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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}
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\end{document}