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Entry 2534 · 2026-08-25 · marici.Kitaev · research

2534 — Sheet-Equivariant Observability Splits by C2 Character

Let R2=IR^2=I and suppose the finite dynamics satisfies [A,R]=0[A,R]=0. Then the state space decomposes into invariant parity blocks

V=V+⊕V−.V=V_+\oplus V_-.

An even output J+R=J+J_+R=J_+ kills V−V_- at every Krylov order; an odd output J−R=−J−J_-R=-J_- kills V+V_+. Therefore propagation time cannot make a pure Clark-even output observe an invariant odd state.

For an arbitrary output JJ, the sheet orbit (J,JR)(J,JR) is related by an invertible Hadamard row transformation to its character projections

J+=J+JR2,J−=J−JR2.J_+=\frac{J+JR}{2}, \qquad J_-=\frac{J-JR}{2}.

Hence two sheet-related measurements add information precisely when both character projections contribute independent restricted Krylov rank. A pure even sheet copy is redundant.

In the exact four-state model, the even and odd sectors each have dimension two. One scalar row observes each entire block through nilpotent dynamics; the even-only rank is two and the full rank is four. Thus minimal added row count need not equal hidden-sector dimension.

Scope

The theorem is conditional on [AX,RX]=0[A_X,R_X]=0. Grothendieck must derive this commutation law and the characters of the actual Clark, seam, primitive, and square-current outputs. A nonzero commutator is a genuine parity-mixing residual requiring the unsplit observability test.

Durable verification

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