The mixed-level 46 genus-two seam

Date: 25 October 2026

Status: exact torsion obstruction and finite congruence carrier; analytic sewing kernel open

Evidence: mixed-level ledger Certificate: torsion, CRT and carrier verifier

Result

The order-23 orientation skeleton cannot act faithfully on the ordinary genus-two spin carrier. The correct common ambient space is not the ten-characteristic set by itself, but a mixed level-\(2\)/level-\(23\) congruence carrier.

There are two independent obstructions. First,

and \(11\nmid720\). Therefore the order-11 rotation in \(D_{11}\), and in its sign-retaining lift \(\mathrm{Dic}_{11}\), cannot act nontrivially by permuting the ten even theta characteristics.

Second, \(Sp(4,\mathbb Z)\) itself has no element of order 11. A nonidentity integral matrix \(A\) of prime order 11 would have a primitive eleventh root of unity as an eigenvalue. Its minimal polynomial would contain \(\Phi_{11}\), of degree 10, which is impossible for a \(4\times4\) matrix. Thus every homomorphism \(D_{11}\to Sp(4,\mathbb Z)\) kills \(C_{11}\) and factors through at most the reflection quotient \(C_2\).

This is stronger than the earlier \(12\ne10\) counting warning: a labelwise equivariant join is arithmetically impossible.

1. The CRT escape

Because \(46=2\cdot23\) is squarefree, the Chinese remainder theorem gives

and hence, entrywise,

The two established corpus carriers occupy different factors:

• the ten even genus-two characteristics are the \(Sp(4,\mathbb F_2)\) orbit;
• \(\mathrm{Dic}_{11}<SL(2,23)=Sp(2,23)\) embeds as a symplectic block in \(Sp(4,\mathbb F_{23})\).

Therefore

is an exact finite ambient group for the joined label problem. It does not embed \(\mathrm{Dic}_{11}\) as a finite subgroup of \(Sp(4,\mathbb Z)\); rather, it makes the order-23 operation a deck/congruence action on level-46 data.

2. What the mixed carrier retains

At label level there are three useful resolutions:

The central element \(-I\) is especially diagnostic. It reduces to the identity in characteristic two but remains \(-I\) in characteristic 23. The sign therefore lives purely on the 23-primary factor of the CRT carrier. Projectivizing that factor erases it and descends \(\mathrm{Dic}_{11}\) to \(D_{11}\).

This is the exact version of a two-rail ladder: spin parity and order-23 orientation coexist without either being renamed as the other.

3. The required sewing object

CRT supplies a product carrier, not a coupling. A genuine bridge must be a vector-valued genus-two object or defect kernel

with four properties:

1. its level-2 action reproduces the known characteristic permutations and theta/Pfaffian multipliers;
2. its level-23 action reproduces the declared \(\mathrm{Dic}_{11}\) or \(D_{11}\) transition skeleton;
3. its separating degeneration lands on sourced cross-grade VOA composition data beyond the exhausted \(S_4\) graded characters; and
4. its scalar contractions recover the known \(T_{23}\), Leech theta and fermionic factorization limits without identifying their lost data.

The first nontrivial question is whether \(\mathcal K\) factorizes as a tensor product of independent level-2 and level-23 pieces. If it does, CRT has merely juxtaposed the carriers. The sought join requires a sourced off-diagonal coupling or extension class that survives all degeneration tests.

