. Therefore stable bound configurations exhibit indefinite non-decay in the sense of \cref{def:non-decay}.
\end{proof}

\begin{proposition}[Radioactive presentation: memoryless channel realization]
\label{prop:radioactive-empirical}
Radioactive bound configurations exhibit lawful-rate channel realization with rates fixed by structural content and independent of accumulated proper time.
\end{proposition}

\begin{proof}
A radioactive bound configuration possesses, by \cref{def:radioactive}, at least one admissible spontaneous decay channel in its lawful content. By \cref{def:decay-channel}, the channel's lawful rate $\lambda$ is fixed by the structural content. By \cref{def:realization}, the probability of channel realization in proper-time interval $d\tau$ is $\lambda\, d\tau$. The expected surviving fraction of an initial population by proper time $\tau$ is $e^{-\lambda\tau}$, and the expected fraction that has realized the channel is $1-e^{-\lambda\tau}$. This is the standard exponential-decay law of radioactivity~\cite{Krane}.

In the experimentally dominant exponential regime, the memoryless property is empirical: a radioactive nucleus that has not yet realized its decay channel has decay probability per unit proper time equal to $\lambda$, the same for a nucleus that has existed for one second and for one that has existed for billions of years. The decay rate does not depend on the nucleus's accumulated proper time along its worldline. This property is the hallmark of structural content fixing the rate, rather than a clock running down inside the nucleus.

Standard quantum mechanics predicts small deviations from pure exponential decay at extreme regimes: quadratic survival probability at very short times (the regime in which the Quantum Zeno effect operates) and power-law tails at very long times when the energy spectrum is bounded from below~\cite{Khalfin1957,Fonda1978}. These deviations are themselves features of the lawful content: they are derived from the spectral properties of the time-independent Hamiltonian and its coupling to the continuum. They are fixed by structural content and reproducible across freshly-prepared identical nuclides regardless of preparation epoch. They do not introduce substance-aging into the realization statistics.

The canonical example of $\lambda$ being fixed by structural content is the Gamow derivation of alpha decay~\cite{Gamow1928}: the alpha-decay constant emerges as the product of an attempt frequency inside the nuclear potential well and the transmission coefficient through the Coulomb barrier, both determined by the static structural configuration of the nucleus. Beta and gamma decays are determined by different mathematical apparatus (weak-interaction matrix elements and electromagnetic transition amplitudes, respectively), but in each case the decay rate is a function of static structural features of the nucleus rather than of accumulated proper time. The reading offered here is the structural reading of this standard mathematical content: the rate is content; the channel is content; the realization is registration.

Therefore radioactive bound configurations exhibit lawful-rate channel realization with rates fixed by structural content and independent of accumulated proper time.
\end{proof}

\begin{corollary}[Joint empirical content]
\label{cor:joint-empirical}
Stable and radioactive bound configurations jointly exhibit: lawful content fixed independent of accumulated proper time; registration of state changes (or absence thereof) governed by content; no cumulative modification of structural content traceable to exposure to time.
\end{corollary}

\begin{proof}
By \cref{prop:stable-empirical}, stable configurations have no admissible channel in their content and register no state changes across cosmological timescales. By \cref{prop:radioactive-empirical}, radioactive configurations have at least one channel in their content and register channel realizations at rate $\lambda$ fixed by content. In neither case does the lawful content evolve with accumulated proper time. In neither case is the rate of realization (or non-realization) a function of accumulated age. The joint empirical pattern is: lawful content fixes registration statistics; lawful content is itself fixed; nothing in the bound configuration's structural content tracks exposure to time.
\end{proof}

\section{Joint Structural Reading}

\begin{proposition}[Joint structural reading]
\label{prop:joint-reading}
Stable and radioactive bound configurations are two presentations of the same structural reading: each is timeless lawful structure in force, registered when called on, with registrations governed by the lawful content. They differ in whether the content includes a decay channel; they do not differ in ontological category.
\end{proposition}

\begin{proof}
By \cref{cor:joint-empirical}, the joint empirical pattern across stable and radioactive configurations is: lawful content fixed independent of accumulated proper time; registration of state changes governed by content; no cumulative modification of structural content traceable to exposure to time. The matter-side timelessness reading~\cite{Atom} describes precisely this pattern: a bound material system is the lawful availability of a stationary structure, in force at the region of spacetime supporting the bound state, registered when called on; the structural content is timeless; registrations occur in spacetime when the structure is invoked or when a channel realizes; between registrations, no internal process occurs.

The stable case fits this reading directly: the lawful content includes no admissible spontaneous decay channel; no channel realization registrations occur; the configuration persists indefinitely as the in-force lawful structure that it is. The radioactive case fits this reading equally directly: the lawful content includes at least one decay channel; channel realizations occur statistically at rate $\lambda$ fixed by content; between realizations, the configuration persists as the in-force lawful structure that it is, with the channel as part of its content but not as an internal process.

The two cases share the ontological category (timeless lawful structure in force) and the registration discipline (registrations governed by content). They differ only in whether the lawful content includes a decay channel. Stability and radioactivity are content-level distinctions, not ontological categories.
\end{proof}

\begin{corollary}[No substance-aging in either case]
\label{cor:no-aging-either}
Neither stable nor radioactive bound configurations undergo substance-aging in the sense of \cref{def:substance-aging}.
\end{corollary}

\begin{proof}
Substance-aging requires cumulative modification of a system's properties as a function of accumulated proper time, attributable to persistence through time. By \cref{cor:joint-empirical}, no such cumulative modification of structural content is observed in either case. Stable configurations register no state changes traceable to exposure to time; radioactive configurations register channel realizations at rates fixed by content rather than by accumulated age. The substance-aging picture is required by neither case.
\end{proof}

\begin{remark}[The two faces of bound matter]
\label{rem:two-faces}
A bound configuration has two structurally distinct aspects, both consistent with the timelessness reading. The first is the worldline-internal accrual of proper time~\cite{Accrual}: the configuration's worldline accrues $\tau$ locally, governed by the local metric. The second is the time-independent structural content: the lawful content of the configuration is the same at every event on the worldline, with registrations (or their absence) governed by that content. The two faces are not in conflict; they are how lawful structure participates in timelessness while being registered in spacetime. Worldline-internal accrual belongs to the spacetime-registration side; time-independent structural content belongs to the timeless lawful-structure side. The atom (stable or radioactive) is in time in the worldline-internal sense and out of time in the structural-content sense. Both are true. The two-face structure is what participation in timelessness looks like for a bound system.
\end{remark}

\begin{remark}[Matter-side instance of the inversion]
\label{rem:matter-side-inversion}
The two-faced structure of bound matter is the matter-side instance of the architectural rule established in the inversion paper~\cite{Inversion}: timeless lawful relation is the general condition, and spacetime is the special case. On the matter side, the lawful content of a bound configuration belongs to the general timeless condition; the worldline-internal accruals and (where present) channel realizations are spacetime-side registrations in which time has values. The bound configuration is one lawful entity registering across both sides --- timeless on the lawful-structure side, registered with proper-time values on the spacetime side. The present paper does not establish the rule; it exhibits the matter-side case under it.
\end{remark}

\begin{remark}[The interval $d\tau$ is a registration window, not a metric of structural wear]
\label{rem:dtau-window}
The appearance of the proper-time interval $d\tau$ in the channel-realization probability $\lambda\, d\tau$ does not reintroduce time into the lawful content. The interval $d\tau$ is a geometric segment of the worldline: a registration window on the spacetime-registration side within which a channel realization may register with density $\lambda$. It is not a metric of accumulated physical wear, and it is not a parameter along which the lawful content evolves. The rate $\lambda$ is content; the interval is registration-side. The memoryless property states the point directly: $\lambda$ is identical whether the worldline has accrued one second or ten billion years of proper time, so the structural content possesses no parameter that the elapsed interval could drive. The interval belongs to the registration side of the ledger; the content does not evolve along it.
\end{remark}

\begin{remark}[Stability and radioactivity are content distinctions, not ontology distinctions]
\label{rem:content-distinction}
The current paper's reading rejects a common implicit assumption: that ``stable'' and ``unstable'' name fundamentally different kinds of physical entities. They do not. They name two ranges of lawful content. A stable configuration has no admissible spontaneous decay channel under the operative description. A radioactive configuration has at least one. The first leaves the configuration's worldline free of decay-event registrations; the second produces statistical decay-event registrations at rate $\lambda$. The configurations themselves are not made of different stuff and are not different in their participation in the timelessness condition. The distinction is structural, not ontological.
\end{remark}

\begin{remark}[Channel realization is registration, not internal self-selection]
\label{rem:realization-not-selection}
Channel realization for a radioactive bound configuration is the statistical registration of an admissible lawful transition, not an internal act of self-specification by the closed nucleus. The lawful content fixes the channel's admissibility and its rate $\lambda$; admissibility is structural. Which specific moment of proper time along a given nucleus's worldline carries the registration is not specified by the closed lawful content of the nucleus and is not selected by the nucleus from within its own closed description. That question --- the actualization of one admissible continuation among admissible alternatives --- belongs to the structural no-go on internal self-selection by closed physical systems and is outside the scope of the present paper. The present paper makes no positive claim about what does the selecting. It claims only that channel realization is registration governed by lawful statistics, not substance-aging.
\end{remark}

\section{Defense Against Substance-Aging}

The joint structural reading is the affirmative claim. This section defends it against the residual move that might attempt to reinstate substance-aging despite the joint reading.

\subsection*{The frictionless-substance escape}

A residual escape might attempt to redefine substance as a frictionless primitive material that endures through time without accumulating any exposure-effects whatever. On this redefinition, substance-aging would not produce observable accumulation; the absence of accumulation in stable configurations and the memoryless rate in radioactive configurations would then be consistent with a (frictionless) substance reading.

\begin{proposition}[Frictionless-substance redefinition has no empirical content]
\label{prop:frictionless}
A redefinition of substance as frictionless primitive material immune to accumulated exposure-effects adds no predictive content over the time-independent lawful structure already in use.
\end{proposition}

\begin{proof}
The time-independent Hamiltonian formalism that successfully describes both stable and radioactive bound configurations~\cite{Atom} contains no parameter, no sub-structural degree of freedom, and no hidden index in which a substance could carry an accumulated exposure-value or in which substance-aging could be encoded as a structural feature. A posited frictionless-substance reading therefore adds no predictive content over the time-independent lawful structure already in use. The substance term in the redefinition does no predictive or explanatory work that the lawful structure is not already doing.

A frictionless primitive substance immune to all accumulated exposure-effects is, by stipulation, indistinguishable from the time-independent lawful structure the matter-side reading already describes: it has the same predictions, the same registrations, the same statistics, the same memoryless rates. The substance term is then a redundant interpretive label, not an alternative ontology with empirical cash value. The redefinition does not refute the joint reading; it relabels the joint reading without adding content.
\end{proof}

\subsection*{Deviations from pure exponential decay are structural, not substance-aging}

A second residual move might appeal to the small deviations from pure exponential decay predicted by standard QM at extreme regimes (Quantum Zeno at very short times, power-law tails at very long times). Such deviations might be read as evidence of some residual substance-aging not captured by the pure exponential rate.

\begin{remark}[Deviations are content-fixed]
\label{rem:deviations}
The deviations from pure exponential decay at extreme regimes are themselves derived from the spectral properties of the time-independent Hamiltonian and its coupling to the continuum~\cite{Khalfin1957,Fonda1978}. They are features of the lawful content of the configuration, not signatures of a material substance aging through time. The deviations are reproducible across freshly-prepared identical nuclides regardless of preparation epoch; their functional form is fixed by spectral structure, not by accumulated proper time. They narrow the strict-memorylessness claim to the experimentally dominant exponential regime; they do not introduce substance-aging. Both the dominant exponential regime and the extreme-regime deviations are presentations of timeless lawful content with a decay channel.
\end{remark}

\subsection*{Ensemble statistics are cumulative counting, not substance-aging}

A third residual move might appeal to the population level. A bulk sample of radioactive material exhibits a parent-to-daughter ratio that evolves measurably as a function of accumulated proper time: at $\tau = 0$ the sample is 100\% parent; at $\tau \gg 1/\lambda$ the sample is dominated by daughter products. A skeptic might say: ``individual nuclei may not age, but the bulk material does --- the population's macroscopic composition changes with time, and that is substance-aging at the ensemble level.''

\begin{remark}[Ensemble-level ratio change is integrated counting of individual realizations]
\label{rem:ensemble-counting}
The parent-to-daughter ratio in a bulk sample at proper time $\tau$ is the integrated record of channel realizations across the worldlines of the population members up to $\tau$. Each realization is itself a registration of an admissible lawful transition along one nucleus's worldline (\cref{def:realization}), not a substance-modification of that nucleus prior to the realization. An un-realized member of the population at proper time $\tau$ has structural content identical to that of a freshly-prepared member of the same nuclide; its decay probability per unit further proper time is $\lambda$, the same as at $\tau = 0$. The bulk-level decay curve $N(\tau) = N_0 e^{-\lambda\tau}$ is a description of how many population members have registered the channel by $\tau$, not a description of how much the un-decayed members have changed. The ratio change is cumulative counting, not cumulative wear. Substance-aging at the population level is therefore not licensed by population-level ratio change: the change is an artifact of integrating worldline-level registrations across the ensemble, each of which is itself not substance-aging. The further question of which specific population member realizes its channel at which event is the actualization question already set aside in \cref{rem:realization-not-selection}: it belongs to the structural no-go on internal self-selection by closed systems and is not answered by the bulk statistics, which fix only the rate, not the individual selections.
\end{remark}

\section{What This Does Not Touch}

\begin{proposition}[No formal revision]
\label{prop:no-formal}
The present paper alters no equations of atomic physics, nuclear physics, or quantum mechanics.
\end{proposition}

\begin{proof}
The argument introduces no new equations and no modifications to the Schr\"odinger equation, the Dirac equation, nuclear shell-model Hamiltonians, electroweak selection rules, the standard exponential-decay law, or any other predictive content. It reads the standard empirical and theoretical content of atomic and nuclear physics through the structural ontology already established. No formal content is added or removed.
\end{proof}

\begin{proposition}[No empirical claim]
\label{prop:no-empirical}
The present paper makes no new empirical predictions.
\end{proposition}

\begin{proof}
The reading of stable and radioactive cases as a joint empirical pattern supporting the matter-side timelessness reading does not predict new experimental outcomes. Atomic and nuclear stability measurements, half-lives, decay constants, ground-state energies, spectral lines, and stability predictions proceed with the same numerical content. The reading concerns the ontological status of the established facts, not their numerical content.
\end{proof}

\begin{proposition}[No denial of radioactive decay]
\label{prop:no-decay-denial}
The present paper does not deny the reality of radioactive decay.
\end{proposition}

\begin{proof}
Radioactive decay is a real, registered, predictively-described phenomenon. The present paper denies only the substance-aging reading of radioactive decay. Channel realization events occur, register at detectors, and follow lawful statistics with rates fixed by the lawful content of each nuclide. Reading these events as channel realizations rather than substance-wear changes no operational content of radiochemistry, nuclear medicine, geochronology, or any other application of decay measurements.
\end{proof}

\begin{proposition}[No denial of matter]
\label{prop:no-matter-denial}
The present paper does not deny matter.
\end{proposition}

\begin{proof}
Matter remains the ordinary regime of bound configurations registered by detectors. Atoms, molecules, condensed matter, nuclei, and other bound systems retain their standard physical roles: they have energies, spectra, response functions, decay channels (or none), and interaction structures. The present paper denies only that these features require a primitive substance underlying the registrations.
\end{proof}

\section{Falsifier}

The joint reading fails only if either of the following is exhibited: (i) a stable bound configuration whose structural properties drift cumulatively as a function of accumulated proper time, in a manner traceable to substance-exposure rather than to environmental interaction; or (ii) a radioactive bound configuration whose channel realization rate depends measurably on the configuration's accumulated proper time along its worldline, beyond the spectral-structural deviations from pure exponential decay already predicted by standard quantum mechanics~\cite{Khalfin1957,Fonda1978}.

Neither has been observed. Stable configurations are described as governed by time-independent Hamiltonians with no admissible spontaneous decay channel under the operative description; their properties do not drift. Radioactive configurations exhibit the memoryless property: $P(\text{realization in } d\tau) = \lambda\, d\tau$, with $\lambda$ fixed by content. Environmental interactions can register state changes through external coupling, but these are ordinary lawful interactions, not substance-aging of an isolated material.

\section{Conclusion}

Stable and radioactive bound configurations are two presentations of the same structural reading. Each is timeless lawful structure in force, registered when called on, with registrations governed by the lawful content. They differ in whether the content includes a decay channel; they do not differ in ontological category. Stability and radioactivity are content-level distinctions, not ontological categories. The reading covers both equally.

Stable atoms do not age. Radioactive atoms do not age either; they realize channels present in their lawful content. The lawful content is fixed in both cases; what differs is whether that content includes a decay channel. The memoryless property of exponential decay confirms the reading: a nucleus's decay probability per unit proper time is $\lambda$, set by content, independent of accumulated age. An unrealized radioactive nucleus is, at every event on its worldline prior to channel realization, the same lawful structure it was at every prior event.

The matter-side instance of general timelessness is now supported by two structurally independent grounds: the formal time-independence of stationary-state structural content~\cite{Atom} and the joint empirical pattern of indefinite non-decay in stable configurations and memoryless channel realization in radioactive configurations. Both grounds converge on the same conclusion. Bound matter participates in the general timelessness condition established for lawful structure in the architectural inversion~\cite{Inversion}: spacetime registration of bound configurations occurs in the special-case regime, while the lawful structure registered is in the general timeless condition. The two faces of bound matter --- its worldline-internal proper time accruals and its time-independent structural content --- are not in conflict; they are how lawful structure participates in timelessness while being registered in spacetime.

The substance-aging picture is required by neither stable nor radioactive cases. Both are timeless lawful structure with content. Stability is the absence of a decay channel in the content. Radioactivity is the presence of one. Neither is a substance aging through time.

\section*{TLM Summary}

The Timeless Light Model reads matter as the lawful availability of a stationary structure, in force at the region of spacetime supporting the bound state, registered when called on. The Atom paper~\cite{Atom} grounds this reading on the formal time-independence of stationary-state structural content. The present paper supplies the empirical companion: stable and radioactive bound configurations jointly exhibit the matter-side instance of general timelessness. Stable configurations have no admissible spontaneous decay channel under the operative description; nothing registers; they exhibit indefinite non-decay absent external disruption. Radioactive configurations have at least one decay channel in their lawful content; channel realizations register along the worldline at rate $\lambda$ fixed by content, independent of accumulated proper time. The two cases share the ontological category (timeless lawful structure in force) and the registration discipline (registrations governed by content). They differ only in whether the lawful content includes a decay channel. Stability and radioactivity are content distinctions, not ontology distinctions. Both presentations participate in the general timelessness condition established in the architectural inversion. Neither involves substance-aging.

\section*{Glossary}

\begin{description}
    \item[Bound configuration] A bound atomic or nuclear system describable by a stationary state of a time-independent Hamiltonian under the applicable atomic, electroweak, and strong-interaction selection rules.

    \item[Channel realization] A spacetime-side registration event in which a decay channel present in a configuration's lawful content statistically transitions the configuration to a lower-energy state; statistics governed by the channel's lawful rate $\lambda$.

    \item[Decay channel] An admissible spontaneous state-change pathway present in a bound configuration's lawful content, by which the configuration transitions to a configuration of lower total energy under the applicable selection rules. Structural, not dynamical.

    \item[Indefinite non-decay] Persistence of a stable bound configuration, absent external disruption, across open-ended timelike intervals without registering any state change traceable to exposure to time.

    \item[Lawful content] The structural specification of a bound configuration: time-independent Hamiltonian, applicable selection rules, admissible transitions, conservation principles, and decay channels (if any). Structural, not dynamical; does not evolve in proper time.

    \item[Lawful structure in force] A lawful structure is in force at a region of spacetime if the admissibility conditions for its application are met at that region. Being in force is not itself a spacetime event.

    \item[Memoryless decay] The empirical property of exponential decay: the probability of channel realization in the next interval of proper time does not depend on the system's accumulated age. The lawful rate is fixed by structural content alone.

    \item[No intrinsic age-marker] The property, shared by stable and radioactive configurations, of carrying no structural feature that records elapsed proper time. Two configurations in the same state are the same lawful configuration regardless of formation epoch; for the radioactive case, the realization rate is independent of accumulated age (memoryless decay); for the stable case, structural content is identical regardless of how long the configuration has existed.

    \item[Persistence] The non-lapse of a lawful structure across timelike ordering. A bound configuration persists across an interval if its governing lawful structure remains in force across the interval.

    \item[Radioactive bound configuration] A bound configuration whose lawful content contains at least one admissible spontaneous decay channel.

    \item[Stable bound configuration] A bound configuration whose lawful content contains no admissible spontaneous decay channel under the operative description.

    \item[Substance-aging] The cumulative modification of a system's properties as a function of accumulated proper time, attributable to the system being a primitive material persisting through time. Distinct from channel realization.

    \item[Two faces of bound matter] The two structurally distinct aspects of a bound configuration: its worldline-internal proper time accrual (spacetime-registration side) and its time-independent structural content (timeless lawful-structure side). The two faces together are how lawful structure participates in timelessness while being registered in spacetime.
\end{description}

\begin{thebibliography}{9}

\bibitem[McKinley(2026a)]{Bedrock}
J.~C.~W.~McKinley.
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\newblock \doi{10.5281/zenodo.19167403}.

\bibitem[McKinley(2026b)]{Atom}
J.~C.~W.~McKinley.
\newblock \emph{The Atom Is Available When Called On: Stationary States as the Matter-Side Instance of General Timelessness}.
\newblock Zenodo (2026).
\newblock \doi{10.5281/zenodo.20114822}.

\bibitem[McKinley(2026c)]{Accrual}
J.~C.~W.~McKinley.
\newblock \emph{Local Accrual as the Only Intrinsic Time-Quantity: A Positive Structural Statement for the Massive Regime}.
\newblock Zenodo (2026).
\newblock \doi{10.5281/zenodo.20225645}.

\bibitem[McKinley(2026d)]{Inversion}
J.~C.~W.~McKinley.
\newblock \emph{Timelessness Is the General Condition: Spacetime Is the Special Case}.
\newblock Zenodo (2026).
\newblock \doi{10.5281/zenodo.19771925}.

\bibitem[ParticleDataGroup(2024)]{ParticleDataGroup}
S.~Navas et al. (Particle Data Group).
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\bibitem[Krane(1987)]{Krane}
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\bibitem[Gamow(1928)]{Gamow1928}
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\bibitem[Khalfin(1957)]{Khalfin1957}
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\newblock \emph{Zh.~Eksp.~Teor.~Fiz.} \textbf{33}, 1371 (1957);
\newblock English translation: \emph{Sov.~Phys.~JETP} \textbf{6}, 1053 (1958).

\bibitem[Fonda et al.(1978)]{Fonda1978}
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\end{thebibliography}

\end{document}

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\lhead{Propagation, Admissibility, Redescription, Reassignment}
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\title{\textbf{Propagation, Admissibility, Redescription, and Reassignment Applied to a Prior No-Go --- No Closed Physical System Internally Fixes the Onset and Direction of a New Causal Chain}}
\author{John C. W. McKinley\,\orcidlink{0009-0005-7097-5035}}
\date{May 19, 2026}

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\footnotetext{This version published at \href{https://doi.org/10.5281/zenodo.20253135}{https://doi.org/10.5281/zenodo.20253135}.}
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\begin{abstract}
A prior result is taken as proven \cite{mckinley_9A}: no closed physical system internally fixes the onset and direction of a new causal chain among multiple lawful possibilities. The present paper develops that result. It distinguishes propagation of processes underway from onset of a new causal chain, admissibility of multiple lawful paths from determination of the path taken, and redescription of a given onset or direction from determination of that onset or direction; and it establishes a reassignment lemma: reassigning the onset or direction of a new causal chain to a prior internal step merely pushes the determination question to that prior step without discharging it.
\end{abstract}

\section{Introduction}

The prior result establishes a structural no-go: no closed physical system internally fixes the onset and direction of a new causal chain among multiple lawful possibilities \cite{mckinley_9A}. This result is taken as proven here.

