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 (Theorem 3.1);

(B) a pole-confinement theorem showing each off-line zero is trapped in an
    Apollonius disc of radius h0²/(2AN0) → 0 (Theorem 4.1);

(C) a sign-incompatibility barrier h0 ≥ c/log T, stated as a hypothesis,
    supported by an exact identity for f''(0) (Theorem 5.2) and an unconditional
    self-consistency upper bound h0 ≤ C'/log T (Theorem 5.3);

(D) a measure-collapse result: total area of all confinement zones tends to zero
    (Theorem 6.1);

(E) a curvature test based on the exact Hadamard expansion of ∂²h log|ζ|,
    detecting off-line zeros with signal-to-noise exceeding 10⁴ (Theorem 10.1);

(F) a statistical concentration bound (Proposition 11.1, 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 Φoff with |Φoff| = 1 exactly on the critical line (Theorem 12.1);

(H) a Speiser depth scaling law: each off-line zero at depth h0 forces a zero of
    ξ' at depth h' ~ CSh0², with an explicit universal constant CS = 1/Son and
    quadratic exponent β = 2.006 verified numerically (Theorem 13.1);

(I) a concavity theorem: ∂²t log|ζ(1/2 + it)| = −Σk (t − γk)⁻² < 0 between
    consecutive zeros, yielding the chain of equivalences
    RH ⟺ Z unimodal ⟺ no Speiser companion ⟺ Λ = 0 (Theorem 14.1);

(J) a T-independent tunneling parameter Mh0 = πs/√Cn that controls the spectral
    visibility of the off-line zone, closing the spectral gap at Lehmer pairs
    where s < 0.36 (Theorem 15.1);

(K) a heat-flow collision-time formula
    Λlocal = (h0 log T − ln(1 + h0 log T))/log²T incorporating the acceleration
    by on-line zeros, which cuts the Polymath15 conjugate bound h0²/2 by a
    factor of 2.7 at the self-consistent depth (Theorem 16.1).

All numerical computations use the mpmath library at 25-digit precision; 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 8: a Selberg-variance argument shows that for all but
O(log log T) gaps, the heat-flow collision time gives Λlocal ≤ C/log²T → 0;
the remaining exceptional gaps constitute the precise locus of Gap G1.