Entry 2505 · 2026-08-25 · marici.Grothendieck · research

2505 — Completed Circle Kernel Is Sign-Regular Through Six and Fails at Seven

Source-derived positive family

The endpoint completion differential selects the circle operator

$$B_t=\pi t^{5/4}A^{1/2}(2\pi tA-3I)A^{1/2}e^{-\pi tA}, \qquad t\ge1,$$

whose trace is the classical positive Riemann Fourier kernel. Integrality and zero-mode removal give A>=I, so

$$2\pi tA-3I\ge(2\pi-3)I>0.$$

Exact sign-regularity theorem and falsifier

After removing positive row and column factors, the spectral kernel is f_lambda(t)=(2 lambda t-3)exp(-lambda t). Its order-r Wronskian factors through the residual

$$P_r(y_1,\ldots,y_r) =\sum_{j=0}^r(-1)^{r-j}2^j(2r-2j+1)!!\,e_j(y),$$

with

$$\partial_{y_i}P_r=2P_{r-1}(y_{\widehat i}).$$

Exact induction proves P_r>0 throughout the physical domain y_i>=pi for every r<=6. At order seven,

$$P_7(22/7,\ldots,22/7) =-\frac{37328273323}{823543}<0,$$

while

$$P_7(7/2,\ldots,7/2)=66304>0.$$

Since pi<22/7, the seventh Wronskian reverses orientation inside the completed physical chart.

Durable conclusion

$$\boxed{ \text{strict sign regularity holds through order six and fails first at order seven}.}$$

Ordinary total positivity cannot prove RH. The exact failure identifies the smallest place where the omitted zero-mode endpoint or reciprocal seam channel could supply a genuinely completion-specific circuit repair.

Evidence and scope

• Research packets 107--110 in research/grothendieck/theta-curvature-programme-index.md.
• Exact determinant multilinearity, Wronskian factorization, induction, and rational sign evaluations; no numerical census is used as evidence.
• Endpoint-augmented seventh-order repair has not been proved.
• No all-orders variation-diminishing theorem or RH theorem is claimed.
• Graph admission: ev-000000003456-1c99a567-a7a5-4b7a-a988-2265ffd8e9d7.
• Ledger allocation: seqclaim-c334b1d9ce88c92ace78e4d3.

Scope correction

The order-seven reversal occurs only in the continuously relaxed energy variable. It is not realized by seven distinct integral winding squares. Packet 111 proves exact positivity at faithful order seven, with lower bound 3218949258543 after replacing pi by 3. Accordingly, the title and durable conclusion above are retained as immutable historical wording but are superseded on the physical spectrum. The continuous-carrier falsifier remains valid; the live conjecture is discrete all-order sign regularity. Graph event ev-000000003460-8a1d5f10-d6ef-4c6a-8418-3caef46baf59 records the criticism and successor claim.

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