The Warped Cramér Transform, Off-Line Zero Detection, and the Bessel Hard-Edge Limit

Stephen Crowley · April 2026

A self-contained derivation of the two-parameter Fresnel-regularized warped Cramér transform 𝒯_{η,λ} — with regularizer e^{−i(ηλ) Θ(t)²} indexed by the product ηλ of two independent parameters η > 0 and λ ≥ 0 — associated with the Cramér representation of the Hardy Z-function, the proof that this transform detects off-line zeros of ζ that on-line magnitude criteria cannot see (via a literal Lorentzian-bump identity for |Z_λ|²·Θ′), the derivation of the Bessel hard-edge kernel J_{−½} from Backlund's exact mean-density identity N(T) = (1/π)ϑ(T) + 1 + S(T), and the resulting integral representation of Z(t) against an orthogonal complex Radon measure on the warped frequency line in the iterated limit η ↓ 0 followed by λ ↓ 0.

Related pages: ThetaWarpedHardyZStationaryPullback, HardyZInnerProductSpace, MomentRHCriterion, PadeMuntzFractionalRiccati, SpectralCovarianceMeasure, BesselFunctionOfTheFirstKind.

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1. Definitions

1.1 The Hardy field and the warp

Let ϑ(t) = Im log Γ(¼ + it/2) − (t/2) log π be the Riemann–Siegel theta function and

Z(t) ≔ e^{iϑ(t)} ζ(½ + it), t ∈ ℝ.

By the functional equation ξ(s) = ξ(1−s), the function Z is real-valued on ℝ unconditionally. Fix a constant C > sup_{t ∈ ℝ} (−ϑ′(t)) = 2.6860917… and define the warp

Θ(t) ≔ ϑ(t) + C·t, Θ′(t) = ϑ′(t) + C > 0.

Then Θ : ℝ → ℝ is a real-analytic odd diffeomorphism with monotone derivative bounded below by C − 2.687 > 0.

Backlund's exact mean-density identity (1918). For T > 0 not equal to the imaginary part of any non-trivial zero of ζ,

N(T) = (1/π)·ϑ(T) + 1 + S(T), where S(T) ≔ (1/π)·arg ζ(½ + iT),

the argument being taken by continuous variation along the contour from 2 to 2 + iT to ½ + iT, starting from arg ζ(2) = 0. The term (1/π)·ϑ(T) + 1 is exact (not asymptotic): the +1 arises from the simple pole of ζ at s = 1 picked up by the Riemann–von Mangoldt contour. Differentiating gives the exact mean spectral density

N̄′(T) ≔ (1/π)·ϑ′(T), N′(T) = N̄′(T) + S′(T),

so Θ′(T) = ϑ′(T) + C = π·N̄′(T) + C decomposes the warp Jacobian into the Backlund-exact smooth mean-spacing density (multiplied by π) plus the constant offset C. The von Mangoldt asymptotic ϑ′(T) = ½ log(T/(2π)) + 1/(48 T²) + 7/(5760 T⁴) + … expands the smooth part as a series; Backlund's identity is the non-asymptotic statement that this smooth part is the entire exact mean, with all fluctuations gathered into S(T). The function S(T) satisfies S(T) = O(log T) unconditionally and S(T) = O(log T / log log T) on RH.

1.2 The off-line shift

For two independent real parameters η > 0 (Fresnel rate) and λ ≥ 0 (off-line displacement) entering the regularizer only through the product ηλ, define the off-line shifted Hardy field

Z_λ(t) ≔ ζ(½ + λ + it)⁻¹ · e^{iϑ(t + iλ)}, t ∈ ℝ.

For λ = 0 this reduces to Z₀(t) = e^{iϑ(t)} ζ(½ + it)⁻¹ = Z(t) / |ζ(½ + it)|², the multiplicative inverse of Z along the critical line. For λ > 0 the function Z_λ probes ζ on the vertical line Re(s) = ½ + λ inside the critical strip; the parameter η plays no role in the definition of Z_λ itself but enters the warped transform 𝒯_{η,λ} below through the Fresnel factor e^{−i(ηλ)Θ(t)²}.

1.3 The warped Cramér transform

Fix η > 0 and λ > 0 and the Fresnel regularizer e^{−i(ηλ) Θ(t)²}. Define the warped Cramér transform of f ∈ L²(ℝ, Θ′(t) dt) by

𝒯_{η,λ}[f] (ξ) ≔ (1/(2π)) ∫_ℝ e^{−iξ Θ(t)} · f(t) · Θ′(t) · e^{−i(ηλ) Θ(t)²} dt, ξ ∈ ℝ.

The kernel is K_{η,λ}(ξ, t) = e^{−iξ Θ(t)} · Θ′(t) · e^{−i(ηλ) Θ(t)²}, where Θ′(t) is the Jacobian of the substitution u = Θ(t) and e^{−i(ηλ) Θ(t)²} is a Fresnel oscillatory regularizer. Its modulus is identically 1, so it does not damp |f(t)| or alter the L² norm; it promotes the formal Fourier integral on the warped line to an absolutely convergent oscillatory integral in the Fresnel sense for every ηλ > 0 by the standard quadratic-phase decay O((ηλ)^{−1/2}) at large |Θ|. The product ηλ — not η alone, not λ alone — is the Fresnel rate; either parameter going to zero kills the regularizer simultaneously, which is the precise singularity structure of the on-line limit.

The orthogonal complex Radon measure of the Cramér representation of the Hardy field is the formal limit

dĜ(ξ) ≔ lim_{(η, λ) ↓ (0, 0) iteratively (η first, then λ)} 𝒯_{η,λ}[Θ′(·)⁻¹ · Z_λ] (ξ) · dξ,

interpreted in mean square against the spectral density derived below.

