ZqPhaseStructureStatus
plain-language theorem explainer
Status ledger for the quotient-first phased path sum at fixed complexity cap. Four green flags mark proved structure (phase model, bounded phased Zq, pairing cancellation, B=2 non-vacuity); three red flags stay open (continuum limit, oscillatory regulator removal, substrate-derived phase). Gravity auditors cite it to read the Lane D3 closure state at a glance. Pure data structure: no proof body.
Claim. A seven-field Boolean status record for the oscillatory phase structure on the quotient-first path sum $Z_q$: (1) an explicit relabeling-invariant phase model is defined; (2) the phased $Z_q$ is well-defined and norm-bounded at fixed complexity cap $B$; (3) exact pairing cancellation (and a strict improvement over the triangle inequality) is proved under a stated pairing hypothesis; (4) that hypothesis is discharged by an explicit $B=2$ witness; (5)--(7) continuum limit, oscillatory regulator removal, and substrate derivation of the phase remain open (recorded as false).
background
Lane D3 of the Seven Gaps program works on the quotient-first object $Z_q$: a finite complex-weighted sum over triangulation classes at complexity cap $B$, with measure factors $\mu$ descending from labeled complexes. This module equips that sum with an oscillatory phase.
A phase model is a real-valued function on labeled bounded complexes that is invariant under relabeling equivalence, hence descends to a well-defined class phase. The phased weight is $\exp(i\cdot\mathrm{phase})$; it has unit modulus. The phased path sum is then the finite sum of $\mu\cdot w_q$ over classes. At fixed $B$ one already has $|Z_q|\le\mathrm{totalClassMass}(B)\le|\mathrm{TriangulationClass},B|$.
Under an exact-opposite pairing hypothesis (an injection from a subfamily $s$ to its complement whose measured summands cancel), the paired terms drop out exactly and the modulus bound improves strictly whenever $s$ is nonempty. The cancellation mechanism is not derived from the substrate; it is supplied by the hypothesis and discharged concretely at $B=2$.
proof idea
No proof: this is a structure of seven Booleans. Each field is a named status slot whose meaning is fixed by the field doc-comments and grounded by the sibling theorems Zq_phased_wellDefined, Zq_pairing_decomposition, Zq_pairing_beats_triangle, and the $B=2$ witness chain. The canonical inhabitant zqPhaseStructureStatus assigns true to the four proved slots and false to the three open (RED) slots.
why it matters
Gives a single machine-readable closure snapshot for Lane D3 of the Seven Gaps gravity program. Downstream, zqPhaseStructureStatus is the canonical record: phase model, fixed-cap boundedness, pairing cancellation, and $B=2$ non-vacuity are marked proved; continuum limit of $Z_{\mathrm{RS}}$, oscillatory regulator removal, and substrate derivation of the phase stay false.
That split matches the module claim: structure theorems at fixed complexity cap are closed; the continuum and the origin of the phase remain open. The zero-phase regulator-removal route is already refuted elsewhere (RegulatorRemovalNoGo); the oscillatory route is not decided here. In the broader Recognition chain this sits on the quotient-first gravity path, not on T5--T8 forcing, but it inherits the eight-tick/Clifford and RS-native unit scaffolding used by the surrounding monolith.
Anyone auditing how much of the phased path-sum story is theorem versus model input should read this record first.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.