The Arf transform is the lightning route to the Baby-Monster seam
Date: 25 October 2026
Status: exact finite reconstruction map and conjugated \(K\) multiplier certified; its sixteen order-23-twined Monster background amplitudes remain source-gated
Evidence: machine-readable Arf-transform ledger
Certificate: finite Fourier verifier
Result
The Baby-Monster genus-two vector need not be constructed only by sewing \(VB^\natural\) directly. Fermionization gives an exact alternative starting on the Monster rail.
For a genus-\(g\) bosonic theory with a non-anomalous \(\mathbb Z_2\) symmetry, Lin and Shao write the fermionic partition functions as an Arf-weighted finite Fourier transform of bosonic amplitudes with flat \(\mathbb Z_2\) backgrounds. At genus two, choose an even reference spin structure \(\rho\). For \(S,t\in H^1(\Sigma_2,\mathbb F_2)\cong\mathbb F_2^4\),
The coefficient matrix is a signed Walsh--Hadamard matrix \(A/4\). The certificate checks exactly that
The transform is invertible: no information is lost between the sixteen \(2A\)-background amplitudes and the sixteen fermionic spin amplitudes.
1. Add the order-23 rail
Let \(z\in2A\) and let \(b\in23AB\) commute with it. The order-46 junction provides precisely such a pair:
The Arf kernel depends only on the \(z\)-background and spin structure, so an order-23 insertion is a spectator label:
The exact cross-rail input is therefore a sixteen-component bosonic vector: the genus-two Monster amplitude twined by \(b\), with every flat \(2A\) background \(t\in\mathbb F_2^4\).
2. Extract the Baby-Monster factor
The fermionized theory factorizes pointwise on the shared spin structure:
The six odd-spin free-Majorana amplitudes vanish by the zero mode. The transformed Monster vector must therefore obey six exact zero constraints. On each of the ten even spin structures the free amplitude is nonzero, so
This reconstructs exactly the ten-component Baby-Monster vector required by the relation-defect calculation.
3. The explicit defect word
Once the sixteen bosonic functions are supplied with compatible conventions, apply \(K=S^2R^{-11}\) either before or after the Arf transform. The Arf-conjugation audit shows that the multiplier itself is already fixed: the Baby factor is the inverse free phase and the complete \(2A\) action is the pure \(3^3\,1\) permutation.
46C/D root h -> commuting pair (b=h^2 in 23AB, z=h^23 in 2A) -> 16 b-twined Monster amplitudes with flat z backgrounds -> invertible Arf/Fourier transform -> 16 fermionic spin amplitudes -> 6 odd-sector zero tests + 10 even sectors -> divide by known free Majorana -> Baby-Monster amplitude vector with fixed K multiplier
4. Remaining source object
The transform does not conjure the sixteen bosonic amplitudes from the five genus-one centralizer traces. It converts the construction problem into one native to the existing Monster/order-46 branch. The remaining source object is either:
- direct twined Baby-Monster sewing data; or
- the sixteen genus-two Monster amplitudes with simultaneous order-23 insertion and all flat \(2A\) backgrounds.
The second packet is tied to the certified \(46C/D\) pair and carries six exact odd-spin null tests.
Boundary
The finite Arf transform and conjugated multiplier are exact. The twined Monster background vector is not yet constructed. Genus-one McKay--Thompson series do not determine it.