Yang–Mills Mass Gap via Residual Topological Density $\delta_\phi$: From Moduli-Space Spectral Gap of $\Delta_{\mathcal{M}}$ to a Physical Gap, with OS/BRST Reconstruction and Operator Domination $H^2 \ge c\,\Delta_{\mathcal{M}} - C$ (v5.0)

We present a constructive, gauge-invariant route to the Yang–Mills mass gap based on a residual topological density $\delta_\phi$ on the moduli space $\mathcal{M}=\mathcal{A}/\mathcal{G}$. A strictly positive $\delta_\phi$ yields geometric rigidity and a spectral gap for the Laplace–Beltrami operator $\Delta_{\mathcal{M}}$; via OS/BRST reconstruction and an operator-domination inequality $H^2 \ge c\,\Delta_{\mathcal{M}} - C$, this transfers to a positive physical mass gap $m_0 \ge \sqrt{\,c\,\lambda_{\min}(\Delta_{\mathcal{M}}) - C\,}$. Lattice implementations and IR boundary schemes are provided to operationalize and test the framework.

This v5.0 set contains:
- Main paper. Global overview of the chain $\delta_\phi>0 \Rightarrow$ geometric rigidity $\Rightarrow$ spectral gap of $\Delta_{\mathcal{M}} \Rightarrow$ physical mass gap via OS/BRST.
- Auxiliary A. Definitions and equivalences for $\delta_\phi$ (compact and non-compact settings) and sufficient conditions for $\delta_\phi>0$ in the vacuum sector $k=0$. Core formula (compact $M$):
  $\displaystyle \delta_\phi^2 := \frac{1}{\mathrm{Vol}(M)} \inf_{[A]\in\mathcal{A}/\mathcal{G}}\Big( \|F_A\|_{L^2(M)}^2 - 8\pi^2\,|Q_{\mathrm{top}}([A])| \Big),$
  with SD/ASD and Pontryagin-density forms proved equivalent. For $\mathbb{R}^4$, an averaged-density definition is used and boundary Chern–Simons terms vanish per unit volume.
- Auxiliary B. OS/BRST reconstruction, time-zero cylinder algebra, and a Dirichlet-form comparison establishing on a dense core
  $H^2 \ge c\,\Delta_{\mathcal{M}} - C,$
  which yields the gap transfer
  $m_0 \ge \sqrt{\,c\,\lambda_{\min}(\Delta_{\mathcal{M}}) - C\,}.$
  The domination constants are gauge-independent under bounded-geometry hypotheses.
- Supplement 1. A proof of $\delta_\phi>0$ on $\mathbb{R}^4$ under decay in a based gauge, Uhlenbeck-type compactness, and an IR boundary (holonomy/CS-distance) condition.
- Supplement 2. Constructive details for the OS axioms and BRST positivity on cohomology, the semigroup $e^{-tH}$, and the identification of the physical Hilbert space.
- Supplement 3. Lattice pipeline and renormalization stability: three equivalent IR schemes (twist, Polyakov-holonomy constraint, CS-proxy), gradient flow, and the estimator
  $\displaystyle \delta_\phi^2(a,L;t) := \frac{1}{V}\big( \langle E_t \rangle - 8\pi^2 \langle |Q_t| \rangle \big),$
  with continuum $a\to 0$ and infinite-volume $L\to\infty$ limits commuting under reflection positivity and locality.

Assumptions and scope. Compact simple gauge group $G$, spacetime $S^4$ or $\mathbb{R}^4$ with decay and based gauge at infinity, bounded geometry of the slice $\mathcal{M}$, OS axioms (reflection positivity, Euclidean invariance, clustering), BRST positivity on cohomology, and chartwise measure comparability $d\nu = \rho\, d\mu_{\mathcal{M}}$ with $\rho, \rho^{-1}$ locally bounded. Under these conditions one obtains $\lambda_{\min}(\Delta_{\mathcal{M}}) \gtrsim \kappa\,\delta_\phi^{\,2}$ and hence a strictly positive mass gap whenever $\kappa\,\delta_\phi^{\,2} > C$.

Version note (v5.0). This release consolidates the equivalence of $\delta_\phi$ definitions on $S^4$ and $\mathbb{R}^4$, provides a complete operator-domination proof $H^2 \ge c\,\Delta_{\mathcal{M}} - C$, and adds lattice-ready IR prescriptions and estimators to facilitate numerical tests of $\delta_\phi>0$ and the associated gap bound.

Keywords. Yang–Mills; mass gap; residual topological density $\delta_\phi$; moduli space $\mathcal{M}$; Laplace–Beltrami $\Delta_{\mathcal{M}}$; OS axioms; BRST; spectral gap; $SU(N)$; lattice gauge theory.