phasedZq_eq_exactComplexityCutoff

Module P2.3 builds the missing carrier bridge between the capped triangulation quotient at bound $B$ and the exact-shell complexes with shell index $n\le B$. A bounded complex has a unique exact complexity $\max(n_V,n_E,n_T)\le B$; conversely an exact complex in shell $n\le B$ becomes bounded by reattaching the three cap proofs. Both maps preserve incidence data and relabeling witnesses, so they descend to inverse equivalences of the two quotient carriers and preserve automorphism cardinality, hence the class measure $1/|\mathrm{Aut}|$.

An arbitrary exact-shell phase transports to a phase model at every cap. The phased capped sum is the finite sum over triangulation classes of class measure times a complex phase weight; the exact complexity cutoff is the matching sum written on the exact-shell side through shell $B$ (range $B+1$). No continuum limit, substrate phase, or physical interpretation is assumed: only finite reindexing of discrete quotient sums.

Unfold the phased capped sequence to the quotient sum $Z_q$ with the phase-model weight at cap $B$. Rewrite $Z_q$ as an explicit finite sum over triangulation classes of $\mu(\mathrm{out},q)$ times the phased weight. Reindex that sum along the cap–shell carrier equivalence via Fintype.sum_equiv, using that class measure is preserved under the shell map and that the phased weight at the transported model is $\exp(i\cdot\mathrm{phase})$. The resulting sum over shells up to $B$ is identified with the exact complexity cutoff by the prior shell-sum lemma.

This is the headline finite-sum reindexing of Seven Gaps P2.3. Downstream, capShellCompatibility is a one-constructor wrapper: every exact-shell phase has a canonically transported capped phase family satisfying the previously missing ZqContinuumBlocker.CapShellCompatibility. Closing that interface removes a blocker between discrete exact-shell bookkeeping and the capped continuum-side quotient sums used in the gravity analysis stack. The module explicitly disclaims target-sum equality, convergence, and physical continuum claims; the contribution is purely the carrier and measure-preserving reindexing that lets the two presentations of the same finite sum match.