Arithmetic Non-Realizability of Off-Line Blaschke Factors in the Riemann Zeta Function: Constraint Metric Tensor and the Bound Λ ≤ 0.200

We study constraints on hypothetical zeros of the Riemann zeta function lying o the
critical line Re(s) = 1/2 , using only the partial-fraction structure of ζ′/ζ, the Levinson lower
bound N0(T) ≥ cT log T, the Ingham zero-density estimate, Titchmarsh bounds, and the
Hadamard factorization of ξ. The framework is organized into three layers.
Layer A (rigorous). We establish, as theorems, an Apollonius disc connement of
vanishing radius, a self-consistency upper bound h0 ≤ Ln/(2√Cn), measure collapse, mono-
tone decrease of an auxiliary functional, antisymmetry of ξ′/ξ, the Hadamard curvature
identity, a Blaschke inner-function characterization with explicit obstruction remark, a con-
cavity equivalence chain on the critical line, and the unconditional bound Λ ≤ 0.200 on the
de BruijnNewman constant via an independent two-zero collision-time argument matching
PlattTrudgian.
Layer B (numerical observations). A 5 × 5 Gram tensor Gij of ve independent
prime-side weight classes is introduced; the structural inequality E(T)/B(T) → ∞ between
constraint energy and Poisson capacity is rigorous, while observations of universal collision-
time ratio η = 0.459 ± 0.003, square-root Speiser scaling, and Cn ∈ [1.002, 2.306] across all
1999 tested gaps are supported by computations at 25-digit precision with 2000 zeros.
Layer C (conjectural). We formulate Arithmetic Non-Realizability: nontrivial o-line
Blaschke factors of ξ lie outside the Euler-realizability set Rζ . This reformulates the obstruc-
tion at the prime side, motivated by Theorem 12.1 which shows that analytical methods on
the critical line cannot distinguish Φoff ≡ 1 from Φoff ̸≡ 1