The present paper develops the prior result. It distinguishes propagation, admissibility, and redescription from internal fixation, and it establishes a reassignment lemma covering both onset and direction.


\section{Definitions}

\begin{definition}[Closed physical description]
A \emph{closed physical description} is a description containing only the laws, state-terms, and causal resources internal to the causal chain under discussion.
\end{definition}

\begin{definition}[New causal chain]
A \emph{new causal chain} is a chain whose onset is the issue under discussion, rather than a process already underway and simply continuing under known laws.
\end{definition}

\begin{definition}[Onset]
The \emph{onset} of a new causal chain is the definite point at which that chain begins rather than not yet beginning.
\end{definition}

\begin{definition}[Direction]
The \emph{direction} of a new causal chain is the particular lawful path taken among multiple physically admissible alternatives.
\end{definition}

\begin{definition}[Internal fixation]
A closed physical description \emph{internally fixes} onset or direction if it determines the relevant item from within the closed description itself.
\end{definition}

\begin{definition}[Redescription]
\label{def:redescription}
A \emph{redescription} of an actual onset or direction is a re-expression of that onset or direction in alternative or finer-grained internal vocabulary within the same closed physical description, without the introduction of any further causal resource.
\end{definition}

\section{Propagation, Admissibility, and Redescription}

\begin{remark}[Propagation is not onset]
\label{remark:propagation-not-onset}
A closed physical description may govern the evolution of a process once the process is underway. That is not the same as determining the onset of a new process.
\end{remark}

\begin{remark}[Admissibility is not actual direction]
\label{remark:admissibility-not-direction}
A closed physical description may contain multiple lawful paths, and it may structure, weight, pre-order, or differentiate them. None of this is the same as determining which path is taken.
\end{remark}

\begin{remark}[Redescription is not determination]
A redescription, in the sense of \cref{def:redescription}, may express the actual onset or direction in more detailed internal vocabulary without thereby determining onset or direction from within the closed description.
\end{remark}

\section{The Reassignment Lemma}

Let $S(t)$ be a closed physical description of a new causal chain over an interval during which multiple lawful continuations remain open. Denote its onset by $t^\ast$ and its realized path by the continuation actually taken.

\begin{lemma}[Prior-step reassignment does not determine onset or direction]
\label{lemma:reassignment}
Reassigning the determination of onset or direction to a prior internal step does not supply internal fixation.
\end{lemma}

\begin{proof}
If the proposed fixing step lies on the new causal chain itself, it is not earlier than that chain's onset; it cannot fix what it is part of. If the proposed fixing step is earlier than the chain, it lies outside the closed chain under discussion and is therefore a non-internal contribution rather than internal fixation. In neither case has the onset or direction of the new causal chain been internally fixed.
\end{proof}

\section{The No-Go in Fuller Form}

\begin{proposition}[No-go for internal fixation, fuller form]
\label{prop:nogo-fuller}
Let $S(t)$ be a closed physical description of a new causal chain. Then $S(t)$ does not internally fix both the onset and direction of that chain, and reassignment to prior internal structure does not supply internal fixation.
\end{proposition}

\begin{proof}
The structural no-go is established in \cite{mckinley_9A} and is taken as proven here. The present claim adds that reassignment to prior internal structure does not supply internal fixation. By \cref{remark:propagation-not-onset}, propagation is not onset. By \cref{remark:admissibility-not-direction}, admissibility is not actual direction. By \cref{lemma:reassignment}, reassigning onset or direction to a prior internal step does not determine onset or direction. Therefore the closed description does not internally fix both the onset and direction of a new causal chain, and reassignment to prior internal structure leaves that conclusion in force.
\end{proof}

\section{Falsifier}

The fuller-form claim of this paper fails only if a closed physical description is exhibited that internally determines both the onset of a new causal chain and its actual direction among multiple lawful alternatives.

\section{Conclusion}
A closed physical description does not internally fix both the onset and direction of a new causal chain. Reassignment to prior internal structure does not supply internal fixation.


\begin{thebibliography}{9}

\bibitem{mckinley_9A}
J. C. W. McKinley, \emph{No Closed Physical System Internally Fixes the Onset and Direction of a New Causal Chain}, Zenodo, \href{https://doi.org/10.5281/zenodo.19464780}{10.5281/zenodo.19464780} (2026).

\end{thebibliography}

\end{document}

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\lhead{No Causal Chain Proceeds Without Initiation}
\rhead{John C. W. McKinley}
\cfoot{\thepage}

\newtheorem{proposition}{Proposition}
\newtheorem{definition}{Definition}

\title{\textbf{No Causal Chain Proceeds Without Initiation:\\ A Structural No-Go Result}}
\author{John C. W. McKinley\,\orcidlink{0009-0005-7097-5035}}
\date{May 17, 2026}

\begin{document}
\maketitle

\blfootnote{\scriptsize This version published at \url{https://doi.org/10.5281/zenodo.20263607}.}

\begin{abstract}
This note states a structural no-go. No causal chain proceeds without an initiation: an occurrence within the chain that is not derived from any prior occurrence within the chain. Derivation transmits occurrence; it does not generate it. A chain in which every occurrence is derived supplies no occurrence to derive from.
\end{abstract}

\section{Definitions}

\begin{definition}[Causal chain]
A \emph{causal chain} is a sequence of events in which each event's occurrence is derived from the occurrence of its predecessor in the chain.
\end{definition}

\begin{definition}[Derivation]
An event's occurrence is \emph{derived} from a prior event's occurrence when the prior event's occurrence brings the later event's occurrence about. Derivation transmits occurrence; it does not generate it.
\end{definition}

\begin{definition}[Initiation]
The \emph{initiation} of a causal chain is an occurrence within the chain that is not derived from any prior occurrence within the chain.
\end{definition}

\begin{definition}[Proceed]
A causal chain \emph{proceeds} when its events occur in sequence.
\end{definition}

\section{The Structural Claim}

\begin{quote}
\textbf{Core Thesis.} No causal chain proceeds without initiation.
\end{quote}

\begin{proposition}[No-go on uninitiated chains]
No causal chain proceeds without an initiation.
\end{proposition}

\begin{proof}
Assume for contradiction that a causal chain proceeds without an initiation. Then every occurrence in the chain is derived from a prior occurrence within the chain.

Derivation transmits occurrence; it does not generate it. Each occurrence therefore presupposes a prior occurrence from which it is derived, and that prior occurrence is itself derived from a further prior occurrence, without end.

Every occurrence in the chain is therefore a derivation, with no underived occurrence at any point. A totality composed only of derivations supplies no underived occurrence from which any derivation can begin. The chain supplies no occurrence to derive from, contradicting the assumption that it proceeds as a causal chain without initiation.

Therefore no causal chain proceeds without an initiation.
\end{proof}

\section{Falsifier}
The claim fails only if a causal chain is exhibited whose events occur in sequence, each occurrence derived from a prior occurrence within the chain, yet no occurrence in the chain is underived.

\section{Conclusion}

No causal chain proceeds without an initiation. A chain of pure transmission supplies no occurrence to transmit.
\end{document}

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\lhead{The Atom Is Available When Called On}
\rhead{John C. W. McKinley}
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\title{\textbf{The Atom Is Available When Called On}\\
\large Stationary States as the Matter-Side Instance of General Timelessness}
\author{John C. W. McKinley\,\orcidlink{0009-0005-7097-5035}}
\date{May 16, 2026}

\begin{document}

\maketitle

\blfootnote{\scriptsize This version published at \url{https://doi.org/10.5281/zenodo.20114823}.}

\begin{abstract}
This paper states the matter-side instance of general timelessness on standard-physics territory. It introduces no new equations and alters no predictive machinery of special relativity, general relativity, quantum mechanics, or quantum field theory. The claim is that the lawful structure underlying bound matter is timeless in a sense that standard physics already grants. Stationary states have no time-dependence in their structural content; energy eigenstates evolve by an overall phase that contributes to no observable; ground states are the canonical case of time-independent structure; the Hamiltonian governing a bound system is itself time-independent under the conditions in which bound states are defined. These features are not interpretive additions. They are how standard quantum mechanics describes bound systems. Null proper time is the photon's route to timeless lawful structure; stationary-state time-independence is the atom's. Timelessness is the general condition of lawful structure; spacetime registration is the special case. The present paper draws the ontological consequence the formalism already permits: the bound state is in force as a timeless lawful structure, registered in spacetime when invoked by lawful coupling with an admissible interactant. Persistence is the non-lapse of the lawful structure across timelike ordering. The \(c\)-bound governs invocation propagation, not any hidden internal process. No traveler-substance occupies the interval between detector interactions; no granule sits beneath the registration; no component action is required to maintain the state. The atom is the lawful availability of a stationary structure, registered when called on. The claim is interpretive only.
\end{abstract}

\section{Introduction}

The photon-side argument of the Timeless Light Model rested on a structural feature of standard relativity: the photon has null proper time, no rest frame, and no rest-frame spatial predicates. There is therefore no internal temporal structure inside which a photon-traveler biography could be defined. The photon is read instead as a lawful charge-state relation, registered in spacetime as paired endpoint events, with no traveler between them~\cite{Bedrock}.

The matter case appears different. Massive systems possess nonzero proper time, rest-frame descriptions, and timelike worldlines. Bound systems persist across detector interactions, accumulate proper time, and respond to repeated probing. None of this is null. The photon's structural argument does not extend to matter by simple analogy.

The present paper argues that general timelessness nevertheless reaches bound matter, and that it does so on territory standard quantum mechanics has already claimed for a century. The lawful structure underlying a bound system is time-independent in its structural content. This is not a new claim. It is the canonical formal status of stationary states, energy eigenstates, and ground states. The bound-state wavefunction's magnitude is time-independent; its expectation values are time-independent; its physically meaningful structure is time-independent. The phase that evolves contributes to no observable.

This is the bound-matter instance of the architectural inversion already established in the corpus~\cite{Inversion}: timelessness is the general condition of lawful structure, and spacetime registration is the special case in which the structure is invoked. The photon case exhibits this inversion in single-event form; the present paper places the matter-side instance under the same umbrella. Null proper time is the photon's route to timeless lawful structure; stationary-state time-independence is the atom's.

The ontological consequence is what this paper draws. If the structural content of the bound state is time-independent in the formalism standard physics uses, then the lawful structure underlying the bound state is timeless in the same sense the photon's lawful structure is timeless: it does not have a temporal interior. The bound state is in force at a region of spacetime under the relevant admissibility conditions. Detector interactions register according to the state's structure. Between detector interactions, no internal process occurs. The state is not doing anything. The law is in force.

\begin{quote}
The atom is the lawful availability of a stationary structure, registered when called on.
\end{quote}

This paper does not deny that atoms have energies, that bonds have strengths, that bound systems respond to perturbations, or that detector interactions yield registrations whose statistics depend on the state's structure. It does not deny proper time, rest frames, worldlines, or the standard predictive machinery of quantum mechanics. It denies only the substance-reading: the claim that some primitive material is sitting at the atomic location, persisting through time, and being detected when sampled. There is no such material. There is the lawful structure, in force, and there are the registrations the structure licenses when invoked.

\section{Minimal Background}

The present paper assumes the bedrock statement of the Timeless Light Model: the photon is a lawfully admissible charge-state relation whose spacetime appearance is a lawful change, not a particle in transit~\cite{Bedrock}. The present paper does not re-argue that result.

The present paper also assumes the architectural inversion result: timelessness is the general condition of lawful structure, and spacetime registration is the special case in which the structure is invoked~\cite{Inversion}. The present paper does not re-argue that result either; it supplies the matter-side instance.

The present paper also assumes the published matter-side no-go: timebound systems are not licensed as travelers by quantum admissibility~\cite{Timebound}. That paper closes the bead-path inference; the present paper supplies the affirmative structural account consistent with that no-go.

The present paper also assumes that standard quantum mechanics describes bound systems through stationary states, energy eigenstates, and time-independent Hamiltonians. The canonical references are standard~\cite{Sakurai,Griffiths}. The structural content of a bound state is time-independent in the formalism. The time-dependence that appears is a phase \(e^{-iEt/\hbar}\) that contributes to no expectation value, no probability, no observable. These are textbook facts, not interpretive moves.

The interpretive question the present paper takes up is what the time-independence of the structural content means ontologically. Standard physics is largely agnostic on this; the formalism succeeds without taking a position. The present paper takes the position that time-independence in the structural content is the spacetime-side appearance of an underlying lawful structure that is itself timeless. That position is consistent with the formalism and with the architectural inversion already established in the corpus.

\section{The Stationary State as Standard Physics' Timeless Territory}

\begin{definition}[Stationary state]
A stationary state is a quantum state whose structural content is time-independent. For an energy eigenstate \(\ket{\psi_E}\), time evolution acts only as an overall phase: \(\ket{\psi_E(t)} = e^{-iEt/\hbar}\ket{\psi_E(0)}\). The probability density \(|\psi_E(x)|^2\), the expectation values \(\langle\hat{O}\rangle\) for time-independent observables, and the structural content of the state are independent of \(t\).
\end{definition}

This definition is not a TLM construction. It is the standard quantum-mechanical definition of a stationary state. The hydrogen atom's ground state is a stationary state. The bound states of molecular systems under the Born-Oppenheimer approximation are stationary states. The electronic states used in band theory are built from stationary-state solutions under time-independent lattice Hamiltonians. The standard formal apparatus for describing bound matter is built around stationary states.

\begin{proposition}[Structural time-independence of bound states]
The structural content of a bound stationary state is time-independent under the conditions in which the state is defined.
\end{proposition}

\begin{proof}
A bound state in quantum mechanics is an eigenstate of a time-independent Hamiltonian. The Schr\"odinger equation \(\hat{H}\ket{\psi_E} = E\ket{\psi_E}\) determines the spatial structure of the state; the time-dependent Schr\"odinger equation gives the time evolution as \(\ket{\psi_E(t)} = e^{-iEt/\hbar}\ket{\psi_E(0)}\). All physical observables computed from the state---probability densities, expectation values of time-independent observables, transition matrix elements with other stationary states---are independent of \(t\). The overall phase does not contribute to any of these. Therefore the structural content of a bound stationary state is time-independent.
\end{proof}

\begin{remark}
This is canonical. The proof is in every introductory quantum mechanics textbook. The present paper does not extend it or modify it. The present paper uses it as the standard-physics foundation on which the ontological reading rests.
\end{remark}

\section{The Ontological Reading}

The interpretive move is to read the time-independence of the structural content as evidence that the lawful structure underlying the bound state is itself timeless.

\begin{definition}[Timeless lawful structure]
A lawful structure is timeless if it is not a spacetime object: it does not have a worldline, a proper time, a rest frame, or a location in spacetime. It is the rule under which certain spacetime registrations are admissible.
\end{definition}

\begin{definition}[In force]
A lawful structure is in force at a region of spacetime if the admissibility conditions for its application are met at that region. The lawful structure being in force is not itself a spacetime event; it is the condition under which spacetime events admissible to the structure can occur.
\end{definition}

\begin{definition}[Registered when called on]
A lawful structure that is in force at a region registers when called on: when the structure is invoked by lawful coupling with an admissible interactant (a detector, a probe, another bound system, or other physically eligible system), the invocation produces a registration in spacetime whose content is determined by the structure. Between invocations, the structure remains in force; no registration is produced because no invocation occurs.
\end{definition}

\begin{proposition}[Matter-side counterpart]
A bound material system is the registered availability of a timeless lawful structure: the structure is in force at the region of spacetime supporting the bound state, and detector invocations yield registrations consistent with the structure's content. There is no traveler-substance, no granule, and no component process maintaining the state between invocations.
\end{proposition}

\begin{proof}
Standard quantum mechanics describes the bound system through a time-independent stationary state. The state's structural content does not evolve in time. Reading this time-independence as the spacetime-side appearance of a timeless lawful structure adds no new equations and modifies no predictive content; it states the ontological consequence the formalism already permits.

A detector interaction with the bound system invokes the lawful structure at the spacetime region where the detector and the system overlap. The invocation produces a registration whose statistics are governed by the state's structure: the probability density, the matrix elements, the energy spectrum. Between detector interactions, no further registration occurs, because no further invocation occurs. The bound system is not idle in the sense of waiting for something to happen; the lawful structure is in force, and would register if invoked.

The substance-reading---according to which some primitive material is sitting at the atomic location, persisting through time---is an interpretive addition to the formalism. The formalism does not require such material. The bound state is computed without it. The detector registrations are predicted without it. The substance-reading is an ontological gloss, not a formal requirement. The published matter-side no-go on traveler ontology~\cite{Timebound} has already established that timebound systems do not license bead-path ontology by way of quantum admissibility; the present reading is the affirmative structural account consistent with that no-go.

Therefore the bound material system is best read as the registered availability of a timeless lawful structure, with detector invocations producing registrations and no traveler-substance underlying them.
\end{proof}

\begin{remark}
This is the matter-side instance of general timelessness. The photon case exhibits it in single-event form: lawful structure is timeless because null structure leaves no interior. The bound atom exhibits it in continuous in-force availability: lawful structure is timeless because standard quantum mechanics describes its structural content as time-independent. The two cases differ in regime---photon as single-event registration, atom as continuous in-force availability---but share the same architectural inversion: timelessness is the general condition; spacetime registration is the special case.
\end{remark}

\section{Why the Electron Is Not Orbiting}

The clearest case of the timeless reading is the canonical bound state of hydrogen. The electron in the ground state of hydrogen is not orbiting the proton. This is not a novel claim; it has been the standard quantum-mechanical understanding for a century. The Bohr orbit picture was retired with the introduction of wave mechanics in 1926. The ground state of hydrogen is described by a wavefunction whose magnitude is spherically symmetric and time-independent. The electron has no trajectory, no path, no motion in the classical sense. Its expectation values for position, momentum, and angular momentum are determined by the state's structure, not by a sequence of intermediate positions through which it moves.

Standard physics already says: the electron is not orbiting. The present paper draws the ontological consequence: the electron is not doing anything at all, in the sense of internal kinetic process, because the state's structural content is time-independent. The lawful structure governing the bound state is in force at the region of the atom. Detector interactions with the atom---spectroscopic measurements, ionization events, scattering---invoke the structure and yield registrations. Between such interactions, no internal process occurs. The atom is not running; it is available.

\begin{proposition}[Hydrogen ground state]
The ground state of hydrogen is a stationary state. Its structural content is time-independent. The electron's spatial distribution is determined by the state's structure, not by intermediate positions through which it moves. The atom does not require an internal kinetic process to persist.
\end{proposition}

\begin{proof}
The ground state of hydrogen is an eigenstate of the time-independent Coulomb Hamiltonian \(\hat{H} = \hat{p}^2/2m - e^2/r\) (in Gaussian units), with ground-state energy \(E_1 = -13.6\) eV (negative, since the state is bound). The eigenstate wavefunction \(\psi_{100}(\mathbf{r}) = (1/\sqrt{\pi a_0^3})\, e^{-r/a_0}\) is independent of \(t\) in its structural content. Time evolution introduces only the phase \(e^{-iE_1 t/\hbar}\), which contributes to no observable. The electron's expectation values are determined by integrals over \(|\psi_{100}|^2\), which are time-independent. No internal kinetic process is invoked in the calculation; the state is solved by separation of variables in the time-independent Schr\"odinger equation. Therefore the ground state's structural content is time-independent, and the standard formalism does not invoke an internal kinetic process to maintain it.
\end{proof}

\begin{remark}
The kinetic energy operator has a nonzero expectation value in the ground state: \(\langle \hat{T} \rangle = |E_1| > 0\). This is sometimes read as evidence that the electron is ``moving'' inside the atom. The reading is not required. The expectation value is a feature of the state's structure, not evidence of a sequence of motions in time. The virial relation \(2\langle\hat{T}\rangle = -\langle\hat{V}\rangle\) is a structural identity of the stationary state, not a time-average of a dynamical process; it holds because of the form of the Coulomb potential, not because anything is being averaged over time. The state is stationary; its kinetic energy expectation value is a constant of the state, not a record of dynamical activity.
\end{remark}

\section{What \texorpdfstring{\(c\)}{c} Does in the Matter Case}

The information speed limit \(c\) plays a structural role in the corpus: it bounds the propagation of lawful invocation between spacelike-separated spacetime points~\cite{cBound}. In the photon case, \(c\) bounds the spacetime relation between emission and absorption events. In the matter case, \(c\) bounds the propagation of detector invocation across the relevant separation.

\begin{proposition}[\(c\)-bound on matter-side invocation]
The propagation of lawful invocation between a detector at spacetime point \(B\) and a bound system at spacetime region \(A\) is bounded by \(c\). The registration at \(B\) consistent with invocation of the bound system at \(A\) is therefore \(c\)-bounded from \(A\).
\end{proposition}

\begin{proof}
The bound system at region \(A\) is described by a stationary state whose structural content is in force at \(A\). A detector at point \(B\) interacts with the bound system through the relevant gauge interaction (electromagnetic, for the cases most directly at issue). The interaction is mediated by the relevant gauge field, whose propagation in spacetime is bounded by \(c\). Therefore the registration at \(B\) consistent with invocation of the bound system at \(A\) is bounded by \(c\) along the relevant null or timelike interval from \(A\) to \(B\). The \(c\)-bound applies to the propagation of invocation, not to any hidden internal process of the bound system.
\end{proof}

\begin{remark}
This is consistent with the registration-bound reading of \(c\)~\cite{cBound}. The bound system itself is not bounded by \(c\) in any internal process, because no internal process occurs. The bound system is in force; invocation from elsewhere is \(c\)-bounded in its propagation. The matter-side application of the \(c\)-bound is symmetric with the photon-side application: in both cases, \(c\) bounds invocation propagation in spacetime, not internal traveler-process.
\end{remark}

\section{Persistence}

Persistence in the present reading is the non-lapse of the lawful structure across timelike ordering. A bound system persists not because some primitive material remains in place across time, but because the lawful structure that licenses the bound state remains in force across the relevant region of spacetime.

\begin{definition}[Persistence]
A bound material system persists across a timelike interval if the lawful structure governing its stationary state remains in force across the interval. Persistence is the non-lapse of the lawful structure, not the continued existence of an underlying material.
\end{definition}

\begin{proposition}[Operational consequence of persistence]
For a bound system, persistence is the operational mark of the lawful structure remaining in force: detector interactions across the relevant timelike interval continue to register consistently with the same stationary state.
\end{proposition}

\begin{proof}
A bound system whose lawful structure remains in force across a timelike interval is invokable at any spacetime point within that interval. Detector interactions at different points produce registrations whose statistics are governed by the same stationary state. The detector at \(t_1\) and the detector at \(t_2\) both register consistently with the same structural content. Therefore persistence---the consistency of registrations across the timelike interval---is the operational mark of the lawful structure remaining in force.
\end{proof}

\begin{remark}
Persistence in this reading is not a process. The bound system is not doing something repeatedly across time. The lawful structure is in force; detector interactions register consistently because the structure is the same one across the interval. If at some point the lawful structure no longer applies---if the bound state is broken by ionization, by absorption of sufficient energy, by chemical reaction---the system does not persist beyond that point. Persistence is the in-force condition of the lawful structure, not the continued existence of an underlying substance.
\end{remark}

\section{What This Does Not Change}

\begin{proposition}[No formal revision]
The present proposal does not modify the equations of special relativity, general relativity, quantum mechanics, or quantum field theory.
\end{proposition}

\begin{proof}
The proposal introduces no new equations, no new postulates, and no modifications to the Schr\"odinger equation, the Dirac equation, the standard treatment of stationary states, the role of \(c\), the metric structure of spacetime, or any predictive relation in standard physics. It concerns only the ontological reading attached to bound states, persistence, and matter. Therefore it introduces no formal revision.
\end{proof}

\begin{proposition}[No empirical claim]
The present proposal makes no new empirical prediction.
\end{proposition}

\begin{proof}
Reading the bound state as the registered availability of a timeless lawful structure does not predict new experimental outcomes. The expectation values, spectra, transition probabilities, and other observables computed from stationary-state quantum mechanics remain unchanged. The proposal is interpretive only.
\end{proof}

\begin{proposition}[No denial of matter]
The proposal does not deny matter.
\end{proposition}

\begin{proof}
Matter remains the ordinary regime of bound systems registered by detectors. Atoms, molecules, condensed matter, and other bound systems retain their standard physical roles: they have energies, spectra, response functions, and interaction structures. The proposal denies only that these features require a primitive substance underlying the registrations. The registrations are real; the lawful structure that licenses them is real; the substance underneath is not required.
\end{proof}

\section{Scope and Limitations}

The present paper is restricted to bound matter: atoms, molecules, bound condensed-matter systems, and other systems describable by stationary states. The treatment of free particles, scattering states, wave packets, and unbound matter raises additional issues---propagation, dispersion, wave-packet structure, scattering amplitudes---that the bound-state case avoids. The unbound case is left to a separate treatment.

