Three-mode intrinsic seam-flux preflight

Date: 24 October 2026 Status: exact loop invariant and verified directional transfer preflight; hardware execution and modular binding open Evidence: machine-readable preflight Certificate: loop verifier

Result

A three-resonator ring is the smallest device that can carry an intrinsic coupling phase. It also repeats the corpus's central detector lesson:

the spectrum sees the unoriented loop, while directional coherent transport sees its orientation.

Let the three Hermitian hopping amplitudes have equal magnitude \(g\) and phases \(\phi_{12},\phi_{23},\phi_{31}\). Their gauge-invariant flux is

Local rephasings redistribute the three edge phases but cannot change \(\Phi\).

What the spectrum retains

For equal onsite frequencies, the characteristic polynomial depends on

Consequently the unordered eigenfrequency set is the same for \(+\Phi\) and \(-\Phi\). A resonance-frequency or density-of-states measurement detects \(\cos\Phi\), not the oriented sign of the flux.

This is the loop analogue of replacing \(\chi_5\) by an orientation-blind square.

What directional transfer retains

Attach coherent ports to two vertices and include the measured linewidths. The resolvent

contains two paths between the ports: the direct edge and the two-edge path through the third resonator. Their relative phase contains \(\Phi\).

The directional statistic

is odd under \(\Phi\mapsto-\Phi\) when loss and port coupling make the interference observable. It vanishes at the time-reversal-invariant values \(\Phi=0,\pi\).

The stronger complex statistic

retains both amplitude and phase and should be the primary record.

Run matrix

Program:

For each flux:

1. realize at least four different distributions of the same total phase among the edges;
2. measure all three reciprocal port pairs;
3. sweep through the full three-mode resonance band;
4. repeat after interchanging input and output cables;
5. acquire each individual edge with the third edge disabled;
6. randomize the order of flux signs and phase distributions.

Decision rules

An intrinsic loop result requires:

• spectra and complex transfer functions invariant under edge-phase redistribution at fixed \(\Phi\);
• spectra identical within uncertainty for \(+\Phi\) and \(-\Phi\);
• directional complex transfer reversing under flux reversal;
• the directional effect vanishing when any edge is opened;
• one calibrated coupled-mode model predicting held-out phase distributions without refit.

A signal that follows one edge phase instead of the loop sum is externally referenced leakage, not intrinsic flux.

Relation to the corpus

This device realizes the same type of object as a Wilson/Bargmann loop phase. It does not by itself realize:

• the genus-two \(\chi_5\) line;
• the Monster \(2A\) duality defect;
• the Rule-60/Golay deck bit;
• the response-root Pfaffian sheet.

Those identifications require source-derived intertwiners. The device supplies a controlled place for such an intertwiner to land.

Frontier value

The three-mode ring is a stronger platform than the two-mode experiment because its phase survives removal of the external basis convention. The immediate engineering result is a calibrated detector of intrinsic synthetic flux. The frontier test begins only when a modular or lattice calculation predicts \(\Phi\), edge magnitudes or a defect response without fitting them to the device.