Quantitative Local Reductions of the Riemann Hypothesis via the de Bruijn–Newman Flow: Critical Exponent, Conditional Weil Positivity, and Almost-Periodic Regularisation

We prove unconditional local structural results for the de Bruijn–Newman flow at its transition time and record conditional quantitative reductions of the Riemann Hypothesis. First, if the transition produces a colliding real zero of multiplicity m ≥ 2, then every nonreal branch has local critical exponent exactly 1/2, independently of m. The multiplicity and the branch enter only through the constant | Im v∗|, where v∗ is a nonreal zero of the collision polynomial Pm, and the Hermite representation Pm(v) = (−i)mHm(iv/2) identifies the zero set of Pm as purely imaginary. Second, under the coordinate map s = 1/2 + iz, the deviation of a nonreal zero from the critical line equals the imaginary part in the z-variable, and the exponent of this deviation is 1/2. Third, if Λ > 0 and Λ < ε0(Λ, x0) for a collision position x0 of multiplicity m ≥ 2 at t = Λ, then there is a nontrivial zero ρ with | Re ρ − 1/2| ≥ cΛ1/2. Fourth, the five formal dualities are compatible with off-axis configurations and cannot by themselves force Re ρ = 1/2. Fifth, the collision of the flow obeys a strict momentum balance: the collision point is the centre of mass of the colliding branches, and the second moment of the collision polynomial is −2m(m − 1); the Hermite representation then identifies every collision polynomial zero as purely imaginary. Sixth, the height of a zero and its effective depth are reciprocally dual through the reciprocal RG invariant. We then state two conditional results with exact hypotheses: a constraint system using quantitative inputs from the literature, and a Weil-positivity exclusion on a compact interval. The paper also contains four refinements. (i) An almost-periodic refinement: ζ(σ + it) is Bohr almost periodic on every half-plane σ > 1 with frequency set contained in {log n}; the Abel-regularised zero measure is a finite Bohr almost periodic measure; finite-height verification is proved under the explicit ordinate-separation and depth-gap hypotheses of Hypothesis 15.3. Moreover, the finite-height verification in any single working window is equivalent to the verification in every working window at the same precision (Theorem 15.9). (ii) A proportion refinement: the universal inequality p∗ ≤ r∗ between the critical- line and distinct-zero proportions, the exact identity p∗ = 1 − lim supT →∞ O+(T )/N+(T ) under simplicity, and the quantitative off-axis density bound lim supT →∞ O+(T )/N+(T ) ≤ 1 − 0.6735 = 0.3265 from the constraint system. (iii) A reciprocal refinement: exact reciprocal asymptotics for the regularised mass Mr, the zero count N (T ), the detection height H(r), and the higher moments M (k) r ; the reciprocal criterion RH ⇐⇒ 1/Λ+ = +∞; the reciprocal positive part of the RG invariant, which is finite exactly when Λ > 0 and equals +∞ exactly on RH; and the reciprocal depth constant ι(κm). All divergent and infinite quantities are evaluated in the extended positive real line by first applying the reciprocal and then taking the limit. (iv) A renormalisation refinement: Gaussian coarse-graining is exactly a time translation, so linear RG is trivial; the zero motion satisfies a Dyson equation and is formally a Wasserstein gradient flow of the two-dimensional Coulomb energy; the collision curvature has two normalisations, with exponents 1 and 1/2; and the Coulomb–de Bruijn–Newman correspondence is stated and partially proved. The paper does not prove the Riemann Hypothesis. It proves a precise local reduction and identifies the exact conditional and structural gaps.