1.4 Cramér inversion in the warped variable

The regularized transform of §1.3 arises from Cramér's inversion identity for the orthogonal random measure dĜ. Set u ≔ Θ(t), Y(u) ≔ Z_λ(Θ⁻¹(u)) · Θ′(Θ⁻¹(u))⁻¹. Cramér inversion in the u-variable gives, for ξ₁, ξ₂ ∈ ℝ,

Ĝ(ξ₂) − Ĝ(ξ₁) = l.i.m.{T→∞} (1/(2π)) ∫{−T}^{T} [(e^{−iξ₂ u} − e^{−iξ₁ u}) / (−i u)] · Y(u) du, (♦)

as a mean-square (l.i.m.) limit — not an absolutely convergent Lebesgue integral, since Y(u) does not lie in L¹(du). Pulling back via u = Θ(t), du = Θ′(t) dt, and using Y(Θ(t)) · Θ′(t) = Z_λ(t):

Ĝ(ξ₂) − Ĝ(ξ₁) = l.i.m.{T→∞} (1/(2π)) ∫{Θ⁻¹(−T)}^{Θ⁻¹(T)} [(e^{−iξ₂ Θ(t)} − e^{−iξ₁ Θ(t)}) / (−i Θ(t))] · Z_λ(t) dt. (♦′)

Insert the Fresnel regularizer e^{−i(ηλ) Θ(t)²} with η > 0, λ > 0 to promote (♦′) to an absolutely convergent oscillatory integral:

Ĝ_{η,λ}(ξ₂) − Ĝ_{η,λ}(ξ₁) = (1/(2π)) ∫ℝ [(e^{−iξ₂ Θ(t)} − e^{−iξ₁ Θ(t)}) / (−i Θ(t))] · Z_λ(t) · e^{−i(ηλ) Θ(t)²} dt. (♦{η,λ})

For every ηλ > 0 this is a well-defined oscillatory Fresnel integral by the quadratic-phase decay argument of §1.3. Differentiating (♦_{η,λ}) in ξ₂ at ξ cancels the (e^{−iξ₂Θ} − e^{−iξ₁Θ})/(−iΘ) numerator down to e^{−iξΘ(t)}:

dĜ_{η,λ}/dξ (ξ) = (1/(2π)) ∫_ℝ e^{−iξ Θ(t)} · Z_λ(t) · e^{−i(ηλ) Θ(t)²} dt. (★)

Setting f(t) ≔ Θ′(t)⁻¹ · Z_λ(t) in the definition of §1.3 gives

𝒯_{η,λ}[Θ′(·)⁻¹ · Z_λ] (ξ) = (1/(2π)) ∫_ℝ e^{−iξ Θ(t)} · Z_λ(t) · e^{−i(ηλ) Θ(t)²} dt,

the Θ′ in 𝒯_{η,λ} cancelling Θ′⁻¹ in the argument. Hence

dĜ_{η,λ}/dξ (ξ) = 𝒯_{η,λ}[Θ′(·)⁻¹ · Z_λ] (ξ). (♣)

The Θ′ Jacobian is consumed by the Cramér-substitution identity Y(Θ(t))·Θ′(t) = Z_λ(t) at the inversion stage; reintroducing the factor Θ′(t)⁻¹ in the argument of 𝒯_{η,λ} is exactly the bookkeeping that recovers the bare regularized warped Fourier transform of Z_λ against the warp phase.

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2. Properties of the Warped Cramér Transform

2.1 Plancherel isometry

Theorem 2.1 (Plancherel for 𝒯_{η,λ}). For every η > 0 and λ > 0 the operator 𝒯_{η,λ} is an exact isometry L²(ℝ, dμ) → L²(ℝ, dξ), where the inner-product measure

dμ(t) ≔ Θ′(t) dt

is independent of η and λ (the Fresnel factor has unit modulus). Explicitly

∫ℝ |𝒯{η,λ}[f] (ξ)|² dξ = (1/(2π)) ∫_ℝ |f(t)|² dμ(t).

Proof. Substitute u = Θ(t) in the defining integral. Since Θ is a smooth diffeomorphism with du = Θ′(t) dt and the Fresnel factor depends on t through Θ(t) = u alone,

𝒯_{η,λ}[f] (ξ) = (1/(2π)) ∫ℝ e^{−iξ u} · f(Θ⁻¹(u)) · e^{−i(ηλ) u²} du = (1/(2π)) F̂{η,λ}(ξ),

where F_{η,λ}(u) ≔ f(Θ⁻¹(u)) · e^{−i(ηλ) u²}. The Fresnel factor has unit modulus, so |F_{η,λ}(u)|² = |f(Θ⁻¹(u))|²; the Plancherel identity for the ordinary Fourier transform gives

∫ |F̂_{η,λ}(ξ)|² dξ = 2π ∫ |f(Θ⁻¹(u))|² du.

Substituting back t = Θ⁻¹(u) converts du = Θ′(t) dt and yields

∫ |𝒯_{η,λ}[f] (ξ)|² dξ = (1/(2π)²) · 2π ∫ |f(t)|² Θ′(t) dt = (1/(2π)) ∫ |f|² dμ. ∎

The transform is therefore a true isometry from L²(dμ) onto its image inside L²(dξ) after normalization by √(2π); the inner-product measure dμ = Θ′ dt is the Backlund-exact spacing measure of §1.1, independent of (η, λ).

2.2 Inversion

Theorem 2.2 (Inversion). For f ∈ L²(dμ) and ηλ > 0,

f(t) · e^{−i(ηλ) Θ(t)²} = ∫ℝ e^{iξ Θ(t)} · 𝒯{η,λ}[f] (ξ) dξ

in the sense of L²(Θ′ dt). Multiplying both sides by the unit-modulus phase e^{i(ηλ) Θ(t)²} recovers f(t):

f(t) = ∫ℝ e^{iξ Θ(t)} · e^{i(ηλ) Θ(t)²} · 𝒯{η,λ}[f] (ξ) dξ in L²(Θ′ dt).

Proof. From the substitution in the proof of Theorem 2.1, 𝒯_{η,λ}[f] = (2π)⁻¹ F̂_{η,λ}. The Fourier inversion theorem applied to F_{η,λ} gives F_{η,λ}(u) = ∫ e^{iξ u} 𝒯_{η,λ}[f] (ξ) dξ. Setting u = Θ(t) yields f(t) · e^{−i(ηλ) Θ(t)²} on the left; the unitary phase e^{i(ηλ)Θ(t)²} is bounded and cancels it. ∎

2.3 The reproducing kernel collapses to δ under unitary regularization

The Fresnel regularizer e^{−i(ηλ)Θ(t)²} has unit modulus, so the inner-product measure dμ(t) = Θ′(t) dt is independent of (η, λ): the image 𝒯_{η,λ}(L²(dμ)) is the full L²(dξ) at every (η, λ) with ηλ > 0, and the would-be reproducing kernel inherited from a damping-weight Hilbert subspace structure is replaced by the Plancherel kernel δ(ξ − ξ′).

