The order-23 projective-centralizer no-go
Date: 25 October 2026
Status: genus-one centralizer representation route exhausted; higher-genus and cross-grade structure open
Evidence: projective-centralizer ledger Certificate: cohomology and character-degree verifier
Result
The complete scalar census leaves an apparent escape: perhaps the \(g\)-twisted module carries a genuinely projective \(S_4\) representation whose central extension retains the \(23A/23B\) orientation even though every individual trace agrees. The anomaly arithmetic rules out that escape.
For \(g\) of order 23,
Abstractly,
because a cyclic group has trivial Schur multiplier and the cross term \(C_{23}\otimes C_2\) vanishes. The nontrivial class would lift \(S_4\) to a Schur double cover \(2.S_4\), with spin irreducible dimensions \(2,2,4\).
But generalized moonshine does not choose an arbitrary element of this \(C_2\). The projective class in the \(g\)-twisted sector is the transgression, or slant product, of the Monster anomaly \([\omega]\in H^3(\mathbb M,U(1))\), whose order is 24.
On \(C_{23}\times S_4\), positive-degree cohomology from the first factor is 23-primary and that from the second is supported only at primes 2 and 3. All mixed tensor and Tor terms therefore vanish. The restricted anomaly splits into factor classes. Its \(C_{23}\) component is zero because its order must divide both 24 and 23. Its remaining \(S_4\) component is inflated from the second factor, and slanting it with \(g\) in the first factor is trivial. Consequently
The \(g\)-twisted centralizer action is linearisable.
1. Ordinary character determinacy finishes the argument
The ordinary irreducible dimensions of \(S_4\) are
There are five irreducibles and five conjugacy classes. At every conformal grade, the five scalar traces on the identity, double-transposition, transposition, 3-cycle and 4-cycle classes therefore determine the complete complex \(S_4\)-module multiplicity vector by character-table inversion.
The centralizer census has already shown that the corresponding A/B trace series agree in all five product families:
After one compatible linearization, the 23A and 23B twisted graded \(S_4\)-characters are identical. Their graded centralizer representations are isomorphic as complex \(S_4\)-modules. A change of basis cannot be an orientation observable.
2. Why multi-insertion traces do not reopen genus one
For an ordinary representation \(\rho\), a trace of a word
is the character evaluated on the single product element. Since every word in \(S_4\) is already in one of the five conjugacy classes, arbitrary scalar word traces add no information beyond the character table.
Matrix elements in a chosen basis can distinguish presentations, but the basis is gauge unless it is fixed by an external VOA operator, defect junction or sewing map. Thus “retain noncommuting relations” is not by itself a new invariant. The source must supply extra compositional structure across states or grades.
3. What remains genuinely open
The following routes survive:
- genus-two sewing: two handles can retain a bilinear/junction datum not reducible to one group word or one character value;
- cross-grade VOA structure: intertwining operators, fusion and vertex products can distinguish graded algebras whose graded group modules agree;
- externally marked orientation: an independently sourced lattice or defect marking may choose one Galois rail before scalar quotienting.
The ordinary \(S_4\) multiplicity calculation need not be performed as an orientation test: its answer is already fixed by the five scalar series. The binary-octahedral degrees \(2,2,4\) remain mathematically real, but that projective sector is not activated by the order-23 Monster anomaly transgression.
4. Consequence for the mixed-level seam
The level-\(2\)/\(23\) CRT carrier remains the correct genus-two ambient space, but its level-23 boundary input cannot be “a missing projective \(S_4\) character.” It must be a sewn defect or cross-grade operation. The quarter-turn \(C_4\) character is likewise not a dormant genus-one cocycle. Its proposed finite-\(C_4\) scalar metaplectic promotion is ruled out for the one-block embedding. A genuine eta line with eighth-root quarter-turn value survives, so genus-two sewing must first decide its phase on \(K\) before a vector-valued action is required.
The next decisive object is therefore a kernel \(\mathcal K_{\delta,\lambda}(\Omega)\) whose separating degeneration includes composition data, not merely the already exhausted graded characters.
Sources and corpus links
- Johnson-Freyd, The Moonshine Anomaly
- Gaberdiel, Persson and Volpato, Generalised Moonshine and Holomorphic Orbifolds
- Carnahan, Generalized Moonshine IV: Monstrous Lie algebras
- Order-23 centralizer exhaustion
- Mixed-level 46 genus-two seam
- Quarter-turn metaplectic coupling gate
- Integral relation rail and metaplectic scalar no-go
- Dedekind eta relation phase