Track1MixedAxisLhsTranslationReductionEndpoint
plain-language theorem explainer
Records the Track 1.B Session-209 reduction: full vanishing of the corrected N=5 mixed-axis residual coefficients follows from translation invariance of the mixed explicit-fiber LHS coefficient model alone. Gravity auditors cite it as the LHS-only handoff leaf after the RHS translation step. It is a pure implication Prop, not a proved theorem.
Claim. The full residual coefficient certificate (for all pairs of vertices $u,v$ in the $N=5$ stencil, the mixed-axis residual coefficient vanishes) is implied by translation invariance of the mixed explicit-fiber LHS coefficient model: for all $u,v$, the LHS coefficient at $(u,v)$ equals the coefficient at the origin relative to the translated vertex pair.
background
Module Gravity.MasterTheoremHandoffIntegration is the Track 7 integration-lane receipt for parallel fork handoffs (Tracks 1.B stationarity, physical residual/Bianchi, many-body amplitude lift, Page capacity, dark-energy $w(z)$, and falsifier sensitivity). It records endpoints without upgrading the discovery claim, and keeps remaining Track 1 displacement-class leaves open.
The corrected $N=5$ axis-stencil residual is audited by a finite coefficient certificate: every mixed-axis residual coefficient on Vertex5 pairs must vanish. That full certificate is the last finite gate before converting the audit into the canonical periodic mixed-hinge deficit target at $N=5$.
After the RHS translation theorem, one heavy finite reindexing bridge remains: translation invariance of the mixed explicit-fiber LHS coefficient model, equating each pair's LHS coefficient to the origin-relative pair. This definition packages that single remaining implication.
proof idea
Definitional abbreviation of an implication Prop. The antecedent is the mixed-axis LHS translation-invariance statement; the consequent is the universal vanishing of mixed-axis residual coefficients on Vertex5. No tactics or lemmas live in the body; the companion theorem discharges it by applying the named reduction fullResidualCoeffCert_of_lhs_translationInvariant.
why it matters
Session 209 LHS-only translation-reduction endpoint consumed by Track 7. It feeds the companion holding theorem and appears in the Fork handoff integration certificate structure alongside Schläfli reduction, disp0 base-vertex and stationary reductions, many-body amplitude-linear lift, and Track 6 sensitivity packaging.
In the Recognition gravity lane this is bookkeeping for the corrected axis-stencil coefficient audit at $N=5$, not a closure of open Schläfli or displacement-class leaves. The structural master theorem still uses structural witnesses where the master plan requires them; this endpoint only records that the full residual certificate now hinges on one LHS reindexing bridge after RHS translation is done.
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