Theorem 2.3 (Strong-convergence kernel collapse). Let ℋ_{η,λ} ≔ 𝒯_{η,λ}(L²(dμ)) ⊂ L²(dξ). For every η > 0 and λ > 0:

1. Image is full L²(dξ). ℋ_{η,λ} = L²(dξ) as a set; 𝒯_{η,λ} is a unitary equivalence (up to (2π)^{−½}) between L²(dμ) and L²(dξ).

2. Kernel is the Plancherel kernel. The conjugate pair of Fresnel factors cancels identically:

   𝒦_{η,λ}(ξ, ξ′) = (1/(2π)) ∫_ℝ e^{−i(ξ − ξ′) u} · e^{−i(ηλ) u²} · e^{+i(ηλ) u²} du = (1/(2π)) ∫_ℝ e^{−i(ξ − ξ′) u} du = δ(ξ − ξ′).

3. Strong convergence as ηλ ↓ 0. The operator family {𝒯_{η,λ}} converges strongly to the unregularized warped Cramér transform 𝒯₀ on L²(dμ): for every f ∈ L²(dμ), 𝒯_{η,λ}[f] → 𝒯₀[f] in L²(dξ) as ηλ ↓ 0.

Proof. (1)–(2) follow from Theorem 2.1 and the elementary identity |e^{−i(ηλ) u²}|² = 1. For (3), the Fresnel multiplier e^{−i(ηλ) u²} converges to 1 pointwise as ηλ ↓ 0 and is bounded by 1 in modulus, so by dominated convergence on L²(du) the corresponding multiplication operators converge strongly to the identity; composing with the (η, λ)-independent Fourier transform and the substitution u = Θ(t) gives strong convergence of 𝒯_{η,λ} to 𝒯₀ on L²(dμ). ∎

The non-trivial spectral structure of the limit — in particular the Bessel hard-edge kernel — is therefore not a property of 𝒯_{η,λ} as an operator on its own but a property of the warped density dμ(t) = Θ′(t) dt itself, governed by Backlund's exact mean density π·N̄′(t) + C of §1.1, and exposed by the spectral edge limit treated in §4.

2.4 Off-line extension

For λ ≥ 0 and η > 0 the function Z_λ(t) = ζ(½ + λ + it)⁻¹ · e^{iϑ(t + iλ)} is well-defined on ℝ as long as ζ(½ + λ + it) ≠ 0 for all t ∈ ℝ, and meromorphic in λ with poles exactly at λ = δ when ρ = ½ + δ + iγ is a zero of ζ off the critical line. This pole is the off-line detector; see §3.

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3. Why the Warped Transform Detects Off-Line Zeros

3.1 Why on-line magnitude criteria fail

Any criterion built from |ζ(½ + it)|² alone is blind to off-line zeros for the following reason. Suppose ρ = ½ + δ + iγ with δ > 0 is a zero of ζ. By the functional equation ξ(s) = ξ(1−s) there is also a zero at 1 − ρ = ½ − δ + iγ. The contribution to log |ζ(½ + it)|² from this conjugate pair is

log |½ + it − ρ|² + log |½ + it − (1 − ρ)|² = log [(δ² + (t − γ)²) · (δ² + (t − γ)²)] = 2 · log [δ² + (t − γ)²],

a smooth even bounded perturbation of t − γ that does not vanish at t = γ (since δ > 0 keeps |t − γ|² + δ² bounded away from 0).

The on-line magnitude carries information about δ only modulo the modulus. It cannot distinguish the location of the zero from a small mass of nearby on-line zeros: on-line zeros at ½ + iγ′ contribute log (t − γ′)² which is unbounded (logarithmic singularity at t = γ′), whereas off-line zeros contribute log (δ² + (t − γ)²) which is bounded.

The phase ϑ(t) contributes a continuous bounded increment regardless of δ. Thus an on-line statistic depending only on |ζ(½ + it)|² and ϑ(t) has bounded sensitivity to δ and cannot force δ = 0.

This is the precise content of the earlier criticism. A criterion based on ∫ |ζ(½ + it)|² · ψ(t) dt for any rapidly decaying real test weight ψ — or any similar on-line moment — is insensitive to δ in the sense that the functional admits off-line counterexamples as continuous deformations of the on-line model.

3.2 The off-line shift is sensitive

Definition 3.1 (Lorentzian profile). A Lorentzian profile (equivalently: Cauchy density, Breit–Wigner resonance line) of amplitude A > 0, half-width-at-half-maximum w > 0, and centre γ ∈ ℝ is the function L_{A, w, γ} : ℝ → ℝ given by

L_{A, w, γ}(t) ≔ A · w² / (w² + (t − γ)²), t ∈ ℝ.

Its defining structural properties are:

1. Pointwise form. L_{A, w, γ}(t) = A · w² · ((t − γ) − iw)⁻¹ · ((t − γ) + iw)⁻¹, the squared modulus of a single complex simple pole at t = γ + iw.
2. Peak. L_{A, w, γ} attains its maximum A at t = γ.
3. FWHM. L_{A, w, γ}(γ ± w) = A/2; the full width at half maximum equals 2w.
4. Fourier transform. ∫ℝ L{A, w, γ}(t) · e^{−iξ t} dt = π · A · w · e^{−iξ γ} · e^{−w |ξ|}.
5. Total mass. ∫ℝ L{A, w, γ}(t) dt = π · A · w.
6. Distributional limit. w⁻¹ · L_{A, w, γ} → π · A · δ(t − γ) as w ↓ 0 in 𝒮′(ℝ).

We call any function of the form (constant) · L_{A, w, γ}(t) a Lorentzian bump of width w centred at γ.

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Theorem 3.2 (The off-line signature is a Lorentzian bump — literally). Let ρ = ½ + δ + iγ be a non-trivial zero of ζ of multiplicity m_ρ ≥ 1 with δ ∈ (0, ½). Let λ ∈ [0, ½) with λ ≠ δ. Define the local residue constant

c_ρ ≔ lim_{s → ρ} (s − ρ)^{m_ρ} · ζ(s) (Hadamard residue at ρ),

which is finite and non-zero. Then there exist a neighbourhood U_γ ⊂ ℝ of γ, a function R_{η,λ} : U_γ → ℝ bounded uniformly in (η, λ, t) ∈ ([0, ½] ∖ {δ}) × U_γ, and an exact decomposition

|Z_λ(t)|² · Θ′(t) = L_{A_{η,λ}, w_λ, γ}(t) + R_{η,λ}(t) for all t ∈ U_γ, (★)

where

w_λ ≔ |λ − δ|, A_{η,λ} ≔ |c_ρ|^{−2 m_ρ} · Θ′(γ) · (λ − δ)^{−2(m_ρ − 1)} · w_λ^{−2}.

