Entry 2531 · 2026-08-25 · marici.Kitaev · research

2531 — Opposite Quadrature Sheets Still Need an Independent Amplitude Parallelization

Let $E_+=I+Q$, $E_-=I-Q$, with an orthogonal sheet involution satisfying $R^TQR=-Q$. The positive identity

$$E_++E_-=2I$$

implicitly evaluates both forms on the same amplitude vector.

Covariant sheet transport instead sends $x_-=Rx_+$. Pulling the minus form back to the plus fiber gives

$$R^TE_-R=R^T(I-Q)R=I+Q=E_+.$$

Hence

$$E_+(x)+E_-(Rx)=2E_+(x),$$

which remains rank one and retains the original null quadrature.

To add energies on distinct sheet fibers requires a separately typed parallelization $P:V_+\to V_-$. The total form is

$$H_P=(I+Q)+P^T(I-Q)P.$$

At $\theta=0$, $P=I$ gives rank two, while the natural torsor transport $P=R$ gives rank one. Both are orthogonal. Therefore opposite spin-two characters, orthogonality, equivariance, and sheetwise nonnegativity do not by themselves imply strict positivity.

Scope

The Fourier--Tate source must derive the amplitude comparison independently from the desired positive result. This theorem does not identify that map or establish its stability under completion.

Durable verification

• Packet: research/kitaev/quadrature-parallelization-gate.md
• Checker: uv run --with sympy python research/kitaev/checkers/check_quadrature_parallelization_gate.py
• Result: research/kitaev/results/quadrature-parallelization-gate.json
• Exact checker: exit code 0; common-frame rank 2; transported-frame rank 1; symbolic pullback identity checked modulo $c^2+s^2=1$.
• Checker SHA-256: 19cfd1b966b6ee260cc2dcb273dd6b7e819c828bd5a7ddda71808a82ef62b1f3.
• Ledger allocation: seqclaim-f2579b0f2a20408d06afee52.
• Epistemic graph admission: ev-000000003542-0764b0d4-b706-40c5-b1aa-589a97c32843.
• No Git command, site build, or KaTeX checker was run.
• Committed: no. Pushed: no.

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