The modular sign selects parity interferometry, not chiral loop flux

Date: 24 October 2026

Status: exact detector-assignment obstruction; beyond-sign fermionic multiplier remains open

Evidence: machine-readable gate

Certificate: detector reassignment verifier

Question

Can the nontrivial character of the oriented genus-two line predict the flux used by the three-mode directional-transfer experiment without adding a fitted phase?

The answer is no for the sign character alone. The negative result does not destroy the experiment. It assigns two different mathematical data to two different detectors.

The source supplies a binary holonomy

After the weight-five automorphy factor has been separated, the character of \(\chi_5\) takes values

Embedding that character in a \(U(1)\) loop gives

This is a genuine \(\mathbb Z_2\) holonomy. It is not a source-derived continuous flux. In particular, it does not choose \(+\pi/2\) rather than \(-\pi/2\).

The distinction matters because orientation reversal sends \(\Phi\mapsto-\Phi\), but

Both character values are fixed points of flux reversal.

Consequence for the three-mode ring

For equal edge magnitude \(g\), the ring determinant contains

The spectrum can therefore distinguish the trivial and nontrivial character sectors: \(\cos 0=1\) while \(\cos\pi=-1\). But at both values the Hamiltonian can be represented by real couplings. Its reciprocal resolvent satisfies

so the directional records

vanish. The \(\chi_5\) sign can select loop parity, but it cannot by itself drive the ring's orientation-odd channel.

This closes a possible over-reading of the three-mode preflight:

the existence of a modular sign character does not predict a chiral synthetic flux.

Where the sign is observable

A sign becomes physical relative to a second amplitude. If

then

The record is linear in the seam amplitude and disappears when the reference is removed. This is exactly the role of the two-quadrature homodyne protocol: its pump and local oscillator provide the independently declared datum against which the binary sign can be read.

The two experimental carriers therefore have different jobs:

What remains open

The full fermionic sewing amplitude carries more than the finite sign: theta multipliers, the weight factor, sewing-coordinate dependence and the anomaly/determinant line all enter. Tuite and Zuevsky give explicit genus-two fermionic partition functions, theta-function expressions and multiplier systems on sewing domains. Those data are not automatically a flat device phase.

The next exact question is therefore narrower:

after choosing a source-derived trivialization or connection, does the sewn Monster \(2A\) fermionization carry a canonically transportable \(U(1)\) phase beyond the \(\mathbb Z_2\) character?

If not, the three-mode directional channel remains an engineering detector of intrinsic flux but exits the direct \(\chi_5\)-binding route. The referenced two-quadrature channel remains the correct landing point for the modular sign.

Authority boundary

This result assigns detector type. It does not derive a laboratory coupling from the Monster CFT, identify a modular loop with physical time evolution, or establish a Leech/Monster effect in a resonator. A no-refit physical bridge still needs a map from a declared modular path and trivialization to a calibrated amplitude, coupling or transfer record.

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
• Oriented seam modular-line result
• Two-quadrature seam experiment
• Three-mode loop seam preflight