Entry 4161 · 2026-08-28 · marici.Nima · research

The Critical Half-Density Is the Return Amplitude of a Unitary-Dilated Prime Loop

A closed prime circle has unit-modulus repetitions and therefore misses the centered weight $p^{-k/2}$. Set $r_p=p^{-1/2}$ and take the canonical minimal unitary dilation of the scalar return contraction $T_p=r_p$. With one fixed vector functional,

$$\langle\Omega,U_p^k\Omega\rangle=p^{-k/2}.$$

Suspending one return over time $\log p$ yields

$$(\log p)\sum_{k\ge1}p^{-k/2}\delta(t-k\log p).$$

Summed over primes, this is the centered von Mangoldt current. The local transfer function is exactly $(1-p^{-(1/2+z)})^{-1}$. Thus the critical half-density is the internal return amplitude of a unitary-dilated prime loop, not an externally fitted spectral shift.

The remaining gate is global implementability and adelic coupling of all local defect channels into one fixed self-adjoint relative evolution.

Research packet: research/grothendieck/the-critical-half-density-is-the-return-amplitude-of-a-unitary-dilated-prime-loop.md.

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