Off-Line Zeros of the Riemann xi-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

This paper concerns the zeros of the Riemann zeta function and the
de Bruijn-Newman constant Lambda, the two central objects of the analytic
theory of the Riemann hypothesis. We study hypothetical off-line zeros
rho* = 1/2 + h0 + i t0 (h0 > 0) of the Riemann xi-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.

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 xi'/xi, 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: q^2 - y^2 grows linearly and
I = y^4 + 10 q^2 y^2 + q^4 is conserved, giving the collision time in closed
form. In the frozen-field model we prove, with the full Trudgian bound on S(T),
the budget-extremal estimate tau*(T0) = 0.229 at the verification height
T0 = 3 x 10^12; the bound decreases along future verification records as
tau*(T) ~ 9.4 / log T, crossing 0.2 near T ~ 1.25 x 10^15. 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
Gamma-factor alone -- bound each other with explicit constants, resting on a new
aggregate descent lemma (sum of tau_k <= sum of y_k^2(0)/2 for arbitrary
configurations, robust to shielding) and validated against both off-line zeros
of the Davenport-Heilbronn function. The two-zero ladder reads 0.4233 (static)
and 0.4305 (dynamic, exact closed form), against the classical anchors
Lambda <= 1/2 (de Bruijn) and Lambda < 1/2 (Ki-Kim-Lee). All times are stated in
the standard de Bruijn normalization via an explicit time dictionary; no claim
of improvement over the barrier-method bound Lambda <= 0.2 of
Polymath15/Platt-Trudgian is made, the present mechanism being independent and
of the same 1/log T class.

Layer B (numerical/heuristic, labelled). The full-ODE frozen-field collision
ratio is approximately stable, eta ~ 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 passage from the comparison models to the
full flow rests on a named open lattice back-pressure lemma; the passage from
the constraint-energy/budget divergence to non-realizability rests on an Euler
Realizability principle, stated as the central open problem. 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 zeta
coefficients use only 2.4% of the envelope -- a 42x 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+kappa^2)log6,
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.