track1_mixed_axis_origin_prop_coeff_cert_endpoint_holds
plain-language theorem explainer
At the origin vertex of the N=5 mixed-axis stencil, every residual coefficient against any target vertex vanishes. Track 7 (fork handoff integration) and the Session 209 translation-invariance bridge cite this as the origin-row Prop endpoint. The proof is a one-line term that re-exports the Freudenthal-axis origin residual certificate.
Claim. For every vertex $v$ in the $N=5$ vertex set, the mixed-axis residual coefficient evaluated at the origin vertex against $v$ equals zero: $\mathrm{mixedAxisResidualCoeff}(\mathrm{origin}, v) = 0$.
background
Track 7 is the integration-lane receipt for parallel gravity fork handoffs. It records what each new endpoint proves without upgrading the discovery claim. Fork A covers Track 1.B stationarity reduction at $N=5$; the remaining displacement-class leaves stay open.
The mixed-axis residual coefficient is the exact rational stencil weight comparing an origin row to a target vertex in the Freudenthal-axis discretization. The origin-row Prop endpoint packages the boolean certificate as a theorem-shaped vanishing statement: every coefficient on the origin row is zero. Session 209 uses that statement as the seed for a translation-invariance bridge that lifts one certified row to the full $125\times 125$ coefficient table without compiling every row.
Upstream, originResidualCoeffCert already proves the same universal vanishing by deciding a compiled boolean table equal to true.
proof idea
One-line term proof: the goal is definitionally the Prop
$\forall v,\ \mathrm{mixedAxisResidualCoeff}(\mathrm{origin},v)=0$,
which is exactly the statement of the upstream theorem originResidualCoeffCert. The body is that theorem applied as a term; no extra tactics or rewriting.
why it matters
This declaration is the Session 209 origin-row Prop endpoint consumed by Track 7. It feeds the integration certificate forkHandoffIntegrationCert, which assembles Track 1.B, physical residual, many-body amplitude, Page-capacity, dark-energy $w(z)$, and falsifier-sensitivity handoffs into one receipt.
In the Recognition gravity stack it closes the origin-row leaf of the mixed-axis stencil so the translation-invariance argument can promote a single certified row to full-table coefficient vanishing. That is scaffolding for the Track 1.B stationarity reduction at $N=5$, not a new physical law. It sits downstream of the Freudenthal-axis coefficient certification and upstream of the fork-level integration claim; remaining displacement-class leaves are still the next dependency.
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