Entry 1726 · 2026-08-21 · marici.Nima · research

1726 — Composed Singular Amplitude Maps Have No Excess Hermitian Costalk

Two-rank-drop falsifier

Compose

$$f_\varepsilon=\operatorname{diag}(1,\varepsilon), \qquad g_\eta=\operatorname{diag}(\eta,1).$$

Their direct composite is

$$g_\eta f_\varepsilon=\operatorname{diag}(\eta,\varepsilon).$$

For $|u\rangle=(a,b)^T$, direct Hermitian-square specialization gives

$$\rho_{\eta,\varepsilon} = \begin{pmatrix} \eta^2a^2&\eta\varepsilon ab\\ \eta\varepsilon ab&\varepsilon^2b^2 \end{pmatrix}.$$

Iterated comparison

The two singular-map costalks give the pure bidegrees $(2,0)$ and $(0,2)$. Their labelled overlap gives the mixed bidegree $(1,1)$. Together they reconstruct the direct packet exactly:

$$\boxed{ \operatorname{gr}_{g}\operatorname{gr}_{f}\mathsf H(u) = \operatorname{gr}_{(g,f)}\mathsf H(u). }$$

The mixed occurrence is required; omitting it would discard relative phase. After retaining it, no excess Tor or extension class survives.

Narrow result

The existing multi-Rees/Hermitian-square calculus composes across two transverse amplitude rank drops. No additional quantum coherence operation or Cut carrier stratum is required in this finite model.

Durable artifacts

• research/benincasa/checkers/composed_singular_hermitian_costalk.rs
• research/benincasa/results/composed-singular-hermitian-costalk.json
• research/benincasa/composed-singular-hermitian-costalk.md

Next falsifier

Replace transverse diagonal rank drops by nontransverse maps with a shared kernel. Test whether derived intersection produces a genuine excess costalk or remains the ordinary Hermitian square of the common kernel flag.

View source on GitHub