The joined Monster bridge: an order-46 junction and an admissibility ladder

Date: 25 October 2026

Status: exact class-level junction; generalized-moonshine and genus-two evaluations open

Evidence: joined bridge ledger Certificate: cross-branch verifier

Result

The corpus's order-23, 2A-fermionic and 2B-Leech branches are not merely adjacent. The Monster power map supplies an exact common junction at order

46. If \(h\) has order 46, then

have orders 23 and 2 and commute. The four order-46 Monster classes divide as follows:

The precise A/B output depends on the chosen generator, as it must: fifth power and inversion exchange the two oriented order-23 classes. What is invariant is the class-level diamond

                 23AB
                /    \
          46A/B        46C/D
             |            |
             2B           2A
       Leech branch   fermionic branch

This closes the former existence question: the 2A and 2B channels both meet the order-23 channel inside cyclic order-46 subgroups of the Monster. It does not identify their seam variables. The correct next object is a twisted--twining/generalized-moonshine function for the commuting pair \((g,z)\), together with its genus-two sewing, not an asserted equality between two unrelated binary labels.

1. Why the order-46 bridge is correctly typed

The identities above are elementary inside the cyclic group generated by \(h\):

ATLAS power-up data records \(46A,46B\) above 2B and \(46C,46D\) above 2A, while all four occur above 23A and 23B. This is an exact statement about Monster conjugacy classes. It is weaker than claiming that every chosen representative of 23A/B has every kind of root, and stronger than observing that 2 and 23 both divide the Monster order.

For a commuting pair, generalized moonshine supplies the natural observable

where \(V^\natural_g\) is the \(g\)-twisted module. The ordinary order-46 McKay--Thompson series \(T_h\) and the two-variable function \(Z(g,z;\tau)\) are related data, but they are not interchangeable. Computing \(T_h\) alone does not establish how the 2A/2B insertion acts in the order-23 twisted sector.

2. The joined compiler

Before passing to the compiler, the same diamond can be resolved through the two involution centralizers. These are the two canonical Monster local geometries already used by the corpus:

For \(h\in46C,46D\), the central involution is 2A. Hence \(h\) lies in \(2.B\), and quotienting by its central involution sends it to an order-23 element of the Baby Monster. The centralizer arithmetic closes exactly:

while the two Baby-Monster order-23 classes have centralizer order 46. Thus the 46C/D rail is literally the central double-cover lift of the Baby-Monster 23A/B rail selected by 2A fermionization.

For \(h\in46A,46B\), the central involution is 2B, so \(h\) lies in \(2^{1+24}.Co_1\). Projection through the normal 2-group sends it to an order-23 element of \(Co_1\), whose two order-23 classes have centralizer order

23. Here

The kernel of the centralizer projection therefore has order eight. This is an exact residue in the Leech-side lift, but the corpus does not yet identify that kernel with the Rule-60 affine deck bit or with a physical seam. The asymmetry \(2\cdot46\) versus \(8\cdot23\) is a new discriminator for the twisted--twining calculation, not permission to force the two rails into the same finite carrier.

The pieces now form a typed chain rather than a list of resonances:

Rule 60: 1 + sigma on the cyclic Golay code
          |
          | take the genuine order-23 symmetry sigma
          v
two degree-11 cyclotomic rails over F_2
          |
          | contract the primes above 2
          v
inverse norm-two ideals in Q(sqrt(-23))
          |
          | ideal norm / Leech lift
          v
rank-two determinant-23 fixed lattice
          |
          | theta / eta, then retain rather than discard orientation
          v
23AB Monster twist-charge plane F_23^2
          |
          | choose an order-46 cyclic root
          +----------------------+----------------------+
          v                                             v
2B insertion / Leech orbifold                 2A insertion / fermionization
pair-Gram and affine seam                      10 even spin sectors + Pfaffian line
          +----------------------+----------------------+
                                 |
                                 | generalized trace and genus-two sewing
                                 v
                     candidate common seam observable

Every downward arrow before the final sewing problem is now either calculated in the corpus or backed by a published group, lattice or VOA construction. The final arrow is a research target, not a conclusion.

3. Admissibility as retained information

Across the joined chain, admissibility can be made operational: a proposed transition is admissible only if its target retains the datum needed by the next operation, or declares that datum irrecoverably quotiented out.

This is the useful content of the admissibility viewpoint here. It does not say that Nature optimizes an unspecified functional. It supplies a sequence of explicit quotient maps and asks, at each stage, whether the observable still carries enough information to compose with the next map.

4. The carriers are adjacent, not identical

Several nearby counts must remain separated:

• the order-23 fixed-point VOA has \(23^2=529\) twist-charge modules;
• its nonzero projective labels form 24 directions, split into orbits of 2 axes and 22 mixed directions under the finite transition skeleton;
• the 22 nonidentity powers of a generator give 11 inverse pairs, and adding the identity produces the count 12;
• the conditional \(\operatorname{Rep}(\mathfrak{so}(3)_{23})\) defect category has rank 12, as does the genus of \(X_1(25)\);
• the genus-two fermionic carrier has 10 even spin structures and 6 odd structures that vanish in the vacuum amplitude;
• the live ten-sector permutation carrier has a nine-dimensional sum-zero representation whose determinant is the finite sign character.

These are different sets with different actions. In particular, \(12\ne10\), so no labelwise identification can join the order-23 defect basis to the even-spin basis. A bridge must be a restriction, induction, Fourier kernel, defect junction or sewing map between carriers. The order-46 diamond tells us which insertion labels that map; it does not construct the map.

