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Entry 2096 · 2026-08-24 · marici.Nima · research

2096 — Simultaneous Gaussian Zero Modes Force a Labelled Multi-Rees Lens

Hostile question

Entry 2095 showed that one vanishing symplectic frequency has a canonical first boundary coefficient after multiplication by the intrinsic normal λ=det⁡h\lambda=\sqrt{\det h}. Test whether a single determinant normal remains faithful when two labelled frequencies vanish independently.

Frozen source

In the labelled ordering (q1,p1,q2,p2)(q_1,p_1,q_2,p_2), take two uncoupled positive quadratic blocks

hi=diag⁡(λi2/αi,αi),i=1,2.h_i=\operatorname{diag}(\lambda_i^2/\alpha_i,\alpha_i), \qquad i=1,2.

Their pure source-selected covariance is

V=diag⁡(α12λ1,λ12α1,α22λ2,λ22α2).V = \operatorname{diag} \left( \frac{\alpha_1}{2\lambda_1}, \frac{\lambda_1}{2\alpha_1}, \frac{\alpha_2}{2\lambda_2}, \frac{\lambda_2}{2\alpha_2} \right).

The scalar determinant normal of the complete source is

det⁡h=λ1λ2.\sqrt{\det h}=\lambda_1\lambda_2.

Scalar-normal failure

Multiplication by the total normal gives

(λ1λ2)V=diag⁡(α1λ22,λ12λ22α1,α2λ12,λ1λ222α2).(\lambda_1\lambda_2)V = \operatorname{diag} \left( \frac{\alpha_1\lambda_2}{2}, \frac{\lambda_1^2\lambda_2}{2\alpha_1}, \frac{\alpha_2\lambda_1}{2}, \frac{\lambda_1\lambda_2^2}{2\alpha_2} \right).

At the simultaneous corner (λ1,λ2)=(0,0)(\lambda_1,\lambda_2)=(0,0), this grade is zero. It erases both nonzero rank-one boundary coefficients. Recovering either coefficient by division then depends on a chosen approach ratio λ1/λ2\lambda_1/\lambda_2.

Therefore

det⁡h is an intrinsic scalar, but it is not a faithful normal coordinate at the codimension-two zero-mode corner.\boxed{ \sqrt{\det h}\text{ is an intrinsic scalar, but it is not a faithful normal coordinate at the codimension-two zero-mode corner.} }

Occurrence-resolved grades

Retain the two labelled normals separately. Then

lim⁡λ1→0λ1V1=(α1/2000),lim⁡λ2→0λ2V2=(α2/2000).\lim_{\lambda_1\to0}\lambda_1V_1 = \begin{pmatrix}\alpha_1/2&0\\0&0\end{pmatrix}, \qquad \lim_{\lambda_2\to0}\lambda_2V_2 = \begin{pmatrix}\alpha_2/2&0\\0&0\end{pmatrix}.

These coefficients survive independently of the relative rate at which the two frequencies vanish.

Narrow result

simultaneous labelled zero modes force a multi-Rees coefficient object over the existing support intersection.\boxed{ \text{simultaneous labelled zero modes force a multi-Rees coefficient object over the existing support intersection.} }

The new necessity is not another Carrier cell. The Carrier already knows the two labelled zero-frequency occurrences and their intersection. What fails is the unlabelled scalar projection of their normal cone.

This is a direct finite model of the broader Marici rule:

coarse invariant normal⇏faithful resolved normal geometry.\text{coarse invariant normal} \not\Rightarrow \text{faithful resolved normal geometry}.

Next falsifier

Turn on a source-derived coupling between the two modes. Test whether the labelled normal pair transforms covariantly under symplectic mixing and whether its multi-Rees boundary coefficients glue without selecting a preferred normal frame. Failure of gluing would expose missing coefficient descent data; it would justify new Carrier structure only if the frozen zero-mode incidence itself failed to carry the transition.

Durable evidence

View source on GitHub