The paper is also restricted to non-relativistic bound-state quantum mechanics and to the canonical relativistic generalizations (Dirac equation, bound states in QED). The treatment of strongly relativistic bound states, of bound states in curved spacetime under general relativity, and of bound states in quantum field theory generally is consistent with the present reading but is not developed here.

The present paper does not address the question of how lawful structures come to be in force at particular regions of spacetime, why some regions support bound states and others do not, or how the conditions for the in-force status are themselves established. These questions belong to a more general account of admissibility structure and are not taken up here.

\section{Discussion}

This paper supplies the matter-side instance of general timelessness on standard-physics territory. The architectural inversion result was already established in the corpus~\cite{Inversion}: timelessness is the general condition of lawful structure, and spacetime registration is the special case in which the structure is invoked. The photon case exhibits this inversion in single-event form; the present paper places the matter-side instance under the same umbrella on territory standard quantum mechanics has already claimed for a century.

The general timelessness claim reaches bound matter through different formal terrain than it reaches the photon. The photon case is forced by null proper time, no rest frame, and no rest-frame spatial predicates: there is no temporal interior in which a photon-traveler biography could occur. The bound-matter case is supported by the time-independence of stationary-state structural content: standard physics already describes the bound state as time-independent in its structural content, and the lawful structure underlying that content is read as timeless in the same structural sense as the photon's. The matter-side no-go on traveler ontology~\cite{Timebound} is the negative complement to the present affirmative reading: that paper closes the bead-path inference; this paper supplies the structural account of what is in force instead.

Two grounds, one architecture. The photon's lawful structure is timeless because null structure leaves no interior for it. The bound state's lawful structure is timeless because standard quantum mechanics describes its structural content as time-independent. In both cases, the lawful structure is in force; registrations occur in spacetime when the structure is invoked; no traveler-substance occupies the interval between invocations.

The formal ingredients are standard. The ontological reading is not. Standard quantum mechanics supplies lawful bound-state structure; the present paper reads matter as the registered availability of that structure rather than as primitive substance beneath registration. The bound system is real; the lawful structure is real; the registrations are real. What is not required, and not supplied by the formalism, is a substrate sitting beneath the registrations as their hidden material.

The affirmative form of the claim is direct. The atom is the lawful availability of a stationary structure. The structure is in force at the region of spacetime supporting the bound state. Detector interactions register according to the structure. Between detector interactions, no internal process occurs. The atom is not running, vibrating at a hidden rate, or maintaining itself by an internal process. It is available. When called on, it registers.

This reading clarifies several features that the substance-reading complicates. The electron is not orbiting because the state is stationary; standard physics already says so. The bound state does not require maintenance by an internal process because its structural content is time-independent; the formalism does not invoke such a process. The \(c\)-bound applies to detector invocations propagating between spacelike-separated points, not to an internal traveler in the atom. The atom persists because the lawful structure remains in force, not because some primitive material remains in place.

\section{Conclusion}

This paper has supplied the matter-side instance of general timelessness on standard-physics territory. Standard quantum mechanics describes bound systems through stationary states whose structural content is time-independent. The present paper draws the ontological consequence: the lawful structure underlying the bound state is timeless. The bound state is in force at the region of spacetime supporting it. Detector invocations register according to the structure. Persistence is the non-lapse of the structure across timelike ordering.

Two grounds, one architecture: the photon's lawful structure is timeless because null structure leaves no interior for it; the bound state's lawful structure is timeless because standard quantum mechanics describes its structural content as time-independent. In both cases, timelessness is the general condition; spacetime registration is the special case; no traveler-substance occupies the interval between registrations.

The atom is the lawful availability of a stationary structure, registered when called on.

\section*{TLM Summary}

The Timeless Light Model (TLM) is a minimal interpretive framework. It treats the photon as a lawful charge-state relation registered in spacetime at paired endpoint events, not as a particle in transit. The present paper places the matter-side instance of the same timelessness claim within the architectural inversion: stationary-state time-independence is the atom's route to timeless lawful structure, as null proper time is the photon's. Stationary states have time-independent structural content. The lawful structure underlying a bound state is timeless: it is not a spacetime object, has no temporal interior, and is in force at the region of spacetime supporting the bound state. Detector interactions invoke the structure and yield registrations. Persistence is the non-lapse of the structure across timelike ordering. The \(c\)-bound applies to the propagation of detector invocation between spacelike-separated points. The atom does not orbit, vibrate at a hidden rate, or maintain itself by an internal process. The atom is the lawful availability of a stationary structure, registered when called on.

\section*{Glossary}

\begin{description}
    \item[In force] A lawful structure is in force at a region of spacetime if the admissibility conditions for its application are met at that region. Being in force is not itself a spacetime event.

    \item[Lawful admissibility] The physical constraint structure under which an outcome, interaction, registration, or appearance is realizable, including conservation principles, coupling structure, boundary conditions, and other relevant constraints.

    \item[Matter] The registered availability of a timeless lawful structure governing a bound system. The structure is in force at the region of spacetime supporting the bound state; detector invocations yield registrations consistent with the structure's content.

    \item[Persistence] The non-lapse of a lawful structure across timelike ordering. A bound system persists across an interval if its governing lawful structure remains in force across the interval.

    \item[Registered when called on] The condition under which a lawful structure in force at a region produces a registration in spacetime: when the structure is invoked by lawful coupling with an admissible interactant (a detector, a probe, another bound system, or other physically eligible system), a registration occurs. Between invocations, the structure remains in force; no registration is produced because no invocation occurs.

    \item[Registration] A spacetime-side event in which a lawful state change is recorded as a definite physical occurrence.

    \item[Stationary state] A quantum state whose structural content is time-independent. For an energy eigenstate, time evolution acts only as an overall phase that contributes to no observable.

    \item[Timeless lawful structure] A lawful structure that is not a spacetime object: it has no worldline, no proper time, no rest frame, and no location in spacetime. It is the rule under which spacetime registrations admissible to it can occur.
\end{description}

\begin{thebibliography}{9}
\bibitem[McKinley(2026a)]{Bedrock}
J.~C.~W.~McKinley.
\newblock \emph{A Minimal Structural Statement of the Timeless Light Model}.
\newblock Zenodo (2026).
\newblock \doi{10.5281/zenodo.19167403}.

\bibitem[McKinley(2026b)]{cBound}
J.~C.~W.~McKinley.
\newblock \emph{c Is a Registration Bound, Not a Traveler's Speed: A Registration-Based Interpretation of the Information Speed Limit}.
\newblock Zenodo (2026).
\newblock \doi{10.5281/zenodo.20114176}.

\bibitem[McKinley(2026c)]{Inversion}
J.~C.~W.~McKinley.
\newblock \emph{Timelessness Is the General Condition: Spacetime Is the Special Case}.
\newblock Zenodo (2026).
\newblock \doi{10.5281/zenodo.19771925}.

\bibitem[McKinley(2026d)]{Timebound}
J.~C.~W.~McKinley.
\newblock \emph{Timebound Does Not Mean Traveler: A No-Go on Deriving Bead-Path Ontology from Quantum Admissibility}.
\newblock Zenodo (2026).
\newblock \doi{10.5281/zenodo.20114078}.

\bibitem[Sakurai and Napolitano(2017)]{Sakurai}
J.~J.~Sakurai and J.~Napolitano.
\newblock \emph{Modern Quantum Mechanics}.
\newblock Cambridge University Press, 2nd edition (2017).

\bibitem[Griffiths and Schroeter(2018)]{Griffiths}
D.~J.~Griffiths and D.~F.~Schroeter.
\newblock \emph{Introduction to Quantum Mechanics}.
\newblock Cambridge University Press, 3rd edition (2018).

\end{thebibliography}

\end{document}

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\title{Worldlines Produce; Dilation Reports\\
\large A No-Go on Treating Consequence-Reality as Evidence of Intrinsic Dilation}
\author{John C. W. McKinley\,\orcidlink{0009-0005-7097-5035}}
\date{May 17, 2026}

\begin{document}
\maketitle
\blfootnote{\scriptsize This version published at DOI: \doi{10.5281/zenodo.20225757}.}

\begin{abstract}
Relativistic worldline differences produce real, measurable, decision-relevant physical consequences. Muons reach the ground. The traveling twin is younger at reunion. GPS clocks require correction. These consequences are facts, not appearances. They are commonly described in the language of time dilation, and the reality of the consequences is sometimes treated as evidence that the dilated frame's clock has been slowed in itself --- that dilation is therefore an intrinsic property of the frame after all. This paper states a narrow no-go: the inference from consequence-reality to intrinsicness is not licensed. Consequences follow from the structure of worldline-internal accruals together with between-frames registration relations; neither component requires an intrinsic dilation of any frame's clock. The reality of dilation as a registration relation is fully preserved. What is denied is the further claim that the relation must be located inside one of the parties to it.
\end{abstract}

\section{Introduction}

A standing objection to the registration-interface treatment of dilation runs as follows. If dilation were merely a between-frames relation and not a property of either frame, then dilation would be a kind of appearance. But the consequences of dilation are real: muons reach the ground, the twin ages asymmetrically, GPS depends on the correction. Real consequences require a real underlying cause. Therefore dilation must be intrinsic to the frame whose clock has been slowed.

The objection is intuitive and wrong. It rests on a buried inference from ``real consequence'' to ``intrinsic property of one party.'' The inference is unlicensed.

This paper states the no-go: the reality of the consequences does not entail that any frame's clock has been slowed in itself. Consequences in relativity follow from the joint structure of worldline-internal proper-time accruals and between-frames registration relations~\cite{einstein1905,taylor_wheeler,wald}. The same structural decomposition appears in the conventional-vocabulary treatments of dilation~\cite{mckinley_illusion,mckinley_massspeed}: in-flight reciprocity is a frame-dependent comparison; the reunion difference is a frame-invariant path integral; environmental conditions (mass, speed) act on the integrand rather than on any clock in itself. Neither component is an intrinsic dilation of any frame.

The complementary results are stated separately. The negative no-go on intrinsic dilation~\cite{mckinley_nointrinsic} rules out the claim that a frame's clock is slowed in itself. The positive structural claim~\cite{mckinley_accrual} identifies locally accrued proper time as the only intrinsic time-quantity associated with a massive object. The general registration framing~\cite{mckinley_creg} establishes the regime-level reading of $c$ as a registration bound. The null-case companion~\cite{mckinley_nocount} states the corresponding photon-side result. The present paper closes the standing objection by showing that consequence-reality does not reopen the intrinsic-dilation reading.

\section{Definitions}

\begin{definition}[Physical consequence of dilation]
\label{def:consequence}
A physical consequence of dilation is an outcome in spacetime --- a survival event, an age difference, a clock offset, a phase shift --- whose occurrence and magnitude depend on relativistic-kinematic quantities. The consequence is registrable by appropriate apparatus in at least one frame.
\end{definition}

\begin{definition}[Real, in the relevant sense]
\label{def:real}
A consequence is real when it is a fact in spacetime: a registration occurs, an outcome obtains, an integrated quantity has the value it has. Reality in this sense is opposed to subjective impression. A real consequence is not abolished by changing frames; what changes between frames is the registration of the consequence, not the consequence itself.
\end{definition}

\begin{definition}[Intrinsic dilation]
\label{def:intrinsic-dilation}
An intrinsic dilation of frame $B$'s clock would be a modification of $B$'s clock's accrual rate definable without reference to any frame other than $B$. That is, $B$'s clock would tick more slowly than $\int d\tau$ along $B$'s worldline.
\end{definition}

\begin{definition}[Consequence-from-intrinsicness inference]
\label{def:inference}
The consequence-from-intrinsicness inference is the move from the premise ``dilation produces real consequences'' to ``therefore $B$'s clock is intrinsically dilated.'' Schematically:
\[
\text{Real consequence } \Rightarrow \text{ intrinsic property of one frame.}
\]
\end{definition}

\begin{definition}[Worldline-internal quantity]
\label{def:worldline-internal}
A worldline-internal quantity is one defined using only the worldline geometry and the local metric along the worldline, with no reference to a second frame, coordinate system, synchronization convention, or external apparatus.
\end{definition}

\begin{definition}[Between-frames registration relation]
\label{def:between-frames}
A between-frames registration relation is the quantitative mapping between proper-time accruals on one worldline and coordinate-time separations assigned by another frame's apparatus. For dilation, this is $\Delta t_A = \gamma\, \Delta \tau_B$ with $\gamma = (1 - v^2/c^2)^{-1/2}$.
\end{definition}

\section{The Structure of a Dilation Consequence}

A dilation consequence has two structural components. The first is a worldline-internal accrual. The second is a between-frames registration relation. The consequence emerges from their joint operation.

\begin{lemma}[Two-component structure]
\label{lem:two-component}
Every standardly cited consequence of dilation decomposes into (i) a worldline-internal accrual along one or more worldlines and (ii) a between-frames registration relation describing how the accruals are reported across frames.
\end{lemma}

\begin{proof}
Consider the standard cases.

\emph{Muon survival.} The muon's worldline accrues proper time from production to either decay or ground impact, whichever event occurs first along the worldline. Whether the muon decays before reaching the ground is determined by comparison of two worldline-internal quantities: the proper time accrued and the proper time required for decay. This is component (i). Component (ii) is the Earth frame's registration of the muon's flight as taking longer than the proper lifetime, which is the dilation relation $\Delta t_{\text{Earth}} = \gamma\, \Delta \tau_{\text{muon}}$.

\emph{Asymmetric twin aging.} Each twin's worldline accrues its own proper time between departure and reunion. The age at reunion is the worldline-internal accrual on that twin's worldline up to the reunion event. This is component (i). Component (ii) is the dilation relation each twin's apparatus would register were either to track the other's clock during separation.

\emph{GPS clock correction.} The GPS satellite's worldline accrues proper time in its orbital trajectory; the ground clock's worldline accrues proper time at the surface. The two accruals differ. This is component (i). The metrology-side correction --- the formula by which ground observers translate satellite signals into consistent positional data --- is the registration relation, accounting for both special-relativistic and gravitational components. This is component (ii)~\cite{ashby}.

In each case, the consequence emerges from worldline-internal accruals together with the registration relation. Neither component supplies an intrinsic dilation.
\end{proof}

\begin{proposition}[Consequences are worldline-internal facts]
\label{prop:consequences-internal}
The occurrence and magnitude of every standardly cited consequence of dilation is determined by worldline-internal quantities. The registration relation reports the consequence across frames; it does not produce the consequence.
\end{proposition}

\begin{proof}
By \cref{lem:two-component}, each consequence decomposes into a worldline-internal component and a between-frames component. Component (i) supplies the values: the integrated proper time on each worldline, the proper time required for decay, the age accrued. Component (ii) supplies the mapping by which these values are translated into the coordinate-time language of a chosen frame.

The mapping does not produce the values. The values are fixed by the worldline geometry and the local metric. A different frame applying a different mapping would assign different coordinate-time separations to the same events but would not alter the worldline-internal accruals. The consequence --- the muon's arrival, the twin's age, the GPS clock offset --- is determined by component (i) regardless of which frame's mapping is used to describe it.

Therefore consequences are worldline-internal facts. The registration relation is how they are reported; it is not their cause.
\end{proof}

\section{The No-Go}

\begin{proposition}[Consequence-reality does not license intrinsicness]
\label{prop:no-go}
The reality of the consequences does not license the inference to intrinsic dilation of any frame.
\end{proposition}

\begin{proof}
By \cref{prop:consequences-internal}, every standardly cited consequence of dilation is determined by worldline-internal accruals together with the between-frames registration relation. The worldline-internal component is, by \cref{def:worldline-internal}, defined without reference to any other frame; it is not an intrinsic dilation of a frame's clock, since it is precisely $\int d\tau$ along the worldline, with no modification. The between-frames component is, by \cref{def:between-frames}, a relation between two frames; it is not a property of either frame alone.

Suppose, for contradiction, that the reality of the consequences licenses the inference to intrinsic dilation of some frame $B$. Then there would exist an intrinsic dilation of $B$'s clock --- a modification of $B$'s accrual rate definable without reference to any other frame --- whose existence is required to explain the consequences. But the consequences are already fully explained by component (i) together with component (ii), neither of which is such an intrinsic dilation. The supposed intrinsic dilation does no explanatory work. Its addition is redundant with the worldline-internal accrual it would supposedly correct.

Therefore the inference from consequence-reality to intrinsicness is not licensed.
\end{proof}

\begin{corollary}[The objection fails]
\label{cor:objection-fails}
The standing objection that real consequences require an intrinsic underlying dilation fails.
\end{corollary}

\begin{proof}
The objection assumes that real consequences require a real underlying property of one frame. By \cref{prop:no-go}, this is not the structure of dilation consequences in relativity. The consequences are real; the worldline-internal accruals are real; the registration relations are real; but no frame's clock is intrinsically dilated. The reality of the consequences is fully accounted for without the intrinsicness assumption. The objection therefore fails.
\end{proof}

\section{Why the Inference Is Tempting}

The consequence-from-intrinsicness inference is tempting because it tracks a sound pattern in non-relativistic settings. If a chemical reaction produces a real outcome, one looks for a real property of the reagents that explains it. If a mechanical collision produces a real momentum change, one looks for real masses and velocities. The pattern --- real consequence implies real underlying property --- works in regimes where the relevant properties are intrinsic to single objects.

The pattern breaks in relativity because the relevant properties are partly relational. The dilation relation is real; it is not intrinsic to either frame. Relativistic consequences are produced by joint operations on intrinsic worldline data and relational registration data. The pattern that worked for chemistry and Newtonian mechanics, transferred to relativity unmodified, mislocates the relational component as intrinsic.

The misreading is not a failure to take consequences seriously. It is a failure to recognize that some real components of the explanation are not properties of single objects. Once the structural distinction in \cref{lem:two-component} is in view, the temptation dissipates: the relational component supplies what the supposedly intrinsic dilation was being recruited to supply, and the recruitment becomes unnecessary.

\section{Standard Cases Reread}

The standard cases are reread without loss.

\emph{Muon survival.} The muon's clock ticks at its proper rate along its worldline. The muon's proper lifetime is what it is. The muon's worldline reaches the ground event with less proper time accrued than the proper-lifetime threshold; the muon arrives intact. The Earth frame's apparatus registers the muon's flight using the dilation relation, but the survival outcome itself is fixed by the muon's worldline-internal accrual. No intrinsic dilation of the muon's clock is required.

\emph{Asymmetric twin aging.} Each twin's clock ticks at its proper rate along its worldline. The two worldlines have different geometric shapes between the common departure and reunion events. The two integrated proper times accordingly differ. At reunion, the difference manifests as a difference in biological age, atomic clock reading, or any other proper-time-driven process. No twin's clock has been slowed in itself. The age asymmetry is a difference between two worldline-internal accruals.

\emph{GPS clock correction.} The satellite's clock and the ground clock each tick at their proper rates along their respective worldlines. The two worldlines differ in altitude (gravitational potential) and in velocity (orbital speed). The two integrated proper times accordingly differ, with both special-relativistic and gravitational components. The metrological correction is a between-frames registration mapping that translates satellite-clock readings into a frame the ground apparatus can use. The system works because the worldline-internal accruals are what they are and the registration mapping is correctly specified. Neither clock has been intrinsically dilated.

In all three cases, the consequences are real and the standard predictive machinery is correct~\cite{einstein1905,wald,ashby,mckinley_massspeed}. The intrinsic-dilation interpretive overlay is not required, and adding it is structurally redundant.

\section{What Is Real, Clarified}

Five distinct things are real in any dilation case. Distinguishing them removes the residual force of the standing objection.

\begin{enumerate}
\item Each worldline's locally accrued proper time. Real, intrinsic, worldline-internal.
\item The proper-time-required threshold for any internal process, such as particle decay. Real, intrinsic, worldline-internal.
\item The geometric shape of each worldline. Real, intrinsic, worldline-internal.
\item The between-frames registration relation. Real, relational, between-frames.
\item The spacetime-side registration events themselves: detector clicks, timestamps, comparisons at meeting events. Real, registration-side.
\end{enumerate}

What is \emph{not} on the list, and is not real, is intrinsic dilation: a modification of any frame's accrual rate definable without reference to any other frame. Its absence does not subtract from the reality of items 1 through 5. Each of those is fully present in the standard formalism. The consequence-from-intrinsicness inference treated intrinsic dilation as a sixth real item, supposedly required to explain the others. It is not required, and it is not licensed~\cite{mckinley_nointrinsic}.

\section{Conservativeness}

The present paper introduces no new formal content. Special Relativity, General Relativity, and the standard predictive machinery are preserved. No new equation, constant, or postulate is introduced.

The contribution is interpretive: the reality of the consequences is reconciled with the no-go on intrinsic dilation by showing that the structure of relativistic consequences does not require intrinsicness. Worldline-internal accruals and between-frames registration relations jointly suffice. The standard cases are recovered without modification.

\section{Conclusion}

Relativistic worldline differences produce real consequences. Muons reach the ground. The traveling twin ages asymmetrically. GPS depends on the correction. These are facts in spacetime, not appearances. They are commonly described in the language of time dilation, but the consequences themselves are produced by worldline geometry and proper-time accrual, not by any intrinsic slowing inside a clock.

The reality of the consequences does not license the inference to intrinsic dilation of any frame. Consequences decompose into worldline-internal accruals together with between-frames registration relations. The worldline-internal component is each clock ticking at its proper rate along its worldline; the between-frames component is the registration relation by which one frame's apparatus reports another frame's accrual. Neither component is an intrinsic dilation, and no intrinsic dilation is required to produce the consequences.

The standing objection --- that real consequences must be backed by a real intrinsic property of one frame --- transfers a sound non-relativistic pattern to a regime where some real components of the explanation are not properties of single objects. Once the two-component structure is recognized, the objection dissolves.

Dilation is not the physical cause; it is the registration description. The physical difference is the worldline difference. What is real is the accrual, the worldline, the registration relation, and the registration events. What is not real, and not required, is a slowed clock in itself.

\begin{thebibliography}{9}

\bibitem{einstein1905}
A.~Einstein. Zur Elektrodynamik bewegter K\"orper. \textit{Annalen der Physik} \textbf{17}, 891--921 (1905). \doi{10.1002/andp.19053221004}.

\bibitem{taylor_wheeler}
E.~F.~Taylor and J.~A.~Wheeler. \textit{Spacetime Physics: Introduction to Special Relativity} (2nd ed.). W.~H.~Freeman (1992).

\bibitem{wald}
R.~M.~Wald. \textit{General Relativity}. University of Chicago Press (1984).

\bibitem{ashby}
N.~Ashby. Relativity in the Global Positioning System. \textit{Living Reviews in Relativity} \textbf{6}, 1 (2003). \doi{10.12942/lrr-2003-1}.

\bibitem{mckinley_creg}
J.~C.~W.~McKinley. \textit{c Is a Registration Bound, Not a Traveler's Speed: A Registration-Based Interpretation of the Information Speed Limit}. Zenodo (2026). \doi{10.5281/zenodo.20114176}.

\bibitem{mckinley_nocount}
J.~C.~W.~McKinley. \textit{Spacetime Changes Can Be Counted; Photons Cannot: A No-Go Result on Photon-Object Inventory}. Zenodo (2026). \doi{10.5281/zenodo.20113982}.

\bibitem{mckinley_accrual}
J.~C.~W.~McKinley. \textit{Local Accrual as the Only Intrinsic Time-Quantity: A Positive Structural Statement for the Massive Regime}. Zenodo (2026). \doi{10.5281/zenodo.20225645}.

