The congruence defect becomes a three-cycle seam operator
Date: 25 October 2026
Status: exact level-\(2\), free-Majorana and Arf-reconstructed Baby-Monster multipliers; amplitude sewing remains open
Evidence: machine-readable seam-action ledger
Certificate: characteristic-action verifier
Result
The explicit relation defect
was constructed because it is invisible modulo 23:
It is not invisible to the level-\(2\) genus-two seam. For the symmetric lift \(R=(\begin{smallmatrix}5&23\\23&106\end{smallmatrix})\),
an element of order three in \(SL(2,2)\). Embed this matrix on the first symplectic pair and fix the second pair. Its action on the ten even genus-two theta characteristics has cycle type
Nine characteristics form three 3-cycles and the odd--odd characteristic is fixed. The permutation is even, so its determinant is \(+1\).
This locates the relation defect precisely:
level 23 labels: K is invisible
level 2 even characteristics: K moves 9 of 10 sectors
determinant/sign line: K is invisible again
eta multiplier line: K has phase iThe seam operator therefore cannot generally be replaced by the scalar eta phase \(i\). Before characteristic-dependent phases are included, its minimum form is already monomial:
where \(P_K\) has cycle type \(3^3\,1\). The free-Majorana part of \(D_K\) is now computed from the certified \(T_1/S_1\) action. After eta normalization, all three cycle products are \(1\), the fixed odd--odd sector has eigenvalue \(-1\), and the determinant is \(-1\). Arf conjugation forces the Baby-Monster phase matrix to be its edgewise inverse. The complete \(2A\) multiplier is therefore the pure \(3^3\,1\) permutation with determinant \(+1\).
1. Exact characteristic action
Write a genus-two characteristic as
with even parity \(a_1b_1+a_2b_2=0\). In the standard \((q_1,q_2,p_1,p_2)\) basis, the one-block lift of \(\bar K=(\begin{smallmatrix}1&1\\1&0\end{smallmatrix})\) is
Using the affine symplectic action on characteristics, the cycles are
The fixed characteristic is odd on each handle but even in genus two. It is also the characteristic whose theta constant vanishes on the separating divisor because each genus-one odd theta constant vanishes at zero. The fixed point therefore has a clear sewing interpretation rather than being an unlabelled combinatorial residue.
2. What the determinant forgets
Three 3-cycles have permutation sign
Hence neither the ordinary permutation determinant nor the existing \(C_2\) seam sign detects \(K\). This is the same admissibility pattern seen elsewhere in the corpus:
- the full sector vector records a transformation;
- its determinant quotient records only orientation parity;
- an even permutation can therefore be dynamically nontrivial while leaving the scalar sign unchanged.
The eta line adds a phase \(i\), but that scalar and the characteristic permutation are independent pieces of the lifted action.
3. Revised next calculation
Both matrix multipliers are complete. The remaining source-gated target is:
- retain the computed eta-normalized free-Majorana monomial action;
- construct the fermionic Baby-Monster order-23 amplitude vector carrying the reconstructed inverse multiplier;
- test its six odd-spin zeros and separating degeneration;
- compare the resulting sewn functions, not only their monomial action.
A scalar phase match is necessary but no longer sufficient. Failure of the permutation or diagonal phase comparison rules out the seam even if both sides separately contain an \(i\).
Boundary
The cycle calculation uses the declared one-block embedding and the standard affine action on theta characteristics. It does not compute the full genus-two theta multiplier, construct the Monster sewn defect, or identify the modular operation with laboratory dynamics.