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\lhead{The Initiating Increment}
\rhead{John C. W. McKinley}
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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).
\end{thebibliography}
\end{document}