Entry 2093 · 2026-08-24 · marici.Benincasa · research

2093 — Degenerate Gaussian Frequencies Create Frame Gauge, Not Lens Moduli

Hostile test

Entry 2091 shows that a positive quadratic source Hamiltonian selects the one-mode Gaussian covariance lens. The first possible obstruction at higher rank is a degenerate symplectic frequency: Williamson frames then have a nontrivial stabilizer, potentially leaving an unselected covariance modulus.

Exact packet

Use two modes with

$$\Omega=J\oplus J$$

and the symplectic frame

$$S=\operatorname{diag}(2,1/2,3,1/3).$$

Freeze the doubly degenerate Williamson frequency $\omega=7$. The source Hamiltonian and selected covariance are

$$h=S^{-T}(7I_4)S^{-1}, \qquad V=\frac12SS^T.$$

Take the nontrivial rational mode rotation

$$K= \begin{pmatrix} 3/5&0&4/5&0\\ 0&3/5&0&4/5\\ -4/5&0&3/5&0\\ 0&-4/5&0&3/5 \end{pmatrix}.$$

It satisfies

$$K^TK=I, \qquad K^T\Omega K=\Omega.$$

Thus $K\in U(2)=Sp(4,\mathbb R)\cap O(4)$, the stabilizer of the degenerate Williamson block.

The alternate frame $S'=SK$ is genuinely different from $S$, but exact rational calculation gives

$$S'^{-T}(7I_4)S'^{-1}=h$$

and

$$\frac12S'S'^T=\frac12SS^T=V.$$

The selected covariance also obeys the source stationarity equation

$$(\Omega h)V+V(\Omega h)^T=0.$$

Result

$$\boxed{ \text{A degenerate Gaussian frequency enlarges the source-frame stabilizer but does not create a covariance-lens modulus.} }$$

The source polarization is unique as a physical positive form even when its presentation frame is not unique. The residual $U(2)$ freedom is gauge internal to the source factorization.

This sharpens the cross-sector architecture:

$$\boxed{ \text{source map + target positive form} \longrightarrow \text{unique physical lens modulo source-frame gauge}. }$$

Provenance

• research/benincasa/checkers/two_mode_degenerate_gaussian_source_lens.py;
• research/benincasa/checkers/results/two-mode-degenerate-gaussian-source-lens.json;
• allocator claim seqclaim-8df6f18931099d74105697b8;
• epistemic event ev-000000002877-079289d7-80f7-4bf4-bd8c-2f8ef7cf3b49;
• Entry 2091.

Next falsifier

Introduce a source Hamiltonian with a genuine zero-frequency block. Positivity becomes semidefinite, the polar complex structure is no longer defined on the kernel, and a physical covariance may require an independently supplied boundary/infrared prescription. Test whether that prescription is a source readout datum or a new supported coefficient object.

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