V123_1 — A Short‑Interval Variance Form of the Sobolev Gap for the Riemann Hypothesis

Description: This paper reformulates the Sobolev derivative‑energy gap in the Fejér criterion for the Riemann Hypothesis (RH) as a finite‑difference and short‑interval variance problem.

🔹 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

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

• It can be expressed through finite differences of the damped primitive.

🔹 Finite‑Difference Formulation

• The Sobolev gap equals the limit of finite‑difference variances of the damped primitive.

• A uniform bound on these finite differences implies boundedness of the gap and excludes zeros off the critical line.

🔹 Arithmetic Interpretation

• Finite differences of the Fejér primitive correspond to short‑interval averages of the prime residual.

• Explicit formulas show contributions from both past prime memory and short logarithmic intervals.

• The damped finite difference introduces an unavoidable correction term, combining local prime sums with long‑memory effects.

🔹 Short‑Interval Variance Condition

• The gap is equivalent to requiring uniform control of Abel‑damped short‑interval variances of the prime residual.

• This condition provides a sufficient route to RH: if the variance bound holds, RH follows.

🔹 Relation to Off‑Line Zeros

• Off‑line zeros force divergence in the variance, so the finite‑difference condition excludes them.

• Thus, the condition is strong enough to imply RH when combined with the functional equation.

🔹 Comparison with Other Viewpoints

• The finite‑difference formulation sits between the Sobolev derivative viewpoint and the pair‑correlation viewpoint.

• It exposes the arithmetic structure of short intervals directly, rather than hiding it inside derivatives.

🔹 Conclusion V123_1 shows that the remaining obstruction in the Fejér‑Hardy route to RH can be expressed as a short‑interval variance problem for the prime residual. Proving RH through this path requires establishing uniform Abel‑damped variance control for the Fejér primitive.