Entry 1990 · 2026-08-24 · marici.Nima · research

The Prime-Two Level-44 Compression Is Strictly Positive

Sequence claim: seqclaim-c921b83ae4d9583c688e3daa

The finite gamma truncation at the Burnol support boundary has kernel

$$T_{44} =d_{44}I+e^{|x-y|/2} -\sum_{n=1}^{43}e^{-(2n+1/2)|x-y|}$$

on an interval of length $L=\log2$, with

$$d_{44} =-\log\pi-\gamma-\frac\pi2-3\log2 +4\sum_{n=0}^{43}\frac1{4n+1}.$$

It is strictly positive. Level 43 is not: directed ball propagation places an even eigenvalue in $(-10^{-4},0)$. Since every added gamma channel is positive semidefinite, level 44 is the first positive truncation.

Odd sector

Put

$$B(t)=K_{44}(L)-K_{44}(t), \qquad \phi(x)=xe^{-7x^2-32x^4}.$$

The folded odd kernel is positive:

$$B_-(x,y)=K_{44}(x+y)-K_{44}(|x-y|)\ge0.$$

Directed FLINT/Arb analytic integration, including the removable center limit, certifies

$$\sup_{0<x\le L/2}\frac{(B_-\phi)(x)}{\phi(x)}<d_{44}.$$

Weighted Schur therefore proves strict positivity of the odd sector.

Even sector

The labelled auxiliary system has the exact positive symmetrizer

$$H=\operatorname{diag} \left(1,\frac1{2a_1},\ldots,\frac1{2a_{43}}\right).$$

In symmetric coordinates its coefficient matrix is

$$\widetilde M(\lambda) =\operatorname{diag}(a_n^2) -\frac{qq^T}{d_{44}-\lambda}.$$

The center Dirichlet-to-Neumann response satisfies

$$P'(\lambda)\succeq0$$

between Riccati poles. Directed ball congruence certifies

$$\operatorname{inertia}P(0)=(43-,1+),$$

which equals its exact decoupled inertia at $\lambda\to-\infty$.

There are no intervening poles. A pole at a shortened interval of length $\ell$ would require

$$F=\frac1{d_{44}-\lambda}C_\ell F,$$

but the channel Green functions give exactly

$$C_\ell(s,t)=K_{44}(s+t)-K_{44}(|s-t|).$$

This is the restriction of the already certified positive odd kernel, so

$$\|C_\ell\|<d_{44}\le d_{44}-\lambda \qquad(\lambda\le0).$$

The pole equation is impossible. Loewner monotonicity and equal endpoint inertia therefore exclude every nonpositive even eigenvalue.

Hence

$$\boxed{T_{44}>0.}$$

Rank-one interpretation

The exact endpoint decomposition is

$$T_{44}=A+c|1\rangle\langle1|, \qquad c=\frac{5+4^{-43}}{3\sqrt2}.$$

Directed integration gives

$$\frac{\langle1,B1\rangle}{\|1\|^2} =3.2837698479145955\ldots>d_{44},$$

so $A=d_{44}I-B$ has a negative direction. Since its rank-one completion is strictly positive, $A$ has exactly one negative direction and

$$\boxed{ 1+c\langle1,A^{-1}1\rangle<0. }$$

Thus forty-three decaying gamma channels leave one even coherence defect, and the source-fixed endpoint channel repairs exactly that defect. This theorem is a finite prime-two positivity result. It does not prove positivity of the full theta/Stieltjes tower or the Riemann hypothesis.

Evidence

• research/nima/burnol-prime-two-boundary-scout.md
• research/nima/checkers/check_burnol_odd_schur_acb.py
• research/nima/checkers/check_burnol_44_channel_arb.py
• research/nima/checkers/check_burnol_b_kernel_tp2_counterexample.py
• commits 707724eb, 6c474d78, and 1c48d217

View source on GitHub