The free-Majorana relation-defect operator is fixed

Date: 25 October 2026

Status: complete for the reduced free-Majorana characteristic action and eta normalization; subsequent Arf conjugation fixes the compensating Baby-Monster multiplier

Evidence: machine-readable multiplier ledger

Certificate: word-composition verifier

Result

The corpus's existing first-handle \(T_1/S_1\) phase rules are sufficient to compute the free-Majorana part of the relation-defect operator. The explicit integral lift has the word

and

Composing the certified reduced theta-characteristic actions gives the following result on each of the three 3-cycles:

The fixed odd--odd characteristic has reduced eigenphase \(\zeta_8^2=i\) as well.

The common Dedekind eta multiplier contributes \(v_\eta(K)=i=\zeta_8^2\) to every sector. With that normalization, each 3-cycle has total product

while the fixed odd--odd sector has eigenvalue

Thus the eta-normalized free-Majorana operator has the exact spectral form

Its determinant on the ten even-characteristic sectors is \(-1\).

1. Exact monomial action

Using the characteristic ordering in the seam-permutation audit, the reduced action is

Adding the common eta exponent \(2\) produces normalized edge exponents \((5,0,3)\) on each 3-cycle and exponent \(4\) on the fixed sector.

This is invariantly summarized by the closed-cycle products and fixed-sector eigenvalue; diagonal rephasings of individual sectors cannot change them.

2. Why this is more than the permutation

The bare permutation has sign \(+1\), so the determinant quotient misses \(K\). The reduced phase matrix also has determinant \(+1\): its total exponent is zero modulo eight. The eta normalization changes the ten-sector determinant to

The lost sign is therefore restored only after retaining the metaplectic anomaly line. This is a literal example of the corpus's admissibility rule: the label permutation and phase extension are separately insufficient, while their joined operator has a new invariant.

3. The remaining Monster gate

The \(2A\) fermionization splits into a free Majorana--Weyl factor and a fermionic Baby-Monster factor. This calculation completes only the free factor. Let the unknown Baby-Monster action on the same cycles be \(\rho_{\mathrm{Baby}}(K)\). The complete \(2A\) seam requires

with compatible anomaly trivialization and separating degeneration.

The subsequent Arf-conjugation calculation fixes this multiplier to the edgewise inverse of the free action. What remains source-gated is the genus-two fermionic Baby-Monster amplitude vector, or an equivalent sewn defect module, carrying that multiplier and satisfying the required degeneration.

Boundary

The result is exact for the reduced free-Majorana \(T_1/S_1\) conventions and the declared eta normalization. It is not yet the full Monster \(2A\) multiplier, a \(2B\) Leech-seam result, or a physical detector prediction.

Sources and corpus links

• Tuite and Zuevsky, Genus Two Partition and Correlation Functions for Fermionic Vertex Operator Superalgebras I
• Tuite and Zuevsky, Part II
• Lin and Shao, Duality Defect of the Monster CFT
• Existing 0 boundary
• Relation-defect seam permutation
• Dedekind eta relation phase
• Arf-conjugated Baby-Monster 0 multiplier