status_relabel_invariance
plain-language theorem explainer
Status flag that the path-sum measure's relabeling invariance is recorded as proved. Auditors of the Seven Gaps Lane-2 Z_RS stack cite it when checking honest status tags. The proof is pure reflexivity: the status record's boolean field is definitionally true.
Claim. The path-sum measure status record marks relabeling invariance as proved: the corresponding boolean flag equals $\mathrm{true}$.
background
Lane 2 of the Seven Gaps gravity stack builds a scoped path-sum measure for $Z_{\mathrm{RS}}$ on bounded combinatorial complexes. Configurations live in BoundedComplex B: incidence data at fixed lattice scale (vertices, edges, tets), CDT-style, with geometry carried by incidence maps rather than a metric field.
The measure on a labeled complex $K$ is the symmetry factor $\mu(K)=1/|\mathrm{Aut}(K)|$, which is positive, at most one, and unchanged under relabeling. The path sum is $Z(B,w)=\sum_K \mu(K),w(K)$ over the finite labeled class. Module theorems establish that relabeling is an equivalence relation, the quotient of triangulation classes is finite, and $Z$ itself is invariant under equivalence-preserving bijections of configurations.
This declaration does not restate those analytic facts. It is the boolean status bit on the module's status record that tags the relabel-invariance clause as discharged.
proof idea
One-line term proof by rfl. The status structure's relabel_invariance_proved field is definitionally true, so equality to true holds by reflexivity. No lemmas are applied; no analytic content is re-proved here.
why it matters
The module header promises honest status tiers: proved theorems (zero sorry, zero new axioms) versus model assumptions and open continuum limits. Relabel invariance of $\mu$ and of $Z$ is listed among the proved items; this flag is the machine-checkable tag that the clause is closed in the status record.
It sits beside the finite-type structure on bounded complexes, the setoid of relabelings, finiteness of automorphism groups, and the norm bounds $|Z|\le\sum\mu$ and $|Z|\le|\mathrm{BoundedComplex},B|$. Together they underwrite the honest scoped $Z_{\mathrm{RS}}$ statement for unitary weights $w(K)=e^{iS(K)}$.
No downstream consumers are wired yet (used_by empty). The continuum limit $B\to\infty$ remains explicitly open in the module, so this flag only certifies the finite-scope invariance layer, not a continuum path integral.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.