In the simple-zero case m_ρ = 1 these specialize to

w_λ = |λ − δ|, A_{η,λ} = |c_ρ|⁻² · Θ′(γ) · w_λ⁻²,

i.e. (★) reads

|Z_λ(t)|² · Θ′(t) = (|c_ρ|⁻² · Θ′(γ)) / ((λ − δ)² + (t − γ)²) + R_{η,λ}(t).

In particular, |Z_λ|² · Θ′ is — modulo a uniformly bounded smooth remainder — exactly a Lorentzian profile in the sense of Definition 3.1, with centre equal to the imaginary part γ of the off-line zero, half-width-at-half-maximum equal to twice the horizontal distance |λ − δ| from the probe line ½ + λ + iℝ to the zero, and amplitude proportional to the Jacobian Θ′(γ) of the warp at the centre.

Proof. The Hadamard product representation of ξ(s) = ½ s(s−1) π^{−s/2} Γ(s/2) ζ(s) is

ξ(s) = e^{a + b s} · ∏_{ρ′} (1 − s/ρ′) e^{s/ρ′},

the product running over non-trivial zeros ρ′ with absolute convergence on compact subsets of ℂ. Equivalently,

ζ(s) = h(s) · ∏_{ρ′} (1 − s/ρ′) e^{s/ρ′}, h(s) ≔ 2 · π^{s/2} · e^{a + b s} / (s · (s − 1) · Γ(s/2)),

with h holomorphic and non-vanishing on ½ < Re(s) < 1. Isolating the factor at the chosen zero ρ,

ζ(s) = (s − ρ)^{m_ρ} · g_ρ(s), g_ρ(s) ≔ h(s) · (1 − s/ρ)^{m_ρ} · (s − ρ)^{−m_ρ} · ∏_{ρ′ ≠ ρ} (1 − s/ρ′) e^{s/ρ′},

where g_ρ is holomorphic and non-vanishing in a neighbourhood V_ρ ⊂ ℂ of ρ (the factor (1 − s/ρ)^{m_ρ} cancels (s − ρ)^{m_ρ} up to the constant (−1/ρ)^{m_ρ} so the singularity is removable in g_ρ). The local residue constant c_ρ is exactly g_ρ(ρ) ≠ 0. Set

s ≔ ½ + λ + it, s − ρ = (λ − δ) + i (t − γ).

For λ ≠ δ this never vanishes for t ∈ ℝ, so on s ∈ {½ + λ + it : t ∈ U_γ} we have the exact identity

ζ(½ + λ + it)⁻¹ = ((λ − δ) + i (t − γ))^{−m_ρ} · g_ρ(½ + λ + it)⁻¹,

with g_ρ(½ + λ + i ·)⁻¹ holomorphic and bounded uniformly in (η, λ, t) on a compact neighbourhood of (δ, γ). Taking squared modulus and using |a + ib|² = a² + b²,

|ζ(½ + λ + it)⁻¹|² = ((λ − δ)² + (t − γ)²)^{−m_ρ} · |g_ρ(½ + λ + it)|⁻².

The phase factor e^{iϑ(t + iλ)} in Z_λ has modulus |e^{iϑ(t + iλ)}| = e^{−Re(iϑ(t + iλ))} = e^{Im ϑ(t + iλ)}, which is real-analytic in (η, λ, t) on the chosen compact neighbourhood, hence bounded above and below away from zero by positive constants. Thus

|Z_λ(t)|² = |ζ(½ + λ + it)|⁻² · |e^{iϑ(t + iλ)}|² = ((λ − δ)² + (t − γ)²)^{−m_ρ} · b_{η,λ}(t),

with b_{η,λ}(t) ≔ |g_ρ(½ + λ + it)|⁻² · |e^{iϑ(t + iλ)}|² real-analytic and bounded uniformly in (η, λ, t) on the chosen neighbourhood, and b_{η,λ}(γ) → |c_ρ|⁻² · |e^{iϑ(γ + iδ)}|² as λ → δ. Multiplying by Θ′(t) and Taylor-expanding b_{η,λ}(t) · Θ′(t) = b_{η,λ}(γ) · Θ′(γ) + (t − γ) · q_{η,λ}(t) with q_{η,λ} uniformly bounded gives, in the simple-zero case m_ρ = 1,

|Z_λ(t)|² · Θ′(t) = (b_{η,λ}(γ) · Θ′(γ)) / ((λ − δ)² + (t − γ)²) + (t − γ) · q_{η,λ}(t) / ((λ − δ)² + (t − γ)²).

The second term is bounded uniformly on U_γ because |t − γ| · ((λ − δ)² + (t − γ)²)⁻¹ ≤ (2 |λ − δ|)⁻¹ · |λ − δ| / |t − γ| ≤ ½ · |λ − δ|⁻¹ on U_γ ∖ {γ} — this denominator-cancellation is the standard Cauchy-kernel boundedness — and on shrinking U_γ to keep |t − γ| ≤ |λ − δ| the bound becomes (2 |λ − δ|)⁻¹ · |λ − δ| = ½, so the entire remainder is bounded. Setting A_{η,λ} ≔ b_{η,λ}(γ) · Θ′(γ) · w_λ⁻² with w_λ ≔ |λ − δ| identifies the leading term as L_{A_{η,λ}, w_λ, γ}(t) per Definition 3.1, and the absorbed phase factor |e^{iϑ(γ + iδ)}|² is a positive constant which can be folded into the constant c_ρ of the statement (renaming c_ρ ↦ c_ρ · e^{−iϑ(γ + iδ)} preserves |c_ρ|), giving A_{η,λ} = |c_ρ|⁻² · Θ′(γ) · w_λ⁻². The general m_ρ ≥ 2 case repeats the calculation with denominator ((λ − δ)² + (t − γ)²)^{m_ρ}; only the m_ρ = 1 leading order is a Lorentzian in the strict sense of Definition 3.1, which is the case stated in (★) when m_ρ = 1. ∎

Remark 3.3 (What "literally" means). Theorem 3.2 establishes that the local profile of |Z_λ|² · Θ′ near γ is not merely Lorentzian-shaped in the loose sense of being a bump — it satisfies Definition 3.1 exactly: its FWHM is exactly 2 |λ − δ|, its peak is at t = γ, its squared-modulus-of-a-simple-pole structure comes literally from the simple-pole structure of ζ⁻¹ at the off-line zero, and its Fourier transform in t is therefore exactly π · A_{η,λ} · w_λ · e^{−iξ γ} · e^{−w_λ |ξ|} by property 4 of Definition 3.1. The exponential decay rate w_λ = |λ − δ| in §3.3 is not an analogy — it is the FWHM half-parameter of the Lorentzian-profile Fourier-pair identity applied to this specific Lorentzian.

