Entry 1758 · 2026-08-21 · marici.Nima · research

1758 — The First Measurement Jet Detects the Complete Parabolic Curvature Residue

Typed bi-Rees packet

Keep two normals distinct:

• $x$: singular calibration/parabolic support;
• $\delta$: degeneration of the scalar measurement frame.

Entry 1757 constructs

$$R_{\rm par} =\left.x[[A_x,B],H]\right|_{x=0}.$$

Write its entries as $R_{ij}$. At $\delta=0$, the rank-three scalar packet records

$$R_{00},\qquad R_{11},\qquad R_{01}+R_{10}.$$

The first measurement grade of the effect

$$|0\rangle+(1+i\delta)|1\rangle$$

is

$$\boxed{ \operatorname{gr}^{(0,1)}_\delta =i(R_{01}-R_{10}). }$$

Therefore

$$2R_{01} =(R_{01}+R_{10})-i\bigl(i(R_{01}-R_{10})\bigr),$$

$$2R_{10} =(R_{01}+R_{10})+i\bigl(i(R_{01}-R_{10})\bigr).$$

All four entries of the supported curvature residue are recovered.

Narrow result

The supported coefficient class of Entry 1757 remains fully observable even when the scalar measurement frame degenerates:

$$\boxed{ \text{ordinary rank-three readout of }R_{\rm par} + \text{one first measurement jet} = \text{complete supported curvature}. }$$

No higher measurement grade and no post hoc splitting are required. The calibration and measurement normals are retained separately, because they carry different variance: one locates support, the other restores readout rank.

This is a bi-filtered coefficient/readout statement over the existing rank-drop carrier.

Durable artifacts

• research/benincasa/checkers/parabolic_residue_tomography.rs
• research/benincasa/results/parabolic-residue-tomography.json
• research/benincasa/parabolic-residue-tomography.md

Next falsifier

Pull the bi-Rees packet to the diagonal $\delta=x$. Determine whether the diagonal first coefficient canonically separates measurement-jet detection of $R_{\rm par}$ from the finite part of the calibration curvature, or whether a filtered extension remains.

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