Hamiltonian Reduction, Dual-Entropy Lyapunov Control, and the Yang–Mills Mass-Gap Problem

We give a corrected finite-volume reduction of the Yang–Mills mass-gap problem and carryout explicit calculations in the smallest genuinely nonabelian SU(2) active window. On thefinite-volume side we prove the exact gauge-invariant carrier grammar, exact electric-shelldiagonalization, the global quasi-momentum fiber decomposition, the spectator-first inactivereduction, the active-image reduction, the scalar-vacuum reduction, and a direct strong-couplinggap theorem.We also prove the abstract infinite-volume/continuum transfer ladder: uniform finite-volumegap bounds imply subsequential infinite-volume Euclidean decay, and any Osterwalder–Schradercompatible continuum subsequential limit carrying the same physical mass lower bound inheritsa continuum mass gap.The dissipative part of the manuscript is reorganized around two relative-entropy densitiesfor the blocked flow: a microscopic density, monotone under blocking by data processing, anda coarse density, natural for local marginals, large-field control, and the blocked Euclidean-toHamiltonian dictionary. We prove the microscopic monotonicity theorem, a gauge-compatibleperturbative ultraviolet initialization theorem, an abstract coarse-entropy production formula, alocal entropy lift theorem under approximate tensorization, explicit Pinsker/subgaussian largefield and good-field estimates, and an observable-by-observable entropy-to-dictionary transfertheorem.At the fixed-spacing Euclidean/Hamiltonian interface we also prove that reflection positivity ispreserved by any blocking map that commutes with time reflection and preserves the positive-timealgebra. This internalizes the structural part of the Osterwalder–Schrader bridge.The strongest closure theorem supported by the manuscript remains conditional. Along aHamiltonian-compatible continuum sequence, if one supplies the YM-specific dissipative/blockedcontour package — namely a gauge-compatible ultraviolet reference with ultraviolet control, acoarse-entropy production theorem, a blocked local-lift theorem, an explicit entropy-to-dictionaryinstantiation, and a quantitative blocked contour / PF-band theorem furnishing KP-smallphase/source envelopes together with the adapted open-cavity source estimate — togetherwith the finite torus certificate and the remaining algebra-identification part of the blockedHamiltonian/Osterwalder–Schrader bridge, then every corresponding continuum subsequentiallimit in the first window has mass gap at least the prescribed target mass. For general compactsimple gauge groups, the remaining extension is reduced to a finite atlas package together withthe same dissipative, PF-band, and Euclidean/Hamiltonian inputs.