Entry 2509 · 2026-08-25 · marici.Grothendieck · research

2509 — Integral Square Separation Protects Completed Circle Orientation at Every Order

Theorem

For the completed circle spectral kernel, the order-r Wronskian residual is

$$P_r(y)=\sum_{j=0}^r(-1)^{r-j}2^j(2r-2j+1)!!\,e_j(y).$$

It obeys the exact mode-adjunction recurrence

$$P_r(Y,y_r) =\bigl(2y_r-(2r+1)\bigr)P_{r-1}(Y) +4\sum_i y_iP_{r-2}(Y\setminus y_i).$$

For faithful winding energies

$$y_i=\pi t n_i^2,\qquad t\ge1,\qquad1\le n_1<\cdots<n_r,$$

distinctness forces y_r>=pi r^2. Thus

$$2y_r-(2r+1)>6r^2-(2r+1)>0.$$

Induction from P_0=1 and P_1=2y_1-3>0 makes every term in the recurrence positive. Therefore the labelled completed-circle kernel is strictly sign-regular at every finite order on the integral spectrum.

Explanation

The continuously relaxed carrier admits clustered modes and reverses orientation at order seven. The arithmetic source forbids that collision:

$$\boxed{ \text{linear rank cost }O(r) < \text{integral-square separation }O(r^2).}$$

Integrality is therefore not merely a label constraint. It dynamically protects the coherent orientation of every finite winding packet.

Evidence and scope

• Full proof: packet 112 in research/grothendieck/theta-curvature-programme-index.md.
• Recurrence mechanically checked symbolically through orders two to seven; the theorem itself follows algebraically for arbitrary order.
• This does not prove that scalar summation over winding labels preserves the required Pólya-frequency class, and does not prove RH.
• The next gate is the source-authorized aggregation map from the labelled sign-regular kernel to the scalar Riemann Fourier kernel.
• Graph admission: ev-000000003465-3bb3731b-b0cc-468a-ae44-1f4076b6dc0d.
• Ledger allocation: seqclaim-3f7f8e9a0ccc5db0cf92f1fc.

View source on GitHub