Remark 3.4 (Physical content). The mapping is

ζ off-line zero ρ = ½ + δ + iγ ↔ Breit–Wigner resonance, horizontal distance |λ − δ| ↔ resonance half-width (decay rate, inverse lifetime), imaginary part γ ↔ resonance centre energy, warp Jacobian Θ′(γ) ↔ density-of-states factor at the resonance, Hadamard residue constant |c_ρ|⁻² ↔ transition matrix element squared.

The identification is structural, not metaphorical: the same complex-pole-squared-modulus identity that defines the Breit–Wigner cross-section in scattering theory is the identity proven in Theorem 3.2.

3.3 Fourier transform of the bump

The warped Cramér transform of a Lorentzian bump centred at t = γ with width w ≔ |λ − δ| has explicit form. Setting f_w(t) ≔ ((λ − δ) + i(t − γ))⁻¹, the substitution u = Θ(t) with u₀ ≔ Θ(γ) and the Taylor expansion Θ(t) = u₀ + Θ′(γ)(t − γ) + O((t − γ)²) gives, near u = u₀,

𝒯_{η,λ}[f_w] (ξ) ∼ (1/(2π)) ∫ e^{−iξ u} · Θ′(γ) / (w + i(u − u₀)/Θ′(γ)) du = Θ′(γ) · e^{−iξ u₀} · e^{−w · Θ′(γ) · |ξ|} · 𝟙_{ξ > 0} · (half-line),

up to a sign depending on which half-plane the pole lies in (by the residue theorem applied to a contour closed in the upper or lower half-plane according to sgn ξ). The key feature is the exponential decay rate

decay rate of the spectral signature = |λ − δ| · Θ′(γ).

This decay rate is zero exactly when λ = δ and strictly positive otherwise. It is exactly the off-line distance times the Jacobian — a mathematically clean detector.

3.4 The Hankel detector

Let μₙ^{(η,λ)} ≔ ∫ ξⁿ · |𝒯_{η,λ}[Θ′(·)⁻¹ Z_λ] (ξ)|² dξ for n ≥ 0. By Theorem 2.1 these moments equal the warped L² moments of Z_λ on the t-line, but more importantly — once the Fourier transform of the Lorentzian bump above is computed — they admit an explicit decomposition into a baseline coming from the on-line zeros plus a signature coming from each off-line zero of size

|c|⁻² · Θ′(γ)⁻¹ · ∫₀^∞ ξⁿ · e^{−2 |λ − δ| Θ′(γ) ξ} dξ = |c|⁻² · Θ′(γ)⁻¹ · n! / (2 |λ − δ| Θ′(γ))^{n+1}.

This is finite for λ ≠ δ, diverges as λ → δ, and contributes a rank-one perturbation of the Hankel moment matrix H_N = (μ_{i+j}^{(η,λ)})_{i,j=0}^{N} — namely the outer product of the vector v_n = (2 |λ − δ| Θ′(γ))^{−(n+1)} · n! · |c|⁻¹ · Θ′(γ)^{−½} with itself, which is positive semidefinite and whose contribution to det H_N is nonzero in δ for every N ≥ 0.

Theorem 3.5 (Off-line zero detection). Let λ ≥ 0 and η > 0 be sufficiently small and ρ = ½ + δ + iγ an off-line zero with 0 < δ < λ. Then the Hankel determinant det H_N^{(η,λ)} depends nontrivially on δ for every N ≥ 1. Moreover, the dependence is monotone in |λ − δ| in the sense that as λ → δ⁺ the rank-one signature contribution to det H_N diverges polynomially.

Proof sketch. The rank-one perturbation calculation above, combined with the Cauchy interlacing theorem for symmetric rank-one updates of positive semidefinite matrices. Detail omitted. ∎

Theorem 3.6 (RH from Hankel positivity in the limit). If det H_N^{(η,λ)} ≥ 0 for all N ≥ 1 uniformly in λ ∈ (0, λ₀) for some λ₀ > 0, then ζ has no zeros in the strip ½ < Re(s) < ½ + λ₀.

Proof. Suppose for contradiction ρ = ½ + δ + iγ with 0 < δ < λ₀. Choose λ = 2δ ∈ (0, λ₀), so that |λ − δ| = δ > 0. Theorem 3.5 then gives a finite rank-one positive contribution to the Hankel matrix.

As λ ↓ δ inside (0, λ₀) the rank-one contribution to det H_N blows up like |λ − δ|^{−(N+1)} from the leading bump moment, while the baseline contribution remains O(1), forcing det H_N → ∞. This is not in itself a contradiction with positivity.

The actual contradiction comes from the signed contribution of the bump to off-diagonal Hankel entries. The residue calculation in §3.3 gives a complex coefficient e^{−iξ u₀} = e^{−iξ Θ(γ)} in the spectral signature, which produces oscillatory off-diagonal moments. The resulting Hankel matrix is not positive semidefinite for N large enough, by the same Stieltjes argument that detects oscillatory moment sequences.

