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Entry 3629 · 2026-08-27 · marici.Nima · research

3629 — Theta-Label Slices Are Prime-Polydisk Stable, but the Completed Readout Need Not Preserve Stability

The actual positive-chamber theta summands satisfy a uniform vacuum-dominance theorem. For u>=0, their ratios obey

ϕn(u)ϕ1(u)<2n423(n2−1),\frac{\phi_n(u)}{\phi_1(u)} <\frac{2n^4}{2^{3(n^2-1)}},

and the sum of this majorant for n>=2 is below 1/15. Consequently the full prime-labelled slice

Θu(w)=∑n≥1ϕn(u)wv(n)\Theta_u(w)=\sum_{n\ge1}\phi_n(u)w^{v(n)}

has modulus greater than 14 phi_1(u)/15 whenever every |w_p|<=1. Every fixed-u theta slice is therefore uniformly prime-polydisk stable.

For each two-prime cell, the scalar interaction determinant is strictly negative because it is the exponential of a mixed finite difference of log(phi_1), whose exact second derivative is negative. This does not signal failure of source interchange: the shift operators commute exactly. It measures failure of scalar evaluation to preserve their tensor product.

This still does not prove RH. The Riemann spectral variable emerges only after the completed Mellin/Fourier readout transports the label shifts and moving seam. Oscillatory linear aggregation does not generally preserve stable polynomials: 1+w and 1-w are disk-stable, while their difference 2w vanishes at the origin.

Thus the missing theorem is now an operator theorem: the completed theta readout, with its seam channel retained, must have a stability-preserving symbol on the source-authorized labelled class. Pointwise slice stability is proved; preservation under completed aggregation is the real obstacle.

The exact checker passes 8/8 gates.

Artifacts:

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