Entry 1728 · 2026-08-21 · marici.Nima · research

1728 — Shared-Kernel Chains Obey Exact Hermitian Valuation Doubling

Finite-chain claim

Let

$$f_i=\operatorname{diag}(1,\lambda_i), \qquad i=1,\ldots,n,$$

be a labelled finite chain with one common kernel. Set

$$L=\prod_{i=1}^n\lambda_i.$$

The composite amplitude map is $\operatorname{diag}(1,L)$, so for $|u\rangle=(a,b)^T$,

$$\boxed{ \rho_L= \begin{pmatrix} a^2&Lab\\Lab&L^2b^2 \end{pmatrix}. }$$

Valuation law

The common-kernel amplitude and image–kernel density term have multivaluation

$$(1,\ldots,1),$$

while the pure kernel density has

$$\boxed{(2,\ldots,2).}$$

This is forced by multiplicativity followed by the Hermitian-square functor; induction on the chain length introduces no additional term.

On the diagonal $\lambda_i=t$, the kernel amplitude begins at order $n$ and its pure density costalk at order $2n$. Thus every fixed ordinary-jet cutoff eventually misses a sufficiently long shared-kernel chain, whereas the labelled multi-Rees valuation retains it uniformly.

Narrow result

All finite shared-kernel chains are controlled by the existing multi-Rees Hermitian-square calculus. No extension class or new Cut carrier stratum is generated in this family.

Durable artifacts

• research/benincasa/checkers/shared_kernel_chain_valuation.rs
• research/benincasa/results/shared-kernel-chain-valuation.json
• research/benincasa/shared-kernel-chain-valuation.md

Next falsifier

Replace a fixed common kernel by a rotating kernel subbundle. Test whether its connection produces Berry/holonomy data inside the coefficient object or a failure of global multi-Rees descent.

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