\bibitem{mckinley_nointrinsic}
J.~C.~W.~McKinley. \textit{Dilation Is Not a Property of the Object: A No-Go Result on Intrinsic Time-Dilation}. Zenodo (2026). \doi{10.5281/zenodo.20225720}.

\bibitem{mckinley_illusion}
J.~C.~W.~McKinley. \textit{Illusion and Invariant: Making Sense of Time Dilation --- Reciprocity, Simultaneity, and Proper Time}. Zenodo (2025). \doi{10.5281/zenodo.17083276}.

\bibitem{mckinley_massspeed}
J.~C.~W.~McKinley. \textit{Mass Slows Time. Speed Slows Time. Concept, Derivations, and Evidence}. Zenodo (2025). \doi{10.5281/zenodo.17083288}.

\end{thebibliography}

\end{document}

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\title{Dilation Is Not a Property of the Object\\
\large A No-Go Result on Intrinsic Time-Dilation}
\author{John C. W. McKinley\,\orcidlink{0009-0005-7097-5035}}
\date{May 16, 2026}

\begin{document}
\maketitle
\blfootnote{\scriptsize This version published at DOI: \doi{10.5281/zenodo.20225720}.}

\begin{abstract}
Relativistic worldline differences produce real, measurable physical consequences. These consequences are commonly described in the language of time dilation, and time dilation as a between-frames registration relation is real, but the consequences themselves are not produced by an intrinsic slowing of any clock. A massive object's proper time accrues locally along its worldline at its own proper rate, always. Dilation is the between-frames relation by which one frame's accumulated proper time is registered through another frame's measurement apparatus. The relation is genuine; the registration is genuine; the consequences are genuine. What is absent is any modification of either frame's intrinsic proper time. This paper states a narrow no-go: there is no such thing as a frame's own dilated time. Dilation lives at the registration interface, not inside the object. Standard Special Relativity is preserved throughout. The only claim is that the dilation relation is not licensed as an intrinsic property of either party to it.
\end{abstract}

\section{Introduction}

The phrase ``time dilation'' is unavoidable in relativistic physics. It names a real measurement relation: when frame $A$ registers frame $B$'s clock through its own apparatus, frame $A$ registers $B$'s clock ticks as separated by $\gamma$ times the proper interval of $B$'s clock. The relation is symmetric. Each frame, registering the other, finds the other's clock to register as dilated.

The relation is real. The muons created in the upper atmosphere really do reach the ground; their reaching is a fact in the Earth frame. The traveling twin really is younger at reunion; the age difference is a fact in any frame. The GPS clocks really do require correction; the correction is a fact in the operational metrology.

None of those facts requires that frame $B$'s own clock has been slowed in itself. Each frame's clock ticks at its own proper rate, locally. The clock has no internal awareness of, and no internal modification by, the frame from which it is being registered. What changes between frames is the registration relation, not the intrinsic accrual.

This paper states that distinction as a narrow no-go: a frame's own dilated time is not licensed. There is no such thing as $B$'s clock running slow from $B$'s own standpoint. There is only the between-frames registration relation by which $A$ registers $B$'s clock through $A$'s apparatus.

The argument uses only Special Relativity~\cite{einstein1905}. The positive structural companion --- that locally accrued proper time along a worldline is the only intrinsic time-quantity for a massive object --- has been stated separately~\cite{mckinley_accrual}. The general registration framing on which the present no-go depends has been stated separately~\cite{mckinley_creg}. The null-case companion has been stated separately~\cite{mckinley_nocount}. The same structural claim --- that reciprocal in-flight dilation statements are frame-dependent comparisons while the reunion difference is a frame-invariant path integral --- has been stated in conventional Special Relativity vocabulary in an earlier work~\cite{mckinley_illusion}. The present paper restates that result as a registration-interface no-go anchored to~\cite{mckinley_creg}, in the vocabulary used throughout the Timeless Light Model corpus.

\section{Definitions}

\begin{definition}[Proper time]
\label{def:proper-time}
The proper time $\tau$ of a timelike worldline $\mathcal{W}$ is the quantity
\[
\tau(\mathcal{W}) = \int_{\mathcal{W}} d\tau,
\]
where $d\tau$ is the local clock-tick interval along $\mathcal{W}$. Proper time is reparametrization-invariant and is the time read by a clock following $\mathcal{W}$.
\end{definition}

\begin{definition}[Local accrual]
\label{def:local-accrual}
A frame's proper time accrues locally along its worldline: each segment of the worldline contributes $d\tau$ to the integrated total. The accrual is intrinsic to the worldline. It is computed without reference to any other frame.
\end{definition}

\begin{definition}[Registration]
\label{def:registration}
A registration is a spacetime-side event in which a lawful state change is physically recorded as a definite localized occurrence by a measurement apparatus~\cite{mckinley_creg}.
\end{definition}

\begin{definition}[Between-frames registration relation]
\label{def:between-frames}
The between-frames registration relation is the quantitative mapping by which frame $A$'s apparatus registers frame $B$'s state. For time-dilation in Special Relativity, the relation is
\[
\Delta t_A = \gamma\, \Delta \tau_B,
\]
where $\Delta \tau_B$ is $B$'s locally accrued proper time between two events on $B$'s worldline, $\Delta t_A$ is the coordinate-time separation $A$ assigns to those events using $A$'s apparatus, and $\gamma = (1 - v^2/c^2)^{-1/2}$ with $v$ the relative velocity.
\end{definition}

\begin{definition}[Intrinsic property of a frame]
\label{def:intrinsic}
A property is intrinsic to a frame if it is defined without reference to any other frame. A property is non-intrinsic, or relational, if its definition requires a second frame or a measurement standpoint outside the frame.
\end{definition}

\begin{definition}[Intrinsically dilated time]
\label{def:intrinsic-dilated}
An intrinsically dilated time would be a time-quantity belonging to a frame whose rate of accrual is modified by something other than the frame's own worldline. That is, it would be a $\tau'$ assigned to a worldline such that $\tau' \neq \int d\tau$ along that worldline.
\end{definition}

\section{The Two Distinct Quantities}

The argument turns on keeping two distinct quantities cleanly separated. They have been used loosely in pedagogical exposition for over a century, and the looseness underwrites the misreading this paper aims to foreclose.

The first quantity is $\tau$, the proper time of a worldline, defined in \cref{def:proper-time}. It is intrinsic to the worldline. It is what the worldline's own clock reads, integrated along its actual history. Different worldlines through the same pair of events give different values of $\tau$. The famous twin-paradox asymmetry at reunion is the statement that the two twins' worldlines, having different shapes, accrue different integrated $\tau$ values.

The second quantity is $\Delta t_A$, the coordinate-time separation that frame $A$ assigns, using $A$'s apparatus, to two events on another worldline. It is not intrinsic to either worldline. It is a registration relation between two frames, defined only in the presence of both.

Standard Special Relativity is explicit on this distinction. $\tau$ is reparametrization-invariant; it is the same value computed in any coordinate system. $\Delta t_A$ depends on the choice of frame; a different $A$ assigns a different $\Delta t_A$ to the same pair of events on $B$'s worldline.

\section{The No-Go}

\begin{proposition}[Proper time is unaffected by registration]
\label{prop:tau-unaffected}
The proper time accrued along a worldline is unaffected by any other frame's registration of that worldline.
\end{proposition}

\begin{proof}
By \cref{def:proper-time}, proper time is the integral of $d\tau$ along the worldline. The integral is determined by the worldline's own geometry. No quantity defined on another worldline, and no measurement performed in another frame, appears in the integrand. Therefore the value of $\tau$ along $\mathcal{W}$ is independent of whether any other frame registers $\mathcal{W}$, and independent of how any such registration is performed.
\end{proof}

\begin{proposition}[Dilation is a between-frames relation]
\label{prop:dilation-relational}
The dilation factor $\gamma$ is defined only between two frames in relative motion. It is not defined for a single frame in isolation.
\end{proposition}

\begin{proof}
The factor $\gamma = (1 - v^2/c^2)^{-1/2}$ contains the relative velocity $v$. Velocity is a relation between two frames~\cite{einstein1905,taylor_wheeler}. A single frame in isolation has no $v$ and therefore no $\gamma$. The factor exists only when two frames are specified. Therefore dilation is a between-frames relation, not a property of a single frame.
\end{proof}

\begin{proposition}[Intrinsically dilated time is not licensed]
\label{prop:no-intrinsic-dilation}
A frame's intrinsically dilated time, in the sense of \cref{def:intrinsic-dilated}, is not licensed.
\end{proposition}

\begin{proof}
Suppose, for contradiction, that a frame $B$ possesses an intrinsically dilated time $\tau'_B$, defined as in \cref{def:intrinsic-dilated}: a time-quantity belonging to $B$ such that $\tau'_B \neq \int d\tau$ along $B$'s worldline.

By \cref{def:intrinsic}, an intrinsic property is defined without reference to any other frame. So $\tau'_B$ must be definable using only quantities belonging to $B$.

The only time-quantity defined using only quantities belonging to $B$ is $\int d\tau$ along $B$'s worldline. Any modification of this quantity must come from outside $B$: from a relative velocity to some second frame, from a coordinate choice external to $B$, or from a registration by another apparatus. Each of these requires reference to a frame other than $B$, violating intrinsicness.

Therefore $\tau'_B$ cannot be both intrinsic to $B$ and different from $\int d\tau$. The supposed intrinsically dilated time is not licensed.
\end{proof}

\begin{corollary}[A frame does not register its own time as dilated]
\label{cor:no-self-dilation}
No frame registers its own clock as dilated.
\end{corollary}

\begin{proof}
By \cref{prop:tau-unaffected}, a frame's proper time is unaffected by registration. By \cref{prop:dilation-relational}, dilation is a between-frames relation. A frame registering its own clock is not in a between-frames relation; it is in a self-registration relation. Therefore no dilation factor applies to its own clock from its own standpoint. The clock registers at its proper rate.
\end{proof}

\section{What Is Real}

The no-go denies the licensing of intrinsic dilation. It does not deny the reality of the between-frames relation, the reality of the registration, or the reality of the physical consequences. Each of these is preserved.

The between-frames relation $\Delta t_A = \gamma\, \Delta \tau_B$ is real in the sense that it is the correct quantitative mapping. Any operationally adequate apparatus in $A$, registering events on $B$'s worldline, will register coordinate-time separations consistent with this relation.

The registration itself is real. The detector clicks, the timestamps, the recorded data --- all are spacetime-side facts. They are not subjective impressions.

The physical consequences are real. When a muon is produced in the upper atmosphere and reaches the ground, the reaching is a fact in the Earth frame. When the traveling twin returns younger, the age difference is a fact at the reunion event. The integrated proper times along the two twins' worldlines differ; this is a consequence of geometry, not of anyone's perception.

What is absent in all of this is intrinsic dilation. The muon's clock is not running slow from the muon's standpoint; the muon decays at its own proper rate along its own worldline. The Earth frame registers the muon's flight as taking longer than the muon's proper lifetime, and this registration is the dilation relation. The decay-versus-survival outcome at the ground is determined by integrated proper time along the muon's worldline relative to the integrated proper time required for decay --- a worldline-internal quantity. The dilation language describes the registration; the worldline integrals determine the consequences.

\section{Relation to the Twin Paradox}

The twin paradox is often presented as if it required selecting between symmetric dilation claims, one twin or the other being the one whose clock is ``actually'' running slow. The no-go forecloses that framing.

Neither twin's clock runs slow from its own standpoint. Each twin's clock ticks at its proper rate. Each twin's apparatus, registering the other, registers the other's clock as dilated. Both registrations are correct between-frames relations. Neither is an intrinsic statement.

The asymmetric aging at reunion is not produced by one twin's clock having been slowed in itself. It is produced by the two worldlines having different geometric shapes. The traveling twin's worldline is longer in space but, because of the Minkowski signature, shorter in integrated proper time. The asymmetry is a fact about the worldlines, not about any clock having been slowed by motion.

This reframing dissolves the apparent paradox. The paradox depends on the assumption that one twin's clock must ``really'' have been running slow. The no-go denies that anyone's clock was ever running slow in itself. There is no real underlying dilation to assign to a winner. There are only worldline integrals and registration relations, both of which behave as Special Relativity says.

\section{Standard Physics Preserved}

The present paper does not modify Special Relativity, the Lorentz transformations, the velocity-addition law, or any predictive content of standard relativistic physics. It does not introduce a new equation, a new constant, or a new postulate.

What it does is decline an interpretive overlay. The overlay says: ``and therefore $B$'s clock is really running slow.'' The no-go says: $B$'s clock is registering at its proper rate from $B$'s standpoint; what is registered as slow is registered as slow by $A$'s apparatus, through the between-frames relation. The equations do not require the overlay. The overlay was a verbal convenience that has been mistaken for a physical claim.

\section{Conclusion}

Time dilation is a real, measurable, consequential between-frames registration relation. It is not a property of any frame.

A frame's proper time accrues locally along its worldline. The accrual is the only intrinsic time-quantity belonging to the frame. Dilation, defined as the registration of one frame's clock by another frame's apparatus, is a relation between two frames; it is not a quantity belonging to either frame alone.

The familiar physical consequences described in the language of dilation --- muon survival, asymmetric aging, GPS corrections --- are determined by integrated proper times along worldlines and by the correct between-frames registration relation. They are not consequences of any frame's clock being slowed in itself.

Therefore: a frame's intrinsically dilated time is not licensed. There is no such thing as the frame's own slowed time. The clock ticks at its proper rate. The dilation lives at the registration interface, and the interface is between frames, not inside the object.

\begin{thebibliography}{9}

\bibitem{einstein1905}
A.~Einstein. Zur Elektrodynamik bewegter K\"orper. \textit{Annalen der Physik} \textbf{17}, 891--921 (1905). \doi{10.1002/andp.19053221004}.

\bibitem{taylor_wheeler}
E.~F.~Taylor and J.~A.~Wheeler. \textit{Spacetime Physics: Introduction to Special Relativity} (2nd ed.). W.~H.~Freeman (1992).

\bibitem{mckinley_creg}
J.~C.~W.~McKinley. \textit{c Is a Registration Bound, Not a Traveler's Speed: A Registration-Based Interpretation of the Information Speed Limit}. Zenodo (2026). \doi{10.5281/zenodo.20114176}.

\bibitem{mckinley_nocount}
J.~C.~W.~McKinley. \textit{Spacetime Changes Can Be Counted; Photons Cannot: A No-Go Result on Photon-Object Inventory}. Zenodo (2026). \doi{10.5281/zenodo.20113982}.

\bibitem{mckinley_accrual}
J.~C.~W.~McKinley. \textit{Local Accrual as the Only Intrinsic Time-Quantity: A Positive Structural Statement for the Massive Regime}. Zenodo (2026). \doi{10.5281/zenodo.20225645}.

\bibitem{mckinley_illusion}
J.~C.~W.~McKinley. \textit{Illusion and Invariant: Making Sense of Time Dilation --- Reciprocity, Simultaneity, and Proper Time}. Zenodo (2025). \doi{10.5281/zenodo.17083276}.

\end{thebibliography}

\appendix
\section{Worked Case: Rocket and Passed Planet}

Consider a rocket under constant proper acceleration and a planet at rest in some chosen inertial frame. The rocket's apparatus registers the planet's clock as dilated; the planet's apparatus registers the rocket's clock as dilated.

Both registrations are correct between-frames relations. Neither corresponds to anyone's clock being slowed in itself. The rocket's clock continues to accrue at its proper rate along the rocket's worldline; the planet's clock continues to accrue at its proper rate along the planet's worldline.

The two registrations report each worldline's accrual through the other's apparatus. No reunion event occurs in this scenario, so the question of which clock ``really'' ran slow has no answer to give; what is asymmetric in the twin case (worldline shapes meeting at a common event) is simply absent here. The symmetry of registration, with no preferred frame, is the structurally honest picture.

The rocket case is the cleanest illustration of \cref{prop:no-intrinsic-dilation}. Where the twin case can mislead by suggesting that one twin's clock must ``really'' have been slowed by motion, the rocket-and-passed-planet case has no reunion to anchor such an inference. Each apparatus registers the other's clock through the standard between-frames relation, both registrations are correct, and there is nothing further to say. The supposed intrinsic dilation has no referent.

\end{document}

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\title{Local Accrual as the Only Intrinsic Time-Quantity\\
\large A Positive Structural Statement for the Massive Regime}
\author{John C. W. McKinley\,\orcidlink{0009-0005-7097-5035}}
\date{May 15, 2026}

\begin{document}
\maketitle
\blfootnote{\scriptsize This version published at DOI: \doi{10.5281/zenodo.20225645}.}

\begin{abstract}
For a massive object, the locally accrued proper time along its worldline is the only time-quantity intrinsic to it. Coordinate time, registered dilation, simultaneity assignments, and any other time-like quantity associated with the object are between-frames relations, not intrinsic properties. This paper states that distinction as a positive structural claim. The accrual is the integral of the local clock-tick interval along the worldline; it requires no second frame, no coordinate choice, and no apparatus external to the worldline. Every other time-quantity standardly associated with the object requires such an external reference and is therefore relational. The claim is conservative with respect to Special and General Relativity: it states what the formalism already entails about which time-quantities are worldline-internal and which are between-frames. The contribution is interpretive only.
\end{abstract}

\section{Introduction}

Time-quantities associated with a massive object come in two structurally distinct kinds. The first is the proper time accrued along the object's worldline. The second is any time-quantity that requires a second frame, a coordinate choice, or an external apparatus to define. The two kinds are routinely conflated in expository physics. The conflation is harmless for calculation and harmful for interpretation.

This paper states the structural distinction as a positive claim. The locally accrued proper time is the only time-quantity intrinsic to the object. Everything else --- coordinate time, registered dilation, simultaneity assignments, retardation labels, frame-dependent timestamps --- is a between-frames relation.

The present paper makes the positive structural claim: what \emph{is} intrinsic is the local accrual along the worldline, and only that. The general registration framing on which this claim depends has been stated separately~\cite{mckinley_creg}. The null-case companion, in which no intrinsic time-quantity exists at all, has been stated separately~\cite{mckinley_nocount}.

The same positive content --- proper time as the worldline-internal path integral, environmental factors (mass, speed) as conditions on the integrand, the watch always feeling normal locally, the difference appearing only at comparison --- has been stated in conventional Special Relativity and General Relativity vocabulary in earlier work~\cite{mckinley_massspeed}. The present paper restates that result as a registration-interface structural claim anchored to~\cite{mckinley_creg}, in the vocabulary used throughout the Timeless Light Model corpus, and adds the explicit no-go on competing candidate intrinsic time-quantities.

The argument uses only standard relativistic kinematics~\cite{einstein1905,taylor_wheeler,wald}. No new equations, constants, or postulates are introduced. The claim is interpretive: it states which of the time-quantities licensed by the standard formalism are intrinsic, and which are relational.

\section{Definitions}

\begin{definition}[Worldline]
\label{def:worldline}
A worldline $\mathcal{W}$ is a timelike curve through spacetime, parametrized smoothly, representing the spacetime history of a massive object.
\end{definition}

\begin{definition}[Local clock-tick interval]
\label{def:dtau}
The local clock-tick interval $d\tau$ along $\mathcal{W}$ is the proper-time differential, defined as
\[
d\tau = \sqrt{-g_{\mu\nu}\, dx^\mu\, dx^\nu}\,/\,c
\]
in a metric signature where timelike intervals are negative. The differential is a property of $\mathcal{W}$ and the local metric. It is the interval read by an ideal co-moving clock following $\mathcal{W}$.
\end{definition}

\begin{definition}[Locally accrued proper time]
\label{def:accrual}
The locally accrued proper time of a worldline segment is
\[
\tau(\mathcal{W}) = \int_{\mathcal{W}} d\tau.
\]
The accrual is computed along the worldline using only $d\tau$ along that worldline. No second frame, no external coordinate system, and no apparatus outside the worldline is required for its definition.
\end{definition}

\begin{definition}[Intrinsic time-quantity]
\label{def:intrinsic}
A time-quantity associated with an object is intrinsic if its definition refers only to quantities defined on the object's worldline together with the local metric along that worldline. A time-quantity is relational, or between-frames, if its definition requires reference to a frame, coordinate system, or apparatus external to the object's worldline.
\end{definition}

\begin{definition}[Coordinate time]
\label{def:coord-time}
The coordinate time $t$ assigned to an event by an observer is the time-component of the event's coordinates in the observer's chosen coordinate system. Coordinate time is defined relative to the chosen system; a different system assigns a different coordinate time to the same event.
\end{definition}

\begin{definition}[Registered dilation]
\label{def:dilation}
The registered dilation of frame $B$'s clock by frame $A$'s apparatus is the relation $\Delta t_A = \gamma\, \Delta \tau_B$, where $\Delta \tau_B$ is the proper time accrued on $B$'s worldline between two events and $\Delta t_A$ is the coordinate-time separation $A$'s apparatus assigns to those events, with $\gamma = (1 - v^2/c^2)^{-1/2}$ and $v$ the relative velocity. The relation is defined only when both frames are specified.
\end{definition}

\begin{definition}[Simultaneity assignment]
\label{def:simultaneity}
A simultaneity assignment is a rule by which events at spatially distinct locations are labeled as occurring ``at the same time.'' The rule depends on the choice of reference frame and on the convention used to synchronize distant clocks. Different frames and different conventions yield different simultaneity assignments for the same set of events.
\end{definition}

\section{The Positive Claim}

\begin{proposition}[Local accrual is intrinsic]
\label{prop:accrual-intrinsic}
The locally accrued proper time $\tau(\mathcal{W})$ of an object's worldline is intrinsic to the object.
\end{proposition}

\begin{proof}
By \cref{def:accrual}, $\tau(\mathcal{W})$ is the integral of $d\tau$ along $\mathcal{W}$. By \cref{def:dtau}, $d\tau$ is a property of $\mathcal{W}$ and the local metric along $\mathcal{W}$. Both quantities are defined without reference to any frame, coordinate system, or apparatus external to $\mathcal{W}$. By \cref{def:intrinsic}, the resulting integrated quantity is intrinsic to the object.
\end{proof}

\begin{proposition}[Local accrual is reparametrization-invariant]
\label{prop:reparam}
The value of $\tau(\mathcal{W})$ for a given worldline segment is independent of the coordinate system used to compute it.
\end{proposition}

\begin{proof}
The proper-time differential $d\tau = \sqrt{-g_{\mu\nu}\, dx^\mu\, dx^\nu}/c$ is a scalar under coordinate transformations: $g_{\mu\nu}\, dx^\mu\, dx^\nu$ is constructed from contractions of tensors and is therefore coordinate-invariant. The integral of a scalar along a curve is independent of the coordinate parametrization of the curve. Therefore $\tau(\mathcal{W})$ has the same value in any coordinate system~\cite{wald}.
\end{proof}

\begin{proposition}[Coordinate time is relational]
\label{prop:coord-relational}
The coordinate time $t$ assigned to events on $\mathcal{W}$ is not intrinsic to the object.
\end{proposition}

\begin{proof}
By \cref{def:coord-time}, coordinate time depends on the choice of coordinate system. The coordinate system is external to $\mathcal{W}$: different systems assign different $t$ values to the same event on $\mathcal{W}$. By \cref{def:intrinsic}, a quantity whose definition requires a choice external to the worldline is relational, not intrinsic.
\end{proof}

\begin{proposition}[Registered dilation is relational]
\label{prop:dilation-relational}
The dilation registered for an object's clock is not intrinsic to the object.
\end{proposition}

\begin{proof}
By \cref{def:dilation}, registered dilation is defined only when two frames are specified: the frame $B$ whose clock is being registered and the frame $A$ whose apparatus performs the registration. The factor $\gamma$ contains the relative velocity, which is a between-frames quantity. Therefore the registered dilation is defined only in the presence of a second frame. By \cref{def:intrinsic}, it is relational.
\end{proof}

\begin{proposition}[Simultaneity is relational]
\label{prop:simultaneity-relational}
A simultaneity assignment for events including events on $\mathcal{W}$ is not intrinsic to the object.
\end{proposition}

\begin{proof}
By \cref{def:simultaneity}, a simultaneity assignment depends on the choice of frame and synchronization convention. Both are external to $\mathcal{W}$. Different frames and conventions produce different simultaneity assignments for the same events on $\mathcal{W}$. By \cref{def:intrinsic}, simultaneity is relational.
\end{proof}

\section{The Main Result}

\begin{proposition}[Local accrual is the only intrinsic time-quantity]
\label{prop:main}
Among the time-quantities standardly associated with a massive object, the locally accrued proper time along its worldline is the only one intrinsic to the object. All other standardly associated time-quantities --- coordinate time, registered dilation, simultaneity assignments, retardation labels, frame-dependent timestamps --- are relational.
\end{proposition}

\begin{proof}
By \cref{prop:accrual-intrinsic}, local accrual is intrinsic. By \cref{prop:reparam}, its value is independent of coordinate choice, confirming worldline-internal definition. By \cref{prop:coord-relational}, \cref{prop:dilation-relational}, and \cref{prop:simultaneity-relational}, coordinate time, registered dilation, and simultaneity are each relational. The same reasoning extends to any standardly associated time-quantity whose definition includes a frame choice, coordinate choice, synchronization convention, or external apparatus: each such quantity is, by \cref{def:intrinsic}, relational.

