Off-line zeros of the Riemann ξ-function: a constraint network, an exactly solvable collision model, an energy-budget divergence, and a lifetime-deficit dictionary with negative controls

The de Bruijn–Newman constant Λ, known to satisfy 0 ≤ Λ ≤ 1/2, measures the displacement of the zeros of the Riemann ξ-function under the backward heat flow, and the Riemann Hypothesis is equivalent to Λ = 0. We assemble six interlocking structures around a hypothetical off-line zero. (I) A constraint network: fundamental positivity of an auxiliary function V' outside exceptional discs of total area O(T⁻¹ log²T), Apollonius confinement, the exact Hadamard curvature identity with its 2/h₀² signal, a self-contained inner-function theorem with a modulus-blindness obstruction, and a Speiser threshold mechanism. (II) An exactly solvable collision model: the symmetric four-body system is integrable, with conserved quantity I = y⁴ + 10q²y² + q⁴ and closed-form collision time; with a corrected explicit argument bound |S(T)| ≤ 0.1366·log T + 0.1786·log log T + 1.305 (full derivation in Appendix B), the budget-extremal frozen-field bound τ*(T₀) = 0.2151 is proved at the verification height T₀ = 3×10¹². (III) An energy–budget divergence E/B ≥ (π/2)·log³T in the averaged sense of Lemma 18.2 — pointwise E/B ≥ c·log²T with explicit c > 0 — together with a calibration section explaining why the divergence does not by itself discriminate ζ from the Davenport–Heilbronn counterexample. (IV) A lifetime–deficit dictionary between the flow and windowed Weil positivity, validated at both off-line zeros of the Davenport–Heilbronn function; the translation's isolation hypotheses are quantified, an explicit spill bound for distant quadruples is proved, and a synthetic three-quadruple cluster shows the unqualified aggregate form fails by a factor 4.4 — isolation is necessary, not technical. (V) A conditional arithmetic non-realizability principle, declared as the single conjectural input. (VI) Interference-channel negative controls with an index staircase. Every numerical claim outside the theorems is labelled as a numbered Numerical Observation and kept separate from the proved statements. We do not claim to prove the Riemann Hypothesis; the contribution is the network of proved constraints, the exactly solvable local model, and the corrected, fully quantified dictionary, each reproduced by public verification scripts (repository rh_pf_v18).