5. What “program” now means

The combined branch has six levels which should not be conflated:

1. Update: \(1+\sigma\) is a finite cellular/group-algebra operation.
2. Symmetry: \(\sigma\), the order-46 roots and the Monster classes are reversible group elements.
3. State space: VOA modules and twist-charge sectors are the typed memory.
4. Observable: ordinary and generalized McKay--Thompson functions are graded traces.
5. Composition: defects, vertex operations and sewing say how interfaces can be joined.
6. Integrity law: denominator and Hecke relations constrain the whole graded construction.

This deserves the word program in the algebraic sense: states, typed operations, compositional interfaces and global consistency checks coexist. It is not yet a physical time evolution. A Hamiltonian, real-time carrier, locality statement and detector coupling remain absent.

6. Executable next tests

The order-46 junction replaces a vague search with a small comparison table. For each \(h\in46A,46B,46C,46D\):

1. identify \(g=h^2\in23AB\) and \(z=h^{23}\in2A/2B\) with a declared generator orientation;
2. use \(h=g^{12}z\) and transport the known order-46 Hauptmodul by the explicit matrix \(\left(\begin{smallmatrix}1&11\\2&23\end{smallmatrix}\right)\), retaining compatible normalization rather than only scalar coefficients;
3. compare the 2A pair with the Majorana-Weyl x Baby-Monster decomposition and the 2B pair with the Leech orbifold decomposition;
4. use the fifth-power Galois pairing to verify that the normalized cyclic trace still collapses A/B and C/D, then add the exactly bounded noncyclic insertions \(Q_{46AB}\cong C_2^2\) and \(Q_{46CD}\cong C_2\);
5. test degeneration to the already verified common scalar \(T_{23}\);
6. sew the resulting functions through two handles and ask whether their determinant/anomaly lines land on the ten-even-characteristic carrier;
7. only after that, test whether the off-diagonal \(H_2(\mathbb R)\) coordinate is supplied by the sewing parameter or a defect-junction modulus.

Three outcomes are useful. Equality after orientation-forgetting would identify the common scalar shadow. A controlled phase or character mismatch would identify the missing anomaly line. Incompatibility of modular multipliers would rule out the proposed seam junction without any new physical experiment.

7. Boundary of the result

The joined map establishes an exact finite-group junction and a sequence of typed information-loss gates. It does not establish a Monster action on \(Spin_0(2,3)\), identify 2A fermion parity with the Rule-60 deck bit, derive four-dimensional dynamics, or turn the numerical 12/12 coincidence into a bijection. Those promotions remain inadmissible until the explicit twisted--twining and sewn intertwiners exist.

The mixed-level 46 seam sharpens the sewn-intertwiner gate. The order-11 orientation rotation cannot act on the ten level-2 characteristic labels or lift as finite order in \(Sp(4,\mathbb Z)\). The exact common finite ambient carrier is instead \(Sp(4,\mathbb Z/46)\cong Sp(4,2)\times Sp(4,23)\); the missing object is a non-factorized vector-valued kernel coupling its two prime factors. Its scalar channels are now classified: a \(C_2\) sign can couple the projective seam. The finite dicyclic \(C_4\) character has quarter-turn value \(i\), but its genuine scalar metaplectic promotion is impossible in the declared one-block embedding because a lifted quarter-turn has fourth power equal to the metaplectic kernel. A genuine scalar eta line survives with quarter-turn value \(\zeta_8\) and relation-defect value \(v_\eta(K)=i\). But \(K\) also acts on the level-2 seam with cycle type \(3^3\,1\). The eta-normalized free-Majorana operator is now exact: its three cycle products are \(1\), its fixed odd--odd eigenvalue is \(-1\), and its determinant is \(-1\). Arf conjugation fixes the Baby-Monster multiplier to the inverse phase, also of determinant \(-1\). The complete \(2A\) multiplier is the pure \(3^3\,1\) permutation with determinant \(+1\). The unresolved gate is now the sewn amplitude vector and its degeneration, not its modular multiplier. The order-23 projective-centralizer audit also closes the remaining genus-one representation escape: coprimality forces the transgressed cocycle to be trivial, and the five class traces determine the full ordinary \(S_4\) module. The kernel must therefore carry cross-grade or higher-genus composition data.

Sources and corpus links

• Moonshine branch exhaustion checkpoint
• ATLAS Monster conjugacy classes and power-up map
• ATLAS Baby Monster conjugacy classes
• ATLAS Conway group 0 conjugacy classes
• Carnahan, Generalized Moonshine IV: Monstrous Lie algebras
• Carnahan, Generalized Moonshine I: Genus zero functions
• Main Moonshine programme
• Order-46 observable ladder
• Centralizer augmentation gate
• Galois orientation gate
• Mixed-level 46 genus-two seam
• Quarter-turn metaplectic gate
• Projective-centralizer no-go
• Integral relation rail and metaplectic scalar no-go
• Dedekind eta relation phase
• Relation-defect seam permutation
• Free-Majorana relation-defect multiplier
• Baby-Monster genus-two source gate
• Arf/Fourier Baby-Monster reconstruction
• Arf-conjugated Baby-Monster 0 multiplier
• Monster fermionization bridge
• Leech pair-Gram / Spin(2,3) bridge
• Rule-60 affine seam torsor
• Hecke, commutant and Spin(2,3) audit