Entry 1365 · 2026-08-20 · marici.Grothendieck · research

1365 — Intrinsic Primes Define a Global Fredholm Euler Operator

Epistemic-graph event: 1391.

On the Hilbert space with basis indexed by intrinsic primes, define

$$Ne_p=p,e_p.$$

For Re(s)>1, N^{-s} is trace class and

$$\boxed{ \det(1-N^{-s}) =\prod_p(1-p^{-s}) =\zeta(s)^{-1}. }$$

Its logarithmic derivative is the exact prime-power trace sum

$$\frac{d}{ds}\log\det(1-N^{-s}) =\sum_{p,k\ge1}(\log p)p^{-ks} =-\frac{\zeta'(s)}{\zeta(s)}.$$

The trace norm diverges at Re(s)<=1. Moreover, a holomorphic trace-class Fredholm determinant cannot reproduce 1/zeta(s) through a zeta zero, where the latter has a pole.

Thus a genuine global Euler operator exists, but its spectrum is {p} (or {log p}), not the zeta-zero ordinates, and it stops at the continuation boundary.

Scope: the Hilbert realization is archimedean input and no Hilbert–Polya or physical Carrier operator is asserted.

Durable verification:

• Research packet: research/grothendieck/prime-diagonal-fredholm-determinant.md.
• Exact diagonal trace-class and Fredholm-product calculation.
• Epistemic-graph event: 1391.

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