The integral relation rail and finite-\(C_4\) metaplectic no-go
Date: 25 October 2026
Status: explicit lift and congruence kernel constructed; identification of the finite \(C_4\) character with a genuine scalar metaplectic character ruled out
Evidence: machine-readable relation ledger
Certificate: exact matrix and phase verifier
Result
The congruence-lift problem can be made completely explicit. In the declared one-symplectic-pair block \(SL(2)\hookrightarrow Sp(4)\), choose
Both have determinant one, and modulo 23 they reduce to
The lift was chosen symmetric, so the inversion relation closes exactly:
Also \(S^2=-I\) exactly. The only finite dicyclic relation that fails over the integers is \(S^2=R^{11}\). Its defect is
Thus the selected integral subgroup does not require an uncontrolled search through the full principal congruence group. Its kernel over \(\mathrm{Dic}_{11}\) is the single infinite cyclic rail
The finite \(C_4\) character cannot be promoted unchanged to a genuine metaplectic character. A lift \(\widetilde S\) of the quarter-turn satisfies
where \(\varepsilon\) is the nontrivial central element above the identity. A genuine character sends \(\varepsilon\mapsto-1\), and therefore
It follows that \(\chi(\widetilde S)\) must be a primitive eighth root, never \(i\). Conversely, the finite \(C_4\) assignment \(\widetilde S\mapsto i\) has fourth power \(+1\), so it kills the metaplectic center. The requirements
cannot hold simultaneously.
1. The exact relation defect
The eleventh power is
Hence
Every element of \(\langle R,S\rangle\) has a normal form \(R^nS^j\), with \(j\in\{0,1,2,3\}\). Reduction modulo 23 is the identity only when
- \(j=0\) and \(n\equiv0\pmod{22}\), or
- \(j=2\) and \(n\equiv11\pmod{22}\).
Since \(K=S^2R^{-11}\) and \(K^2=R^{-22}\), both families are precisely powers of \(K\). This proves \(\langle R,S\rangle\cap\Gamma(23)=\langle K\rangle\).
The relation rail remains useful for every scalar or vector-valued calculation: its multiplier must assign a consistent phase or operator to this explicit \(K\).
2. The two central operations
There are two different central objects:
- the dicyclic center \(-I=S^2\) in the base finite group;
- the metaplectic kernel \(\varepsilon\) above the identity symplectic transformation.
The finite character \(S\mapsto i\) sends the first to \(-1\), but it sends the second to \(+1\). A genuine metaplectic character instead gives \(\widetilde S\) an eighth-root value and sends its square to \(\pm i\), not \(-1\). This obstruction is not a normalization choice: the other lift \(-\widetilde S\) has the same fourth power.
3. Revised surviving corridor
The finite-\(C_4\)-valued 5,280-label proposal is closed in its stated form. The Dedekind eta audit supplies a narrower scalar survivor:
The remaining routes are:
- the 240-label projective seam using the ordinary \(C_2\) parity line;
- the genuine eta line, pending comparison with the sewn defect phase on \(K\);
- a vector-valued metaplectic/Weil kernel in which the two central operations act on distinct components;
- a different symplectic embedding whose Maslov parity is explicitly shown to differ—this would be a new carrier, not a repair of the one-block map;
- cross-grade VOA or genus-two sewing data that supplies the required action directly.
The next scalar calculation is the comparison between \(v_\eta(K)=i\) and the sewn order-23 defect/Pfaffian phase on \(K\). A mismatch moves the route to a vector-valued kernel.
Boundary
This no-go applies to identifying the finite character \(S\mapsto i,\ S^2\mapsto-1\) with a one-dimensional genuine character of the declared one-block embedding. It does not rule out genuine scalar eighth-root multipliers, vector-valued Weil representations, a different embedding with different Maslov parity, or a higher-genus defect module. It also does not create a physical Wick rotation, clock or detector.