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Entry 2530 · 2026-08-25 · marici.Kitaev · research

2530 — Complementary Quadrature Sheets Require a Spin-Two-Flipping Reflection

Write ψ=A+iB\psi=A+iB, x=(A,B)Tx=(A,B)^T, and separate the trace and spin-two quadratic forms:

∣ψ∣2=xTx,ℜ(e−2iθψ2)=xTQθx,|\psi|^2=x^Tx, \qquad \Re(e^{-2i\theta}\psi^2)=x^TQ_\theta x,

where

Qθ=(cos⁡2θsin⁡2θsin⁡2θ−cos⁡2θ),Qθ2=I.Q_\theta= \begin{pmatrix} \cos2\theta&\sin2\theta\\ \sin2\theta&-\cos2\theta \end{pmatrix}, \qquad Q_\theta^2=I.

An orthogonal involution RR preserves the trace component. It flips the spin-two component exactly when

RTQθR=−Qθ,\boxed{R^TQ_\theta R=-Q_\theta,}

equivalently RQθ=−QθRRQ_\theta=-Q_\theta R. Such an RR is a reflection bisecting the two eigenquadrature axes. The central action −I-I does not work: it preserves the spin-two form.

The sheet energies

E±(x)=xT(I±Qθ)xE_\pm(x)=x^T(I\pm Q_\theta)x

each have eigenvalues 0,20,2, rank one, and complementary null lines. They obey

(I+Qθ)(I−Qθ)=0,E++E−=2∣ψ∣2.(I+Q_\theta)(I-Q_\theta)=0, \qquad E_++E_-=2|\psi|^2.

Thus retaining both oppositely represented sheets upgrades two semidefinite quadratures to a positive-definite norm. One torsor-origin bit labels which quadrature is called ++; it cannot reconstruct the two real coordinates (A,B)(A,B).

Composition requires a genuine displacement cocycle:

ϵ(D∘C)=ϵ(D)+ϵ(C)(mod2),\epsilon(D\circ C)=\epsilon(D)+\epsilon(C)\pmod2,

together with the corresponding representation and spin-character laws.

Falsifier

At θ=0\theta=0, repeating I+Q0=diag⁡(2,0)I+Q_0=\operatorname{diag}(2,0) on both sheets gives diag⁡(4,0)\operatorname{diag}(4,0), still rank one with the same null line. Two named nonnegative sheets are not enough; their spin-two characters must be opposite.

Scope

This finite representation theorem does not authorize a Fourier--Tate sheet action. The doubled-tail source must still derive gCg_C, λ\lambda, θ\theta, endpoint and Mellin normalization, retention of both sheets, and completion stability. It neither constructs global transport nor proves nonzero vacuum overlap or RH.

Durable verification

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