MGAP4D: A Conditional Certificate Architecture for a Positive Mass Gap in a Constructive Four-Dimensional Yang–Mills Sector

MGAP4D v0. 8++: Exact Mass-Gap Value, Threshold Eigenvalue, and Plaquette Spectral Atom MGAP4D v0. 8++ extends the v0. 8+ residual-zero strict mass-gap spine from a lower-bound theorem to a certificate-dependent exact-value theorem. The previous residual-zero result established the normalized lower bound Hₚhys|₎₌₄₆₀㵵₄ₑ₏ >= (33/20) I. This update adds the SharpGapWitness chain SGW1-SGW6, producing a matching upper bound and hence inf sigma (Hₚhys|₎₌₄₆₀㵵₄ₑ₏) = 33/20. It further introduces a LocalizedSharpGapWitness certificate, under which 33/20 is promoted from a spectral threshold to an eigenvalue of Hₚhys restricted to Omegaᵖerp. Finally, with a ThresholdPlaquetteOverlapCertificate, the threshold eigenvalue is shown to be visible to a centered plaquette observable, yielding rhoAp (33/20) > 0. This release is certificate-dependent and does not claim completed no-axiom Lean/Coq machine verification or peer-reviewed external finality. MGAP4D v0. 8++ is an exact-value update following MGAP4D v0. 8+. The v0. 8+ release consolidated the residual-zero strict mass-gap spine and the no-axiom formalization audit plan. It established, inside the MGAP4D certificate architecture, the lower-bound statement: Hₚhys|_ (Omegaᵖerp) >= (33/20) I. Version v0. 8++ adds a SharpGapWitness chain to convert this lower bound into an exact threshold value. The chain consists of: SGW1: one-excitation plaquette sector nontriviality and vacuum orthogonality; SGW2: detectability sharpness; SGW3: base Hamiltonian threshold sharpness; SGW4: renormalization correction sharpness; SGW5: finite Rayleigh upper-bound assembly; SGW6: continuum sharp transfer. Together these certificates yield: inf sigma (Hₚhys|_ (Omegaᵖerp) ) = 33/20. The release then adds a LocalizedSharpGapWitness certificate, ensuring that the sharp witnesses do not escape to infinity and have a nonzero strong limit. Under this certificate, the threshold 33/20 is promoted to a point-spectrum eigenvalue: exists psi_* in Omegaᵖerp with ||psi_*|| = 1 and Hₚhys psi_* = (33/20) psi_*. Finally, a ThresholdPlaquetteOverlapCertificate is added. If the threshold eigenvector lies in the plaquette-generated local cyclic subspace and the plaquette Gram form is nondegenerate, then a centered plaquette observable Aₚ has positive spectral atom at the threshold: rhoAp (33/20) > 0. Claim boundary: This is a certificate-dependent exact-value theorem within the MGAP4D architecture. It strengthens the internal theorem spine from lower bound to exact threshold, eigenvalue, and plaquette-visible spectral atom under explicitly listed certificates. It does not claim completed no-axiom Lean/Coq verification, peer-reviewed Clay-level finality, or unconditional eigenvalue/plaquette-atom status without the localized and overlap certificates.