Equality-Manifold Certificates for Overlapping Triadic Factor Graphs

A preregistered, exact-arithmetic pilot for the equality-rigidity program arising from projective minimum-production renormalization (M-I) and Navier–Stokes triadic realizability (M-III). For finite symmetric lattice mode sets, helical amplitudes carry all noncollinear Waleffe-active channels; a frozen rank-one Waleffe-aligned force map makes bundle saturation equivalent to equality of input monomials, and on the full-support torus the equality set is a binomial algebraic subgroup determined by an integer exponent matrix. The confirmation protocol (D9) was frozen and SHA-256 hashed before the reported execution; the frozen protocol file and its hash ship in the package. Exact modular minors and Smith normal forms over the integers certify the nullspace structure on the two separated confirmation instances (mode sets with |k|² ≤ 3 and ≤ 5), with the sparse (≤ 2) and boundary (≤ 4) instances serving as negative controls. Four unavoidable character families delimit the physical real locus. All certificates are exact — no floating point in the certified chain. The package includes the independent referee reproduction report, the post-confirmation audit, and the frozen-gate backend correction note. Fourth paper of the M-series (M-I…M-V): projective layer in M-I (doi:10.6084/m9.figshare.33150845), canonical covariance in M-II (doi:10.6084/m9.figshare.33151331), triadic realizability in M-III (doi:10.6084/m9.figshare.33152555). The historical origin of the broader programme (Zenodo preprints v14.8 and v7.4) is cited as prior public disclosure only, never as a theorem source.