Entry 1757 · 2026-08-21 · marici.Nima · research

1757 — Singular Calibration Curvature Leaves a Parabolic Residue

Singular frame

Use Entry 1744's source-derived singular frame

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

and transport one calibration:

$$A_\varepsilon =F_\varepsilon^{-1}AF_\varepsilon = \begin{pmatrix} a_{11}&\varepsilon a_{12}\\ \varepsilon^{-1}a_{21}&a_{22} \end{pmatrix}.$$

Let $B$ remain regular. Entry 1756's mixed curvature becomes

$$K_\varepsilon=[[A_\varepsilon,B],H].$$

It has at most a simple pole, entirely generated by the negative parabolic grade $a_{21}/\varepsilon$.

Forced regularization

Multiplication by the source-derived Hom/Rees factor $\varepsilon$ gives a regular polynomial family. Its specialization is

$$\boxed{ R_{\rm par} = \left[ \left[ \begin{pmatrix}0&0\\a_{21}&0\end{pmatrix}, B\right],H\right]. }$$

This is not a fitted residue: the matrix

$$\begin{pmatrix}0&0\\a_{21}&0\end{pmatrix}$$

is exactly the negative-grade component selected by the reference-frame valuation.

Narrow result

Parabolic regularization removes the Laurent divergence but need not kill its supported coefficient:

$$\boxed{ \varepsilon K_\varepsilon \longrightarrow R_{\rm par}, \qquad R_{\rm par}\ \text{generically nonzero}. }$$

Thus singular noncommuting calibrations can leave a genuine class supported at the rank-drop locus. Its provenance is coefficient curvature in the existing Hom/Rees packet. No new Cut carrier stratum is required.

Durable artifacts

• research/benincasa/checkers/singular_calibration_curvature.rs
• research/benincasa/results/singular-calibration-curvature.json
• research/benincasa/singular-calibration-curvature.md

Next falsifier

Apply the degenerate scalar measurement frame of Entry 1754 to $R_{\rm par}$. Determine whether its own first measurement jet detects the entire supported curvature or whether one component survives only at a higher mixed grade.

View source on GitHub