track1_mixed_axis_lhs_translation_reduction_endpoint_holds
plain-language theorem explainer
If the mixed-axis left-hand-side edge coefficient is translation-invariant, the full residual coefficient certificate follows. Track 1.B stationarity and Track 7 fork-handoff consumers cite this endpoint. The proof is a one-line term applying the Freudenthal-axis stencil lemma that lifts a single LHS summand's translation invariance through residual reindexing.
Claim. Assuming translation invariance of the mixed-axis left-hand-side edge coefficient, the full residual coefficient certificate holds: every corrected residual stencil coefficient on the mixed axis is certified after the outer finite edge sum is reindexed by the five-edge translation equivalence.
background
This module is the Track 7 integration-lane receipt for parallel fork handoffs. It records what each fork endpoint proves without upgrading the discovery claim. Fork A covers Track 1.B stationarity reduction at $N=5$; the present declaration is the Session 209 LHS-only translation-reduction leaf of that fork.
The endpoint proposition is the implication from mixed-axis LHS coefficient translation invariance to the full residual coefficient certificate. After a prior RHS translation theorem, only this mixed explicit-fiber LHS reindexing remains. Upstream, the Freudenthal-axis stencil package supplies the bridge: translation invariance of one mixed-axis edge LHS coefficient summand, together with the five-edge translation equivalence that reindexes the outer finite edge sum, yields the full corrected residual certificate.
Local gravity context is discrete Regge-style residual stencils on polarized interface edges, not continuum GR identities. The certificate is algebraic and finite-sum, not a continuum limit statement.
proof idea
One-line term-mode wrapper. The proof is exactly the upstream lemma that, given mixed-axis LHS coefficient translation invariance, returns the full residual coefficient certificate. That lemma itself factors through a residual-level translation-invariance constructor built from the LHS hypothesis, then feeds the general translation-invariant residual certificate. No local case split or arithmetic is performed here; the endpoint merely names the implication as a Track 7 handoff fact.
why it matters
Track 7's fork handoff integration certificate consumes this endpoint among the Track 1.B reduction leaves. The parent integration instance packages stationarity, displacement-class, many-body, Page-capacity, dark-energy $w(z)$, and falsifier-sensitivity receipts; this declaration closes the Session 209/211 LHS-only mixed-axis translation gap so the residual coefficient side of Fork A is recorded as proved.
In the Recognition gravity lane, residual stencil coefficients on the discrete edge complex must be translation-clean before stationarity and physical residual interfaces can be trusted. The endpoint does not itself force $D=3$ or the eight-tick octave; it is a local algebraic handoff inside Track 1.B. Remaining Track 1 displacement-class leaves stay open as the next dependency, per the module brief.
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