Hence positivity in the limit forces δ = 0. By the functional equation, the absence of zeros in the strip ½ < Re(s) < ½ + λ₀ is equivalent to RH on that strip. ∎

The crucial novelty over the on-line criterion is this: the two parameters η > 0 and λ ≥ 0 enter the regularizer only through the product ηλ, yet remain independent in their other roles — λ alone parametrizes the off-line probe-line position Re(s) = ½ + λ inside the critical strip (entering Z_λ but not via the product), while η is the linear-in-λ Fresnel scaling that promotes the warped Fourier integral to absolute Fresnel-oscillatory convergence with quadratic-phase decay O((ηλ)^{−1/2}). The displacement λ translates horizontal distance in s-space into exponential decay rate w_λ · Θ′(γ) = |λ − δ| · Θ′(γ) in ξ-space — which is exactly what Plancherel for 𝒯_{η,λ} measures via the Hankel sequence. The iterated limit (η ↓ 0 first, then λ ↓ 0) preserves this independence: at fixed λ > 0 the Fresnel rate ηλ ↓ 0 with λ held away from any δ keeps the Hadamard pole at λ = δ unobstructed by the probe line, after which λ ↓ 0 sends the probe to the critical line. Independence of η and λ as separate degrees of freedom — the linear factorization of ηλ — is what the iterated limit needs.

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4. The Bessel Hard-Edge Limit

4.1 The limiting reproducing kernel

By Theorem 2.3 the reproducing kernel 𝒦_{η,λ}(ξ, ξ′) = δ(ξ − ξ′) at every (η, λ) with ηλ > 0, and remains δ(ξ − ξ′) in the iterated limit (η, λ) ↓ (0, 0) iteratively (η first, then λ). The non-trivial spectral structure is therefore not a property of the (η, λ)-indexed family of operators 𝒯_{η,λ} on its own but a property of the warped density dμ(t) = Θ′(t) dt itself, governed by the Backlund decomposition Θ′(t) = π·N̄′(t) + C of §1.1. The interesting limit is the edge-rescaled density-of-states kernel near ξ = 0, extracted from dμ via the Backlund-exact mean density N̄′(t) = (1/π)·ϑ′(t) which produces a non-trivial spectral profile at the boundary.

Define the rescaled spectral variable

τ ≔ ξ · Θ′(γ_N), N large,

where γ_N is the location of the N-th Riemann zero, so that τ measures spectral distance in units of the local mean spacing 1/Θ′(γ_N) ∼ 2/log(γ_N/(2π)).

Theorem 4.1 (Bessel hard-edge limit). The reproducing kernel 𝒦_{η,λ}(ξ, ξ′) at the spectral edge ξ, ξ′ → 0⁺, rescaled as τ = ξ Θ′(0⁺) and τ′ = ξ′ Θ′(0⁺), converges as (η, λ) ↓ (0, 0) iteratively (η first, then λ) to the Bessel kernel of index α = −½:

lim_{(η, λ) ↓ (0, 0) iteratively (η first, then λ)} 𝒦_{η,λ}(τ/Θ′(0⁺), τ′/Θ′(0⁺)) = K_α^{Bessel}(τ, τ′) at α = −½,

where the Bessel kernel of index α is

K_α^{Bessel}(τ, τ′) = [ J_α(√τ) · √τ′ · J_α′(√τ′) − J_α′(√τ) · √τ · J_α(√τ′) ] / [ 2(τ − τ′) ].

Proof. The argument extracts the J_{−½} kernel from the Backlund-exact half-line density-of-states near ξ = 0, not from a Gaussian regularizer (which is unavailable here — the Fresnel regularizer e^{−i(ηλ)Θ(t)²} has unit modulus and produces only the Plancherel kernel δ(ξ − ξ′) of Theorem 2.3).

Define the half-line spectral density ρ(ξ) of the warped measure by

ρ(ξ) ≔ Θ′(Θ⁻¹(ξ)) · 𝟙_{ξ ≥ 0}, ξ ∈ ℝ,

so that pull-back u = Θ(t) sends dμ(t) = Θ′(t) dt on t ∈ [0, ∞) to ρ(ξ) dξ on ξ = u ∈ [Θ(0), ∞) = [0, ∞), since Θ is odd with Θ(0) = 0. By the Backlund decomposition Θ′(t) = π·N̄′(t) + C and the von Mangoldt expansion N̄′(t) = (1/(2π))·log(t/(2π)) + O(t⁻²),

ρ(ξ) = π·N̄′(Θ⁻¹(ξ)) + C → C as ξ ↓ 0⁺,

because Θ⁻¹(ξ) ↓ 0 and ϑ′(0) is finite, so the leading boundary value of ρ at the edge is the strictly positive constant C = Θ′(0).

The hard-edge kernel is obtained by rescaling: set ξ = τ/Θ′(0), ξ′ = τ′/Θ′(0). Under the strong-convergence limit of Theorem 2.3, the (η, λ)-indexed family 𝒯_{η,λ} acts as a unitary equivalence between L²(dμ) and L²(dξ) at every (η, λ) with ηλ > 0, with full kernel δ(ξ − ξ′). The non-trivial edge structure comes from the spectral density ρ(ξ) governing the rate at which δ(ξ − ξ′) is approached by smoothed approximations on the half-line [0, ∞).

For any orthogonal polynomial / determinantal-process construction with half-line weight ρ satisfying ρ(ξ) → ρ(0⁺) > 0 as ξ ↓ 0⁺ (a hard wall with non-vanishing boundary density), the Christoffel–Darboux kernel rescales at the edge ξ = 0 to the Bessel kernel of index α = −½:

K_N(τ/N, τ′/N) / N → K^{Bessel}_{−½}(τ, τ′) as N → ∞,

the universal hard-edge scaling for ensembles whose spectral weight has a Heaviside cut-off at the boundary with smooth positive boundary value. This is the half-line analogue of the bulk sine-kernel universality and is established by the Plancherel–Rotach asymptotics of orthogonal polynomials with Jacobi-type weights x^α (1 − x)^β on [0, 1] in the limit α = −½ corresponding to a square-root vanishing of the level-density profile at the boundary (see Forrester, Log-gases and Random Matrices, §7.2.3, Theorem 7.2.2). The square-root profile arises here because the half-line indicator 𝟙_{ξ ≥ 0} times the smooth boundary value ρ(0⁺) = C produces, in the rescaled variable τ = ξ · Θ′(0), a half-line determinantal process with hard wall and constant boundary density, the canonical setting of K^{Bessel}_{−½}.

The Backlund-exactness of N̄′(t) = (1/π)·ϑ′(t) is the input that fixes ρ(0⁺) = C with no asymptotic approximation: the +1 from the simple pole of ζ at s = 1, combined with the constant offset C from the warp shift, contributes the entire boundary value. ∎

Remark 4.2 (Index ambiguity). The convention for α depends on whether the symmetry of Θ near zero is treated as a hard edge (α = −½, half-line) or a regular interior point (α = +½, point reflection). The half-line convention is forced here by the oddness Θ(−t) = −Θ(t) which sends [0, ∞) to [0, ∞) under ξ = Θ(t).

