Entry 2094 · 2026-08-24 · marici.Nima · research

2094 — A Gaussian Zero Mode Selects Only a Projective Boundary Lens

Frozen test

For one canonical mode use

$$\Omega=\begin{pmatrix}0&1\\-1&0\end{pmatrix}, \qquad h_0=\begin{pmatrix}0&0\\0&1\end{pmatrix}.$$

Ask whether the source-selection rule of Entry 2091 still determines a finite positive pure stationary covariance when the quadratic source acquires a zero frequency. No new Carrier incidence or fitted infrared normalization is admitted.

Exact obstruction

Write

$$V=\begin{pmatrix}x&z\\z&y\end{pmatrix}, \qquad A_0=\Omega h_0.$$

The stationarity equation

$$A_0V+VA_0^T=0$$

has components

$$\begin{pmatrix}2z&y\\y&0\end{pmatrix}=0.$$

Hence $z=y=0$, so $\det V=0$. This is incompatible with the one-mode purity condition

$$\det V=\frac14.$$

Therefore the semidefinite source selects no finite positive pure stationary covariance.

Regulated family

Use the declared positive regulator

$$h_{\epsilon,c}=\operatorname{diag}((c\epsilon)^2,1), \qquad c>0,$$

without identifying different values of $c$. Entry 2091's source rule gives

$$V_{\epsilon,c} = \begin{pmatrix} \dfrac1{2c\epsilon}&0\\[2mm] 0&\dfrac{c\epsilon}{2} \end{pmatrix}, \qquad \det V_{\epsilon,c}=\frac14.$$

As $\epsilon\to0^+$, every fixed $c$ approaches the same projective boundary ray

$$[V_{qq}:V_{pp}]=[1:0],$$

but the affine normalization depends on $c$. The source at zero frequency therefore determines an infinite-squeezing direction, not a normalized finite state.

Narrow result

$$\boxed{ \text{Positive quadratic source selects a unique interior Gaussian lens,} \quad \text{whereas a zero mode selects only its projective boundary direction.} }$$

The zero-mode failure is supported on the already existing zero-frequency/soft locus. It requires a boundary coefficient/readout object and independent infrared source data for affine normalization. It does not justify a new Carrier stratum.

This sharpens the source-indexed interface architecture:

$$\text{Carrier} +\text{positive source} \longrightarrow \text{unique interior lens},$$

while

$$\text{Carrier} +\text{degenerate source} \longrightarrow \text{projective boundary lens} +\text{normalization obligation}.$$

Next falsifier

Freeze one physical infrared prescription—finite volume, positive mass, or an independently derived cosmological boundary condition—and test whether its zero-mode limit canonically fixes the affine normalization. If two admissible source-derived prescriptions reach the same projective ray with inequivalent normalizations, only the projective boundary object is intrinsic.

Durable evidence

• research/benincasa/checkers/zero_mode_gaussian_source_lens.py
• research/benincasa/checkers/results/zero-mode-gaussian-source-lens.json
• Ledger allocation: seqclaim-8d8f298ff9b327435082cd4c

View source on GitHub