classPushforwardStatus
plain-language theorem explainer
Status board for the class-pushforward wave of Seven Gaps Crux-2 (path-sum pillar). It records five proved bookkeeping facts (generic fiber sums, class mass, Z as class pushforward, mass equals fiber card times mu, non-singleton fiber at B=2) and five still-false open items (inverse-Aut class sum, quotient-first object, continuum limit, substrate measure, gap-1 bridge). Auditors of the path-sum convention cite it for the honest flag set. The body is a pure structure literal of boolean constants.
Claim. The class-pushforward status after this module: generic fiber decomposition, class-mass definition, $Z$ equal to the class pushforward, class mass equal to fiber cardinality times $\mu$, and a non-singleton fiber are all marked proved; $Z$ as the inverse-Aut class sum, quotient-first object construction, $Z_{RS}$ continuum limit, substrate-measure derivation, and gap-1 bridge derivation are marked unproved.
background
Module setting is Seven Gaps Crux-2 (pillar 2, path-sum): the class pushforward of the labeled partition function $Z$, under protocol QUOTIENT_BOOKKEEPING. The standing $Z$ of PathSumMeasure sums over labeled complexes with weights $\mu(K)=1/|\mathrm{Aut}(K)|$. The wave decomposes that sum over triangulation classes.
Key objects: for a finite type with a setoid, the sum of an additive function over $\alpha$ factors as a sum over quotient fibers; when the weight is constant on classes it collapses to $\mathrm{fiberCard}(q)\bullet f(\mathrm{rep},q)$. Class mass is $\mathrm{classMass}(q)=\sum_{K:\llbracket K\rrbracket=q}\mu(K)$, and by the fiber identity equals $|\mathrm{fiber}(q)|\cdot(1/|\mathrm{Aut}(\mathrm{rep},q)|)$, not bare $1/|\mathrm{Aut}|$ per class.
The status structure is a pure Bool record (no True shells). Every flag is forced by rfl in the companion flags theorem. The false flag on inverse-Aut class sum encodes the C1 landmine: at $B=2$ a non-singleton fiber makes $\mu(\mathrm{edgeAB})<\mathrm{classMass}(\llbracket\mathrm{edgeAB}\rrbracket)$.
proof idea
Definitional structure instance, not a proof. Each field of ClassPushforwardStatus is assigned a boolean literal (true for the five closed bookkeeping items, false for the five open ones). No tactics, no lemmas applied at this site; the companion theorem later discharges equality of those fields to true/false by rfl.
why it matters
Gives the honest completion board for Crux-2 class pushforward inside the Seven Gaps gravity stack. Downstream, classPushforwardStatus_flags packages the five true flags as a single conjunction, so later waves can gate on proved bookkeeping without re-auditing the module.
Framework role: separates the labeled path-sum $Z$ (this convention) from the exact-shell quotient-sum $Z_{RS,uv}$. The false Z_is_invAut_class_sum flag is the kernel record of the C1 detonation: class mass carries $|\mathrm{fiber}|\cdot(1/|\mathrm{Aut}|)$, so the standing labeled $Z$ is not the inequivalent-class sum with bare $1/|\mathrm{Aut}|$ weights. Open false flags mark the remaining path to continuum limit, substrate measure, and the gap-1 bridge.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.