The quarter-turn metaplectic coupling gate
Date: 25 October 2026
Status: finite character channels classified; genuine scalar metaplectic \(C_4\) lift ruled out for the declared one-block embedding
Evidence: quarter-turn ledger Certificate: abelianization and phase verifier
Result
The mixed-level CRT product does not generate a coupling merely by placing the level-2 and level-23 carriers beside one another. Over characteristic zero, finite-group representations are semisimple, and irreducible representations of a direct product factor as tensor products. The smallest possible scalar couplings are therefore controlled by common characters of the two factors.
At projective level there is exactly one nontrivial common character:
On the left this is the sign/determinant character of the nine-dimensional oriented-seam carrier. On the right it is reflection parity. This gives a unique binary coupling candidate between the Pfaffian orientation line and the projective order-23 transition skeleton.
It cannot retain the dicyclic central sign.
1. Why the sign-retaining lift needs a quarter-turn phase
Use the corpus presentation
where \(r=D_5\), \(s=S\), and \(s^2=r^{11}=-I\). In the abelianization the last relation gives \(r=r^{-1}\), hence \(r^2=1\). Since 11 is odd, \(s^2=r^{11}=r\), so the abelianization is generated by \(s\) and
By contrast,
Any ordinary character shared with the finite \(S_6\) permutation carrier therefore factors \(C_4\to C_2\). It sends the quarter-turn generator to the nontrivial binary sign but sends its square to the identity:
Thus the finite determinant line sees reflection orientation and necessarily forgets the dicyclic central sign.
2. The precise role of \(i\)
A sign-retaining one-dimensional character needs an order-four phase:
This is an exact representation of the abelianized dicyclic relation. It is also the rigorous sense in which the finite quarter-turn behaves like the imaginary unit: the operator \(S(a,b)=(b,-a)\) satisfies \(S^2=-I\), and its minimal faithful scalar character has value \(i\).
This is not yet a physical Wick rotation. Wick rotation analytically continues a time coordinate or contour and must be justified by spectrum, causality and boundary conditions. The result here is a finite/metaplectic phase law. It becomes physically relevant only if the same phase is derived from the sewn anomaly/Pfaffian line.
3. Why the metaplectic carrier is the right target
The ten-characteristic permutation action supplies only \(C_2\), but theta transformation laws carry eighth roots of unity on a metaplectic cover. Their phase group contains the required order-four subgroup
The explicit integral-lift audit shows that the finite \(C_4\) character cannot itself be genuine. For the one-block embedding, every lift \(\widetilde S\) satisfies \(\widetilde S^4=\varepsilon\), the nontrivial metaplectic kernel element. A genuine character has \(\chi(\varepsilon)=-1\), so \(\chi(\widetilde S)^4=-1\): its value is a primitive eighth root, never \(i\). Thus the \(C_4\) phase remains an exact finite character but is not a scalar metaplectic lift.
4. The minimum coupling table
| level-23 carrier | abelian character | level-2 partner | retained information |
|---|---|---|---|
| \(D_{11}\) | \(C_2\) reflection parity | seam determinant / \(\chi_5\) sign | projective orientation only |
| \(\mathrm{Dic}_{11}\) through \(C_2\) | binary quotient of \(C_4\) | seam determinant / \(\chi_5\) sign | central \(-I\) lost |
| \(\mathrm{Dic}_{11}\) through \(C_4\) | \(s\mapsto i\) | ordinary finite character only | central \(-I\) retained, metaplectic kernel lost |
| one-block genuine metaplectic lift | primitive eighth root | vector-valued theta/Weil carrier required | metaplectic kernel retained; no scalar \(C_4\) identification |
The table prevents the \(\chi_5\) sign line from being credited with more information than it has. The scalar \(\chi_5\) character is order two; the order-four lift, if it exists, belongs to the underlying metaplectic vector-valued data.
5. Exhaustion boundary
At the one-dimensional level the coupling classification is complete. There is a unique nontrivial projective \(C_2\) channel. The finite \(C_4\) channel retains the dicyclic center only by killing the metaplectic center, while a genuine scalar channel necessarily takes eighth-root values and cannot send the lifted quarter-turn to \(i\). The sign-retaining scalar route is therefore closed for the declared one-block embedding. A genuine scalar line nevertheless survives with a primitive eighth-root value. The Dedekind eta calculation gives \(S\mapsto\zeta_8\), the lifted base center \(\mapsto i\), the metaplectic kernel \(\mapsto-1\), and the explicit relation defect \(K\mapsto i\). Compatibility with the sewn defect line remains open; failure there moves the target to vector-valued theta/Weil data.
Sources and corpus links
- Wang, Siegel modular forms associated to Weil representations
- Gaberdiel, Persson and Volpato, Generalised Moonshine and Holomorphic Orbifolds
- Mixed-level 46 genus-two seam
- Oriented seam modular line
- Order-46 Galois orientation gate
- Metaplectic congruence-lift gate
- Integral relation rail and metaplectic scalar no-go
- Dedekind eta relation phase