Editing Openai/69596681-3f7c-8006-b9fd-01d9d8191763 (section)

=== Assistant: Below is the complete, self-contained LaTeX source for the manuscript, assembled end-to-end and ready to compile into a… ===
Below is the complete, self-contained LaTeX source for the manuscript, assembled end-to-end and ready to compile into a submission-grade PDF.
It includes: title page, abstract, Methods, Main Theorem + Proof, RG interpretation, Scope, Determinism statement, Worked Example appendix, Referee FAQ appendix, and bibliography stub.

You can copy this into a single file named main.tex and compile with pdflatex (twice).

<syntaxhighlight lang="latex">\documentclass[11pt]{article}

\usepackage[a4paper,margin=1in]{geometry}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{hyperref}
\usepackage{graphicx}
\usepackage{booktabs}
\usepackage{enumitem}

\hypersetup{
  colorlinks=true,
  linkcolor=blue,
  citecolor=blue,
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}

% ---------- Theorem environments ----------
\newtheorem{theorem}{Theorem}
\newtheorem{corollary}{Corollary}

% ---------- Title ----------
\title{\textbf{Infrared-Complete Theories, Precision Scaling, and\\
Asymptotic Viability under Algorithmic Decision Semantics}}
\author{Robert R. Frost}
\date{January 3, 2026}

\begin{document}
\maketitle

% ---------- Abstract ----------
\begin{abstract}
We present an algorithmically governed physical framework whose evaluation is deterministic in the strongest sense used in computation and formal methods: bitwise-reproducible, path-unique, and state-closed. The framework yields a binary verdict under empirical data and explicitly diagnoses its own breakdown under precision refinement. We prove a necessary and sufficient condition for asymptotic viability in terms of the scaling behavior of a structural tolerance parameter and show that naive precision refinement generically induces collapse. The resulting theory is complete and closed at accessible (finite-resolution) scales, with a formally specified domain of authority and explicit non-claims beyond it. Claims of ultraviolet completeness are excluded unless a nonzero tolerance fixed point is independently established.
\end{abstract}

% =====================================================================
\section{Framework and Decision Semantics}

\subsection{Algorithmic Evaluation}

The theory is evaluated by a deterministic, fail-closed decision procedure that maps empirical inputs to a binary verdict
\[
V \in \{\mathrm{STAND},\mathrm{COLLAPSE}\}.
\]
All operators are frozen prior to execution. No learning, optimization, or stochastic updating is permitted. Once \textsc{Collapse} occurs, the verdict is absorbing.

\subsection{Strong Determinism}

The framework is deterministic in the strongest sense used in computation and formal methods:
\begin{itemize}[leftmargin=1.5em]
\item \textbf{Bitwise-reproducible}: identical inputs (data, parameters, thresholds) produce identical outputs at the level of machine words.
\item \textbf{Path-unique}: evaluation follows a single execution path; no branching depends on randomness or optimization.
\item \textbf{State-closed}: no hidden, mutable, or external state influences execution.
\end{itemize}
Repeated executions are therefore equivalent to a single execution; empirical frequency is a counting measure, not a probability.

% =====================================================================
\section{Residual Field and Structural Tolerance}

Let $y(x)$ denote measured observables and $f(x;\theta,b)$ the frozen model prediction with parameters $\theta$ and baseline structure $b$. Define the residual field
\[
r(x;\theta,b) = y(x) - f(x;\theta,b).
\]

Let $\mathcal{N}$ denote the admissible disturbance class (e.g., bounded adversarial noise, calibration envelopes). Define the worst-case residual amplitude
\[
\lvert r\rvert_{\max} := \sup_{x\in\mathcal{N}} \lvert r(x;\theta,b)\rvert.
\]

The theory declares a precision-dependent structural tolerance $\tau(p)$, where $p\in\mathbb{N}_0$ indexes measurement precision. Structural violation occurs iff
\[
\lvert r(x;\theta,b)\rvert > \tau(p)
\quad\Rightarrow\quad
\mathrm{COLLAPSE}.
\]

% =====================================================================
\section{Precision Scaling}

Finite-resolution evaluation corresponds to $p=0$. Precision refinement increases $p$. Naive scaling laws often implicitly assume $\tau(p)\to 0$ as $p\to\infty$. We show that such laws generically induce collapse unless a nonzero infrared fixed point exists.

\subsection{Canonical Tolerance Law (Locked)}

When asymptotic viability is claimed, the tolerance is required to admit a nonzero infrared fixed point:
\begin{equation}
\label{eq:tau-law}
\tau(p) = \tau^\'' + (\tau_0-\tau^\'')\,2^{-p},
\qquad \tau^\* \ge \lvert r\rvert_{\max}.
\end{equation}
Here $\tau^\*>0$ represents irreducible operational spread. Exact observables are forbidden.

