Entry 1723 · 2026-08-21 · marici.Nima · research

1723 — Relative Phase Lives in the Mixed Second Grade of the Amplitude Rees Packet

Two-jet falsifier

Entry 1722 leaves open whether two amplitude directions entering the same zero-probability outcome require higher coherence data. Freeze

$$|\chi(\varepsilon,\eta)\rangle =\varepsilon|u\rangle+\eta|v\rangle.$$

Complete second grade

The unnormalized selected density expands as

$$\boxed{ |\chi\rangle\langle\chi| =\varepsilon^2|u\rangle\langle u| +\varepsilon\eta \bigl(|u\rangle\langle v|+|v\rangle\langle u|\bigr) +\eta^2|v\rangle\langle v|. }$$

The three labelled Rees bidegrees are

$$(2,0),\qquad(1,1),\qquad(0,2).$$

Replacing $|v\rangle$ by $-|v\rangle$ leaves both diagonal positive rays unchanged but reverses the mixed $(1,1)$ coefficient. Hence separate summed positive blocks lose relative phase, while the complete second-grade packet retains it.

Narrow result

No grade beyond total degree two is required in the tested model. The correct exceptional coefficient object is the complete Hermitian square of the labelled amplitude-normal module, including its mixed occurrence. This is the quantum analogue of the complete symmetric-square covariance packet from Entries 1698–1700.

No new Cut carrier stratum appears.

Durable artifacts

• research/benincasa/checkers/two_amplitude_jet_hermitian_square.rs
• research/benincasa/results/two-amplitude-jet-hermitian-square.json
• research/benincasa/two-amplitude-jet-hermitian-square.md

Next falsifier

Test Cut functoriality of the complete Hermitian-square packet. Under a labelled amplitude map $f$, verify that density specialization is $f\otimes\bar f$, including mixed normals and partial trace. Any failure of naturality would require a genuine quantum coefficient coherence map.

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