V124 — A Spectral Short‑Interval Variance Criterion for the Riemann Hypothesis

Description: This paper develops the spectral formulation of the short‑interval variance approach to the Riemann Hypothesis (RH). It identifies the exact spectral multiplier behind the Abel‑damped finite‑difference variance of the Fejér primitive and shows how this connects to the Sobolev gap.

🔹 Fejér Primitive and Riesz Criterion

• The Fejér primitive arises from the critically weighted prime residual.

• The condition that the primitive grows at most linearly is equivalent to the Riesz criterion, and therefore to RH.

🔹 Sobolev Gap and Finite Differences

• The gap between Fejér growth and Hardy mean‑square control is a Sobolev derivative‑energy condition.

• It can be rewritten using finite differences of the damped primitive.

• These finite differences have a short‑interval arithmetic interpretation, involving averages of the prime residual with damping corrections.

🔹 Spectral Formula

• By extending the damped primitive to the whole line and applying Plancherel’s theorem, the finite‑difference variance admits an exact spectral representation.

• The spectral multiplier is a band‑limited approximation to the Sobolev gap multiplier, suppressing high frequencies while matching the Sobolev weight at low frequencies.

🔹 Spectral Sufficient Condition

• A uniform bound on the spectral short‑interval variance implies boundedness of the Sobolev gap.

• This excludes zeros of the zeta function off the critical line, thereby implying RH.

• The paper does not prove the bound unconditionally, but it isolates the precise vertical‑line weighted L2 obstruction that must be controlled.

🔹 Prime‑Side Relation

• The spectral variance corresponds to the prime‑side Abel‑damped short‑interval variance, up to an explicit boundary correction.

• This shows the exact connection between spectral analysis and prime residual variance.

🔹 Conclusion V124 identifies the exact spectral multiplier governing the Abel‑damped short‑interval variance. It reframes the remaining RH obstruction as a vertical‑line weighted L2 estimate for the zeta function. Through this route, proving RH requires establishing uniform spectral variance control, equivalent to uniform Abel‑damped short‑interval variance for the prime residual.