What remains is to verify that no further intrinsic time-quantity is available. An intrinsic time-quantity would have to be definable using only quantities on $\mathcal{W}$ and the local metric. The available constructions from those ingredients are: $d\tau$ itself; integrals of $d\tau$ over worldline segments, which reduce to $\tau(\mathcal{W})$ values; and functions of $\tau$ such as $\tau^2$ or $f(\tau)$, which are reparametrizations of the same underlying quantity. No structurally distinct intrinsic time-quantity is constructible from worldline-internal ingredients alone.

Therefore the locally accrued proper time is the only intrinsic time-quantity.
\end{proof}

\begin{corollary}[Asymmetric aging is a worldline-internal fact]
\label{cor:aging}
The age difference between two objects that meet at a common event after following different worldlines is a difference in their respective locally accrued proper times.
\end{corollary}

\begin{proof}
The age of each object at a given event on its worldline is the locally accrued proper time from a reference event on the same worldline to that event. By \cref{prop:main}, this is an intrinsic worldline-internal quantity. When two worldlines meet at a common terminal event, the difference between their accrued proper times is a difference between two worldline-internal quantities. The asymmetry is determined entirely by the shapes of the two worldlines and the local metric along them. No frame choice, registered dilation, or simultaneity assignment is required to define the difference.
\end{proof}

\begin{corollary}[Decay outcomes are worldline-internal]
\label{cor:decay}
Whether an unstable particle decays before reaching a given event is determined by the proper time accrued along the particle's worldline relative to the proper time required for decay.
\end{corollary}

\begin{proof}
The decay process of an unstable particle is governed by quantities defined on the particle's worldline: its proper lifetime and its locally accrued proper time. By \cref{prop:main}, both are intrinsic worldline-internal quantities. The decay-or-survival outcome at a given event is determined by the comparison of these two intrinsic quantities. No registration by an external frame is required for the outcome itself, though external frames may register the outcome through their own apparatus.
\end{proof}

\section{The Standard Twin and Muon Cases}

The framework is conservative with respect to the standard relativistic results~\cite{mckinley_massspeed}.

In the twin scenario, the traveling twin's worldline and the staying twin's worldline meet at the reunion event. Each worldline has accrued its own locally accrued proper time between the departure event and the reunion event. The traveling twin's worldline has accrued less. The age difference is a fact about the two integrated proper times. By \cref{cor:aging}, the asymmetry is intrinsic to the worldlines and requires no frame-dependent claim about ``whose clock was really running slow.''

In the muon case, the muon's worldline accrues proper time from production in the upper atmosphere to either decay or ground impact, whichever occurs first along the worldline. If the worldline reaches the ground event before the proper-time-required-for-decay has accrued, the muon arrives. By \cref{cor:decay}, the outcome is determined worldline-internally. The Earth frame registers the muon's flight as taking longer than its proper lifetime, which is the standard dilation relation; but the survival outcome is fixed by the muon's own integrated proper time, not by any registration.

Both standard results emerge from the worldline-internal accrual together with the local metric. The registration relations are consistent with the results but do not produce them.

\section{Why the Distinction Matters}

If local accrual is conflated with the relational time-quantities, two errors follow.

The first error is the assumption that a frame's clock has been slowed in itself when registered as dilated by another frame. The present positive claim explains what remains: each frame's clock continues to accrue at its proper rate along its worldline, regardless of how it is registered.

The second error is the assumption that asymmetric outcomes --- aging at reunion, particle survival, GPS clock offsets --- require selecting a privileged frame whose dilation reading is ``correct.'' The present claim removes the need for privilege selection. Asymmetric outcomes are produced by differences between worldline-internal accruals. Different frames register the same outcomes through their respective between-frames relations, all of which are consistent with the worldline-internal facts.

Removing both errors leaves the standard formalism intact and clarifies which quantities in it are properties of objects and which are relations between frames.

\section{Relation to the Null Regime}

The present claim concerns the massive regime. It does not extend without modification to the null regime.

In the null regime, the proper-time differential along a photon-associated null relation vanishes: $d\tau = 0$ everywhere along the relation. The integral of $d\tau$ along such a relation is zero, and no clock can be associated with it. The construction by which an intrinsic time-quantity is defined for a massive object --- integration of $d\tau$ along a timelike worldline --- has no analog. The null companion result~\cite{mckinley_nocount} accordingly states that no intrinsic time-quantity exists for the null case at all; only spacetime-side endpoint registrations are available.

The structural asymmetry between the regimes is therefore: in the massive regime, one intrinsic time-quantity exists (local accrual) and all other time-quantities are relational; in the null regime, no intrinsic time-quantity exists and all available time-related descriptions are between endpoint registrations. The general registration framing~\cite{mckinley_creg} accommodates both cases.

\section{Conservativeness}

The present paper introduces no new formal content. It does not modify Special Relativity, General Relativity, quantum mechanics, or quantum field theory. It does not modify the metric structure of spacetime, the invariance of $c$, or any predictive relation. The proper-time integral is the standard one; the dilation relation is the standard one; the coordinate-time and simultaneity definitions are the standard ones.

The contribution is interpretive: it states which of the standardly recognized time-quantities are worldline-internal and which are between-frames, and it identifies the locally accrued proper time as the unique member of the first category. The empirical content of relativity is unaffected.

\section{Conclusion}

For a massive object, the locally accrued proper time along its worldline is the only time-quantity intrinsic to it. The accrual is computed from $d\tau$ along the worldline and the local metric, using nothing external. Every other time-quantity standardly associated with the object --- coordinate time, registered dilation, simultaneity, frame-dependent timestamps --- requires reference to a frame, coordinate system, synchronization convention, or external apparatus, and is therefore a between-frames relation.

The asymmetric outcomes that relativity correctly predicts --- different ages at reunion, particle survival across long flight paths, GPS clock offsets --- are determined by differences between worldline-internal accruals. Registration relations describe how these worldline-internal facts are reported across frames; they do not produce the facts.

The clean partition is: one intrinsic time-quantity per massive worldline, namely its locally accrued proper time; everything else, relational.

\begin{thebibliography}{9}

\bibitem{einstein1905}
A.~Einstein. Zur Elektrodynamik bewegter K\"orper. \textit{Annalen der Physik} \textbf{17}, 891--921 (1905). \doi{10.1002/andp.19053221004}.

\bibitem{taylor_wheeler}
E.~F.~Taylor and J.~A.~Wheeler. \textit{Spacetime Physics: Introduction to Special Relativity} (2nd ed.). W.~H.~Freeman (1992).

\bibitem{wald}
R.~M.~Wald. \textit{General Relativity}. University of Chicago Press (1984).

\bibitem{mckinley_creg}
J.~C.~W.~McKinley. \textit{c Is a Registration Bound, Not a Traveler's Speed: A Registration-Based Interpretation of the Information Speed Limit}. Zenodo (2026). \doi{10.5281/zenodo.20114176}.

\bibitem{mckinley_nocount}
J.~C.~W.~McKinley. \textit{Spacetime Changes Can Be Counted; Photons Cannot: A No-Go Result on Photon-Object Inventory}. Zenodo (2026). \doi{10.5281/zenodo.20113982}.

\bibitem{mckinley_massspeed}
J.~C.~W.~McKinley. \textit{Mass Slows Time. Speed Slows Time. Concept, Derivations, and Evidence}. Zenodo (2025). \doi{10.5281/zenodo.17083288}.

\end{thebibliography}

\end{document}

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\title{Light Has No Speed, You Have No Speed, Speed Is What Other Objects Have As They Pass You\\
\large A Declarative Restatement of Existing Physics}
\author{John C. W. McKinley\,\orcidlink{0009-0005-7097-5035}}
\date{May 15, 2026}

\begin{document}
\maketitle

\blfootnote{\scriptsize This version published at DOI: \doi{10.5281/zenodo.20225300}.}

\begin{abstract}
This paper introduces no new physics. It is a declarative summary
of what standard relativity already says about ``the speed of light,''
sorted into three cases.

\medskip

\noindent\textbf{Plainly:}
1. Light has no speed. 2. You have no speed. 3. Speed is what
other objects have as they pass you, and the fastest they can pass you
is $c$.

\medskip

\noindent\textbf{Technically:}
For the photon, the kinematic notion of speed is inapplicable, since
the photon has null proper time, no rest frame, and no licensed
intermediate spacetime history. For any massive object, the
coordinate velocity in its own rest frame is zero. The invariant
constant $c$ bounds the relative velocity that any inertial observer
may assign to another inertial frame from that observer's standpoint.

\medskip

\noindent No equations are altered. The content is drawn from standard
special relativity and from prior papers in this series.
\end{abstract}

\section{What This Paper Is}

This paper states a simple clarification.

The phrase ``the speed of light'' is useful shorthand in ordinary
physics language, but it hides three different cases. Once those cases
are separated, the familiar slogan becomes clearer. The photon case,
the massive-object case, and the observer-relative case do not say the
same thing.

Plainly, light has no speed. You have no speed. Speed is what
other objects have as they pass you.

That sounds strange only because ordinary speech treats speed as
something an object carries around, like weight, color, or charge. But
speed is not like that. Speed is not sitting inside the object. Speed
is what appears when one thing is described from the place of another.
You are not moving from your own point of view. A rocket is not moving
from the rocket's point of view. A planet is not moving from the
planet's point of view. What has speed is the other thing, as seen
from where you are.

Technically, the photon has null proper time, no rest frame, and no
licensed intermediate spacetime history. A massive object has zero
coordinate velocity in its own rest frame. The constant $c$ bounds the
relative velocity assignable from one inertial standpoint to another
inertial frame.

No equation is changed. The Lorentz transformations remain. The
metric structure of relativity remains. The operational role of $c$
remains. The correction is interpretive: $c$ is not the private speed
of a photon traveler. It is the invariant causal and registration
bound appearing in relativistic descriptions.

The technical background is developed in prior papers in this series:
the minimal statement of the Timeless Light Model~\cite{tlmBedrock},
the null-proper-time restriction~\cite{nullProperTime}, the no-rest-frame
restriction~\cite{noRestFrame}, the null-curve clarification~\cite{nullCurves},
the wavefunction no-path result~\cite{wavefunctionNoGo}, the timebound
no-traveler result~\cite{timeboundNoGo}, the registration-bound
restatement of $c$~\cite{cRegBound}, the linguistic verdict on
speed-talk for the null case~\cite{soCalledSpeed}, and the observational
no-go on seeing a photon in flight~\cite{seeingNoGo}. The present
paper gives the plain declarative version.

\section{Case 1: Light Has No Speed}

\subsection*{Plainly}

Light has no speed.

The speed of light is not really the speed of light.

That sentence sounds wrong because the phrase is familiar. But the
plain picture is simple. Speed belongs to a thing that is traveling.
Light is not a little object taking a trip from one place to another.
There is no passenger. There is no clock for the passenger. There is
no route the passenger experiences.

So the famous phrase should be read carefully. It names the constant
that governs lightlike appearances in spacetime. It does not name the
speed of a traveling light-object.

\subsection*{Technically}

A speed is the magnitude of a change of position with respect to time
in a specified frame. The expression \emph{the speed of $X$} therefore
requires a referent $X$, a frame in which $X$ is assigned positions,
and a time parameter with respect to which those positions change.

The photon does not satisfy those requirements. It has null proper
time, no rest frame, and no licensed intermediate spacetime
history~\cite{tlmBedrock,nullProperTime,noRestFrame,nullCurves}. The
wavefunction's predictive success does not supply an intermediate
photon path~\cite{wavefunctionNoGo}. Nor does lawful temporal
governance license recovery of a hidden traveler-bead
ontology~\cite{timeboundNoGo}. The linguistic verdict that follows
from these constraints is given in~\cite{soCalledSpeed}: the
self-frame reading of speed fails for the photon because no photon
rest frame exists, and the other-frame traveler reading fails because
no persisting photon-referent in transit is licensed.

The observational side is just as restrictive. No photon has been
seen in flight; every observation of light is a registration at a
detector, never an observation of a photon between
emission and absorption~\cite{seeingNoGo}.

Accordingly, the kinematic notion of speed is inapplicable to the
photon. The constant $c$ remains fully operative. What is denied is
only that $c$ names a self-owned kinematic property of a photon-object.

\section{Case 2: You Have No Speed}

\subsection*{Plainly}

You have no speed.

Neither does anything else. A rocket has no speed. A planet has no
speed. An electron has no speed.

That sounds strange because ordinary speech says things like ``the
rocket is moving fast.'' But the rocket is not moving from the
rocket's point of view. The same is true of every object, including
you. From its own point of view, every object is standing still.

Speed appears only when something else is described from where you
are. From Earth, the rocket is moving. From the rocket, Earth is
moving. Neither one owns the speed as a private property. Speed is the
description of one thing from the place of another.

\subsection*{Technically}

For any massive system with a rest frame, the coordinate velocity of
that system in its own rest frame is zero. This is not an additional
postulate. It is the definition of a rest frame.

Ordinary language often suppresses the relevant frame. It says that
an object ``is moving'' as though motion were a frame-independent
property of the object. Relativity does not say that. Motion is
assigned from one frame to another~\cite{cRegBound}.

Therefore, a massive object does not possess an absolute self-speed.
Its own-frame coordinate velocity is zero. Other observers may assign
it a nonzero relative velocity, including a velocity arbitrarily close
to $c$, but the object's velocity in its own rest frame remains zero.

\section{Case 3: Speed Is What Other Objects Have As They Pass You}

\subsection*{Plainly}

Speed is what other objects have as they pass you.

From your own point of view, you are standing still. Other objects
pass you. Rockets pass you. Planets pass you. Signals and records are
described from where you are. That is where speed shows up.

The fastest anything can pass you is $c$.

That is the right home of the famous constant. It is not the speed of
a photon traveler. It is the limit on how fast anything can be seen to
pass from where you are.

\subsection*{Technically}

From any inertial standpoint, the relative velocity assigned to
another inertial frame is bounded above by $c$. This bound is not a
traveler-owned property. It is a structural feature of relativistic
spacetime and of the transformations relating inertial descriptions.

The bound is symmetric. If observer $A$ assigns observer $B$ a relative
velocity below $c$, observer $B$ assigns observer $A$ a corresponding
relative velocity below $c$. Neither observer possesses an absolute
motion through spacetime. Each is at rest in its own inertial frame;
each registers the other under the invariant bound~\cite{cRegBound}.

Thus $c$ is best read here as a causal and registration bound. It
limits cross-frame assignment. It does not describe the private speed
of a photon-object.

\section{Summary}

The phrase ``the speed of light'' compresses three cases that should
be kept separate.

Plainly:

\begin{enumerate}
  \item Light has no speed.
  \item You have no speed.
  \item Speed is what other objects have as they pass you, and the
  fastest they can pass you is $c$.
\end{enumerate}

Technically:

\begin{enumerate}
  \item For the photon, the kinematic notion of speed is inapplicable
  because the photon has null proper time, no rest frame, and no
  licensed intermediate spacetime history.
  \item For any massive object, the coordinate velocity in its own
  rest frame is zero.
  \item For any inertial observer, the relative velocity assignable to
  another inertial frame is bounded by the invariant constant $c$.
\end{enumerate}

No physics is changed. The clarification is verbal and ontological.
The constant $c$ keeps its full operational role. What is rejected is
the misleading image of a photon-object carrying $c$ as its own
travel-speed.

\section{What This Does Not Change}

This paper does not alter special relativity, general relativity,
quantum mechanics, or quantum field theory. It changes no equation,
no prediction, no measurement rule, and no experimental expectation.

The Lorentz transformations remain. The relativistic metric remains.
The null structure of lightlike intervals remains. The invariant
constant $c$ remains exactly where standard physics places it.

The only change is disciplinary. The familiar phrase ``the speed of
light'' is retained as operational shorthand, but denied as literal
ontology. In the photon case, speed-talk is inapplicable. In the
massive case, own-frame speed is zero. In the observer-relative case,
$c$ is the invariant bound on cross-frame registration.

\begin{thebibliography}{9}

\bibitem{tlmBedrock}
J. C. W. McKinley.
\emph{A Minimal Structural Statement of the Timeless Light Model}.
Zenodo (2026). \doi{10.5281/zenodo.19167403}.

\bibitem{nullProperTime}
J. C. W. McKinley.
\emph{Taking Null Proper Time Seriously: An Interpretive
Clarification of Null Proper Time}.
Zenodo (2025). \doi{10.5281/zenodo.18004632}.

\bibitem{noRestFrame}
J. C. W. McKinley.
\emph{No Rest Frame, No Persistence: A Kinematic Constraint on
Photon Interpretation}.
Zenodo (2025). \doi{10.5281/zenodo.18005884}.

\bibitem{nullCurves}
J. C. W. McKinley.
\emph{Null Curves Without Carriers: Resolving an Ontological
Tension in Relativistic Geometry}.
Zenodo (2025). \doi{10.5281/zenodo.18028886}.

\bibitem{wavefunctionNoGo}
J. C. W. McKinley.
\emph{Wavefunction Prediction Does Not License a Photon Path}.
Zenodo (2026). \doi{10.5281/zenodo.19504772}.

\bibitem{timeboundNoGo}
J. C. W. McKinley.
\emph{Timebound Does Not Mean Traveler: A No-Go on Deriving
Bead-Path Ontology from Quantum Admissibility}.
Zenodo (2026). \doi{10.5281/zenodo.20114078}.

\bibitem{cRegBound}
J. C. W. McKinley.
\emph{c Is a Registration Bound, Not a Traveler's Speed}.
Zenodo (2026). \doi{10.5281/zenodo.20175517}.

\bibitem{soCalledSpeed}
J. C. W. McKinley.
\emph{The So-Called Speed of Light Is Not the Speed of Light: A
No-Go on Speed-Talk for the Null Case}.
Zenodo (2026). \doi{10.5281/zenodo.20193205}.

\bibitem{seeingNoGo}
J. C. W. McKinley.
\emph{No-Go on Seeing a Photon Cross the Sky}.
Zenodo (2026). \doi{10.5281/zenodo.20225004}.

\end{thebibliography}

\end{document}

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\lhead{No-Go on Seeing a Photon Cross the Sky}
\rhead{John C. W. McKinley}
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\title{\textbf{No-Go on Seeing a Photon Cross the Sky}}
\author{John C. W. McKinley \orcidlink{0009-0005-7097-5035}}
\date{May 15, 2026}

\begin{document}

\maketitle

\blfootnote{\scriptsize This version published at \url{https://doi.org/10.5281/zenodo.20225004}.}

\begin{abstract}
No photon has ever been seen in flight. Every observation of light is a
registration at a detector, never an observation of a photon between emission
and absorption. This is built into standard photodetection theory. This paper
states the narrow no-go: no procedure exists, has existed, or is available
within standard physics by which a photon in flight is observed without that
observation itself constituting a registration. The fact is built into the
standard treatment of quantum optics, in which photodetection is modeled as
photon absorption at the detector. The argument introduces no new equations
and no modification to relativity, quantum mechanics, or quantum field theory.
It is an empirical no-go on seeing a photon in flight.
\end{abstract}


\section{Introduction}

The intuitive picture is familiar. Light leaves the sun, crosses ninety-three
million miles of space, and arrives at the eye. The crossing is imagined as a
sight: a thing in flight, traversing the sky, observable in principle if not
in practice.

No such sight has ever been recorded. Every observation of light is a
registration event at a detector---an eye, a photographic emulsion, a
charge-coupled device, a photomultiplier, an atom in an excited state. The
photon, as such, has never been caught in flight. This is built into the
standard treatment of quantum optics \citep{Glauber1963}.

The present paper states the narrow no-go that follows: no procedure exists by
which a photon in flight is observed without that observation itself
constituting a registration. The paper makes no claim about what is or is not
the case between emission and absorption. It claims only that the seeing has
never occurred and that no procedure for it is available.

The constant $c$, Maxwell's equations, quantum electrodynamics, and the full
operational apparatus of optics remain untouched. The numerical predictions of
standard physics are unchanged. The claim is observational.


\section{The No-Go}

\begin{definition}[Registration]
A registration is a detector-side event at which a photon-related outcome is
recorded. Emission and absorption are registrations. Scattering at an
intermediate medium consists of further registrations.
\end{definition}

\begin{definition}[Seeing a photon in flight]
To see a photon in flight is to observe the photon at a spacetime point that
is neither its emission event nor its absorption event, without that
observation itself constituting a registration.
\end{definition}

\begin{proposition}[No photon has been seen in flight]
\label{prop:noflight}
No photon has been observed at a spacetime point between its emission and its
absorption without that observation itself constituting a registration.
\end{proposition}

\begin{proof}
This is built into the standard treatment of photodetection in quantum
optics, in which detection of a photon is modeled as absorption of the
photon at the detector and the detector-side registration is the observation
\citep{Glauber1963}. No auxiliary observational channel is supplied by the
theory: there is no procedure by which a photon between its emission and its
absorption is observed without that observation itself constituting a
registration.
\end{proof}

\begin{remark}
The apparent visibility of light in flight through fog, dust, or smoke is not
a counterexample. Each visible point along the apparent path is a scattering
center at which an incoming photon is absorbed and an outgoing photon is
emitted in a new direction. The photons that reach the side-viewer's eye are
those re-emitted photons, not the originals; the originals terminated at the
scattering centers. The visible streak is therefore a sequence of fresh
emission-absorption events, not an observation of any photon between
registrations.
\end{remark}

\begin{proposition}[The narrow no-go]
\label{prop:nogo}
No procedure exists, has existed, or is available within standard physics by
which a photon in flight is observed without that observation itself
constituting a registration.
\end{proposition}

\begin{proof}
By \Cref{prop:noflight}, no such observation has been recorded. The detection
of a photon proceeds by absorption at a detector; the registration is the
observation. Any procedure that placed a detector at an intermediate spacetime
point would constitute a fresh absorption at that point, terminating the
prior emission-to-detector relation at the new detector rather than producing
an observation of a photon between registrations. No auxiliary observational
channel is supplied by the theory. Therefore the seeing is unavailable.
\end{proof}

\begin{remark}
The premise is free. It is built into the standard treatment. The no-go
follows without added ontology.
\end{remark}


\section{Relation to Other Work}

This note is intentionally narrower than the broader photon-side analysis
developed elsewhere. The null-proper-time, no-rest-frame, no-carrier, and
minimal structural arguments appear in related papers
\citep{McKinleyNullProperTime2025,McKinleyNoRestFrame2025,McKinleyNullCurves2025,McKinleyBedrock2026}.
The present note needs only the observational premise: seeing a photon in
flight is not an available operation.


\section{Scope}

The argument introduces no new equations, no new observables, and no
modification to special relativity, general relativity, quantum mechanics, or
quantum field theory. The value and operational role of $c$ are unchanged.
Maxwell's equations are unchanged. Quantum electrodynamics is unchanged. The
predictions of standard optics are unchanged.

The claim is only that no photon has been seen in flight and that no procedure
for such a seeing is available. The paper takes no position on what is or is
not the case between emission and absorption.


