V155_1 — A Projected Wavelet Prime‑Pair Criterion Equivalent to the Riemann Hypothesis

Description: This paper develops a projected high‑pass wavelet criterion for the Riemann Hypothesis (RH). It reformulates the explicit formula for primes into a prime‑pair covariance problem, using compactly supported mean‑zero wavelets on the logarithmic scale. By construction, the continuous background term is annihilated, leaving only atomic prime sums.

🔹 Wavelet Projection and Background Cancellation

• Compactly supported wavelets are projected so that the continuous background term vanishes exactly.

• This ensures the prime‑side energy is purely atomic, with diagonal contributions uniformly bounded.

• The remaining obstruction is the off‑diagonal prime‑pair covariance.

🔹 Prime Atomic Energy

• The projected energy expands into two parts:

  1. Diagonal term — bounded uniformly.

  2. Off‑diagonal term — a signed covariance of prime pairs, which is the essential obstruction.

🔹 Equivalence Criterion

• For every admissible detector bundle, RH ⇔ uniform boundedness of the Abel‑regularized projected prime‑pair energy.

• Off‑critical zeros force blow‑up in the energy (pole detection).

• Assuming RH, projected kernel estimates and decay properties yield boundedness.

🔹 Arithmetic Obstruction

• The continuous background, polar term, and diagonal atom energy are removed by construction.

• The unresolved problem is a second‑order prime‑pair covariance estimate for the von Mangoldt function under a sign‑changing logarithmic wavelet kernel.

• This reduces RH to proving uniform cancellation in these projected prime‑pair sums.

🔹 Relation to Existing Methods

• Hardy–Littlewood heuristics: projection annihilates the continuous pair main term, leaving fluctuations.

• Vaughan and Heath‑Brown identities: Type I terms are killed by projection; the difficulty lies in Type II bilinear forms.

• Goldston–Montgomery variance, pretentious methods, and spectral large sieve: related in philosophy but not sufficient to reach the endpoint bound.

🔹 Conclusion V155_1 isolates RH as a precise second‑order arithmetic cancellation problem: proving a uniform projected Type II prime‑pair covariance bound. The criterion provides a sharp equivalence, removing irrelevant structures and leaving only the true obstruction.