Arf conjugation fixes the Baby-Monster \(K\) multiplier
Date: 25 October 2026
Status: exact in the canonical Arf convention and the certified anomaly-free \(46C/D\) cyclic channel; amplitude sewing remains open
Evidence: machine-readable multiplier ledger
Certificate: Arf-conjugation verifier
Result
The Baby-Monster relation-defect multiplier can be determined without first evaluating the sixteen genus-two amplitudes. A modular multiplier is representation data, while the amplitudes are vectors on which that representation acts.
On the \(46C/D\) rail, \(b=h^2\) and \(z=h^{23}\) generate the same cyclic order-46 group. The existing anomaly audit finds its Moonshine anomaly restriction trivial. Since \(K\equiv I\pmod {23}\), it fixes the \(b\)-label; on the sixteen flat \(z\)-backgrounds it acts only by the linear mod-2 permutation.
Let \(A/4\) be the certified genus-two Arf/Fourier transform and let \(P_K\) be the inverse background permutation corresponding to the declared characteristic-action convention. Direct conjugation gives
where every nonzero entry on the right is \(+1\). Its restriction to the ten even spin structures is exactly the previously certified cycle decomposition \(3^3\,1\). Thus the complete fermionized \(2A\) action has no residual edge phase in this convention. The Baby-Monster edge phase is forced to be the inverse of the eta-normalized free-Majorana edge phase.
1. Exact Baby-Monster phases
With \(\zeta_8=e^{i\pi/4}\):
| source | target | free exponent | Baby exponent | combined exponent |
|---|---|---|---|---|
| \(00\mid00\) | \(00\mid10\) | 5 | 3 | 0 |
| \(00\mid10\) | \(10\mid00\) | 0 | 0 | 0 |
| \(10\mid00\) | \(00\mid00\) | 3 | 5 | 0 |
| \(00\mid01\) | \(00\mid11\) | 5 | 3 | 0 |
| \(00\mid11\) | \(10\mid01\) | 0 | 0 | 0 |
| \(10\mid01\) | \(00\mid01\) | 3 | 5 | 0 |
| \(01\mid00\) | \(01\mid10\) | 5 | 3 | 0 |
| \(01\mid10\) | \(11\mid00\) | 0 | 0 | 0 |
| \(11\mid00\) | \(01\mid00\) | 3 | 5 | 0 |
| \(11\mid11\) | \(11\mid11\) | 4 | 4 | 0 |
Consequently the Baby-Monster invariants are
the same three trivial cycle holonomies and fixed-sector sign as the free factor. Pointwise multiplication gives the full \(2A\) invariants
2. The determinant lesson
The free monomial operator has determinant \(-1\). The reconstructed Baby-Monster monomial operator also has determinant \(-1\). Their phase products cancel in the pointwise fermionized amplitude, leaving the single even-sector permutation, whose sign is \(+1\). Therefore
This rules out a tempting promotion: the free-factor determinant sign is not a determinant sign of the complete \(2A\) Monster fermionization. What survives is the nontrivial sector permutation itself. Any proposed bridge to the \(\chi_5\) character line must obtain its sign elsewhere; it cannot import the free-Majorana determinant unchanged.
3. What remains open
The multiplier gate is closed, but the amplitude gate is not. The sixteen twined Monster background functions, or equivalent direct Baby-Monster sewing data, are still required to:
- prove the relevant even-sector functions exist with the assumed normalizations;
- test the six odd-sector zero identities;
- verify separating degeneration to the order-23 genus-one traces;
- compare actual sewn functions, rather than only their mapping-class representation.
The next frontier is therefore no longer “find the Baby multiplier.” It is “construct one amplitude vector carrying the now-fixed multiplier and test its degeneration.”
Boundary
The conclusion uses the canonical Arf fermionization convention, the existing trivial-anomaly result for the cyclic \(46C/D\) channel, and the declared one-block embedding of \(K\). A different basis may redistribute edge phases, but cannot change the closed-cycle holonomies, fixed eigenvalue, or determinant conclusions.