A phase-blind positive response necessarily squares the modular seam

Date: 24 October 2026 Status: exact local codimension and invariant-theory obstruction; phase-sensitive enrichment remains open Evidence: machine-readable gate Certificate: gate verifier

Result

Near a separating genus-two surface, the normal sewing coordinate is complex:

The oriented modular form has a linear local zero,

so it retains both the magnitude and phase of the normal displacement. Its scalar square behaves as

while a non-chiral positive quantity behaves as

This distinction reappears whenever the seam is sent into a real positive response.

The invariant-theory gate

Suppose the phase convention of the sewing coordinate is not physical, so

must leave a scalar detector record unchanged. A smooth real invariant has no nonzero linear term. Its leading nonconstant quadratic term is proportional to

Therefore a phase-blind positive response:

• vanishes at least quadratically at separation;
• cannot change sign;
• forgets the oriented normal;
• has the structural type of \(|\chi_5|^2\) or \(\chi_{10}\), not \(\chi_5\).

This is not a technological limitation. It follows from the \(SO(2)\) action on the two real normal coordinates.

Why a real determinant does not solve it unaided

A signed response-root determinant can have a linear term

but only after a reference phase \(\phi_0\) has been selected. Its zero set is the full line perpendicular to that selected direction, not just the codimension-two point \(\epsilon=0\).

Conversely, a real scalar that vanishes only at \(x=y=0\) can begin with \(x^2+y^2\), but then it has no sign. Requiring all three properties—

1. smooth real scalar;
2. zero only at the complex separating locus;
3. linear sign-changing zero—

is locally impossible.

The minimal enrichment

There are two honest ways forward.

Retain two quadratures

The detector records a vector

or a complex coherent amplitude \(X+iY\). Separation is the simultaneous zero of both records. The phase/orientation remains observable through interference with a declared reference.

Select a real normal

A boundary condition, preparation history, local oscillator or other independently sourced datum chooses \(\phi_0\). The experiment then measures \(a_{\phi_0}\). This creates a signed codimension-one crossing, but the selector and its transformation law become part of the physics.

Both routes express the corpus's Jordan--Lie balance. The positive Jordan record naturally keeps \(|\epsilon|^2\); transport of the oriented phase requires an enriched coherent carrier. Neither side alone contains the complete seam.

Detector consequence

A power-only detector will see the squared seam and cannot distinguish \(\epsilon\) from \(-\epsilon\) or \(e^{i\phi}\epsilon\). The minimum discriminating instrument requires:

• two calibrated quadratures or two phase-related ports;
• a stable external or internally generated phase reference;
• coherent rather than incoherent averaging;
• a declared test of whether reversing the seam changes a phase-sensitive record while leaving power unchanged.

The direct observable is

or its real-quadrature counterpart. The null control is \(|A_{\rm seam}|^2\), which must remain orientation blind.

Implication

The failed canonical map \(Q=\operatorname{Im}\Omega\) and this local obstruction together locate the missing datum:

the modular seam cannot become a real positive rank-loss boundary without either being squared or being accompanied by a phase selector.

This does not derive the selector. It states exactly what kind of additional physical information must be found.