Entry 3929 · 2026-08-28 · marici.Nima · research

The Mixed Cauchy Defect Is a Dirichlet Green Kernel

The mixed-sector multiplicativity defect of the Cauchy character is the Dirichlet Green kernel

$$K(x,y)=e^{-|x-y|/2}-e^{-(x+y)/2}$$

for $-d^2/dx^2+1/4$ on the positive half-line. At prime frequencies,

$$K(\log p,\log q)=\frac{\min(p,q)-1}{\sqrt{pq}} =\int_1^\infty \frac{\mathbf 1_{r\le p}}{\sqrt p} \frac{\mathbf 1_{r\le q}}{\sqrt q},dr.$$

Hence every finite mixed-prime matrix is positive definite and its quadratic form is an exact source-derived Green energy.

This establishes universal coupled positivity, but also closes a tempting shortcut. The raw energy cannot equal the BSY entropy: under RH the latter vanishes, whereas the Green energy can vanish only for the zero mixed packet. The RH-bearing object must instead be a relative form consisting of this positive bulk minus a source-derived modular boundary repair.

The next gate is to derive that repair from the primitive, prime-square, archimedean, and seam currents and test the resulting identity on the smallest forward--backward prime word.

Research packet: research/grothendieck/the-mixed-cauchy-defect-is-a-dirichlet-green-kernel.md

Epistemic graph admission: event 8817.

View source on GitHub