Pfaffian Phase Geometry of Residual Gauges in Port-expanded Network Quotients

Version 1.1 (17 September 2026): census reproducibility update. This version adds a provenance-labelled reconstruction of the census workflow that regenerates the deposited 1,851-row phase table byte-for-byte and reproduces the published 1,512-case five-port census (1,074 R2, 438 PF, 0 MIX). It also adds a separate SHA-256-seeded sensitivity census. No published phase count or manuscript conclusion is changed. Boundary observation of a weighted network can leave a residual gauge: a family of distinct interior weightings producing identical observations. This paper classifies the algebraic structure of those residual gauges at low port count and derives the sensor consequences. The gauge directions of a two-port support form a subspace of bivectors, and the Klein quadric supplies an exact classifier. The inertia of a dual Gram matrix in the Pfaffian form determines whether the fibre is generated by simple rank-two rotors, contains irreducible Pfaffian directions, or sits on a degenerate wall. The classification is proved by Witt decomposition, and verified against 1,851 exhaustive cases, of which 1,512 are five-port, with zero mismatches. Phase is not a graph invariant. A same-support positive-weight example switches phase, proving that topology alone is insufficient and that the weights carry classificatory information. The degenerate wall is shown to be non-generic: it requires a vanishing determinant and therefore occupies a proper algebraic subset of the positive weight cone. Port expansion is required because ordinary module quotients can erase distinct external ports through which residual constraints still act. Under numbered hypotheses the port-expanded quotient preserves the residual gauge space and its Fisher geometry, with counterexamples showing each hypothesis is necessary. A Type-B family has a proved arbitrary-size gauge dimension and minimum sensor law, and a two-sided Schur bound quantifies the conditioning cost of greedy sensor selection. Real sparse infrastructure networks tested under the complete-transfer gauge assay were rigid, which is reported as a negative translation result rather than omitted. This deposit contains the manuscript in PDF and DOCX form together with the complete evidence archive: the 1,851-case validation table, the 51-support wall audit, exact phase certificates, the same-support collision, the Type-B derivation, 636 Schur-bound checks, cocircuit transversals, benchmark summaries and SHA-256 checksums.