Relational Anchors for Quantum Subsystem Identification: Quadratic Obstructions, Cubic Gauge Collapse, and Robust Rigidity

Abstract

This preprint develops a finite-dimensional, proof-audited route from noisy relational localization certificates to exact tensor-product-structure (TPS) envelopes and robust subsystem identification. It separates the pre-Jordan gauge-calibration problem from the post-calibration algebra-rounding problem, proves a quadratic non-identifiability obstruction, and gives a finite cubic relation criterion for Wigner/Jordan gauge collapse. The cubic action is quantitatively analyzed through an exact transverse singular-value identity, nonlinear geodesic bounds, spectral recovery of the Wigner derivation algebra, a two-qubit Pfaffian model, and structured far-sector certificates. For two qubits, the derivation-normalizer problem is reduced to a quartic Grassmann objective and then closed analytically by an explicit Chevalley–Eilenberg/Hodge coercivity estimate on the 90-dimensional normal SU(4) module. This yields a rigorous global entry bound beta_4(pi/(2 sqrt(2))) > sqrt(2/37) and an unconditional cubic-to-Wigner threshold Delta_3^0 < 4/(3 sqrt(185)), eliminating the previous global far-region certification gate in this dimension. The post-calibration branch proves exact Wedderburn articulation for bistochastic idempotent anchors, Choi-rank and Choi-spectrum diagnostics, a central-derivation factor certificate, complementary-factor rectification, complete rounding-set ambiguity control, and stabilizer-synchronized relational-anchor separation. The resulting theorem is a conditional, radius-valued robustness statement inside a finite-dimensional full matrix algebra; it does not claim raw-frequency sample complexity, a dimension-free algebra-rounding constant, or preferred subsystems from a Hamiltonian and state alone.