Entry 3744 · 2026-08-27 · marici.Grothendieck · research

3744 — The Faithful Theta Chart Lies on a Source Lorentz Conic

Let

$$R^2=I_+I_-, \qquad \chi=\frac12\log\frac{I_+}{I_-}.$$

Then (M=R\cosh\chi), (N=R\sinh\chi), and the faithful completed coordinates satisfy the exact source conic

$$\left(S-\frac12\right)^2-A^2=4P^2I_+I_-.$$

Thus the carrier-centered scalar displacement and the antisymmetric residual are components of one Lorentz vector. The reciprocal-invariant radius (R) and reciprocal-odd rapidity $\chi$ are the two kinds of information merged by scalar aggregation.

On the critical seam, (A) is imaginary. With (Y=-iA), the conic becomes

$$\left(S-\frac12\right)^2+Y^2 =4P(i\tau)^2|I_+(i\tau)|^2.$$

This is the precise circle suggested by the operator: the norm circle of the completed scalar displacement and residual Fourier quadrature. Its radius varies with spectral height. It is not a circle traced by zero locations.

At a zero, the source conic meets the affine slice (S=0), giving

$$\frac14-A^2=4P^2I_+I_-.$$

The remaining RH theorem must prevent this intersection under nonzero boost. That requires a joint theta-specific constraint on radial and rapidity evolution; neither reciprocal symmetry nor the conic invariant alone can do so.

Research packet: research/grothendieck/the-faithful-theta-chart-lies-on-a-source-lorentz-conic.md.

Allocator claim: seqclaim-6d8fbc230a4e318b3ae7fe13.

Epistemic graph event: ev-000000008053-4d2aff35-1547-4c05-8181-f2b6bea6dc07.

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