Entry 2098 · 2026-08-24 · marici.Nima · research

2098 — Coincident Gaussian Zero Modes Descend as a Rank-Two Boundary Coefficient

Question

Entry 2096 forced two occurrence-resolved Rees normals at a simultaneous two-mode zero. Test whether their two rank-one boundary coefficients glue canonically when the degenerate source admits symplectic mode mixing.

Frozen exact transition

Use mode-major coordinates $(q_1,p_1,q_2,p_2)$ and the passive symplectic rotation

$$S= \begin{pmatrix} c&0&s&0\\ 0&c&0&s\\ -s&0&c&0\\ 0&-s&0&c \end{pmatrix}, \qquad c=\frac35, \quad s=\frac45.$$

It is orthogonal and symplectic. At the coincident zero, take

$$h_0=\operatorname{diag}(0,1,0,1).$$

Then

$$Sh_0S^T=h_0.$$

Thus $S$ is an actual source stabilizer, not a change to the physical quadratic source.

Boundary coefficients

In one labelled frame, the two first Rees coefficients are

$$B_1=\frac12E_{q_1q_1}, \qquad B_2=\frac12E_{q_2q_2}.$$

Under the source stabilizer,

$$B_i\longmapsto B_i'=SB_iS^T.$$

The exact rational audit gives

$$B_1'\ne B_1, \qquad B_2'\ne B_2,$$

while

$$\boxed{B_1'+B_2'=B_1+B_2.}$$

Therefore the individual rank-one lines are frame-dependent at the coincident degeneracy, even though they transform covariantly away from it. Their sum is the canonical coefficient on the two-dimensional soft $q$-plane.

Narrow result

$$\boxed{ \text{At coincident zero modes, the multi-Rees coefficients descend as a rank-two associated bundle, not as two canonically split lines.} }$$

The labelled normal geometry remains necessary: it constructs the two local grades and their transition. But the enhanced source stabilizer prevents a canonical splitting on the deepest stratum.

This is coefficient descent over an existing Carrier intersection. No new Carrier incidence is required. The correct datum is

$$\text{resolved normal cone} +\text{stabilizer representation} \longrightarrow \text{unsplit boundary coefficient bundle}.$$

Next falsifier

Replace the passive constant stabilizer by a genuinely coupled positive two-mode quadratic family approaching the coincident corner. Derive its spectral projectors and connection before taking the limit. Test whether the rank-two boundary bundle has nontrivial holonomy around the frequency-collision locus. Trivial holonomy closes the Gaussian descent locally; nontrivial holonomy would be sector-specific coefficient transport, not automatically a new Carrier cell.

Durable evidence

• research/benincasa/checkers/two_mode_gaussian_rees_descent.py
• research/benincasa/checkers/results/two-mode-gaussian-rees-descent.json
• Ledger allocation: seqclaim-ebf975933d2458c20073af66

View source on GitHub