The spin-sector vector contains a beyond-sign metaplectic phase

Date: 24 October 2026

Status: exact theta-transformation result, superseded as an absolute device target by the relative-invariant audit; complete Monster (2A) multiplier and physical transport map remain open

Evidence: machine-readable calculation

Certificate: characteristic-phase verifier

Result

The previous detector reassignment established that the one-dimensional \(\chi_5\) character supplies only a binary sign. That is not the full information carried by the ten-component spin-sector vector.

Apply the genus-one modular transformation \(T:\tau\mapsto\tau+1\) to the first handle of a separating genus-two surface. In binary characteristic notation, the first-handle rule is

with phase

Thus sectors with \(a_1=0\) are exchanged in pairs, while sectors with \(a_1=1\) are fixed and acquire \(e^{i\pi/4}\). Enumeration of the ten even genus-two characteristics gives:

• three two-cycles with unit cycle product;
• four fixed sectors with phase \(e^{i\pi/4}\);
• among those fixed sectors, three have even--even separating factors and one is the odd--odd sector that vanishes at separation.

The free-Majorana spin vector therefore contains a canonical eighth-root phase under this declared generator. Its inverse carries \(e^{-i\pi/4}\). This is precisely the kind of beyond-sign phase that a three-mode directional detector can distinguish.

Why the determinant did not show it

The three transpositions have permutation determinant ((-1)^3=-1). The four fixed-sector phases multiply to

The two signs cancel, so the determinant of this ten-component monomial action is (+1). The vector retains phases that its determinant forgets.

This explains why the previous binary-character calculation was both correct and incomplete. It classified the determinant line; it did not classify the full metaplectic transport of the spin-sector vector.

What can now be predicted

For the free-Majorana factor, the declared pair (T_1^{\pm1}) supplies the convention-fixed phase candidate

A three-mode ring at these two fluxes is isospectral under reversal but has a nonzero orientation-odd directional transfer in the calibrated model. This is a better mathematical target than choosing \(\pm\pi/2\) for convenience.

The (T1/S1) boundary audit sharpens this statement. A common anomaly-line trivialization can shift every phase assigned to one generator, so the absolute (\pm\pi/4) flux is not itself intrinsic. The robust (T1) target is the fixed-sector-square/two-cycle-product ratio (i); the robust (S1) target includes the odd--odd/even--even fixed-sector ratio (i). A physical emulator must compare sectors coherently so that the common phase cancels.

The prediction is not yet a Monster-device bridge. The full fermionized Monster theory is

The Baby-Monster sector can contribute its own multiplier, and a chiral partition amplitude is a section requiring a declared trivialization. The net (2A) phase cannot be inferred from the free factor alone.

The narrowed gate

The remaining calculation is now componentwise rather than determinant-only:

1. compute the (T1) and (S1) multiplier matrices of the ten complete (2A) spin-sector amplitudes;
2. divide only by a source-derived common anomaly factor;
3. calculate gauge-invariant phase products around characteristic cycles;
4. determine whether the three nonvanishing \(T_1\)-fixed sectors retain \(e^{i\pi/4}\), cancel it or split it;
5. only then bind (T_1^{\pm1}) to a calibrated forward/reverse laboratory control loop.

A diagonal rephasing of sector basis vectors can move phases between edges, but it cannot change a closed cycle product or a fixed-sector eigenvalue. Those are the appropriate candidates for transport into an intrinsic device flux.

Authority boundary

This calculation establishes a source-derived phase in the free-Majorana theta representation. It does not establish the multiplier of the complete Monster (2A) fermionization, a physical map from modular transport to laboratory time, or a new-physics signal. It replaces an arbitrary continuous phase with a precise conditional target and names the calculation that can cancel it.

Sources

• Tuite and Zuevsky, Genus Two Partition and Correlation Functions for Fermionic Vertex Operator Superalgebras I
• Tuite and Zuevsky, Part II
• Lin and Shao, Duality Defect of the Monster CFT
• Modular-sign detector reassignment