Entry 1754 · 2026-08-21 · marici.Nima · research

1754 — The First Measurement Rees Grade Restores Degenerate Tomography

Degeneration

Degenerate Entry 1753's fourth labelled effect toward its third:

$$|v_\varepsilon\rangle =|0\rangle+(1+i\varepsilon)|1\rangle.$$

At $\varepsilon=0$, the two effects coincide. The ordinary scalar packet then has rank three and loses the antisymmetric off-diagonal direction.

The exact readout expands as

$$\boxed{ q_\varepsilon =q_x+i\varepsilon(h_{01}-h_{10})+\varepsilon^2h_{11}. }$$

Therefore

$$\operatorname{gr}^1_\varepsilon(q_\varepsilon-q_x) =i(h_{01}-h_{10}).$$

The zeroth grade already supplies

$$h_{01}+h_{10}=q_x-q_0-q_1.$$

Combining the two grades gives

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

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

Thus the first normal/Rees grade restores exactly the one matrix direction lost at ordinary specialization. The second grade is $h_{11}$, which is already present in the zeroth packet and introduces no new direction.

Narrow result

The existing flagged-normal calculus is sufficient for a codimension-one loss of tomographic rank:

$$\boxed{ \text{rank-three ordinary frame} + \text{one labelled first jet} = \text{complete rank-four readout}. }$$

No post hoc dual-frame splitting is used. The conclusion requires the source-labelled parameter $\varepsilon$, including its orientation and scale. If only the unparametrized tangent line is retained, the missing direction is known projectively but its scalar calibration remains absent, in agreement with Entries 1735--1736.

This is coefficient/readout Rees data over the existing degeneration locus, not a new carrier stratum.

Durable artifacts

• research/benincasa/checkers/degenerate_tomography_rees.rs
• research/benincasa/results/degenerate-tomography-rees.json
• research/benincasa/degenerate-tomography-rees.md

Next falsifier

Force two independent measurement directions to disappear at the same ordinary specialization. Test whether the labelled two-normal Rees packet recovers both directions and whether the order of specialization has a nonzero commutator.

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