\section{Conclusion}

The intuitive picture is of light crossing the sky as a thing in flight. The
crossing, as a sight, has never been recorded. No procedure for recording it
is supplied by standard physics. The fact is built into the treatment.

What is denied is only the seeing. The denial is observational. The
operational apparatus of physics, the numerical value of $c$, and the
predictions of optics are unaffected.

No photon has been seen in flight.


\begin{thebibliography}{99}

\bibitem{Glauber1963}
R. J. Glauber, \emph{The Quantum Theory of Optical Coherence},
Physical Review \textbf{130}, 2529--2539 (1963). \doi{10.1103/PhysRev.130.2529}.

\bibitem{McKinleyNullProperTime2025}
J. C. W. McKinley, \emph{Taking Null Proper Time Seriously: An Interpretive Clarification of Null Proper Time},
Zenodo (2025). \doi{10.5281/zenodo.18004632}.

\bibitem{McKinleyNoRestFrame2025}
J. C. W. McKinley, \emph{No Rest Frame, No Persistence: A Kinematic Constraint on Photon Interpretation},
Zenodo (2025). \doi{10.5281/zenodo.18005884}.

\bibitem{McKinleyNullCurves2025}
J. C. W. McKinley, \emph{Null Curves Without Carriers: Resolving an Ontological Tension in Relativistic Geometry},
Zenodo (2025). \doi{10.5281/zenodo.18028886}.

\bibitem{McKinleyBedrock2026}
J. C. W. McKinley, \emph{A Minimal Structural Statement of the Timeless Light Model},
Zenodo (2026). \doi{10.5281/zenodo.19167403}.

\end{thebibliography}

\end{document}

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\title{The So-Called Speed of Light Is Not the Speed of Light\\
\large A No-Go on Speed-Talk for the Null Case}
\author{John C. W. McKinley\,\orcidlink{0009-0005-7097-5035}}
\date{May 14, 2026}

\begin{document}
\maketitle
\blfootnote{This version published at \url{https://doi.org/10.5281/zenodo.20193205}.}

\begin{abstract}
This paper states a narrow interpretive no-go. The phrase ``speed of light''
presupposes that something is moving and that the something has a speed. In
the photon case, the literal reading fails. The photon has null proper time,
no rest frame, and no licensed intermediate spacetime history. A speed is a
rate-of-position-change assigned to a referent in a frame across time. Where
there is no licensed photon traveler, there is no persisting subject whose
motion can be tracked. Where there is no photon rest frame, there is no
photon-owned self-speed either. The constant $c$ remains in the equations
and retains its full operational role. What is denied is only the reading of
$c$ as the speed owned by a photon-object. The so-called \emph{speed of light} is therefore not the speed of light in the literal ontological sense. It is
operational shorthand for the invariant constant $c$, not a speed owned by
a photon-object. This is interpretive only.
\end{abstract}

\section{Introduction}

The phrase \emph{speed of light} is taught early and used everywhere. It
appears in introductory texts, in engineering practice, and in scientific
writing intended for general readers. As an operational shorthand it
identifies a constant that enters Maxwell's equations, the Lorentz
transformations, the relativistic metric, and the dispersion relations
of quantum field theory.

This paper does not contest the constant. It contests the phrase.

The phrase ``speed of $X$'' is a four-place structure: it requires an
$X$, a frame, a time, and a rate at which the position of $X$ changes
in that frame across that time. For ordinary objects each slot is
filled. For the photon, the relevant slots are not filled. The photon
has null proper time and no rest frame~\cite{tlmBedrock,nullProperTime,noRestFrame}.
The wavefunction's predictive success does not supply an intermediate
photon path~\cite{wavefunctionNoGo}. The companion timebound no-go
blocks the broader inference from lawful temporal governance to
traveler-bead ontology~\cite{timeboundNoGo}. The observer-side
restatement of $c$ as a registration bound, rather than a traveler-owned
speed, has been developed separately~\cite{cRegBound}.

The present paper states what follows from those results for the phrase
itself. Where no photon traveler is licensed, there is no persisting
subject whose motion can be tracked. Where no photon rest frame exists,
there is no photon-owned self-speed either. The so-called speed of light
is therefore not the speed of light when the phrase is read literally.
It survives as operational shorthand for a constant whose role in
physics is settled. What does not survive is its literal ontological
reading.

\section{The Structure of Speed-Talk}

\begin{proposition}[Speed has a subject]
\label{prop:subject}
A speed in physics is a property assigned to a thing. The locution
``speed of $X$'' requires an $X$ whose position changes in some frame
across some time.
\end{proposition}

\begin{proof}
The kinematic notion of speed is defined as the magnitude of the rate
of change of position with respect to time, $|d\vec{x}/dt|$, where
$\vec{x}$ is the position assigned to a referent in a frame and $t$ is
a time parameter in that frame. The ratio itself belongs to the frame
description. It does not, by itself, establish the ontology of the
referent being described. To treat the ratio as the speed \emph{of}
$X$ is to treat $X$ as the subject of the assigned position-change.
Without a referent whose position is defined in a frame, the derivative
has no subject as an object-owned property. Therefore speed-talk
requires a subject.
\end{proof}

\begin{proposition}[Speed requires a frame in which the referent has a position]
\label{prop:frame}
A speed assigned to a referent presupposes a frame in which that
referent has a defined position.
\end{proposition}

\begin{proof}
The quantity $|d\vec{x}/dt|$ is frame-relative. Different frames
assign different values. For the quantity to be defined at all as the
speed \emph{of} a referent, the referent must possess a position in
the relevant frame. Where no such position is defined, no speed is
defined as an object-owned property of that referent.
\end{proof}

\begin{remark}
\Cref{prop:subject,prop:frame} are uncontroversial in ordinary
mechanics. They are stated explicitly here because they make the
photon case's failure visible without further argument.
\end{remark}

\begin{remark}
This paper does not deny that a lightlike relation, wave-packet model,
field excitation, null geodesic, or detector-to-detector correlation
can be assigned the invariant value $c$ in an inertial-frame
description. The point is narrower. A successful frame-assigned
description does not by itself license a photon-object that owns that
value as its traveler speed. The ratio may be operationally valid
without becoming the property of a persisting thing in transit.
\end{remark}

\section{The Photon Case}

\begin{proposition}[No subject]
\label{prop:nosubject}
The photon is not licensed by standard physics as a persisting object
in transit between emission and absorption.
\end{proposition}

\begin{proof}
The photon has null proper time and no rest frame. On the Timeless
Light Model reading, those constraints withhold the ordinary timelike
template of internal duration, traversed route, and intermediate
history~\cite{tlmBedrock,nullProperTime,noRestFrame,nullCurves}.
The wavefunction's predictive success does not supply an intermediate
photon path~\cite{wavefunctionNoGo}. The broader timebound no-go
blocks the inference from lawful temporal governance to traveler-bead
ontology~\cite{timeboundNoGo}. Therefore the photon is not licensed
as a persisting object in transit.
\end{proof}

\begin{proposition}[No rest frame]
\label{prop:noselfframe}
The photon has no rest frame.
\end{proposition}

\begin{proof}
Standard relativity withholds a rest frame from any null relation.
A rest frame requires a timelike worldline along which proper time
elapses. The photon's null structure provides no such worldline.
Therefore the photon has no rest frame.
\end{proof}

\begin{proposition}[No photon-owned speed]
\label{prop:nospeed}
The photon has no speed in the literal kinematic sense required by
the phrase \emph{speed of light}.
\end{proposition}

\begin{proof}
There are two possible literal readings of the phrase. First, it may
mean that a photon-object has a speed in its own frame. That reading
is blocked by \Cref{prop:noselfframe}: the photon has no rest frame,
and therefore no own-frame in which a photon-owned self-speed could
be defined.

Second, it may mean that a photon-object has a speed assigned from
another observer's frame. That reading is blocked by
\Cref{prop:subject,prop:frame,prop:nosubject}. Speed requires a
referent whose position changes in a frame across time. The photon is
not licensed as a persisting referent in transit between emission and
absorption. Therefore the frame-assigned value $c$ remains a value
assigned within the observer's description; it does not become a
kinematic property owned by a photon-object.

Thus the self-frame reading fails because there is no photon rest
frame. The other-frame traveler reading fails because there is no
licensed photon traveler. Therefore no kinematic speed is a property
of the photon.
\end{proof}

\section{The So-Called Speed of Light}

\begin{proposition}[The so-called speed of light is not the speed of light]
\label{prop:not-speed-of-light}
The so-called speed of light is not the speed of light in the literal
ontological sense.
\end{proposition}

\begin{proof}
The phrase ``speed of light'' purports, when read literally, to name a
speed belonging to light, read physically as the photon case. By
\Cref{prop:nospeed}, no kinematic speed is a property of the photon.
Therefore the so-called speed of light is not the speed of light in the
literal ontological sense. It is operational shorthand for the invariant
constant $c$.
\end{proof}

\begin{remark}
The phrase remains operationally serviceable. It identifies the
invariant constant $c$ that enters the equations of relativity,
electromagnetism, and quantum field theory. Nothing in this paper
disturbs that role. The error lies in the literal reading, not in the
operational use.
\end{remark}

\section{What $c$ Is Instead}

The companion paper on $c$ as a registration bound gives the positive
restatement~\cite{cRegBound}. The constant $c$ is not a traveler-owned
speed. It is the invariant causal bound governing lightlike relation,
null structure, and observer-side registration in spacetime description.

The present paper adds only the linguistic verdict. Once the photon
traveler is removed, the phrase \emph{speed of light} loses its literal
referent. The operational label survives because the constant survives.
The ontological reading does not.

\section{Scope}

The proposal introduces no new equations, no new observables, and no
modification to special relativity, general relativity, quantum
mechanics, or quantum field theory. The numerical value of $c$ is
unchanged. The role of $c$ in the Lorentz transformations, the metric,
and the dispersion relations is unchanged. Standard calculations using
the phrase ``speed of light'' as operational shorthand remain valid.

The claim is only that the phrase should not be read literally as naming
a speed owned by a photon-object.

\section{Conclusion}

The phrase \emph{speed of light}, read literally, predicates a speed
of a referent that is not licensed by physics as a thing with a speed.
The photon has null proper time, no rest frame, and no licensed
intermediate spacetime history.

The no-go has two sides. There is no photon-owned self-speed, because
there is no photon rest frame. There is no photon traveler-speed
assigned from another frame, because the photon is not licensed as a
persisting object in transit whose position changes across time. The phrase therefore fails as a literal statement: it appears to name a photon-owned speed, but no such photon-owned speed is licensed.

The constant $c$ remains. Its operational role is untouched. What is
removed is only the literal reading under which $c$ was treated as the
speed of a traveling photon. There is no traveler. The so-called \emph{speed of light} is not the speed of light. There is the lawful invariant $c$, and there are lawful
spacetime appearances governed by it.

\begin{thebibliography}{9}

\bibitem{tlmBedrock}
J. C. W. McKinley.
\emph{A Minimal Structural Statement of the Timeless Light Model}.
Zenodo (2026). \doi{10.5281/zenodo.19167403}.

\bibitem{nullProperTime}
J. C. W. McKinley.
\emph{Taking Null Proper Time Seriously: An Interpretive
Clarification of Null Proper Time}.
Zenodo (2025). \doi{10.5281/zenodo.18004632}.

\bibitem{noRestFrame}
J. C. W. McKinley.
\emph{No Rest Frame, No Persistence: A Kinematic Constraint on
Photon Interpretation}.
Zenodo (2025). \doi{10.5281/zenodo.18005884}.

\bibitem{nullCurves}
J. C. W. McKinley.
\emph{Null Curves Without Carriers: Resolving an Ontological
Tension in Relativistic Geometry}.
Zenodo (2025). \doi{10.5281/zenodo.18028886}.

\bibitem{wavefunctionNoGo}
J. C. W. McKinley.
\emph{Wavefunction Prediction Does Not License a Photon Path}.
Zenodo (2026). \doi{10.5281/zenodo.19504772}.

\bibitem{timeboundNoGo}
J. C. W. McKinley.
\emph{Timebound Does Not Mean Traveler: A No-Go on Deriving
Bead-Path Ontology from Quantum Admissibility}.
Zenodo (2026). \doi{10.5281/zenodo.20114078}.

\bibitem{cRegBound}
J. C. W. McKinley.
\emph{$c$ Is a Registration Bound, Not a Traveler's Speed}.
Zenodo (2026). \doi{10.5281/zenodo.20175517}.

\end{thebibliography}

\end{document}

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\title{\textbf{\(c\) Is a Registration Bound, Not a Traveler's Speed}\\
\large An Observer-Side Reading of the Information Speed Limit}
\author{John C. W. McKinley\,\orcidlink{0009-0005-7097-5035}}
\date{May 13, 2026}

\begin{document}

\maketitle

\blfootnote{\scriptsize This version prepared for Zenodo. DOI: \href{https://doi.org/10.5281/zenodo.20175517}{10.5281/zenodo.20175517}.}

\begin{abstract}
This paper makes one narrow interpretive claim. In any observer's own inertial frame, that observer is at rest. Motion appears only as the changing spatial relation between the observer's standpoint and other frames, systems, lightlike relations, or records.
The information speed limit, conventionally denoted \(c\), bounds the registration of those cross-frame relations. On this reading, \(c\) is not the speed of light as a thing; it is not a traveler-owned speed. Nor is \(c\) a maximum self-speed for rockets or other massive objects; in an object's own rest frame, its own speed is zero. It is the invariant causal bound governing how events, lightlike relations, and cross-frame relations are registered in spacetime description. This is especially important once the photon is no longer read as a little object traveling through space. The photon has no rest frame and no internally elapsed proper time; therefore \(c\) cannot be interpreted as the speed of a photon-object in its own frame. The claim introduces no new equations and alters no predictions of special relativity, general relativity, quantum mechanics, or quantum field theory. It is interpretive only.
\end{abstract}

\noindent\textbf{Keywords:} special relativity; speed of light; photon ontology; observer frame; causal structure; Timeless Light Model; registration bound.

\section{Introduction}

The constant \(c\) is conventionally introduced as the speed of light. That phrase is operationally useful. It is also ontologically misleading when it is read as the speed of a little photon-object traveling through spacetime.

The Timeless Light Model has taken the relativistic null invariants seriously. The minimal Bedrock canon compresses the relevant result into the claim that a photon is not a particle in transit, but a lawfully admissible charge-state relation whose spacetime appearance is a lawful change~\cite{Bedrock}. The wavefunction's predictive success does not license an intermediate photon path either~\cite{Wavefunction}. The traveler is gone. What remains is the invariant causal structure of spacetime itself, viewed from local inertial standpoints.

Once the photon traveler is removed, the interpretive status of \(c\) should be restated without traveler vocabulary. The present paper states that restatement from the observer side.

The starting point is standard special relativity: in an observer's own inertial frame, the observer's spatial coordinates are constant, and the observer's own velocity is zero. There is no frame-independent standpoint from which that observer is ``really'' moving. Motion is always assigned from one frame to another.

The standard relativistic background is the invariance of \(c\) in inertial-frame descriptions~\cite{Einstein1905} and the metric treatment of causal structure in spacetime~\cite{Wald}.

Accordingly, what an observer registers as motion is the changing relation between the observer's standpoint and other frames, systems, or records. The information speed limit bounds the registration of those relations. It is not a speed possessed by an object in its own frame, because no object has speed in its own frame. Thus \(c\) is not the maximum speed of a rocket in the rocket's own frame; in the rocket's own frame, the rocket is at rest. In the photon case, the point is stricter: the photon has no rest frame at all, so \(c\) cannot be the speed of a photon-object in the photon's own frame.

The claim is narrow. It does not deny that observers compute speeds for other systems. It does not deny that \(c\) is assigned to lightlike relations in inertial-frame descriptions. It does not deny the metric role of \(c\) in relativity. It denies only the residual traveler reading in which \(c\) is treated as the speed of a photon-object moving through spacetime, or as a maximum self-speed belonging to a massive object in its own frame.

Terminology used in this paper is collected in the Glossary at the end.

\section{The Observer's Standpoint}

\begin{definition}[Standpoint]
An observer's standpoint is the frame-relative position from which the observer registers events and assigns coordinates.
\end{definition}

\begin{definition}[Registration]
A registration is a spacetime-side record of an event, relation, signal, measurement, detection, or other physically definite occurrence in an observer's description.
\end{definition}

\begin{proposition}[Own-frame rest]
In an observer's own inertial frame, the observer is at rest.
\end{proposition}

\begin{proof}
An inertial frame assigns coordinates to events. In the observer's own inertial frame, the observer's spatial coordinate is fixed at the observer's standpoint. Therefore the observer's coordinate velocity in that frame is zero. Hence, in the observer's own inertial frame, the observer is at rest.
\end{proof}

\begin{remark}
This is standard special relativity. The point of foregrounding it is interpretive. The ordinary habit of saying that ``the observer is moving'' always suppresses the question: moving relative to what frame? In the observer's own frame, the observer is not moving. Motion enters as a relation between frames.
\end{remark}

\begin{proposition}[Motion is cross-frame relation]
What an observer registers as motion is the changing spatial relation between the observer's standpoint and other frames, systems, or records.
\end{proposition}

\begin{proof}
In the observer's own inertial frame, the observer's own velocity is zero. Any nonzero velocity assigned in that frame is therefore assigned to another frame, system, signal, or record relative to the observer's standpoint. Such assigned motion is not absolute motion. It is the changing relation between the observer's frame and what is described from that frame. Therefore what the observer registers as motion is cross-frame relation.
\end{proof}

\begin{remark}
The driving example illustrates the point. In the driver's frame, the driver and car are at rest, and the roadside trees are assigned motion relative to the car. In the roadside frame, the trees are at rest, and the car is assigned motion relative to the road. Both descriptions are frame-relative. Neither is an absolute motion story.
\end{remark}

\section{The Information Speed Limit}

\begin{definition}[Information speed limit]
The information speed limit is the invariant causal bound conventionally denoted \(c\).
\end{definition}

This definition does not change the numerical value of \(c\), nor does it alter the formal role \(c\) plays in relativity. It names the familiar invariant causal bound while refusing to interpret that bound as the speed of a traveler-object.

\begin{proposition}[\(c\) bounds registration]
The information speed limit bounds observer-side registration of cross-frame relations.
\end{proposition}

\begin{proof}
An observer registers events, lightlike relations, detections, and frame-relative relations from within the observer's own frame. Relativity imposes an invariant causal bound on such registrations: no causal update or information-bearing record is assigned superluminal propagation in local inertial description. That invariant bound is \(c\).  
Since the observer's own motion is zero in the observer's own inertial frame, motion registered in that frame is motion assigned to other frames, systems, lightlike relations, or records relative to the observer's standpoint. Therefore \(c\) bounds observer-side registration of cross-frame relations.
\end{proof}

\begin{remark}
The rocket-and-planet case gives the clean symmetry. From the planet frame, the planet is at rest and the rocket is assigned motion. From the rocket frame, the rocket is at rest and the planet is assigned motion. Thus \(c\) is not the maximum speed of the rocket in the rocket's own frame. In the rocket's own frame, the rocket is at rest; the planet is assigned motion relative to the rocket. The bound \(c\) governs the rate at which each observer can register updates about the other's assigned motion, not a self-speed possessed by either. Each observer registers the other's cross-frame relation under the same invariant causal bound. Neither description requires an absolute moving object viewed from nowhere.
\end{remark}

\section{\(c\) Is Not a Traveler-Owned Speed}

\begin{proposition}[\(c\) is not an object's speed in its own frame]
The invariant \(c\) is not the speed of an object in that object's own frame.
\end{proposition}

\begin{proof}
For any massive object with a rest frame, the object's velocity in its own rest frame is zero. Therefore \(c\) cannot be the speed of such an object in its own frame. For a photon, the situation is stricter: the photon has no rest frame. Therefore there is no photon-own-frame in which \(c\) could be interpreted as the speed of a photon-object. In both cases, \(c\) is not an object-owned speed. It is the invariant causal bound appearing in spacetime descriptions of lightlike relation, null structure, and observer-side registration.
\end{proof}

\begin{remark}
This preserves ordinary operational language while disciplining its ontology. One may continue to say that light is measured at speed \(c\) in inertial-frame description. What does not follow is that there exists a photon-object possessing \(c\) as its traveler speed through an internally lived journey. Nor does it follow that a rocket possesses \(c\) as a maximum self-speed in its own frame. In the rocket's own frame, the rocket's own speed is zero. Even as a massive object's assigned speed in another frame approaches \(c\), the bound \(c\) continues to govern cross-frame registration rather than any self-speed in the object's own rest frame.
\end{remark}

\begin{remark}
The photon case is the decisive application. A traveler-speed interpretation requires a traveler. A photon, on the minimal TLM reading, has null proper time, no rest frame, and no internally defined spatial traversal~\cite{Bedrock}. The wavefunction's predictive structure does not supply an intermediate photon route~\cite{Wavefunction}. Therefore the photon is not licensed as a traveler whose speed is \(c\). What remains is the spacetime-side invariant causal bound governing lightlike registration and null relation.
\end{remark}

\section{Why the Usual Phrase Survives Operationally}

The phrase ``speed of light'' remains useful in ordinary calculation. A radar signal, a laser pulse, a null geodesic calculation, or an electromagnetic field solution all use \(c\) in standard ways. Nothing in this paper changes that usage.

The correction is ontological. The phrase ``speed of light'' should not be inflated into the claim that a photon-object owns a speed while traveling through an internally unfolding route. The phrase ``maximum speed of a rocket'' should not be inflated into the claim that a rocket possesses such a speed in its own frame. In its own frame, the rocket is at rest. The operational phrases survive. The traveler picture does not.

\begin{proposition}[Operational shorthand is not ontology]
The operational use of the phrase ``speed of light'' does not license photon-traveler ontology, and the operational use of \(c\) as a speed limit does not make \(c\) a maximum self-speed.
\end{proposition}

\begin{proof}
Operational shorthand identifies how a constant functions in measurement, calculation, and frame description. Ontology identifies what kind of thing is being claimed to exist. The fact that \(c\) appears as the invariant value assigned to lightlike relations in inertial-frame descriptions does not establish that a photon is a persisting object with an internal route. That further claim requires traveler ontology. Since the null photon lacks the rest-frame and proper-time structure required for such a traveler, the operational phrase does not license the ontological picture. Likewise, the fact that \(c\) bounds speeds assigned to massive objects from other frames does not make \(c\) a self-speed possessed by those objects. In an object's own rest frame, its own speed is zero. Therefore operational shorthand does not license traveler-owned speed ontology.
\end{proof}

\section{Invariance and Regime Status}

\begin{proposition}[Invariance supports the regime reading]
The invariance of \(c\) is naturally read as a property of the spacetime registration regime rather than as a property owned by a traveler-object.
\end{proposition}

\begin{proof}
The invariant \(c\) appears in the transformation structure relating inertial frames and in the causal structure of spacetime. It applies uniformly across inertial standpoints. No observer has privileged access to an absolute motion background. Since every observer registers events from within a frame-relative standpoint, and since the same invariant causal bound governs those registrations, \(c\) is naturally read as a regime property of spacetime description. It is not naturally read as a traveler-owned property of a photon-object, because the photon has no own-frame in which such ownership could be defined. It is not naturally read as a maximum self-speed for a massive object either, because a massive object with a rest frame has zero velocity in that frame.
\end{proof}

\begin{remark}
This is interpretive, not formal revision. The second postulate of special relativity remains untouched. The metric structure of general relativity remains untouched. Standard quantum and field-theoretic calculations remain untouched. What changes is the story attached to the constant: not traveler speed, not maximum self-speed, but registration bound.
\end{remark}

\section{Symmetry Strengthens the Reading}

\begin{proposition}[Symmetry of standpoint]
For any two inertial observers, each is at rest in their own frame, and each registers the other under the same invariant causal bound.
\end{proposition}

\begin{proof}
Each inertial observer has zero velocity in their own inertial frame. Each assigns relative motion to the other. For each observer, records, lightlike relations, and frame-relative relations are registered under the same invariant causal bound \(c\). Therefore the symmetry holds: each observer is at rest from their own standpoint, and each registers the other under that same bound.
\end{proof}

\begin{remark}
The symmetric application is the payoff. The two observers do not need a hidden third standpoint telling them who is really moving. Each is at rest in their own frame. Each assigns motion to the other. Each is governed by the same registration bound. That is exactly the structure a registration-bound reading expects.
\end{remark}

\section{Relation to the Timebound No-Go}

The companion timebound no-go result blocks a broader mistake: temporal governance does not imply traveler-bead ontology~\cite{Timebound}. A massive quantum system may be time-governed, countable, detectable, and relativistically constrained without being licensed as a little bead carrying a travel diary between records.

