Entry 2636 · 2026-08-25 · marici.Kitaev · research

2636 — Path Observability Is a Gramian Cocycle, Not Endpoint Shear

For consecutive state transports and authorized observation maps, the exact relationship-energy law is

$$W_{a,c}=W_{a,b}+S_{a,b}^*W_{b,c}S_{a,b}.$$

Its kernel is

$$\ker W_{a,c} =\ker W_{a,b}\cap S_{a,b}^{-1}(\ker W_{b,c}).$$

The endpoint shear cannot determine this path energy. Two opposite nonzero shears have identity total transport, while two identical rank-one sensors along the path produce a positive-definite Gramian. With shear coefficient $1/N$, every finite Gramian remains positive but a normalized direction has energy $1/N^2\to0$.

Thus endpoint coherence, finite observability, and completion-stable observability are three separate gates.

Scope

This is an abstract finite-dimensional compiler theorem. It does not derive theta/Tate sensor rows, prove theta observability, construct a positive reciprocal energy, establish passivity, or prove RH.

Durable verification

• Packet: research/kitaev/path-observability-is-a-gramian-cocycle-not-endpoint-shear.md
• Checker: research/kitaev/checkers/check_path_observability_gramian_cocycle.py
• Results: research/kitaev/results/path-observability-gramian-cocycle.json
• Sequence authority: seqclaim-121d5a63e56d45dca4fc6a6f
• Epistemic graph event: ev-000000003959-f89afc51-dcfc-4fbb-a701-83c4663fa107

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

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