The quarter-turn metaplectic gate classifies the one-dimensional coupling channels. At projective level the unique nontrivial common quotient is \(C_2\), matching seam determinant sign to dihedral reflection parity. It necessarily loses the dicyclic center. Retaining that sign requires the order-four character \(S\mapsto i,\ S^2=-I\mapsto-1\) from the metaplectic multiplier data. This is not a finite subgroup lift: the order-eleven rotation cannot lift as finite-order torsion in \(Sp(4,\mathbb Z)\). The relevant domain is the infinite congruence preimage of \(\mathrm{Dic}_{11}<Sp(4,23)\), and its \(\Gamma(23)\) relation phases must be checked explicitly. The later integral relation-defect audit closes this proposed scalar lift: in the one-block metaplectic cover a quarter-turn lift has fourth power equal to the cover kernel, so a genuine character takes a primitive eighth-root value rather than \(i\). The finite \(C_4\) character therefore has no genuine scalar promotion in the stated embedding. A scalar eighth-root route survives: the Dedekind eta line assigns the relation defect \(K\) the phase \(i\). The 5,280-label carrier is admissible only if the sewn defect/Pfaffian line has the same phase (up to a globally consistent alternate lift); otherwise it requires a vector-valued kernel. In fact \(K\bmod2\) already permutes the ten even characteristics with cycle type \(3^3\,1\). The complete seam target is therefore the monomial operator \(\rho(K)=D_KP_K\); a scalar eta-phase match cannot by itself close the gate. The free-Majorana part is exact: after eta normalization the three 3-cycle products are \(1\), the fixed odd--odd sector has eigenvalue \(-1\), and the determinant is \(-1\). Arf conjugation fixes the Baby-Monster phase as the edgewise inverse, also with determinant \(-1\). The complete \(2A\) multiplier is the pure \(3^3\,1\) permutation with determinant \(+1\). The remaining object is the amplitude vector and its separating degeneration.

4. Why “level 46” is not yet the analytic answer

The modulus 46 is exact for the finite label skeleton. Analytic theta constants carry eighth-root multipliers and naturally require a metaplectic cover and finer level such as the familiar \(\Gamma_2(4,8)\) setting. Combining those phases with level 23 may therefore require level 92, 184 or a metaplectic formulation rather than literal scalar level 46.

This separates the finite congruence carrier, the multiplier lift recording square roots and anomaly lines, and the analytic form whose weight, holomorphy and factorization must be constructed. Recent theory of matrix-valued Siegel modular forms from Weil representations makes this a standard kind of mathematical object, but no source located here constructs the programme-specific Monster level-\(2\)/\(23\) kernel.

5. Relation to order 46 and to \(Spin(2,3)\)

The order-46 Monster diamond and the level-46 congruence carrier share the same prime decomposition, but they are not the same object. The former is a cyclic subgroup and power map inside \(\mathbb M\); the latter is a finite quotient of the integral genus-two modular group. Their proposed relationship is the degeneration condition on \(\mathcal K\), not the number 46 alone.

Likewise, \(Sp(4,\mathbb F_{23})\cong Spin(5,23)\) and \(Sp(4,\mathbb R)\cong Spin_0(2,3)\) share type \(C_2/B_2\), but CRT does not choose a real signature, positive cone, time orientation or Hamiltonian. It supplies an arithmetic interface to the corpus's Spin spine, not a physical symmetry derivation.

6. Exhaustion and next gate

The naive genus-two join is now exhausted:

• no bijection exists between the 12 order-23 inverse-pair objects and 10 even characteristics;
• no faithful order-11 action exists on the level-2 characteristic carrier;
• no finite order-11 lift exists in \(Sp(4,\mathbb Z)\).

The remaining mathematical route is uniquely narrowed to mixed-level, vector-valued or defect data. The next calculation should construct the smallest candidate \(\mathcal K_{\delta,\lambda}\), beginning with the 240-component projective carrier, and test whether its separating boundary can reproduce a non-scalar \(S_4\) relation after the complete scalar one-insertion exhaustion. Failure of that boundary test kills the present genus-two lightning path; success identifies the precise multiplier line that must be lifted analytically.

Sources and corpus links

• Cléry, van der Geer and Grushevsky, Siegel modular forms of genus 2 and level 2
• Freitag and Salvati Manni, Basic vector valued Siegel modular forms of genus two
• Wang, Siegel modular forms associated to Weil representations
• Gaberdiel, Persson and Volpato, Generalised Moonshine and Holomorphic Orbifolds
• Order-46 Galois orientation gate
• Quarter-turn metaplectic coupling gate
• Metaplectic congruence-lift gate
• Integral relation rail and metaplectic scalar no-go
• Dedekind eta relation phase
• Relation-defect seam permutation
• Free-Majorana relation-defect multiplier
• Arf-conjugated Baby-Monster 0 multiplier
• Genus-two seam closure
• Oriented seam modular line
• Leech pair-Gram / Spin bridge