A Fourier-Analytic Constraint Architecture for Off-Line Zeros of the Riemann xi-Function: Annular Spectral Dichotomy, an Exactly Solvable Collision Model, a Lifetime-Deficit Dictionary for Weil Positivity, and an Interference-Channel Negative Control

We develop a Fourier-analytic framework for the off-line zeros of
the Riemann xi-function, in which the analytic obstruction to the
Riemann hypothesis is organized through three concrete spectral
structures: an annular spectral dichotomy on the critical equator
(on-line zeros contribute a slowly decaying 1/n Fourier mode,
off-line zeros an exponentially suppressed e^{-pi |h_0| n}/n mode),
a lifetime-deficit dictionary relating the de Bruijn-Newman backward
heat flow 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 Fourier-analytic direction, with the
screw-function and spectral-triple programmes of Suzuki and of
Connes-Consani-Moscovici.

Concretely, we study hypothetical off-line zeros rho* = 1/2 + h_0 + i t_0
(h_0 > 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, 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 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*(T_0) = 0.229
at the verification height T_0 = 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_k tau_k <= sum_k 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; the present mechanism is
independent of, and of the same 1/log T class as, the barrier-method
bound Lambda <= 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, 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 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 h_0 > 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. 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, 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.