The present paper applies the same discipline to \(c\). A causal bound in spacetime description does not automatically identify a traveler who owns that bound as its speed. Relativity supplies frame structure and causal constraint. Quantum theory supplies admissibility and records. Neither supplies a little photon traveler whose internal story is a trip at speed \(c\). Nor does either supply a frame-independent rocket speed belonging to the rocket itself. Massive objects are assigned speeds relative to other frames; in their own rest frames, they are at rest. The invariant bound belongs to the registration structure, not to a traveler-object.

\section{What This Does Not Change}

\begin{proposition}[No formal revision]
The present proposal does not modify the equations of special relativity, general relativity, quantum mechanics, or quantum field theory.
\end{proposition}

\begin{proof}
The proposal introduces no new equations, no new observables, and no modifications to the metric structure of spacetime, the invariance of \(c\), the propagator structure of quantum field theory, or any predictive relation in standard physics. It concerns only the ontological reading attached to \(c\). Therefore it introduces no formal revision.
\end{proof}

\begin{proposition}[No new empirical prediction]
The present proposal makes no new empirical prediction.
\end{proposition}

\begin{proof}
Reading \(c\) as the bound on observer-side registration does not change the formulas in which \(c\) appears. The same constant remains in the same equations with the same numerical value and the same empirical role. Therefore the proposal is interpretive only.
\end{proof}

\section{Discussion}

The point of this paper is not that physics lacked \(c\). Physics has long calculated with \(c\). The point is that the traveler interpretation of \(c\) should not survive the removal of the photon traveler.

In that precise sense, \(c\) is not the speed of light. It is the invariant registration bound under which lightlike relations appear in spacetime description. The ordinary phrase survives as operational shorthand, but its traveler ontology does not.

Nor is \(c\) the maximum speed of a rocket in the rocket's own frame. In the rocket's own frame, the rocket is at rest. The bound applies to the rate at which another observer registers the rocket's frame-relative motion, and reciprocally to the rate at which the rocket observer registers the other frame's assigned motion. Thus \(c\) is not a maximum self-speed. It is the invariant bound on cross-frame registration.

In an observer's own inertial frame, the observer is at rest. Motion is assigned to other frames, systems, lightlike relations, or records relative to that standpoint. Records, lightlike relations, and frame-relative relations are registered under the invariant causal bound \(c\). That is the observer-side reading.

The driving example remains useful when stated carefully. In the driver's frame, the driver and car are at rest, and the roadside trees are assigned motion relative to the car. In the roadside frame, the trees are at rest, and the car is assigned motion relative to the road. Neither description has a privileged absolute standpoint. The registration of cross-frame relation is what each observer has.

The photon case sharpens the point. The photon has no rest frame. It has null proper time. It has no internally lived middle. A speed normally belongs to a thing as described in some frame, but the photon has no own-frame in which a traveler-speed story could be grounded. Thus the phrase ``speed of light'' remains an operational shorthand, while the traveler ontology attached to that phrase is refused.

The rocket case generalizes the point to massive systems. A rocket has a rest frame, and in that rest frame the rocket's own speed is zero. Other observers assign speed to the rocket from their frames; the rocket assigns speed to them from its frame. What is bounded is not a self-speed owned by the rocket, but the cross-frame registration of relative motion. Even in the ultra-relativistic limit, the assigned speed belongs to another frame's description; the rocket's own rest-frame speed remains zero.

The astounding feature of relativity, on this reading, is not that objects become strange at high speed. It is that speed itself is frame-relative registration, while \(c\) is the invariant bound governing that registration. The bound is real. The traveler story is not required.

\section{Conclusion}

In an observer's own inertial frame, the observer is at rest. Motion appears as the changing relation between that observer's standpoint and other frames, systems, lightlike relations, or records. The information speed limit, conventionally denoted \(c\), bounds the registration of those relations.

This does not change the equations. It changes the ontology attached to the usual phrase. In the precise ontological sense defended here, \(c\) is not the speed of light. It is not the speed of a photon traveler. The photon has no rest frame, no internal proper time, and no internally traversed route. Nor is \(c\) the maximum self-speed of a rocket or other massive object. In a massive object's own rest frame, its own speed is zero. Therefore \(c\) is better read as the invariant causal bound of spacetime registration: the limit governing how lightlike relations, null structure, and cross-frame motion appear in observer-side description.

The traveler is gone; the registration bound remains. The bound is real and invariant. The traveler story was never required.

\section*{Glossary}

\begin{description}
    \item[Information speed limit] The invariant causal bound conventionally denoted \(c\), read here as the bound on observer-side registration.

    \item[Registration] A spacetime-side record of an event, relation, signal, measurement, detection, or other physically definite occurrence in an observer's description.

    \item[Standpoint] The frame-relative position from which an observer registers events and assigns coordinates.

    \item[Cross-frame relation] The relation between one observer's standpoint and another frame, system, signal, or record as described from that observer's frame.

    \item[Traveler-owned speed] A speed interpreted as belonging to a persisting object moving through an internally lived route. This paper denies that \(c\) has this status for the photon.

    \item[Maximum self-speed] A speed interpreted as belonging to an object in its own rest frame. This paper denies that \(c\) has this status for massive objects; in a massive object's own rest frame, its own speed is zero.
\end{description}

\begin{thebibliography}{9}

\bibitem[McKinley(2026a)]{Bedrock}
J.~C.~W.~McKinley.
\newblock \emph{A Minimal Structural Statement of the Timeless Light Model}.
\newblock Zenodo (2026).
\newblock \doi{10.5281/zenodo.19167403}.

\bibitem[McKinley(2026b)]{Wavefunction}
J.~C.~W.~McKinley.
\newblock \emph{Wavefunction Prediction Does Not License a Photon Path}.
\newblock Zenodo (2026).
\newblock \doi{10.5281/zenodo.19504772}.

\bibitem[McKinley(2026c)]{Timebound}
J.~C.~W.~McKinley.
\newblock \emph{Timebound Does Not Mean Traveler: A No-Go on Deriving Bead-Path Ontology from Quantum Admissibility}.
\newblock Zenodo (2026).
\newblock \doi{10.5281/zenodo.20114078}.

\bibitem[Einstein(1905)]{Einstein1905}
A.~Einstein.
\newblock Zur Elektrodynamik bewegter K{\"o}rper.
\newblock \emph{Annalen der Physik} \textbf{17}, 891--921 (1905).
\newblock \doi{10.1002/andp.19053221004}.

\bibitem[Wald(1984)]{Wald}
R.~M.~Wald.
\newblock \emph{General Relativity}.
\newblock University of Chicago Press (1984).

\end{thebibliography}

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\title{\textbf{Timebound Does Not Mean Traveler}\\
\large A No-Go on Deriving Bead-Path Ontology from Quantum Admissibility}
\author{John C. W. McKinley\,\orcidlink{0009-0005-7097-5035}}
\date{May 10, 2026}

\begin{document}

\maketitle

\blfootnote{\scriptsize This version prepared for Zenodo. DOI: \href{https://doi.org/10.5281/zenodo.20114078}{10.5281/zenodo.20114078}.}

\begin{abstract}
A quantum system may be governed by temporal evolution and relativistic constraints without being licensed as a little object traveling along a hidden classical path. This note states a narrow interpretive no-go result: timebound admissibility does not imply traveler-bead ontology. The electron supplies the clean example. Unlike the photon, the electron has rest mass and may be counted as part of a physical system. But an electron in an atomic orbital is not licensed by standard quantum theory as a miniature planet orbiting a nucleus, and a free-electron state is not licensed as a bead secretly occupying a definite hidden route between records. Quantum theory supplies laws of admissible outcomes: states, amplitudes, conservation rules, transition rules, and detection probabilities. Relativity constrains admissible records by causal structure. Neither supplies a classical biography of a small object in transit. Detection gives a record. Quantum theory gives admissibility. Neither gives a travel diary.
\end{abstract}

\section{Introduction}

The photon case removes the traveler most severely. A photon has null proper time and no rest frame. On the Timeless Light Model reading, those standard constraints block photon object-location, intermediate occupancy, trajectory, and transit~\cite{McKinleyBedrock}.

The electron case is different. The electron has rest mass. The electron is countable. A neutral carbon atom contains six electrons. Electron number is part of ordinary physical accounting. The electron is therefore not a timeless null relation in the photon sense.

But this does not restore the classical bead.

The old picture says that the electron is a tiny thing moving around the nucleus, or a tiny thing moving through space along a hidden route until detection. Standard quantum mechanics does not supply that picture. An atomic orbital is not a planetary path. A quantum state is not a little object biography. A detector record is not a revelation of the route the electron secretly traveled.

The present paper states a narrow no-go result:

\begin{quote}
Timebound does not mean traveler.
\end{quote}

The claim is not that no interpretation could add hidden path structure by additional postulate. The claim is that ordinary quantum admissibility does not supply such structure, and that time-governed behavior does not entail it.

The point is not that massive quantum systems are timeless. The point is that temporal and relativistic constraint do not, by themselves, license a bead-path ontology. A quantum system may be governed by time-dependent law, may respect relativistic causal structure, and may yield countable records, without being licensed by those facts as a tiny object following a continuous classical route between those records.

\section{Definitions}

\begin{definition}[Timebound system]
A timebound system is a physical system whose admissible descriptions, records, or state evolution are governed by temporal or relativistic structure.
\end{definition}

\begin{definition}[Traveler-bead ontology]
Traveler-bead ontology is the interpretation according to which a quantum referent is a small persisting object occupying a continuous sequence of definite positions between records.
\end{definition}

\begin{definition}[Quantum admissibility]
Quantum admissibility is the lawful structure governing possible outcomes: states, amplitudes, observables, conservation laws, transition rules, exclusion rules, and detection probabilities.
\end{definition}

\begin{definition}[Record]
A record is a spacetime-side registration of an outcome, such as a detector event, measured transition, absorption event, scattering result, or other physical registration.
\end{definition}

\begin{definition}[Classical biography]
A classical biography is a hidden object-story in which a quantum referent is assigned a continuous sequence of occupied positions, velocities, and intermediate states between records.
\end{definition}

\section{The Mistake}

The mistake targeted here is the inference:

\[
\text{time-governed behavior}
\quad \Rightarrow \quad
\text{little traveler with a continuous path}.
\]

That inference is not licensed.

Temporal structure can govern admissible records without supplying a bead-path. Relativistic structure can constrain possible outcomes without supplying a hidden route. Quantum theory can evolve a state without assigning the system a classical itinerary.

This distinction matters because the rejection of traveler ontology is often treated as special to photons. It is not. The photon case is the strictest case because the photon has no proper time and no rest frame. But quantum theory already teaches a broader lesson: physical records do not automatically imply hidden classical object-biographies.

\section{The Electron Case}

An electron has rest mass. A free electron admits massive-particle descriptions. Electrons may be counted in atoms, ions, solids, currents, and scattering experiments. None of this is denied.

What is denied is the extra claim that the electron must therefore be pictured as a small bead following a hidden classical route between records.

In atomic physics, the electron in an orbital is not a tiny planet orbiting the nucleus. The orbital is a lawful quantum state structure. It fixes admissible energies, angular momentum structure, transition rules, amplitudes, and detection probabilities. It does not supply a little path around the nucleus.

The same warning applies to a free electron. A free-electron state may be represented in different formal bases. Those representations are not themselves material waves, hidden bead-locations, or classical travel diaries. They are structures used to compute admissible records.

Hidden-variable interpretations may add additional structure by postulate. That is not the target here. The target is the unlicensed inference from ordinary quantum admissibility to traveler-bead ontology. Quantum theory supplies the record structure; it does not, by itself, supply a bead-path.

Thus the electron supports the following principle:

\begin{quote}
A countable massive quantum referent need not be a traveler-bead.
\end{quote}

\section{The No-Go Result}

\begin{proposition}[Timebound does not mean traveler]
A quantum system may be governed by temporal or relativistic constraints without being licensed as a persisting bead-like object traveling along a hidden classical path between records.
\end{proposition}

\begin{proof}
Temporal and relativistic constraints govern admissible descriptions, state evolution, records, transitions, and causal relations. They restrict what outcomes may occur and how those outcomes may be related. But these constraints do not, by themselves, assign a continuous sequence of occupied positions to a quantum referent. A traveler-bead ontology requires more than temporal governance: it requires a licensed object occupying intermediate locations along a classical route. Quantum admissibility supplies possible records and probabilities, not a hidden classical biography. Therefore being timebound does not imply being a traveler.
\end{proof}

\begin{corollary}[Countability does not imply bead-path ontology]
The fact that a quantum referent can be counted does not imply that it follows a hidden classical path.
\end{corollary}

\begin{proof}
Counting establishes an inventory or record within a specified physical context. A hidden classical path requires a continuous sequence of definite intermediate positions. The former does not supply the latter. Electrons may be counted, but an electron in an orbital is not thereby licensed as a tiny object orbiting the nucleus. Therefore countability does not imply bead-path ontology.
\end{proof}

\begin{corollary}[Relativistic constraint does not imply classical itinerary]
The fact that a quantum system respects relativistic causal structure does not imply that it possesses a classical itinerary between records.
\end{corollary}

\begin{proof}
Relativistic causal structure constrains which records, transitions, and correlations are physically admissible. A classical itinerary is a further claim: that the system occupied a determinate sequence of intermediate positions. The causal constraint does not supply that sequence. Therefore relativistic constraint does not imply classical itinerary.
\end{proof}

\section{Detection Is Not Biography}

A detection record is real. It is a physical event. It may be localized, time-stamped, measured, compared, and counted.

But a detection record is not a biography.

If an electron is detected at a location, the record does not establish that the electron was a tiny bead traveling along a hidden path to that point. If an electron is later detected elsewhere, the pair of records does not, by itself, fill in a classical route between them. Quantum theory supplies lawful transition amplitudes and admissible records. It does not supply a miniature travel diary unless an additional classical model is imposed.

\begin{proposition}[Detection does not supply hidden route]
A localized detection record does not, by itself, establish a hidden classical route prior to detection.
\end{proposition}

\begin{proof}
A localized detection record establishes that a physical registration occurred under specified measurement conditions. It does not establish that the registered quantum referent occupied a continuous sequence of definite positions before the record. The route is an additional classical interpretation, not a consequence of the record itself. Therefore detection does not supply hidden route.
\end{proof}

\begin{remark}
This point is familiar from atomic orbitals. The electron is detected through records and transitions, but the orbital is not a planetary track. The record is real; the classical path is not supplied.
\end{remark}

\section{Relation to Photon Ontology}

The photon case remains stricter than the electron case.

For the photon, standard relativity gives null proper time and no rest frame. The Timeless Light Model treats those facts as ontologically restrictive: no photon rest frame, no photon object-location, no intermediate location, no trajectory, no transit. The photon is a lawfully admissible charge-state relation whose spacetime appearance is a lawful change~\cite{McKinleyBedrock}.

For the electron, the result is different. The electron has rest mass and may be counted. The no-go is not that electrons cannot be counted. The no-go is that counting, mass, temporal evolution, and relativistic constraint do not restore traveler-bead ontology.

The two results therefore form an asymmetric hierarchy:

\[
\text{timebound}
\not\Rightarrow
\text{traveler-bead ontology},
\]

\[
\text{null/timeless}
\Rightarrow
\text{traveler-bead ontology foreclosed}.
\]

The photon case removes the traveler by null structure. The electron case removes only the inference from timebound governance to traveler ontology: mass, countability, temporal evolution, and relativistic constraint do not themselves supply a hidden classical route. A further interpretation may add additional structure, but it is not delivered by quantum admissibility alone.

\section{Admissibility Is Not Itinerary}

Quantum admissibility is a rule-structure for outcomes. It is not an itinerary.

An itinerary says where a thing went. An admissibility structure says which records are allowed, which transitions are possible, which amplitudes enter, and which conservation rules apply.

The difference is decisive. A lawful outcome structure can be time-dependent without becoming a travel story. The Schrödinger equation, the Dirac equation, and path-integral formulations give lawful structure for state descriptions, amplitudes, and records~\cite{Schrodinger1926,Dirac1928,Feynman1948}. They do not, by themselves, turn the quantum referent into a bead occupying every intermediate step of a hidden route.

\begin{proposition}[Admissibility is not itinerary]
A law of admissible quantum outcomes is not a hidden itinerary of a traveling object.
\end{proposition}

\begin{proof}
A law of admissible outcomes specifies the conditions under which records, transitions, interactions, or measurements may occur. A hidden itinerary specifies a continuous sequence of occupied positions by a persisting object. These are different structures. The first constrains possible records. The second narrates a classical path. The first does not entail the second. Therefore admissibility is not itinerary.
\end{proof}

\section{What Survives}

The no-go removes only the traveler-bead inference. It leaves ordinary physics intact.

\begin{enumerate}
    \item Electrons remain countable.
    \item Electron rest mass remains.
    \item Atomic spectra remain.
    \item Orbital state descriptions remain.
    \item Detector records remain.
    \item Relativistic causal constraints remain.
    \item Quantum dynamics remains.
\end{enumerate}

What fails is the unauthorized extra picture:

\begin{quote}
Because the system is timebound, massive, countable, or detected, it must be a tiny traveler with a hidden path.
\end{quote}

That conclusion does not follow.

\section{Main Result}

\begin{proposition}[Timebound quantum systems do not require traveler-bead ontology]
A timebound quantum system may possess mass, countability, temporal evolution, and relativistic constraint without being licensed as a bead-like traveler following a continuous classical route between records.
\end{proposition}

\begin{proof}
Mass and countability establish physical features of the system. Temporal evolution and relativistic constraint establish lawful structure governing admissible descriptions and records. None of these establishes a continuous sequence of occupied intermediate positions. Traveler-bead ontology requires such a sequence. Therefore a timebound quantum system does not require traveler-bead ontology.
\end{proof}

\begin{corollary}[Electron orbital no-go]
An electron in an atomic orbital is not licensed by standard quantum theory as a tiny object orbiting the nucleus along a hidden classical path.
\end{corollary}

\begin{proof}
An atomic orbital supplies a quantum state structure governing admissible energies, amplitudes, transitions, and detection probabilities. A hidden classical orbit would require a definite path around the nucleus. The orbital structure does not supply such a path. Therefore the electron in an atomic orbital is not licensed by standard quantum theory as a tiny object orbiting the nucleus.
\end{proof}

\section{Conclusion}

Timebound does not mean traveler.

The electron is massive. The electron may be counted. The electron participates in time-governed quantum descriptions and relativistic causal structure. But none of this licenses, from standard quantum admissibility alone, the image of a tiny bead traveling along a hidden classical route between records.

Quantum theory supplies laws of admissible outcomes. Relativity supplies causal constraint. Detection supplies records. None supplies a travel diary.

The no-go is therefore simple:

\[
\text{timebound} \neq \text{traveler-bead}.
\]

The photon case forecloses the traveler through null structure. The electron case blocks the inference from timebound quantum admissibility to classical route. Together they show the same discipline: ordinary lawful physical structure is not a license for cartoon transit ontology.

\begin{thebibliography}{9}

\bibitem[McKinley(2026)]{McKinleyBedrock}
J.~C.~W. McKinley.
\newblock \emph{A Minimal Structural Statement of the Timeless Light Model}.
\newblock Zenodo (2026).
\newblock \doi{10.5281/zenodo.19167403}.

\bibitem[Einstein(1905)]{Einstein1905}
A.~Einstein.
\newblock Zur Elektrodynamik bewegter K{\"o}rper.
\newblock \emph{Annalen der Physik} \textbf{17}, 891--921 (1905).
\newblock \doi{10.1002/andp.19053221004}.

\bibitem[Schrödinger(1926)]{Schrodinger1926}
E.~Schrödinger.
\newblock Quantisierung als Eigenwertproblem.
\newblock \emph{Annalen der Physik} \textbf{79}, 361--376 (1926).
\newblock \doi{10.1002/andp.19263840404}.

\bibitem[Dirac(1928)]{Dirac1928}
P.~A.~M. Dirac.
\newblock The quantum theory of the electron.
\newblock \emph{Proceedings of the Royal Society A} \textbf{117}, 610--624 (1928).
\newblock \doi{10.1098/rspa.1928.0023}.

\bibitem[Feynman(1948)]{Feynman1948}
R.~P. Feynman.
\newblock Space-time approach to non-relativistic quantum mechanics.
\newblock \emph{Reviews of Modern Physics} \textbf{20}, 367--387 (1948).
\newblock \doi{10.1103/RevModPhys.20.367}.

\end{thebibliography}

\end{document}

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\lhead{Spacetime Changes Can Be Counted; Photons Cannot}
\rhead{John C. W. McKinley}
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\title{\textbf{Spacetime Changes Can Be Counted; Photons Cannot}\\
\large A No-Go Result on Photon-Object Inventory}
\author{John C. W. McKinley\,\orcidlink{0009-0005-7097-5035}}
\date{May 10, 2026}

\begin{document}

\maketitle

\blfootnote{\scriptsize This version prepared for Zenodo. DOI: \href{https://doi.org/10.5281/zenodo.20113982}{10.5281/zenodo.20113982}.}

\begin{abstract}
Spacetime-side changes are countable. Photon-objects are not. A detector may register events, an apparatus may record absorptions, and a field-state description may assign an occupation number. But these are counts of records, changes, or representation-relative quantities. They are not counts of persisting photon-objects. In standard relativistic physics, the photon has null proper time and no rest frame. The Timeless Light Model treats those standard constraints as ontologically restrictive rather than as harmless formal curiosities: no photon rest frame means no defined photon object-location, no intermediate location, no trajectory, and no transit. This paper states a narrow two-pronged no-go result: photon-object inventory is not licensed by the null constraints; and even if licensed, any closed photon-object quantity would import a census frame foreign to the null case. Ordinary photon-counting practice is preserved. Endpoint records remain countable. State-relative photon-number descriptions remain valid. What fails is the stronger claim that there exists a completed stock of photon-things in flight.
\end{abstract}

\section{Introduction}

The phrase ``one photon'' is useful. The phrase ``two detector events'' is useful. The phrase ``an \(N\)-photon state'' is useful. None of these expressions should be erased from physics.

The question is what these expressions count.

This paper argues that spacetime-side changes may be counted, but photons are not countable as persisting objects. The distinction is not semantic. It follows from taking the standard null constraints seriously. In standard relativistic physics, a photon has null proper time and no rest frame~\cite{Einstein1905,Wald1984}. TLM refuses to treat those facts as decorative formalities. If the photon has no rest frame, then no photon object-location, intermediate position, trajectory, or transit is licensed. The photon is therefore not a particle in transit, but a lawfully admissible charge-state relation whose spacetime appearance is a lawful change~\cite{McKinleyBedrock}.

The present paper does not modify Special Relativity, General Relativity, quantum mechanics, or quantum field theory. It imposes an interpretive restriction:

\begin{quote}
Do not convert countable spacetime-side changes into a completed inventory of photon-objects.
\end{quote}

The no-go result is narrow. It does not deny detections. It does not deny absorptions. It does not deny emission records. It does not deny state-relative photon-number descriptions. It denies only the extra ontology of a counted photon-stock in between.

The purpose of this note is therefore to preserve ordinary photon-counting practice while blocking a stronger ontological inference. Detector records, absorption events, source-side changes, and state-relative occupation numbers may be counted. A persisting photon-object in spacetime may not be inventoried, because standard null constraints license no such object.

