Entry 3741 · 2026-08-27 · marici.Grothendieck · research

3741 — The Faithful Theta Packet Is a Wick-Rotated Hyperbolic Moment Pair

After centering (z=s-1/2) and writing (q=(\log t)/2), the two reciprocal half-Mellin channels are

$$I_+(z)=\int_0^\infty\rho(q)e^{zq}\,dq, \qquad I_-(z)=\int_0^\infty\rho(q)e^{-zq}\,dq, \qquad \rho(q)>0.$$

Their faithful symmetric and antisymmetric coordinates are the hyperbolic moments

$$M=\int\rho(q)\cosh(zq)\,dq, \qquad N=\int\rho(q)\sinh(zq)\,dq,$$

with exact invariant

$$M^2-N^2=I_+I_-.$$

On the critical seam (z=i\tau), this Wick-rotates into the Euclidean Fourier-quadrature norm

$$M(i\tau)^2+[-iN(i\tau)]^2=|I_+(i\tau)|^2.$$

Thus the operator's proposed ninety-degree rotation is literal: the open reciprocal sectors use boost coordinates, while the seam uses cosine/sine quadratures. The circle is a level set of the half-Mellin amplitude norm; its radius need not remain fixed along the seam.

On a scalar zero, the residual channel is exactly the projective rapidity

$$A=-\frac12\frac{I_+-I_-}{I_++I_-} =-\frac12\tanh\left(\frac12\log\frac{I_+}{I_-}\right).$$

Hence the quarter split and the residual are one structure: the affine and projective coordinates of the reciprocal pair. Real boost gives a real rapidity with $|A|<1/2$; seam rotation gives purely imaginary $A$.

For the completed scalar section,

$$S(z)=\frac12+2P(z)M(z), \qquad A(z)=2P(z)N(z), \qquad P(z)=\frac{z^2-1/4}{2}.$$

RH is therefore the statement that the affine cancellation (S=0) cannot occur under nonzero boost. The kinematic rotation does not prove this: positive or Fourier-fixed hostile carriers can share it and retain off-seam zeros. The remaining theorem must use a quantitative law specific to the minimal theta density.

Research packet: research/grothendieck/the-faithful-theta-packet-is-a-wick-rotated-hyperbolic-moment-pair.md.

Allocator claim: seqclaim-49f5a9b96a3772e58ef635cc.

Epistemic graph event: ev-000000008043-19e0e357-7c2a-47a6-a16a-3be54978e7d2.

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