IndisputableMonolith.Verification.RecognitionStabilityAudit.BackEnd
Back-end pinch layer of the Recognition Stability Audit: a Schur-bounded holomorphic map Ξ on a preconnected open Ω that hits the unit-circle boundary must be identically 1. Complex analysts and RS verifiers cite it to rule out nontrivial boundary contact. The argument chains maximum-modulus style identities from the Core interface into constancy and then into a no-boundary-hit corollary for nontrivial maps.
claimIf $\Xi$ is holomorphic on a preconnected open set $\Omega\subset\mathbb{C}$ and Schur-bounded on $\Omega$ (i.e. $|\Xi|\le 1$), then any boundary hit forces $\Xi\equiv 1$ on $\Omega$. Equivalently, a nontrivial such $\Xi$ never meets the unit-circle boundary on $\Omega$.
background
The Recognition Stability Audit (RSA), documented in papers/tex/Recognition_Stability_Audit.tex, is organized as a Lean compiler from analytic hypotheses to stability certificates. Its Core module supplies the RL-friendly interface: Schur-boundedness, boundary-hit predicates, and the holomorphic setup on open regions.
This BackEnd module sits one layer above Core. It packages the classical pinch: on a preconnected open $\Omega$, a holomorphic Schur map that attains modulus 1 at an interior point is constant of modulus 1; under the RSA normalization that constant is 1. Sibling declarations name the steps: boundary hit implies value 1, equality to the constant-1 function, absence of boundary hits for nontrivial maps, and the assembled back-end certificate.
Notation follows Core: $\Xi$ is the audited map, $\Omega$ the preconnected open domain, and "boundary hit" means contact with the unit circle inside $\Omega$ in the sense fixed by the Core predicates.
proof idea
Not a single theorem: a short lemma stack. First, a boundary hit plus Schur holomorphy yields $|\Xi|=1$ at a point, hence $\Xi$ equals the constant 1 by the identity/maximum principle on preconnected $\Omega$. That identity is recorded as constancy-from-boundary-hit. The contrapositive gives: nontrivial Schur-holomorphic $\Xi$ has no boundary hit. The top declaration packages those facts as the back-end certificate consumed by the RSA umbrella. Dependencies are Core predicates and standard complex-analysis facts imported via Mathlib; no new analytic engine is built here.
why it matters in Recognition Science
RSA needs a clean separation between front-end measurement hypotheses and back-end rigidity. This module is that rigidity layer: once Core has phrased Schur bounds and boundary contact, BackEnd forces the only allowed contact case to be the constant-1 map, and therefore certifies that nontrivial audited maps stay strictly inside the disk.
The umbrella module RecognitionStabilityAudit re-exports Core and imports this BackEnd so downstream verifiers can write a single import and obtain both the interface and the pinch. In the broader Recognition framework the lemma underwrites stability audits of recognition maps (phase/coherence observables) against boundary collapse, complementary to the forcing chain's structural uniqueness results (J-uniqueness, $\varphi$, eight-tick) by policing analytic deformations rather than deriving constants.
scope and limits
- Does not define Schur-boundedness or boundary-hit; those live in Core.
- Does not treat disconnected $\Omega$; preconnectedness is essential for global constancy.
- Does not identify the constant when the Schur bound is strict elsewhere than the RSA normalization to 1.
- Does not prove front-end measurement or physical identification of $\Xi$.
- Does not address non-holomorphic or merely measurable recognition maps.