Prime-Only No-Gap Theorem

Abstract

The prime-only jump Laplacian has nonnegative nonzero-mode energies whose infimum vanishes throughout the absolutely convergent half-plane.

Theorem 1.1 (Prime-only spectral coefficients have no positive uniform gap).

$$sigma>1\Rightarrow \inf \left(\{n\in Z\mid n\ne 0\},C_n\right)=0.$$

Proof. Machine-checked in Lean as D5/S3/Weil/PrimeOnly/PrimeOnlyNoGap.numberField_prime_only_no_gap (✓ std3). ∎

Source. Repository-derived.

Commentary.

The jump indices are positive powers of genuine prime ideals in a number field. Dedekind-zeta convergence for sigma greater than one makes their weights summable.

Compact recurrence in every finite product of regulator circles gives a nonzero integer mode simultaneously close to the identity for any finite collection of prime-power shifts. No irrationality premise on the shifts is needed.

A finite-tail split then makes the Fourier jump energy arbitrarily small. Nonnegativity supplies the reverse bound, so the infimum over the subtype of nonzero integer modes is exactly zero.

References

• Truth anchor: D5/S3/Weil/PrimeOnly/PrimeOnlyNoGap.numberField_prime_only_no_gap