The metaplectic phase lives on a congruence extension
Date: 25 October 2026
Status: direct finite lift obstructed; explicit relation rail constructed; genuine scalar \(C_4\) lift ruled out
Evidence: congruence-lift ledger Certificate: finite-lift obstruction verifier
Result
The quarter-turn phase (S\mapsto i) is algebraically available on (\mathrm{Dic}{11}^{ab}\cong C4), and eighth-root theta multipliers exist on the genus-two metaplectic cover. Those two facts do not yet define a common finite subgroup upstairs.
The level-23 instruction group sits in
Let reduction modulo 23 be
and let (\widehat D=\pi_{23}^{-1}(\overline D)). Then
This extension cannot split. A section would embed (\mathrm{Dic}{11}) in (Sp(4,\mathbb Z)), and its order-eleven rotation would give an order-eleven integral (4\times4) matrix. That is impossible: a nontrivial primitive eleventh root has cyclotomic degree (\deg\Phi{11}=10>4).
Thus the level-23 transition skeleton has no faithful finite-order integral lift. Its integral preimage is an infinite congruence extension.
1. What this changes about the (C_4) test
Composing (\widehat D\to\mathrm{Dic}{11}\to C4) still gives an abstract character on the congruence preimage. The obstruction is subtler: the metaplectic cover pulls back to
The initially desired scalar assignment would send the metaplectic center to (-1) and the quarter-turn lift to (i). The subsequent explicit lift proves that these requirements are inconsistent: a quarter-turn lift has fourth power equal to the metaplectic center, whereas \(i^4=1\).
The finite abelianization (C_4) fixes only the quotient values. It says nothing about the phase accumulated by relations that close only modulo the principal congruence subgroup. That kernel phase is exactly the missing datum.
2. The corrected next calculation
Do not search for a finite embedding (\mathrm{Dic}_{11}<Mp(4,\mathbb Z)); none can project faithfully to (Sp(4,\mathbb Z)). Instead:
The explicit relation audit completes the first two steps. It supplies a symmetric lift for which the inversion relation closes exactly and the selected congruence kernel is generated by one element \(K=S^2R^{-11}\). The scalar test then fails algebraically before a cocycle evaluation: no genuine character can have \(\widetilde S\mapsto i\). The Dedekind eta calculation then evaluates the canonical genuine scalar survivor: \(\widetilde S\mapsto\zeta_8\) and \(K\mapsto i\). The corrected next calculation is to compare that exact \(K\)-phase with the sewn order-23 defect line. Incompatibility moves the target to a vector-valued action.
3. Admissibility consequence
This is another instance of the corpus's central rule: agreement after a quotient does not provide a lift. The finite (C_4) phase names what must be retained; the congruence extension records the additional relations that must also be admitted. A genus-two bridge is not established until both agree.