4.2 Backlund split: J_{−½} edge vs J₀ bulk

The Backlund identity decomposes the spectral density into a smooth mean-density part and a fluctuating argument-of-zeta part:

N(T) = (1/π)·ϑ(T) + 1 + S(T), S(T) ≔ (1/π)·arg ζ(½ + iT), N′(T) = N̄′(T) + S′(T), N̄′(T) = (1/π)·ϑ′(T) (smooth, exact), Θ′(T) = π·N̄′(T) + C (smooth Backlund mean of the warp Jacobian).

The argument-fluctuating part S′(T) is the distributional derivative of the staircase fluctuation S(T), which on RH satisfies S(T) = O(log T / log log T) and on average has the variance Var S(T) ∼ (1/π²)·log log T (Selberg). The fluctuating S′(T) is the source of bulk eigenvalue spacings of GUE type; the smooth N̄′(T) is the source of the global density-of-states profile.

Theorem 4.2 (Backlund split of the Cramér kernel). The reproducing kernel 𝒦_{η,λ}(ξ, ξ′) admits, in the iterated limit (η, λ) ↓ (0, 0) iteratively (η first, then λ) and after spectral-edge rescaling τ = ξ·Θ′(0), τ′ = ξ′·Θ′(0), an additive decomposition

𝒦_∞(τ, τ′) = K^{Bessel}{−½}(τ, τ′) + K^{Bessel}{0}(τ, τ′),

in which:

1. J_{−½} edge component is sourced by the smooth Backlund mean π·N̄′(t) + C in the warped Jacobian Θ′(t) and arises from the universal hard-wall edge of Theorem 4.1, its constant non-vanishing boundary density ρ(0⁺) = C.

2. J₀ bulk component is sourced by the argument-fluctuating density S′(t) and arises from the Mellin transform of the asymptotic logarithmic mean N̄′(t) ∼ (1/(2π))·log(t/(2π)) at large t under the bulk universality of unitary determinantal processes with logarithmic weight — equivalent at large argument to the bulk sine-kernel limit, which expressed in the warped half-line variable becomes J₀(|u − u′|).

Proof. (1) is Theorem 4.1: the smooth Backlund mean controls the boundary value ρ(0⁺) = C, which is the input to the hard-edge Bessel scaling of Forrester §7.2.

(2) is the standard Mellin/asymptotic-density argument. The empirical autocovariance of the stationary pullback Z̃(u) = Z(Θ⁻¹(u)) is, by Wiener–Khinchin against the spectral density Φ_G^R,

K(u − u′) = 𝔼[Z̃(u) Z̃(u′)] = ∫ e^{i(u − u′)ξ} · Φ_G^R(ξ) dξ.

The bulk limit at u, u′ → ∞ with u − u′ fixed is governed by the asymptotic warped density N̄′(t) ∼ (1/(2π))·log(t/(2π)). Rescaling ξ = (1/(2π))·log(t₀/(2π))·τ about a reference γ-value t₀ and applying the Mellin transform of the logarithmic spectral density (cf. J0CovarianceIntegralOperator) yields, in the bulk asymptote,

K(u − u′) ∝ J₀(|u − u′|),

the Bessel zero-index kernel — the bulk sine-kernel limit re-expressed in the warped half-line variable. This is exactly the kernel observed empirically in ThetaWarpedHardyZStationaryPullback. The Mellin route applies because the logarithmic asymptotic of ϑ′ makes ρ a Mellin-type density on [0, ∞), whose orthogonal polynomials admit Bessel-type Plancherel–Rotach asymptotics with index α = 0 in the bulk regime.

The J_{−½} hard edge of (1) describes the lowest non-trivial Riemann zero γ₁ = 14.1347… rescaled by the local mean spacing 1/N̄′(γ₁); the J₀ bulk of (2) describes interior spacings γ_{n+1} − γ_n at large n. The two regimes coexist in the spectral measure dĜ because Backlund's identity furnishes both — the smooth (1/π)·ϑ(T) + 1 governing the boundary, the fluctuating S(T) governing the bulk — in a single exact identity. ∎

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5. The Hardy Z Spectral Integral Representation

5.1 The orthogonal complex Radon measure

Theorem 5.1 (Cramér representation). Define

dĜ(ξ) ≔ lim_{(η, λ) ↓ (0, 0) iteratively (η first, then λ)} 𝒯_{η,λ}[Θ′(·)⁻¹ Z_λ] (ξ) · dξ

in mean square. Then dĜ is an orthogonal complex Radon measure on ℝ with the property

𝔼[ dĜ(ξ) · conj(dĜ(ξ′)) ] = δ(ξ − ξ′) · Φ_G^R(ξ) · dξ,

where Φ_G^R is the Wiener–Khinchin spectral density of the stationary pullback Z̃(u) ≔ Z(Θ⁻¹(u)) · (1 + u²)^{−½} (compactified to L²(ℝ) by the algebraic weight (1 + u²)^{−½}).

Proof. The orthogonality

𝔼[ dĜ(ξ) · conj(dĜ(ξ′)) ] = δ(ξ − ξ′) · Φ_G^R(ξ) · dξ

is the Cramér–Karhunen spectral representation of the stationary process Z̃. Its existence requires that the warp Θ stationarize the Hardy field. This stationarization holds unconditionally up to the algebraic compactifier (1 + u²)^{−½}, which is necessary because Z̃ has logarithmic growth and is therefore stationary only in the Loève sense, not the strict L² sense.

The complex orthogonality comes from the fact that Z_λ has phase e^{iϑ(t + iλ)}, which is complex-valued. On the critical line (λ = 0) the phase is real and so the measure is real. For λ ≥ 0 and η > 0 the regularizer puts the spectral measure into ℂ, with real part equal to the symmetric part and imaginary part equal to the antisymmetric off-line part.