\section{Definitions}

\begin{definition}[Spacetime appearance]
A spacetime appearance is the manifestation of a lawfully admissible charge-state relation as a lawful change within spacetime description.
\end{definition}

\begin{definition}[Spacetime-side change]
A spacetime-side change is the emission-side, absorption-side, detector-side, or measured transition through which the spacetime appearance is described.
\end{definition}

\begin{definition}[Spacetime-side record]
A spacetime-side record is a discrete registration of a spacetime-side change, including a detector click, absorption record, emission-side label, measured transition, or state-relative number assignment.
\end{definition}

\begin{definition}[Photon-object]
A photon-object is a supposed persisting item located in spacetime, possessing intermediate presence between emission and absorption.
\end{definition}

\begin{definition}[Photon-object inventory]
A photon-object inventory is a closed stock of persisting photon-things:
\[
|\Gamma|=N,
\]
where \(\Gamma\) is treated as a bounded set of photon-objects.
\end{definition}

\begin{definition}[Census frame]
A census frame is the standpoint required to treat a domain as a completed inventory. It supplies a boundary, an inclusion rule, an exclusion rule, and a completed count condition.
\end{definition}

\begin{definition}[Charge-state relation]
A charge-state relation is the lawful relation underlying what spacetime description renders process-wise as emission, absorption, or transfer.
\end{definition}

\section{Spacetime-Side Changes Are Countable}

\begin{proposition}[Spacetime-side changes may be counted]
Spacetime-side changes may be counted as detector records, endpoint events, or representation-relative state quantities.
\end{proposition}

\begin{proof}
A detector record is a spacetime-side event. If an apparatus records \(N\) such events, then \(N\) records have occurred within the measurement arrangement. Likewise, a specified field-state description may assign an occupation number relative to its chosen representation. In both cases, the count attaches to a spacetime-side record, change, or formal state description. Therefore spacetime-side changes may be counted.
\end{proof}

\begin{remark}
The claim is not that photon-counting language is useless. The claim is that one must identify what is being counted.
\end{remark}

\begin{proposition}[Counting spacetime-side changes does not imply counting photon-objects]
A count of spacetime-side changes does not imply a count of persisting photon-objects.
\end{proposition}

\begin{proof}
A spacetime-side change is a recorded event, measured transition, or state-relative description. A photon-object is a supposed persisting item located between endpoints. The first is supplied by spacetime-side measurement or representation. The second is an additional ontology. Therefore a count of spacetime-side changes does not imply a count of photon-objects.
\end{proof}

\section{The Null Constraint}

The central physical point is not proprietary to TLM. Standard relativistic physics already assigns the photon null proper time and no rest frame. TLM's contribution is the interpretive discipline of refusing to reinsert an object-in-flight after those constraints have removed the conditions for one.

The minimal chain is:

\[
\begin{aligned}
d\tau=0 \;&\Rightarrow\; \text{no photon rest frame} \\
&\Rightarrow\; \text{no defined photon object-location} \\
&\Rightarrow\; \text{no intermediate location} \\
&\Rightarrow\; \text{no trajectory} \\
&\Rightarrow\; \text{no transit}.
\end{aligned}
\]

The first two links are standard. The later links state the interpretive consequence enforced here. A located object requires location predicates. A persisting traveler requires intermediate states. A trajectory requires ordered occupancy. A transit story requires a carrier to be in transit. The null case supplies none of these for the photon as an object.

\begin{proposition}[No located photon-object]
The photon is not licensed as a located object in spacetime.
\end{proposition}

\begin{proof}
A located object in spacetime requires spatial predicates sufficient for location, intermediate position, and persistence across a sequence of states. In standard relativity, the photon has null proper time and no rest frame. Without a rest frame, the photon does not possess object-location predicates in the sense required for a persisting item. Without object-location predicates, no intermediate location is licensed. Without intermediate location, no trajectory is licensed. Without trajectory, no transit is licensed. Therefore the photon is not licensed as a located object in spacetime.
\end{proof}

\begin{corollary}[What has no object-location is not here]
If the photon has no defined location as an object in spacetime, then it is not a thing here, there, or halfway.
\end{corollary}

\begin{proof}
The predicates ``here,'' ``there,'' and ``halfway'' are location predicates. If location is not defined for the photon as an object, those predicates do not apply to the photon as an object. Therefore the photon is not here, there, or halfway.
\end{proof}

\begin{remark}
This does not deny that spacetime descriptions contain endpoint labels, null relations, field modes, detector records, or measured changes. It denies that such descriptions license an occupied intermediate history.
\end{remark}

\section{The First No-Go: No Photon-Object Count}

\begin{proposition}[Photon-objects cannot be counted]
Photon-objects cannot be counted because the photon-object is not licensed.
\end{proposition}

\begin{proof}
To count photon-objects, there must be photon-objects. A photon-object, as defined here, is a persisting located item in spacetime. By the preceding result, the photon is not licensed as a located object in spacetime. Therefore the required object of the count is absent. Hence photon-objects cannot be counted.
\end{proof}

\begin{corollary}[There is no photon stock]
There is no completed stock of photon-objects between emission and absorption.
\end{corollary}

\begin{proof}
A stock is an inventory of persisting items. Photon-objects are not licensed as persisting located items. Therefore no photon stock exists between emission and absorption.
\end{proof}

\begin{remark}
The point is stronger than saying that a photon stock is hard to observe. The point is that the photon-stock ontology is not licensed by the null constraints.
\end{remark}

\section{The Second No-Go: Closed Photon Quantity Imports a Census Frame}

Even if the first no-go is ignored, a second no-go remains. A completed photon quantity imports a census frame. A census frame imports temporal or quasi-temporal structure.

The problem appears whenever one says:

\[
|\Gamma|=N,
\]
where \(\Gamma\) is treated as the completed set of photon-things.

That statement is not merely a count. It is a closed inventory. A closed inventory requires one of the following hidden structures.

\subsection{Case 1: There are \(N\) photons now}

If the claim means

\[
|\Gamma(t)|=N,
\]
then the count is explicitly indexed to a time \(t\). The word ``now'' supplies a census moment.

\begin{quote}
There are \(N\) photons now.
\end{quote}

This is a time-indexed inventory. It belongs to spacetime-side accounting, not to the photon as a null charge-state relation.

\subsection{Case 2: There were \(N-1\) photons before}

If the count changes from \(N-1\) to \(N\), then the account introduces before and after:

\[
|\Gamma(t_1)|=N-1,
\qquad
|\Gamma(t_2)|=N,
\qquad
t_1<t_2.
\]

This is not timeless. It is a temporal population history.

\subsection{Case 3: There are always \(N\) photons}

If the claim is that there are always \(N\) photons, then the claim becomes:

\[
\forall t,\ |\Gamma(t)|=N.
\]

The word ``always'' is not timeless. It quantifies across time. It treats the inventory as stable throughout a temporal range.

\subsection{Case 4: A source produced \(N\) photons}

If the claim is that a source produced \(N\) photons, then the account introduces production sequence:

\[
\text{not-yet-produced}
\quad \longrightarrow \quad
\text{produced}.
\]

This is again before/after structure. It may be acceptable as source-side spacetime description, but it does not define a photon-object in the middle.

\begin{lemma}[Completed quantity imports a census frame]
A completed quantity of photon-objects requires a census frame.
\end{lemma}

\begin{proof}
A completed quantity says that the total inventory is \(N\), no more and no less. Such a claim requires a boundary around the relevant domain, an inclusion rule, an exclusion rule, and a condition under which the count is complete. These jointly form a census frame.
\end{proof}

\begin{lemma}[A census frame imports time-like structure]
A census frame imports time-like structure through a census moment, a before/after comparison, an always-condition, or a production sequence.
\end{lemma}

\begin{proof}
A completed inventory must be final relative to some standpoint. If the standpoint is ``now,'' a census moment is introduced. If the inventory changes, before/after comparison is introduced. If the inventory is said to be permanently fixed, an always-condition is introduced. If the inventory is produced, production sequence is introduced. Each route imports temporal or quasi-temporal structure.
\end{proof}

\begin{proposition}[Closed photon-object quantity is a category error]
A closed quantity of photon-objects is a category error.
\end{proposition}

\begin{proof}
A closed quantity of photon-objects requires photon-objects and a census frame. Photon-objects are not licensed, because the photon is not a persisting located item in spacetime. A census frame also imports temporal or quasi-temporal structure, which is incompatible with the photon understood as a null charge-state relation without internal time. Therefore a closed quantity of photon-objects is a category error.
\end{proof}

\section{What Is Counted Instead}

The denial of photon-object count does not erase physics. It relocates the count to the correct level.

\begin{enumerate}
    \item Detector records may be counted.
    \item Absorption-side changes may be counted.
    \item Emission-side changes may be counted.
    \item State-relative occupation numbers may be assigned.
    \item Endpoint correlations may be described.
    \item Lawful charge-state changes may be described in spacetime.
\end{enumerate}

These are all counts or descriptions of records, changes, events, or formal states. They are not inventories of photon-things.

\begin{proposition}[Photon-counting is record-counting or state-relative number assignment]
What is called photon-counting is, strictly, counting spacetime-side records or assigning state-relative photon-number quantities.
\end{proposition}

\begin{proof}
A physical count occurs through records, detections, absorptions, emissions, or representation-relative state descriptions. None of these requires a persisting photon-object in the interval. Therefore photon-counting is strictly record-counting or state-relative number assignment, not photon-object inventory.
\end{proof}

\section{Endpoint Accounting}

The clean description is endpoint accounting:

\[
\text{emission-side lawful change}
\quad \longleftrightarrow \quad
\text{absorption-side lawful change}.
\]

The photon is the name attached to the admissible charge-state relation. The relation has a spacetime appearance; the endpoint records are physical; the intermediate traveler is not added.

\begin{center}
\begin{tikzpicture}[scale=1.05, font=\footnotesize]
    % axes
    \draw[->] (-0.2,0) -- (7,0) node[right] {$x$};
    \draw[->] (0,-0.2) -- (0,4.5) node[above] {$ct$};

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    \draw[thick,dashed] (0.9,0.7) -- (5.7,3.9);

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\section{Relation to Photon Number in Quantum Theory}

Quantum theory permits photon-number language. Quantum field theory permits occupation-number language. This paper does not dispute either practice.

The restriction is ontological. A number in a specified state representation does not force a stock of tiny travelers. The formal count belongs to the formal context in which it is defined. It does not restore a photon rest frame, a photon location, a photon path, or a photon census in the middle.

\begin{proposition}[Occupation number does not restore photon-object ontology]
A photon-number state does not by itself establish a population of persisting photon-objects in spacetime.
\end{proposition}

\begin{proof}
A photon-number state is a representation-relative formal description. A population of persisting photon-objects requires located items with intermediate presence. The formal description supplies the former. It does not supply rest-frame location, internal proper time, intermediate trajectory, or object persistence. Therefore occupation number does not restore photon-object ontology.
\end{proof}

\begin{remark}
This is the same interpretive discipline applied elsewhere in the Timeless Light Model: successful formal representation does not license a forbidden middle.
\end{remark}

\section{Why the Distinction Matters}

If one treats spacetime-side records as photon-objects, pseudo-questions appear:

\begin{itemize}
    \item Where is the photon halfway?
    \item How does it know where to go?
    \item Which route did it really take?
    \item How many photon-things were in the interval?
    \item Did the photon wait, choose, update, or correct its path?
\end{itemize}

Each question assumes a located traveler. The TLM answer is not to add hidden process. The answer is to reject the premise.

The photon is not a thing traveling through spacetime. It is a lawful charge-state relation whose spacetime appearance is a lawful change. What is counted is the record, change, or representation-relative quantity, not a passenger.

\section{Main Result}

\begin{proposition}[Spacetime-side changes can be counted; photons cannot]
Spacetime-side changes can be counted as records, endpoint events, or representation-relative quantities. Photons cannot be counted as photon-objects, because no photon-object is licensed by the standard null constraints.
\end{proposition}

\begin{proof}
Spacetime-side changes are recorded or represented inside spacetime descriptions. They therefore admit counts relative to those descriptions. A photon-object would be a persisting located item in spacetime. But in standard relativistic physics, the photon has null proper time and no rest frame. Under the TLM interpretation, those constraints block photon object-location, intermediate location, trajectory, and transit. Therefore the photon-object is not licensed. Consequently spacetime-side changes can be counted, while photons cannot be counted as photon-objects.
\end{proof}

\begin{corollary}[No completed quantity for null charge-state relations]
A null charge-state relation cannot be assigned a completed quantity as a stock of things.
\end{corollary}

\begin{proof}
A completed stock of things requires countable objects. A null charge-state relation is not a persisting object in spacetime. Therefore it cannot be assigned a completed quantity as a stock of things.
\end{proof}

\section{Conclusion}

Spacetime-side changes are countable. Photons are not countable as objects.

A detector may count events. A field description may assign an occupation number. A source-side or absorption-side process may be described in ordinary spacetime language. But the photon itself is not a located item in spacetime. Standard relativity gives the photon null proper time and no rest frame. TLM takes those facts seriously: no photon rest frame means no photon object-location, no intermediate location, no trajectory, and no transit.

Therefore the photon is not a tiny thing with unusual properties. It is not here. It is not halfway. It is not in the interval. It is a lawfully admissible charge-state relation whose spacetime appearance is a lawful change.

The completed inventory question therefore fails twice. First, there is no photon-object to inventory. Second, any attempt to impose a closed photon quantity imports temporal structure through a census moment, a before/after comparison, an always-condition, or a production sequence.

The clean result is:

\[
\text{countable spacetime-side change} \neq \text{countable photon-object}.
\]

Null means null. The photon is counted only where a record, change, or state-relative quantity is available in spacetime accounting. It is not counted as a thing in flight, because there is no thing in flight.

\begin{thebibliography}{9}

\bibitem[McKinley(2026)]{McKinleyBedrock}
J.~C.~W. McKinley.
\newblock \emph{A Minimal Structural Statement of the Timeless Light Model}.
\newblock Zenodo (2026).
\newblock \doi{10.5281/zenodo.19167403}.

\bibitem[Einstein(1905)]{Einstein1905}
A.~Einstein.
\newblock Zur Elektrodynamik bewegter K{\"o}rper.
\newblock \emph{Annalen der Physik} \textbf{17}, 891--921 (1905).
\newblock \doi{10.1002/andp.19053221004}.

\bibitem[Wald(1984)]{Wald1984}
R.~M. Wald.
\newblock \emph{General Relativity}.
\newblock University of Chicago Press (1984).

\end{thebibliography}

\end{document}

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\lhead{Hawking Radiation Is an Exterior Result}
\rhead{John C. W. McKinley}
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\title{\textbf{Hawking Radiation Is an Exterior Result}\\
\large A Short Interpretive No-Go on Interior Ontology}
\author{John C. W. McKinley \orcidlink{0009-0005-7097-5035}}
\date{May 12, 2026}

\begin{document}

\maketitle

\blfootnote{\scriptsize This version prepared for Zenodo. DOI: \href{https://doi.org/10.5281/zenodo.20100229}{10.5281/zenodo.20100229}.}

\begin{abstract}
Hawking radiation is a result about exterior observables. The standard derivation relates an initial vacuum state to late-time exterior particle content through quantum field modes on a collapsing black-hole spacetime. Its measurable outputs are the thermal flux detected at future infinity and the corresponding decrease of the black hole's asymptotic mass. This note states a narrow interpretive no-go result: the Hawking derivation does not, by itself, license an interior ontology. It does not establish what an infalling observer experiences, what localized field configuration exists behind the horizon, or whether a negative-energy particle literally falls inward and reduces the black hole's mass. Those are additional interpretive or theoretical claims. The exterior calculation licenses exterior radiation and exterior mass loss. It does not license interior narration.
\end{abstract}

\section{Introduction}

Hawking radiation is commonly described as if the calculation tells a complete story about both sides of the event horizon. A virtual pair appears; one member escapes; the other falls into the black hole with negative energy; the black hole loses mass. This story is useful as a teaching image, but it says more than the standard derivation establishes.

The standard Hawking result is formulated in terms of field modes, observer-relative particle content, thermal flux at future infinity, and mass measured by the asymptotic geometry. These are exterior or asymptotic quantities. They do not, by themselves, settle what occurs behind the horizon.

This note states a narrow no-go result. The claim is not that black-hole interiors do not exist. The claim is not that no future theory can describe the interior. The claim is not that semiclassical gravity is false. The claim is narrower: the standard Hawking derivation does not license interior ontology. The issue here is not merely whether a particle travels from inside the black hole to the exterior; it is whether an exterior calculation licenses any interior story at all.

An exterior derivation licenses exterior observables. It does not, by itself, license an interior story.

\section{Exterior Structure of the Hawking Result}

In the standard treatment, one studies a quantum field on a spacetime that begins with regular asymptotic structure and later forms a black hole by collapse. The field is assigned an initial vacuum state, usually described at past null infinity. Late-time observers at future null infinity decompose the field into outgoing modes and assign particle content to the state using the corresponding mode basis.

The Hawking effect arises because the early and late mode decompositions do not match. A state that is vacuum relative to the early basis is not vacuum relative to the late exterior basis. The Bogoliubov transformation between the two descriptions yields nonzero late-time occupation numbers and a thermal spectrum.

For a Schwarzschild black hole, the exterior temperature measured at infinity is
\[
T_H = \frac{\hbar c^3}{8\pi G M k_B}.
\]
The outgoing radiation carries positive energy to infinity. In the standard semiclassical extension of Hawking's result, the corresponding black-hole mass, as measured by the asymptotic geometry, decreases.

These statements are physical. The radiation is real. The mass loss is real. The no-go concerns only the inference from these exterior facts to an interior ontology.

\section{Definitions}

\begin{definition}[Exterior observable]
An exterior observable is a quantity defined for observers or measurements outside the black-hole horizon, especially at asymptotic infinity, such as outgoing flux, exterior particle content, and ADM mass.
\end{definition}

\begin{definition}[Interior ontology]
Interior ontology is a claim about what exists, occurs, or is experienced behind the event horizon, including claims about localized particles, negative-energy objects, field configurations, or infalling-observer experience.
\end{definition}

\begin{definition}[Interior narration]
Interior narration is the explanatory move of converting an exterior result into a story about events or objects behind the horizon.
\end{definition}

\begin{definition}[Frame restriction]
Frame restriction is the interpretive rule that a result defined in one descriptive regime cannot be used, without additional argument, to assert ontology in another regime.
\end{definition}

\section{The No-Go Result}

\begin{proposition}[The Hawking derivation is exterior-asymptotic]
The standard Hawking derivation establishes late-time exterior radiation by relating early and late field-mode decompositions. It does not require an independently specified interior particle history.
\end{proposition}

\begin{proof}
The derivation begins with an initial field state and compares mode decompositions associated with past and future asymptotic descriptions. The thermal result is read by late-time exterior observers, especially at future infinity. The calculation yields exterior particle content and outgoing flux. None of these steps requires the identification of a localized particle configuration behind the horizon. Therefore the standard derivation is exterior-asymptotic in its operative content.
\end{proof}

\begin{proposition}[ADM mass loss is an exterior statement]
The decrease of black-hole mass in Hawking evaporation is a real physical statement about the asymptotic geometry, but it is not an interior narrative.
\end{proposition}

\begin{proof}
The relevant black-hole mass in the standard evaporation statement is defined by the exterior spacetime, in particular by the mass parameter measured at infinity. In the standard semiclassical extension of Hawking's result, if outgoing radiation carries positive energy to infinity, then conservation requires a corresponding decrease in the mass attributed to the black hole by the asymptotic geometry. This establishes real exterior mass loss. It does not specify a localized interior mechanism, an infalling object, or an interior observer's account of that loss. Therefore ADM mass loss is an exterior statement, not an interior ontology.
\end{proof}

\begin{proposition}[Negative-energy infall is not licensed ontology]
The common statement that a negative-energy particle falls into the black hole is not established as literal interior ontology by the Hawking derivation.
\end{proposition}

\begin{proof}
The negative-energy-infall story is a heuristic device for representing energy conservation in the pair-creation picture. The standard field-theoretic result does not require a localized negative-energy particle behind the horizon. It requires an outgoing exterior flux and a corresponding decrease in the black hole's asymptotic mass. A bookkeeping narration that helps preserve conservation in a visual model does not establish the existence of the narrated interior object. Therefore negative-energy infall is not licensed ontology.
\end{proof}

\begin{proposition}[Interior experience is not determined by exterior flux]
The exterior detection of Hawking radiation does not determine what an infalling observer experiences behind the horizon.
\end{proposition}

\begin{proof}
Particle content is observer-relative in quantum field theory on curved spacetime. A late-time observer at infinity assigns particle content using an exterior mode decomposition. An infalling observer uses a different local description. In the standard semiclassical picture, an infalling observer crossing the horizon detects locally regular vacuum, not the thermal flux registered at infinity. The fact that radiation is detected at infinity therefore does not, by itself, determine the infalling observer's particle content, field description, or horizon-crossing experience. Thus exterior flux does not determine interior experience.
\end{proof}

\begin{proposition}[Interior ontology requires additional theory]
Claims about the black-hole interior during evaporation require assumptions or formalisms beyond the standard exterior Hawking derivation.
\end{proposition}

\begin{proof}
Questions about the interior include the field state behind the horizon, the fate of infalling information, the experience of an infalling observer, the role of backreaction, and the ultimate resolution of the singularity. The standard Hawking derivation supplies an exterior thermal flux and a corresponding asymptotic mass decrease. It does not resolve these interior questions. Therefore any determinate interior ontology requires additional theoretical commitments beyond the exterior derivation itself.
\end{proof}

\section{The Negative-Energy Story Revisited}

The negative-energy story is the most common way interior ontology enters public explanations of Hawking radiation. The story says that a virtual pair appears at the horizon, the positive-energy member escapes, and the negative-energy member falls inward, reducing the black hole's mass.

This narration is stronger than the derivation. The derivation requires an exterior flux and exterior mass loss. It does not require a literal interior particle doing the subtraction.

The black hole's mass loss is real. The outgoing radiation is real. What is denied is the extra claim that the mass loss has been explained by a localized negative-energy object traveling inward behind the horizon.

The clean statement is:

\begin{quote}
The black hole loses mass as measured from the exterior. The standard Hawking derivation does not convert that exterior fact into an interior particle story.
\end{quote}

\section{Relation to Interior Debates}

The no-go stated here does not solve the black-hole information problem. It does not decide between complementarity, firewall proposals \cite{AMPS2013}, holographic reconstruction, entanglement-based interior proposals \cite{MaldacenaSusskind2013}, or other approaches to black-hole interiors. Its point is narrower and prior: the standard Hawking derivation itself does not supply the interior ontology that those debates contest.

This is why the existence of those debates is significant. If the exterior Hawking calculation already fixed the interior story, there would be no need for competing accounts of horizon crossing, interior reconstruction, information recovery, or breakdown of semiclassical smoothness. The persistence of those disputes reflects the limited scope of the original result.

The no-go is therefore conservative. It does not deny that interior questions matter. It denies only that the standard exterior calculation has already answered them.

\section{Conclusion}

Hawking radiation is an exterior result.

The standard derivation establishes thermal radiation detected by late-time exterior observers and a corresponding decrease in black-hole mass as measured by the asymptotic geometry. These are real physical results. But they do not, by themselves, establish what exists, occurs, or is experienced inside the horizon.

The negative-energy particle story is therefore not licensed as literal interior ontology. It is an explanatory narration attached to an exterior result.

The no-go is simple: exterior radiation and exterior mass loss do not license interior narration.

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S.~W. Hawking.
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\newblock \doi{10.1007/BF02345020}.

\bibitem[Unruh(1976)]{Unruh1976}
W.~G. Unruh.
\newblock Notes on black-hole evaporation.
\newblock \emph{Physical Review D} \textbf{14}, 870--892 (1976).
\newblock \doi{10.1103/PhysRevD.14.870}.

\bibitem[Almheiri et al.(2013)]{AMPS2013}
A.~Almheiri, D.~Marolf, J.~Polchinski, and J.~Sully.
\newblock Black holes: complementarity or firewalls?
\newblock \emph{Journal of High Energy Physics} \textbf{2013}, 62 (2013).
\newblock \doi{10.1007/JHEP02(2013)062}.

\bibitem[Maldacena and Susskind(2013)]{MaldacenaSusskind2013}
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\newblock Cool horizons for entangled black holes.
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\newblock \doi{10.1002/prop.201300020}.

\end{thebibliography}

\end{document}