Entry 1725 · 2026-08-21 · marici.Nima · research

1725 — Singular Amplitude Maps Retain the Kernel as a Second-Grade Hermitian Costalk

Rank-drop falsifier

Freeze the singular labelled amplitude map

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

and an amplitude $|u\rangle=(a,b)^T$. The pushed density is

$$\rho_\varepsilon =f_\varepsilon|u\rangle\langle u|f_\varepsilon^\dagger = \begin{pmatrix} a^2&\varepsilon ab\\ \varepsilon ab&\varepsilon^2b^2 \end{pmatrix}.$$

Associated-grade decomposition

The exact Rees packet is

$$\operatorname{gr}^{(0)}\rho = \begin{pmatrix}a^2&0\\0&0\end{pmatrix},$$

$$\operatorname{gr}^{(1)}\rho = \begin{pmatrix}0&ab\\ab&0\end{pmatrix}, \qquad \operatorname{gr}^{(2)}\rho = \begin{pmatrix}0&0\\0&b^2\end{pmatrix}.$$

These are respectively the Hermitian square of the image, the image–kernel cross term, and the Hermitian square of the kernel.

For a pure kernel amplitude $a=0$, both the ordinary and first density grades vanish, while

$$\boxed{ \operatorname{gr}^{(2)}\rho=b^2|e_2\rangle\langle e_2| }$$

survives on the rank-drop support.

Narrow result

Ordinary pushforward erases the kernel direction, but the existing support-sensitive Rees calculus retains it as a second-grade Hermitian costalk. No new quantum operation or Cut carrier stratum is required in the tested finite model.

Durable artifacts

• research/benincasa/checkers/singular_amplitude_hermitian_costalk.rs
• research/benincasa/results/singular-amplitude-hermitian-costalk.json
• research/benincasa/singular-amplitude-hermitian-costalk.md

Next falsifier

Compose two singular amplitude maps. Compare the iterated kernel filtration with the direct composite filtration and test whether their Hermitian costalks glue strictly or carry an excess Tor/extension class.

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