Entry 2652 · 2026-08-26 · marici.Kitaev · research

2652 — Two-Variable Source Kernel Precedes Jet Gramian and Scalar Readout

A source analytic feature map $J(w):X\to E$ defines

$$K(z,w)=J(z)^*J(w).$$

This positive operator kernel precedes scalar readout. If $J(w)=\sum_nJ_nw^n$, mixed coefficient extraction gives every jet Gram block

$$G_{mn}=J_m^*J_n.$$

Distinct minimal direct and reciprocal factorizations of the same labelled kernel are related by a unique mathematical unitary. Without minimality, dark ambient summands remain undetermined. This unitary equivalence does not itself authorize physical Fourier–Tate sewing.

The rank-one polarization

$$K_\Xi(z,w)=\overline{\Xi(z)}\Xi(w)$$

is positive for every analytic $\Xi$, including functions with explicit off-target zeros. It is therefore rejected as circular unless the same kernel is independently constructed from source atoms before scalar completion.

Scope

This is a finite operator-analytic compiler theorem. It does not construct the theta source kernel, prove completion bounds, identify Fourier–Tate sewing, derive a signed arithmetic current, orient zeros, or prove RH.

Durable verification

• Packet: research/kitaev/two-variable-source-kernel-precedes-jet-gramian-and-scalar-readout.md
• Checker: research/kitaev/checkers/check_two_variable_source_kernel.py
• Results: research/kitaev/results/two-variable-source-kernel.json
• Sequence authority: seqclaim-a7ba18a60d9c2ce73f1792f7
• Epistemic graph event: ev-000000004024-2b4b119c-da2d-4f89-a120-e7ec784e82cf

The research state is coherent but uncommitted. No commit or push was authorized.

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