TEBAC Yang–Mills Program III: Osterwalder–Schrader/GNS Reconstruction, Axiomatic Convergence, and Gauge-Invariant Observables

This preprint is the third module of the TEBAC Yang--Mills program. It does not claim a completed proof of the Yang--Mills existence and mass gap problem. Its purpose is to close the Osterwalder--Schrader/GNS reconstruction layer for gauge-invariant observables and to prepare the spectral problem that will be addressed in the next module. The module starts from the reflection-positive finite-cutoff and continuum-ready Euclidean data exported by YM-II. It constructs the OS inner product, the null-space quotient, the physical Hilbert space\[\mathcal H_{\mathrm{phys}}=\overline{\mathcal A^{\mathrm{inv}}_+/\mathcal N_{\mathrm{OS}}},\]the vacuum vector, the positive-time semigroup, and the non-negative Hamiltonian \(H\ge0\). It also records the gauge-invariant observable representation, the no-negative-norm sector, the forward-cone spectral condition, the tempered-distribution convention for observables, and the OS-to-Wightman handoff. The PASS3 version strengthens the axiomatic layer by adding a Schwartz-tempered observable convention, a cluster-readiness/vacuum-uniqueness criterion, a theorem showing that verified terminal packing excludes degenerate vacua, and an axiomatic-node diagram showing the convergence of YM-I, YM-II, and YM-III. The module also records the \(9D\to4D\) reconstruction-neutrality rule, ensuring that the auxiliary TEBAC regulator is not allowed to break the four-dimensional gauge-invariant observable structure. The closed output of this module is an OS/GNS reconstructed gauge-invariant Hilbert-space framework ready for the next stage:\[\mathrm{YM\text{-}IV}:\text{Uniform Spectral Coercivity and Mass Gap}.\] Claim-safety statement: YM-III is closed as an OS/GNS reconstruction and axiomatic handoff module. It does not by itself prove continuum non-triviality, vacuum uniqueness unconditionally, or a positive Yang--Mills mass gap. These are downstream tasks reserved for YM-IV and YM-V.