Is Evidence Independence an Axiom? Decomposing a Proposed Extension to AX-SH

Decomposition of a proposed fifth axiom for the AX-SH structural-honesty framework. The paper attempts to derive each of the proposal's five clauses from the four existing axioms and reports four different outcomes. One clause, acting-system-independent evidence generation (A5.1), is not derivable and is stated as Axiom 5 (Evidence independence) over a control predicate (WithinAuth), with a finite countermodel establishing independence from A1-A4 and a finite joint-satisfiability witness showing the enlarged set consistent. Causal gating (A5.2) is relocated as a proposed control principle for a control layer rather than the axiom set. Operator-independence (A5.4) is restated as a measured deployment condition over an extended evidence relation. The no-default half of A5.3 and boundary declaration (A5.5) are proved as theorems from the existing axioms; the recording/escalation half of A5.3 is preserved as an open operational requirement. The paper also declares a post-quantum cryptographic boundary (hash-based integrity vs signature-based attribution) and registers ten falsification conditions. All proofs are finite model-theoretic paper proofs verified by clause-by-clause enumeration; no machine-checked verification is claimed.