Entry 1744 · 2026-08-21 · marici.Nima · research

1744 — A Singular Fiber Reference Produces a Parabolic Matrix Readout

Singular-reference test

Let the labelled reference frame degenerate as

$$F_\varepsilon=\operatorname{diag}(1,\varepsilon).$$

For a fiber holonomy matrix $H=(h_{ij})$, Entry 1743's readout is

$$\boxed{ F_\varepsilon^{-1}HF_\varepsilon = \begin{pmatrix} h_{11}&\varepsilon h_{12}\\ \varepsilon^{-1}h_{21}&h_{22} \end{pmatrix}. }$$

Forced parabolic weights

The two reference columns have source valuations $(0,1)$. Therefore the four Hom entries have forced weights

$$(0,1,-1,0).$$

The upper off-diagonal entry survives after dividing by its positive Rees factor $\varepsilon$. The lower entry is represented in the negative parabolic grade by multiplying its pole by $\varepsilon$. Both operations are derived from the source frame valuations, not selected after inspecting $H$.

Thus the parabolic packet recovers

$$(h_{11},h_{12},h_{21},h_{22})$$

exactly. In particular, Entry 1741's $AB$ and $BA$ remain distinct in the two shifted off-diagonal grades.

Narrow result

Ordinary specialization of a singular reference is insufficient: one matrix entry vanishes and the opposite entry develops a pole. The source-derived parabolic/Rees Hom lattice retains the full based holonomy without a post hoc splitting.

This is coefficient/readout filtration data over the existing carrier; no new Cut stratum is required.

Durable artifacts

• research/benincasa/checkers/singular_reference_parabolic_readout.rs
• research/benincasa/results/singular-reference-parabolic-readout.json
• research/benincasa/singular-reference-parabolic-readout.md

Next falsifier

Let two reference columns vanish at the same order so that the associated graded frame loses their ordering. Test whether a labelled flag suffices or a noncanonical splitting reappears at the repeated-weight locus.

View source on GitHub