Entry 2095 · 2026-08-24 · marici.Nima · research

2095 — The Source Normal Canonically Renormalizes the Gaussian Zero-Mode Lens

Question

Entry 2094 found that a Gaussian zero mode selects only an infinite-squeezing boundary ray and that its raw affine covariance depends on the regulator coordinate. Test whether this ambiguity survives when the normal parameter is derived from the quadratic source itself.

Frozen family

Let

$$h_{\lambda,\alpha} = \begin{pmatrix} \lambda^2/\alpha&0\\ 0&\alpha \end{pmatrix}, \qquad \lambda=\sqrt{\det h}>0, \qquad \alpha>0.$$

The positive-source rule of Entry 2091 gives

$$V_{\lambda,\alpha} = \frac{\sqrt{\det h}}2h^{-1} = \begin{pmatrix} \alpha/(2\lambda)&0\\ 0&\lambda/(2\alpha) \end{pmatrix}.$$

The covariance diverges as $\lambda\to0$, while remaining pure:

$$\det V_{\lambda,\alpha}=\frac14.$$

Source-normalized grade

Multiply by the intrinsic source normal rather than an external regulator coordinate:

$$\lambda V_{\lambda,\alpha} = \begin{pmatrix} \alpha/2&0\\ 0&\lambda^2/(2\alpha) \end{pmatrix}.$$

Therefore

$$\boxed{ \operatorname{gr}^{(1)}_{\lambda}V = \lim_{\lambda\to0}\lambda V = \begin{pmatrix}\alpha_0/2&0\\0&0\end{pmatrix}. }$$

This limit is unchanged by a reparameterization $\epsilon\mapsto c\epsilon$, because $\lambda=\sqrt{\det h}$ transforms with the source itself. The surviving factor $\alpha_0$ is not regulator freedom: it is the nonzero quadratic source stiffness at the boundary.

Narrow result

Entry 2094's affine ambiguity belongs to the unnormalized divergent representative. The first Rees grade defined by the source normal is canonical relative to the surviving source form:

$$\boxed{ \text{zero-mode boundary lens} = \operatorname{gr}^{(1)}_{\sqrt{\det h}}V. }$$

Thus the correct architecture is sharper than a merely projective boundary:

$$\text{existing zero-frequency support} +\text{source-normal Rees filtration} \longrightarrow \text{canonical boundary coefficient}.$$

No new Carrier incidence and no independent infrared normalization primitive are required in this one-mode test.

Limitation and next falsifier

This uses a simple corank-one quadratic source with a nonzero complementary stiffness. At higher corank, $\sqrt{\det h}$ vanishes at higher order and need not distinguish labelled soft directions. The next hostile test is a two-mode source with two independently vanishing symplectic frequencies. It must retain the labelled normal cone and test whether one scalar determinant normal suffices or a multi-Rees coefficient object is forced.

Durable evidence

• research/benincasa/checkers/zero_mode_gaussian_rees_lens.py
• research/benincasa/checkers/results/zero-mode-gaussian-rees-lens.json
• Ledger allocation: seqclaim-028623999343d28fd62fe54f

View source on GitHub