Rigidity of the Gauge Signature under PSC: A Sieve Theorem and Residual Classification Program Paper 20 (T3) of the NEMS Suite

A central goal of theoretical physics is to explain why the universe is governed by the specific laws we observe, rather than an infinite landscape of mathematical possibilities. In this paper, we extend the logic of semantic closure to the gauge structure of the universe. We propose that the Standard Model gauge signature is not merely one possible world, but the unique mathematical fixed point for any stable, self-contained record-bearing system in 4D spacetime. We formalize a PSC Sieve on 4D gauge-theory space, consisting of self-containment constraints: structural stability (NM*), semantic admissibility (SA), thermodynamic viability (TV), and anomaly consistency. We prove that these constraints eliminate large classes of theories, leaving a highly constrained residual set. We then state the Residual Classification Conjecture: within this residual set, only the SU (3) SU (2) U (1) signature with =3 survives. Paper 20 proves the sieve structure and residual narrowing results; the final collapse to the Standard Model signature is stated as the Residual Classification Conjecture and supported computationally, not yet elevated to a completed formal theorem. This reframes the landscape problem from an anthropic accident into a falsifiable classification program. All definitions and conditional theorems are formalized in Lean 4, and the residual conjecture is supported by a reproducible computational classification pipeline. This overview presents the core NEMS theorem engine and selected applications; stronger domain-specific derivation and ontological synthesis claims belong to separate release surfaces with their own premise bundles and formal artifacts. Trust boundary. Sieve and residual claims are conditional on the PSC predicate bundle in 4D gauge-theory space; the Residual Classification Conjecture is explicit machine-facing target, not a finished theorem. Cross-suite verification context is nems-lean. See.