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

=== Assistant: Below is a journal-ready LaTeX section containing the Methods, Main Theorem, and Proof, fully aligned with the repaired… ===
Below is a journal-ready LaTeX section containing the Methods, Main Theorem, and Proof, fully aligned with the repaired scope (IR-ToE v1.1), the ANF decision semantics, and the required tolerance fixed-point law. This is drop-in content; notation is consistent and claims are locked.

<syntaxhighlight lang="latex">\section{Methods}

\subsection{Algorithmic Decision Semantics}

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{Residual Field and Disturbance Class}

Let $y(x)$ denote measured observables, $f(x;\theta,b)$ the frozen model prediction with parameters $\theta$ and baseline structure $b$, and 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
\[
\lvert r\rvert_{\max} := \sup_{x\in\mathcal{N}} \lvert r(x;\theta,b)\rvert,
\]
which is fixed once $\mathcal{N}$, $\theta$, and $b$ are fixed.

\subsection{Structural Tolerance and Precision Scaling}

The theory declares a precision-dependent structural tolerance $\tau(p)$, where $p\in\mathbb{N}_0$ indexes measurement precision. Structural violation occurs iff
\[
\lvert r(x;\theta,b)\rvert > \tau(p)
\quad\Rightarrow\quad
\mathrm{COLLAPSE}.
\]
Finite-resolution evaluation corresponds to $p=0$. Precision refinement increases $p$.

\subsection{Canonical Tolerance Law (Locked)}

To ensure asymptotic viability when 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.

\subsection{Verdict Logic}

Evaluation proceeds through fixed gates (data validity, structure, feasibility, evidence, precision). Any gate failure yields \textsc{Collapse}. If no failure occurs, the verdict is \textsc{Stand}. The procedure is deterministic and reproducible.

---

\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 $p_k$,
\[
\lvert r\rvert_{\max} > \tau(p_k),
\]
so a structural violation occurs and the verdict is \textsc{Collapse}. Hence asymptotic viability fails.

(\emph{Sufficiency}) Suppose $\liminf_{p\to\infty}\tau(p) \ge \lvert r\rvert_{\max}$. Then for all $p$,
\[
\tau(p) \ge \lvert r\rvert_{\max},
\]
so no admissible disturbance produces a structural violation at any precision. Therefore \textsc{Collapse} never occurs and the theory is asymptotically viable. \qedhere
\end{proof}

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

\begin{corollary}[Minimal Viable Scaling]
The weakest asymptotically viable tolerance admits a nonzero floor:
\[
\tau(p) = \lvert r\rvert_{\max} + \epsilon(p),
\quad \epsilon(p)>0,\quad \lim_{p\to\infty}\epsilon(p)=0.
\]
\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. Equation~\eqref{eq:tau-law} is the minimal IR-stable flow.

---

\section{Scope and Claims}

\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 and justified.

\paragraph{Classification.}
The theory is an \emph{Infrared-Complete Theory} (IR-ToE): complete and closed at accessible scales, with explicit and testable criteria for asymptotic breakdown.

---

\section{Reproducibility}

All operators are deterministic; all inputs and thresholds are hashed; verdicts are reproducible bitwise under identical inputs.

</syntaxhighlight>

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