status_quotient_finite
plain-language theorem explainer
The path-sum measure status record marks quotient finiteness as proved for the scoped triangulation class. Gravity and Z_RS auditors cite this flag when checking that the Seven Gaps Lane 2 measure is fully discharged on the quotient side. The proof is pure reflexivity against the status definition that sets the field after the finite-quotient theorem.
Claim. In the path-sum measure status record, the boolean flag asserting that the quotient of bounded complexes by relabeling is finite equals $\mathrm{true}$.
background
Lane 2 of the Seven Gaps program builds an honest path-sum measure $Z_{\mathrm{RS}}$ on a scoped class of bounded combinatorial triangulations. Configurations live in BoundedComplex B: incidence data (vertex, edge, tetrahedron counts and maps) with a fixed lattice scale, mirroring Regge-style 3D triangulations without a metric field. Relabeling of labels is an equivalence relation, and the quotient TriangulationClass B collects unlabeled classes.
The module proves that this quotient is finite, that automorphism groups are finite and nonempty, and that the symmetry factor $\mu(K)=1/|\mathrm{Aut},K|$ yields a well-defined finite path sum $Z=\sum_K \mu(K),w(K)$ with modulus bounds. A compact GapStatus record then packages which pieces of that discharge are complete.
The upstream status definition sets every relevant field (count_finite_proved, quotient_finite_proved, measure existence and positivity, modulus bound) to true after those theorems.
proof idea
One-line reflexivity. The status definition hard-codes quotient_finite_proved := true; the theorem is rfl on that field projection. No tactics or lemmas beyond definitional equality.
why it matters
This flag is the audit hook for the quotient-finiteness half of the path-sum measure gap. The module doc lists, among proved content with zero sorry, that relabeling is a genuine equivalence and that TriangulationClass B is finite. Downstream consumers of the status record can branch on this boolean without reopening the finiteness argument.
In the broader Recognition gravity stack, a finite, relabeling-invariant measure on bounded complexes is the combinatorial substrate for a well-defined $Z_{\mathrm{RS}}$ (unitary weights $w=e^{iS}$ included). The flag does not itself close exponential-growth semantics for exact simplicial subclasses; that sharper reading of the old growthBase postulate remains open. It does close the count-and-quotient finiteness content needed for the scoped CDT-style sum.
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