4_ymmg: purge remaining a_N→0 dependencies; fix dangling refs

03_lattice_qft.tex:
- Intro: remove "genuine Yang-Mills continuum limit" phrase → "mean-field Wilson action"
- Wilson action remark: remove dangling thm:small-loop-expansion ref (deleted);
  replace with correct ref to thm:mf-wilson and N→∞ thermodynamic limit
- prop:refinement-additional: rewrite entirely — removed all refs to deleted
  def:refinement; now states that the mean-field plaquette measure inherits
  structure from the label-invariant pair rule χ_t, which must be specified
  as part of the model

04_quantum_ym.tex:
- Abstract second paragraph: replaced old "continuum Wilson-action limit /
  near-identity local measure / formal Euclidean YM path-integral rules" with
  explicit N→∞ thermodynamic limit framing and ℓ_visc regularization claim

Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>

Author: Guillemdb

docs/source/4_ymmg/03_lattice_qft.tex

@@ -118,7 +118,7 @@ \section{Introduction}
 \]
 We then add temporal worldline links and define canonical elementary plaquettes
 from one time step and one persistent interaction link. This restores the local
-2-cell geometry needed for a genuine Yang--Mills continuum limit without
+2-cell geometry needed for the mean-field Wilson action without
 reintroducing any extra edge nomenclature.
 
 This paper supplies all the formal definitions and proofs required for a
@@ -802,8 +802,9 @@ \section{Holonomy and gauge invariance}
 The normalization $1-\frac{1}{3}\Re\tr U_3(P)$ vanishes when $U_3(P)=I$ and is
 non-negative for all $U_3(P)\in SU(3)$, since
 $|\tr U|/3\le1$ by the triangle inequality applied to the three eigenvalues of $U$.
-In the small-loop regime, Theorem~\ref{thm:small-loop-expansion} identifies the
-leading-order term with a Yang--Mills curvature density.
+In the $N\to\infty$ thermodynamic limit, Theorem~\ref{thm:mf-wilson} shows that
+the sum of these plaquette terms converges to the mean-field Wilson action
+$S_{\mathrm W}^\infty$ defined on the stationary measure $\pi_\infty$.
 \end{remark}
 
 \begin{proposition}[Gauge invariance of Wilson action]
@@ -1340,36 +1341,33 @@ \subsection{Thermodynamic limit of graph observables}
 plaquette-level corollary.
 \end{remark}
 
-\begin{proposition}[Why geometric refinement is additional]
+\begin{proposition}[Graph rule is not determined by particle paths]
 \label{prop:refinement-additional}
-The stationary mean-field convergence results above do not by themselves imply
-the geometric refinement hypotheses of Definition~\ref{def:refinement}.
-More precisely, the quantities
+The stationary mean-field convergence results above do not determine the companion
+graph rule $\mathcal P_t^{(N)}$.
+More precisely, the plaquette geometry quantities
 \[
-a_N,\qquad
 \{A_P^{(N)}\}_{P\in\mathcal C_\square^{(N)}},\qquad
 |\mathcal C_\square^{(N)}|,
 \qquad
 \sum_{P\in\mathcal C_\square^{(N)}}
 \Sigma_P^{(N),\mu\nu}\Sigma_P^{(N),\rho\sigma}\,\delta_{z_P^{(N)}}
 \]
-depend on the companion graph rule $\mathcal P_t^{(N)}$ and are not determined
-by the empirical particle path laws alone.
-Consequently, no theorem based only on one-time or path-space propagation of
-chaos can force Definition~\ref{def:refinement}; a separate local graph
-construction or an equivalent geometric sampling hypothesis is necessary.
+depend on $\mathcal P_t^{(N)}$ and are not determined by the empirical particle
+path laws alone.
+Consequently, the mean-field plaquette measure $d\Gamma_\square^\infty$ of
+Definition~\ref{def:mf-plaquette-measure} inherits its structure from the
+label-invariant pair rule $\chi_t$, which must be specified as part of the
+model definition.
 \end{proposition}
 \begin{proof}
 In Definition~\ref{def:run}, the recorded companion sets
 $\{\mathcal P_t^{(N)}\}_{t\in\Tset}$ are part of the lattice input.
 Keeping the same particle trajectories fixed while changing the label-invariant
 companion rule changes the link set $\mcE^{(N)}$, the plaquette family
-$\mathcal C_\square^{(N)}$, the maximal edge length $a_N$, and the associated
-area bivectors and plaquette counts.
+$\mathcal C_\square^{(N)}$, and the associated area bivectors and plaquette counts.
 The empirical one-time and path measures of the particles are unaffected by such
-a change.
-Therefore those mean-field limits do not determine the geometric quantities in
-Definition~\ref{def:refinement}.
+a change, hence the plaquette geometry is not determined by propagation of chaos alone.
 \end{proof}
 
 \subsection{Mean-Field Yang-Mills Action}

docs/source/4_ymmg/04_quantum_ym.tex

@@ -77,18 +77,21 @@
 determinant, and the scalar sector is shown to be integrable with an exact
 Gaussian formula in the free case.
 
-The paper also explains the continuum meaning of the discrete measure.  On each
-edge, Haar measure is shown to be smooth in exponential coordinates and to agree
-to second order with Lebesgue measure on the Lie algebra.  Combined with the
-continuum Wilson-action limit from the companion lattice paper, this yields a
-precise consistency statement: the discrete theory is a genuine ultraviolet
-regularization whose near-identity local measure and local action density match
-the formal Euclidean Yang--Mills path-integral rules.  What is \emph{not} proved
-here is just as important: this paper does not establish an unconditional
-constructive continuum $4$-dimensional Yang--Mills measure, does not verify the
-Osterwalder--Schrader axioms in the continuum, and does not prove a mass gap.
-Those are strictly stronger problems than the finite-lattice quantization solved
-in this paper.
+The paper also establishes the Lie-algebraic structure of the discrete measure.
+On each edge, Haar measure is smooth in exponential coordinates and agrees to
+second order with Lebesgue measure on the Lie algebra.
+The correct continuum limit of this programme is the $N\to\infty$ thermodynamic
+limit, not a mesh refinement $a_N\to0$: in that limit the discrete lattice
+converges to the stationary mean-field law $\pi_\infty$, the bilocal gauge
+field $A(z,z')$ and mean-field Wilson action $S_{\mathrm W}^\infty$ are defined
+directly on $\pi_\infty$, and the natural ultraviolet regularization is provided
+by the viscous kernel length scale $\ell_{\mathrm{visc}}$ rather than a lattice
+spacing.
+What this paper does not prove is equally important: it does not verify the
+Osterwalder--Schrader axioms on the integral-operator continuum theory and does
+not prove a mass gap.
+Those are strictly harder problems that require the full $N\to\infty$ continuum
+structure developed in the companion papers.
 \end{abstract}
 
 \tableofcontents