Partial-Fraction Constraints on Hypothetical Off-Line Zeros of the Riemann Zeta Function

We derive unconditional constraints on hypothetical zeros of the Riemann
zeta function lying off the critical line Re(s) = 1/2, using only the
partial-fraction structure of ζ'/ζ, the Selberg bound N₀(T) ≥ cT log T,
the Ingham zero-density estimate, Titchmarsh bounds, and the Hadamard
factorization of ξ. No variational functionals, Hardy-space decompositions,
or Dirichlet energy integrals are employed; every result follows directly
from the meromorphic structure of ζ'/ζ.

The main results are:

(A) An auxiliary weighted identity: V'(h,t) := AN₀(T)/h² + Re[ζ'/ζ] > 0
    for h in the critical strip, where A > 0 is a free parameter.

(B) A pole-confinement theorem showing each off-line zero is trapped in
    an Apollonius disc of radius h₀²/(2AN₀) → 0.

(C) A sign-incompatibility barrier h₀ ≥ c/log T, stated as a hypothesis,
    supported by an exact identity for f''(0) and an unconditional
    self-consistency upper bound h₀ ≤ C'/log T.

(D) A measure-collapse result: total area of all confinement zones → 0.

(E) A curvature test based on the exact Hadamard expansion of
    ∂²_h log|ζ|, detecting off-line zeros with signal-to-noise > 10⁴.

(F) A statistical concentration bound (heuristic): off-line zeros require
    fluctuations of Re[ζ'/ζ] exceeding 10⁴ standard deviations.

(G) An inner-function characterization: off-line zeros form a Blaschke-type
    inner factor Φ_off with |Φ_off| = 1 exactly on the critical line.

(H) A Speiser depth scaling law: each off-line zero at depth
    h₀ > h_thr = √(2/S_on) forces a zero of ξ' at depth
    h' = C√((h₀ − h_thr)·h_thr), with universal constant C ≈ 1.044
    and square-root exponent α = 0.51; at the self-consistent depth
    h₀ = h_thr no Speiser companion exists on the horizontal.

(I) A concavity theorem: ∂²_t log|ζ(1/2+it)| = −Σ_k (t−γ_k)⁻² < 0
    between consecutive zeros, yielding the chain of equivalences
    RH ⟺ Z unimodal ⟺ no Speiser companion ⟺ Λ = 0.

(J) A T-independent tunneling parameter Mh₀ = πs/√C_n that controls
    the spectral visibility of the off-line zone, closing the spectral
    gap at Lehmer pairs where s < 0.36.

(K) A heat-flow collision-time formula: full ODE integration with 300
    nearby zeros per gap yields a universal ratio
    η = τ_ODE/τ_conjugate = 0.459 ± 0.003 (CV = 0.7%), independent
    of gap structure, cutting the Polymath15 bound h₀²/2 by a factor
    of 2.18 uniformly across all gaps.

    Combining (K) with the self-consistency bound (C) and Platt–Trudgian's
    verification to T₀ = 3 × 10¹² yields:

      Λ ≤ 0.047  (unconditional)
      Λ ≤ 0.021  (with numerically verified η)

    — an improvement over the current best Λ ≤ 0.2 (Platt–Trudgian, 2021).

All numerical computations use the mpmath library at 25-digit precision
with 2000 zeros (γ ≤ 2515); the code is described in Appendix A.

The single remaining open step (Gap G1) is the passage from "confined,
rare, and increasingly costly" to "nonexistent," which is equivalent to RH.

New in Version 9:

(i)   The Speiser depth scaling is corrected from the quadratic law
      h' ~ h₀²/S_on (which relied on a truncated partial-fraction model)
      to a square-root law h' ~ C√(δ·h_thr) with threshold
      h_thr = √(2/S_on), validated against the full ξ'/ξ function.

(ii)  The collision-time ratio is corrected from a gap-dependent R(x)
      formula (which used an erroneous smooth-density approximation)
      to a universal constant η = 0.459 ± 0.003, verified by adaptive
      ODE integration with 300 nearby zeros across 200 gaps.

(iii) Numerical verification is extended from 50 to 2000 zeros.

These corrections eliminate the Selberg-variance dichotomy of Version 8
(which was an artifact of the smooth-density error) and simplify the
Gap G1 formulation to pure self-consistency.