status_measure_defined
plain-language theorem explainer
The Seven Gaps status record for the path-sum measure marks the measure-defined flag as true. Anyone auditing Lane 2 discharge of the Z_RS path-sum gap cites this flag as the bookkeeping witness. Proof is pure reflexivity against the status definition that sets the field.
Claim. In the path-sum-measure gap status record, the Boolean flag asserting that the measure has been defined equals $\mathrm{true}$.
background
Lane 2 of the Seven Gaps program builds an honest path-sum measure for the Recognition Science partition function $Z_{RS}$ on a scoped class of bounded combinatorial complexes. Configurations live in BoundedComplex B (incidence data at fixed lattice scale, CDT-style), the automorphism group of a labeled complex is finite, and the symmetry factor is $\mu(K)=1/|\mathrm{Aut},K|$. The path sum is the finite weighted sum $Z(B,w)=\sum_K \mu(K),w(K)$.
Gap status is tracked by an explicit record whose fields are Booleans: count-finiteness, quotient-finiteness, measure defined, positivity, and modulus bounds. The upstream definition pathSumMeasureStatus sets every field, including measure_defined, to true, reflecting that $\mu$ and $Z$ have been constructed in this module.
proof idea
One-line reflexivity. The theorem projects the measure_defined field of the status record and compares it to true; both sides reduce definitionally to the same constructor argument in pathSumMeasureStatus, so rfl closes the goal.
why it matters
This is bookkeeping, not new analysis: it freezes the claim that the path-sum measure (symmetry factor $\mu$ and the sum $Z$) is defined for the scoped class. The module doc lists the surrounding THEOREM tier: finiteness of BoundedComplex, finite triangulation classes, $0<\mu\le 1$, modulus bounds $|Z|\le\sum\mu$ and $|Z|\le\mathrm{card}$, relabeling invariance, and the unitary instance $w=e^{iS}$ giving a well-defined scoped $Z_{RS}$.
No downstream consumers are wired yet; the flag exists so later gap-closure dashboards and paper-facing status tables can read a single Boolean rather than re-inspect the constructions. The sharper exponential-growth semantics of growthBase for exact simplicial classes remains open, as the module doc states.
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