Yang–Mills Mass Gap via Residual Topological Density: Modular Proof Architecture (v6.1)

This version (v6.1) presents a complete modular architecture for a conditional Clay--style proof framework of the Yang--Mills mass gap problem.  
The main paper and auxiliary modules (Auxiliary C--I) form an integrated logical chain:
\[
\delta\phi > 0 \ \Rightarrow\ \lambda_{\min}(-\Delta_{M}) > 0 \ \Rightarrow\ m_{0} > 0
\]
where:

\(\delta\phi\) is the residual topological density on \(\mathbb{R}^4\), defined as the volume--normalized remainder after subtracting the topological term \(8\pi^2 \, |Q_{\mathrm{top}}|\) from the Yang--Mills action.

\(\lambda_{\min}(-\Delta_{M})\) is the spectral gap of the Laplace--Beltrami operator on the gauge--moduli space \(M\).

\(m_{0}\) is the physical mass gap of the Hamiltonian \(H\) reconstructed from Osterwalder--Schrader (OS) data.

Module roles:

Main Ex --- Geometric definition of \(\delta\phi\), boundary Chern--Simons control, spectral gap theorem, OS/BRST operator domination.

Auxiliary C --- Positivity of \(\delta\phi\) under IR conditions: twisted boundary, Polyakov--holonomy gap, boundary CS--proxy gap.

Auxiliary D --- From OS positivity to uniformly elliptic carre-du-champ, global Gårding inequality, and \(H^2 \ge c\,\Delta_{M} - C\).

Auxiliary E --- Clay--aligned bridges: OS existence window, IR--free positivity of \(\delta\phi\) from center symmetry or Wilson area law.

Auxiliary F --- Conditional Clay--style theorem summarizing assumptions and conclusions.

Auxiliary G/H --- OS window sealing in \(SU(N)\) Yang--Mills via MMLS/SAPZ variational--entropic framework.

Auxiliary I (v3) --- Low--mode entropy--trace module for OS3/OS4 sealing, with explicit constant control.

{Highlights:

Gauge/exhaustion independence of \(\delta\phi\) on \(\mathbb{R}^4\) under decay and based--gauge conditions.

Cellwise coercivity and positive activation density ensure \(\delta\phi \ge c_{\mathrm{IR}} > 0\) under IR constraints.

Uniform ellipticity and operator domination transfer the moduli--space gap to a physical mass gap:
\[
m_{0} \ \ge\ c\,\lambda_{\min}(-\Delta_{M}) - C
\]

OS constants sealed via variational--entropic and low--mode spectral control, with lattice--accessible estimators for \(\delta\phi\).

This v6.1 release serves as a public, citable reference for the full modular structure, prior to the planned v7.0 update with extended verification strategies and numerical demonstrations.