The order-46 observable ladder

Date: 25 October 2026

Status: genus-one rail separation, modular transport and cyclic anomaly status closed; compatible normalization and genus-two sewing open

Evidence: observable-ladder ledger Certificate: arithmetic and cross-ledger verifier

Result

The order-46 junction does more than prove that the 2A, 2B and 23 branches meet. It supplies a two-stage observable ladder.

At order 23, the scalar McKay--Thompson series collapses both the generator orientation and the involution rail:

At order 46, ordinary Moonshine recovers the involution rail while still forgetting orientation within each rail. The published genus-zero groups are

Here \(46+23\) denotes \(\Gamma_0(46)\) extended by the \(W_{23}\) Atkin--Lehner involution, while \(46+\) is the full Atkin--Lehner extension. Consequently

The ordinary order-46 trace therefore distinguishes the 2B/Leech rail from the 2A/Baby-Monster rail, but it does not orient either rail.

1. The root is encoded by the commuting pair

Let

for an order-46 element \(h\). The Chinese-remainder reconstruction is explicit:

because \(g^{12}z=h^{24}h^{23}=h^{47}=h\). Thus the order-46 element is not an auxiliary label added to the \(23\times2\) data. It is the unique combined insertion selected by that commuting cyclic pair.

This explains why \(T_h\) can recover the involution rail even though \(T_g\) cannot. The order-23 square \(g\) forgets the parity component \(z\); the combined insertion \(g^{12}z\) restores it.

2. The desired twisted trace is a modular transport

Generalized moonshine assigns a function \(Z(g_0,h_0;\tau)\) to commuting pairs, with an \(SL_2(\mathbb Z)\) action of the form

where \(\doteq\) allows the projective multiplier fixed by the anomaly and normalization convention. Take

Starting from the ordinary twining pair \((1,h)\), it gives

Therefore

up to the precise direction of the convention and its projective multiplier. The missing order-46 twisted--twining function is not 529 independent input series and not an arbitrary new function. Once a convention is fixed, it is a modular branch of a known order-46 Hauptmodul.

3. The anomaly bound closes, leaving normalization

The order-23 restriction of the Monster anomaly is forced to vanish because its order divides both 24 and 23. The same argument does not trivialize the order-46 restriction:

At this coarse stage only a possible two-torsion obstruction remains. The published Moonshine notation closes it. An anomalous cyclic class is written \(n|h+\cdots\) with \(h>1\), whereas the two order-46 functions are written \(46+23\) and \(46+\). Both therefore have \(h=1\): they are normal, non-anomalous cyclic orbifolds. The restriction of the Monster anomaly to each order-46 cyclic subgroup is cohomologically trivial.

This does not manufacture an absolute phase. A trivialization and the normalization of each generalized-moonshine function must still be chosen consistently, and inverse conjugacy classes may still be identified. But any remaining scalar is a convention/coboundary datum rather than a nontrivial \(\mathbb Z_2\) obstruction. In particular, the arithmetic possibility \(\gcd(24,46)=2\) is not realized by these two cyclic class pairs and cannot be identified with the genus-two Pfaffian sign.

The immediate source-level calculation is therefore:

1. fix compatible anomaly trivializations and generalized-moonshine normalizations for 46A--46D;
2. transport \(T_{46AB}\) and \(T_{46CD}\) by \(\gamma_{2,23}\) (or its inverse under the chosen convention);
3. normalize the transported functions by a common principal-part convention and verify the power-five pairing described below;
4. compare the surviving sign/character with the 2A fermionic determinant line and the 2B order-eight lift kernel;
5. only then sew at genus two.

4. Symmetry and compression count

The index of \(\Gamma_0(46)\) in \(PSL_2(\mathbb Z)\) is

The full Atkin--Lehner group at squarefree level 46 has four elements, while the \(+23\) extension has two. Thus the corresponding indices are 18 and 36. The 46CD function is invariant under the larger group and is consequently the more compressed scalar record. This is a group-theoretic measure of retained information; it is not a count of physical states.

The observable hierarchy is now

order-23 scalar T_23
  1 output: loses involution rail and generator orientation
        |
        | retain the order-2 component through h=g^12 z
        v
order-46 scalar T_h
  2 outputs: 46AB (2B rail) versus 46CD (2A rail)
        |
        | modularly transport within the same cyclic subgroup
        v
generalized pair Z(h^2,h^23)
  2 normalized scalar outputs: the inverse/power-five pairs remain collapsed
        |
        | scalar centralizer insertions are exhausted;
        | the linear/projective S4 action is also exhausted;
        | change compositional level to VOA products or genus-two sewing
        v
candidate seam-sensitive vector

The pairing follows directly. Fifth power exchanges \(46A\leftrightarrow 46B\) and \(46C\leftrightarrow46D\). Under \(h\mapsto h^5\),

The ordinary functions satisfy \(T_h=T_{h^5}\) within each pair. Applying the same matrix \(\gamma_{2,23}\) to the same normalized Hauptmodul therefore gives the same normalized cyclic commuting-pair function for \((g,z)\) and \((g^5,z)\). A convention-dependent scalar cannot serve as an orientation observable. Thus cyclic generalized moonshine recovers the 2A/2B rail but does not recover the 23A/23B generator orientation. The subsequent complete \(C_{23}\times S_4\) census shows that every individual centralizer insertion also generates a cyclic subgroup and is equally scalar-degenerate. The next refinement must therefore change compositional level. The anomaly transgression to the centralizer is trivial, so even the full ordinary \(S_4\) character is determined by those five traces. Only cross-grade VOA operations or higher-genus sewing remain.

5. Consequence for the frontier programme

The first frontier tranche can now be made smaller. There is no need to search for a free-standing order-46 generalized trace table before doing any work. Start from the two known Hauptmoduln, transport them by the explicit matrix and audit the compatible normalization. This is an FPT-002 inverse-detector problem: the order-46 insertion is the minimal extra channel already known to separate the two involution rails. It is also an FPT-019 complexity reduction: the unknown input is cross-grade VOA composition or a higher-genus refinement, not a 529-function oracle or another centralizer representation table.

The physical boundary is unchanged. Modular transport is worldsheet/VOA transport, not real-time evolution, and its normalization phase is not automatically a laboratory parity signal.

Sources and corpus links

• Duncan, Griffin and Ono, Moonshine, table of Monster classes and genus-zero groups
• Carnahan, Generalized Moonshine IV: Monstrous Lie algebras
• Tuite, Generalised Moonshine and Abelian Orbifold Constructions, for the \(n|h\) anomaly convention
• ATLAS Monster conjugacy classes
• Joined order-46 bridge
• Frontier target crosswalk