% =====================================================================
\section{Main Result}

\begin{theorem}[Asymptotic Viability under Precision Scaling]
\label{thm:asymptotic-viability}
Let $\tau(p)$ be the precision-dependent structural tolerance and $\lvert r\rvert_{\max}$ the worst-case residual amplitude. The theory is asymptotically viable (i.e., never collapses for any finite $p$) if and only if
\[
\liminf_{p\to\infty}\tau(p) \;\ge\; \lvert r\rvert_{\max}.
\]
\end{theorem}

\begin{proof}
(\emph{Necessity}) Suppose $\liminf_{p\to\infty}\tau(p) < \lvert r\rvert_{\max}$. Then there exists $\varepsilon>0$ and a subsequence $\{p_k\}$ such that $\tau(p_k) \le \lvert r\rvert_{\max}-\varepsilon$. For each such $p_k$, a structural violation occurs, implying \textsc{Collapse}. Hence asymptotic viability fails.

(\emph{Sufficiency}) Suppose $\liminf_{p\to\infty}\tau(p) \ge \lvert r\rvert_{\max}$. Then for all $p$, $\tau(p)\ge \lvert r\rvert_{\max}$ and no admissible disturbance produces a violation. Therefore collapse never occurs. 
\end{proof}

\begin{corollary}
Any tolerance law with $\lim_{p\to\infty}\tau(p)=0$ is asymptotically non-viable unless $\lvert r\rvert_{\max}=0$.
\end{corollary}

% =====================================================================
\section{Renormalization-Group Interpretation}

Define a resolution scale $\ell := 2^{-p}$. Precision refinement corresponds to $\ell\to 0$. Interpreting $\tau(\ell)$ as an effective tolerance, asymptotic viability is equivalent to the existence of an infrared fixed point
\[
\tau^\* := \lim_{\ell\to 0}\tau(\ell) \;\ge\; \lvert r\rvert_{\max}.
\]
Flows with $\tau(\ell)\to 0$ are ultraviolet-unstable; flows saturating at $\tau^\*>0$ are infrared-stable.

% =====================================================================
\section{Scope and Classification}

\paragraph{Scope.}
The theory asserts completeness only for physically realizable observations at finite operational resolution.

\paragraph{Non-Claims.}
Ultraviolet completeness is not claimed unless a nonzero $\tau^\*$ is independently established.

\paragraph{Classification.}
The framework defines an \emph{Infrared-Complete Theory}: complete and closed at accessible scales, with explicit and testable criteria for asymptotic breakdown.

% =====================================================================
\appendix
\section{Worked Example}

Let $y_0=0.5$, $\theta=0$, $b=0$, and bounded adversarial noise $\lvert\varepsilon\rvert\le 0.4$. Then
\[
\lvert r\rvert_{\max} = 0.9.
\]

With $\tau_0=3.0$, the structural regime ($p=0$) survives. Under naive scaling $\tau(p)=3\cdot2^{-p}$, collapse occurs at $p=2$. Under the viable law~\eqref{eq:tau-law} with $\tau^\*=0.9$, collapse never occurs.

\begin{center}
\begin{tabular}{c|c|c}
$p$ & $\tau(p)$ & Verdict \\
\midrule
0 & 3.00 & STAND \\
1 & 1.95 & STAND \\
2 & 1.43 & STAND \\
$\infty$ & 0.90 & STAND \\
\end{tabular}
\end{center}

% =====================================================================
\section{Referee FAQ}

\begin{enumerate}[leftmargin=1.5em]
\item \textbf{Is this a Theory of Everything?} Infrared-complete, not ultraviolet-complete.
\item \textbf{Why forbid exact measurements?} Exact observables imply $\tau^\*=0$, guaranteeing collapse.
\item \textbf{Is this an EFT?} No; it is governed by decision semantics, not perturbative expansion.
\item \textbf{How is falsification achieved?} Any admissible dataset with $\lvert r\rvert_{\max}>\tau^\*$ induces collapse.
\end{enumerate}

% =====================================================================
\section*{Acknowledgments}
The author acknowledges no external funding.

\begin{thebibliography}{9}
\bibitem{turing}
A.~M.~Turing, \emph{On Computable Numbers}, Proc. London Math. Soc. (1936).

\bibitem{wilson}
K.~G.~Wilson, \emph{The Renormalization Group}, Rev. Mod. Phys. (1975).
\end{thebibliography}

\end{document}

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===== <syntaxhighlight lang="bash">pdflatex main.tex =====
pdflatex main.tex

</syntaxhighlight>

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