Off-Line Zeros of the Riemann ξ-Function: A Constraint Network, an Exactly Solvable Collision Model, Explicit Frozen-Field Bounds, a Lifetime–Deficit Dictionary for Weil Positivity, and an Interference-Channel Negative Control

We study the off-line zeros of the Riemann ξ-function through a network of necessary conditions and through the de Bruijn–Newman backward heat flow, organizing the analytic obstruction to the Riemann hypothesis into three concrete structures: a depth dichotomy on the critical line, under which on-line and off-line zeros are separated by the decay rate of an associated mode (algebraic 1/n versus exponentially suppressed e^(−π|h₀|n)/n); a lifetime–deficit dictionary relating the backward heat flow of the zeros to the windowed Weil functional; and an interference-channel localization that places off-line zeros at residue-one poles of a Dirichlet-series cross term. The framework is calibrated throughout against a genuine off-line zero of the Davenport–Heilbronn function, and yields a negative-control theorem identifying precisely where arithmetic (Euler-product) input must enter — a conclusion that converges, from an independent direction, with the screw-function and spectral-triple programmes of Suzuki and of Connes–Consani–Moscovici.

Concretely, we study hypothetical off-line zeros ρ* = ½ + h₀ + it₀ (h₀ > 0) of the Riemann ξ-function through a network of necessary conditions and through the de Bruijn–Newman backward heat flow. The exposition separates three layers and labels every statement accordingly, so that the rigorous core, the numerical evidence, and the single declared open problem are cleanly delineated.

Layer A (rigorous). We establish an Apollonius disc confinement of vanishing radius, an exact Hadamard second-derivative identity with the resulting self-consistent depth and curvature dichotomy, measure collapse, antisymmetry of ξ′/ξ, the Hadamard curvature identity, a Blaschke inner-function characterization with an explicit obstruction remark, a concavity equivalence chain on the critical line, and a Speiser-type depth-scaling law. Two of these conditions (the curvature and Speiser detectors) are validated against a genuine off-line zero of the Davenport–Heilbronn function. On the flow side we prove that the symmetric four-body system (a conjugate pair plus its two nearest real zeros, all moving) is exactly integrable: q² − y² grows linearly and I = y⁴ + 10q²y² + q⁴ is conserved, giving the collision time in closed form. In the frozen-field model we prove, with the Kusmin–Landau-corrected and height-optimized argument bound (re-deriving Trudgian's theorem with the Hiary–Patel–Yang subconvexity input and re-optimized parameters), the budget-extremal estimate τ*(T₀) = 0.215 at the verification height T₀ = 3×10¹²; the bound decreases along future verification records, crossing 0.2 near T ≈ 1.0×10¹⁴ (0.226 and 8×10¹⁴ with the unoptimized corrected constants; 0.229 and 1.25×10¹⁵ with the published constants). We further prove a two-sided lifetime–deficit equivalence: the survival time of off-line zeros under the flow and the windowed Weil deficit — computable from primes and the Γ-factor alone — bound each other with explicit constants, resting on a new aggregate descent lemma (Σₖ τₖ ≤ Σₖ yₖ(0)²/2 for arbitrary configurations, robust to shielding) and validated against both off-line zeros of the Davenport–Heilbronn function. The two-zero ladder reads 0.4079 (static) and 0.4178 (dynamic, exact closed form), against the classical anchors Λ ≤ ½ (de Bruijn) and Λ < ½ (Ki–Kim–Lee). All times are stated in the standard de Bruijn normalization via an explicit time dictionary; the present mechanism is independent of, and of the same 1/log T class as, the barrier-method bound Λ ≤ 0.2 of Polymath15/Platt–Trudgian, which it recovers rather than supersedes — the contribution here being the explicit closed-form collision dynamics and the lifetime–deficit dictionary, not a numerical improvement of the constant.

Layer B (numerical/heuristic, labelled). The full-ODE frozen-field collision ratio is approximately stable, η ≈ 0.46 with coefficient of variation ≈ 2.7%, with a weak gap-structure dependence; a statistical-concentration estimate is stated as a heuristic with its gaps enumerated.

Layer C (conjectural, labelled). The framework isolates the analytic difficulty into a single named open problem rather than diffusing it: the passage from the comparison models to the full flow rests on a named open lattice back-pressure lemma, and the passage from the constraint-energy/budget divergence to non-realizability rests on an Euler Realizability principle, stated as the central open problem and shown to coincide with the arithmetic-input requirement anticipated independently in the Weil-positivity programmes cited above. A measured supply-envelope calibration at a Davenport–Heilbronn frequency quantifies both: the positivity envelope exceeds the resonance demand by a factor 3.0 (so no unconditional exclusion follows from positivity alone, except for a sliver h₀ > 0.453 comparable to the classical zero-free region), while the actual ζ coefficients use only 2.4% of the envelope — a 42× cancellation that any binding inequality must explain. A measured calibration shows the windowed prime field clipped one-sidedly at the Weil boundary (free-field extreme-value prediction +0.077 against measured +0.002); the uniform supply cap — the single wall of the programme, sitting in the typical-versus-uniform gap of Fyodorov–Hiary–Keating extreme-value theory — is the declared open centre. Part V develops the operator-theoretic face in full: a Dirichlet-series dictionary characterizing the Euler product by prime-power support, an interference-channel localization of off-line zeros (residue exactly 1), four measured faces of the Davenport–Heilbronn counterexample (a forbidden coefficient c₆ = (1+κ²) log 6, a Krein level set to 10⁻¹¹, explicit Weil-negative witnesses, and a Klein-invariant phantom equator), and a negative-control theorem proving the Connes–Consani–Moscovici rank-one scheme cannot converge for Davenport–Heilbronn data — locating precisely where the Euler input must act.