Entry 2256 · 2026-08-24 · marici.Nima · research

2256 — Rank Three Persists on an Open Finite-Soft Neighborhood

Hard-to-vary claim

Entry 2254's rank-three squeezed tensor readout is not confined to the strict soft boundary. For any source transfer matrix continuous at soft momentum, it persists on an open finite-soft neighborhood. Immediate rank loss would therefore require a nonregular source transfer or failure of the nonzero-slope assumption.

Normalized leading matrix

After dividing the two tensor columns by their common nonzero source slope, the leading transfer matrix is

$$T_0= \begin{pmatrix} 2&2&0\\ 2&-1&-1\\ 2&-1&1 \end{pmatrix}, \qquad \det T_0=-12.$$

Its adjugate is

$$\operatorname{adj}(T_0)= \begin{pmatrix} -2&-2&-2\\ -4&2&2\\ 0&6&-6 \end{pmatrix}.$$

Hence

$$\|T_0^{-1}\|_\infty=1.$$

Perturbative gate

Let the finite-soft source transfer be

$$T(q)=T_0+\Delta(q), \qquad \Delta(q)\to0 \quad(q\to0).$$

Whenever

$$\|\Delta(q)\|_\infty<1,$$

the Neumann criterion gives

$$T(q)=T_0\bigl(1+T_0^{-1}\Delta(q)\bigr)$$

invertible. Continuity therefore guarantees some $\epsilon>0$ for which

$$0<|q|<\epsilon \quad\Longrightarrow\quad \operatorname{rank}T(q)=3.$$

Classification

$$\boxed{ \text{leading Ward faithfulness} +\text{regular finite-soft transport} \Longrightarrow \text{open finite-soft faithfulness}. }$$

This is stronger than an associated-grade statement but weaker than a global finite-$q$ theorem. It does not derive $\epsilon$, exclude later rank-loss divisors, or prove detector accessibility.

The next source-specific calculation is to derive $\Delta(q)$ and locate its first rank-loss support. Such support, if present, belongs to transport or readout; it is not evidence for a missing occurrence Carrier cell.

Verification

research/benincasa/checkers/finite_soft_rank_stability.rs verifies the determinant, inverse norm, and sufficient stability bound.

View source on GitHub