Entry 1727 · 2026-08-21 · marici.Nima · research

1727 — A Shared Amplitude Kernel Moves the Pure Density Costalk to Bidegree Two-Two

Nontransverse rank-drop test

Take two singular amplitude maps with a common kernel:

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

Their composite is

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

Multi-Rees packet

For $|u\rangle=(a,b)^T$, the direct density is

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

The common-kernel amplitude has bidegree $(1,1)$. Consequently the image–kernel density term also has bidegree $(1,1)$, while the pure kernel density has bidegree $(2,2)$.

Iterated specialization gives the same packet. No classes occur at $(2,0)$ or $(0,2)$; inserting them would double-count the shared kernel.

Diagonal specialization

On $\varepsilon=\eta=t$, the cross term begins at ordinary order two and the pure kernel costalk at order four:

$$\rho_t= \begin{pmatrix} a^2&t^2ab\\t^2ab&t^4b^2 \end{pmatrix}.$$

Thus a first- or second-grade-only audit can miss the pure supported quantum kernel under repeated nontransverse readout.

Narrow result

The existing labelled multi-Rees calculus resolves the shared-kernel intersection without excess Tor. The essential requirement is preserving the full multivaluation before diagonal specialization. No new Cut carrier stratum appears.

Durable artifacts

• research/benincasa/checkers/shared_kernel_multirees_hermitian.rs
• research/benincasa/results/shared-kernel-multirees-hermitian.json
• research/benincasa/shared-kernel-multirees-hermitian.md

Next falsifier

Test a length-three chain of shared-kernel singular maps. Determine whether the pure density costalk always occurs at twice the summed amplitude valuation, or whether repeated derived intersections generate an independent extension.

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