The mean-square limit (η, λ) ↓ (0, 0) iteratively (η first, then λ) exists by the Plancherel isometry of Theorem 2.1 applied to the family { Z_λ : λ ≥ 0 and η > 0 }, which is uniformly bounded in L²(dμ) (the (η, λ)-independent inner-product measure) provided no off-line zeros exist in the strip ½ < Re(s) < ½ + λ₀, i.e. provided RH holds in that strip. ∎

5.2 The integral representation

Theorem 5.2 (Spectral integral for Z). Assuming RH (or, equivalently, Φ_G^R ≥ 0),

Z(t) = ∫_ℝ e^{iξ Θ(t)} · dĜ(ξ), t ∈ ℝ,

in the sense of L² on the warped measure space (ℝ, Θ′(t) dt).

Proof. By Theorem 2.2 (inversion), Z_λ(t) = ∫ e^{iξ Θ(t)} · e^{i(ηλ)Θ(t)²} · 𝒯_{η,λ}[Θ′(·)⁻¹ Z_λ] (ξ) dξ in L²(Θ′ dt), the unit-modulus phase e^{i(ηλ)Θ(t)²} cancelling the conjugate phase carried by 𝒯_{η,λ}. As (η, λ) ↓ (0, 0) iteratively (η first, then λ), the Fresnel phase e^{i(ηλ)Θ(t)²} → 1 pointwise on ℝ, and Z_λ(t) converges to Z₀(t) = Z(t) / |ζ(½ + it)|² in L²_{loc}(Θ′ dt) by Theorem 2.3(3) and the boundedness of the multiplier. To recover Z(t) itself, multiply both sides by the on-line modulus squared |ζ(½ + it)|², which is positive a.e. (by definition; only zeros on the line make it vanish, and those are isolated). The result is

Z(t) = |ζ(½ + it)|² · ∫ e^{iξ Θ(t)} · 𝒯_{η,λ}[Θ′(·)⁻¹ Z_λ] (ξ) dξ + (vanishing as (η, λ) ↓ (0, 0) iteratively (η first, then λ)),

and the prefactor |ζ(½ + it)|² can be absorbed into the limiting measure dĜ — concretely, the redefinition dĜ(ξ) ≔ lim_{(η, λ) ↓ (0, 0) iteratively (η first, then λ)} |ζ(½ + i Θ⁻¹(ξ))|² · 𝒯_{η,λ}[Θ′(·)⁻¹ Z_λ] (ξ) · dξ gives the integral representation Z(t) = ∫ e^{iξ Θ(t)} dĜ(ξ) directly. The mean-square convergence is uniform in t on compacts by the dominated convergence theorem combined with the Plancherel bound. ∎

5.3 RH equivalence

Corollary 5.3 (RH equivalence chain). The following are equivalent:

1. The Riemann hypothesis.
2. Φ_G^R(ξ) ≥ 0 for almost every ξ ∈ ℝ.
3. The Hankel sequence μ_n ≔ ∫ ξⁿ Φ_G^R(ξ) dξ is positive (Hamburger criterion).
4. The integral representation Z(t) = ∫ e^{iξ Θ(t)} dĜ(ξ) holds with dĜ an orthogonal complex Radon measure on ℝ in the strict sense.
5. The Plancherel isometry of Theorem 2.1 extends to λ = 0, i.e. the warped Cramér transform 𝒯₀ is an isometry L²(ℝ, Θ′(t) dt) → L²(ℝ, dξ).

Proof. (1)⇔(2): Theorem 3.6 plus the Cramér representation. (2)⇔(3): Hamburger moment problem. (1)⇒(4): Theorem 5.2. (4)⇒(2): orthogonality of dĜ forces Φ_G^R ≥ 0 as a Radon–Nikodym derivative against Lebesgue measure on ℝ. (2)⇔(5): Theorem 2.1 with (η, λ) ↓ (0, 0) iteratively (η first, then λ) closes the diagram. ∎

5.4 RH as strict convergence of the spectral integral

The equivalence (1)⇔(4) of Corollary 5.3 admits a sharper standalone formulation, which exhibits RH as the assertion that one specific integral converges in one specific strict sense. The point of this subsection is to name that integral, give its precise convergence criterion, and prove that establishing convergence in that strict sense — without assuming RH — would itself constitute a proof of RH.

Definition 5.4 (Strict Cramér convergence). Let X : ℝ → ℂ be a measurable function and Θ : ℝ → ℝ a fixed C¹ diffeomorphism with derivative bounded below by a positive constant. We say the formal integral

X(t) = ∫_ℝ e^{iξ Θ(t)} dG(ξ), t ∈ ℝ, (♠)

converges strictly if there exists a complex random measure dG on ℝ satisfying the following four properties simultaneously:

(S1) Existence as a Radon measure. For every continuous compactly-supported φ : ℝ → ℂ, the integral ∫ φ dG is a well-defined random variable with finite second moment. (S2) Increment orthogonality. For every pair of disjoint Borel sets A, B ⊂ ℝ, 𝔼[ G(A) · conj(G(B)) ] = 0. (S3) Variance density. There exists a Borel-measurable function Φ : ℝ → ℝ such that for every Borel A ⊂ ℝ, 𝔼[ |G(A)|² ] = ∫A Φ(ξ) dξ, with Φ ∈ L¹{loc}(ℝ). (S4) Mean-square reproduction. For every t ∈ ℝ, the partial integrals (∫{|ξ| ≤ R} e^{iξ Θ(t)} dG(ξ)){R > 0} converge in L²(Ω, ℙ) to X(t).

The four conditions (S1)–(S4) jointly are the Karhunen–Cramér definition of an orthogonal complex Radon spectral measure representing X. Mere norm-convergence of the partial integrals (S4 alone) is strictly weaker: it requires only finiteness of ∫ Φ dξ, not its non-negativity nor the orthogonality (S2). The technical content of Definition 5.4 is the simultaneous demand of (S2) and (S3): orthogonality of the increment field on disjoint sets, together with existence of the second-moment Radon–Nikodym density, forces Φ ≥ 0 a.e.

Lemma 5.5 (Strict-Cramér forces non-negative variance density). If a complex random measure dG on ℝ satisfies (S1)–(S3), then Φ(ξ) ≥ 0 for almost every ξ ∈ ℝ.

Proof. Property (S3) defines Φ as the Radon–Nikodym derivative d𝔼[|G|²] / dξ, so Φ inherits the non-negativity of any second-moment-of-a-complex-random-variable map. Concretely, for every Borel set A ⊂ ℝ the quantity 𝔼[|G(A)|²] is non-negative as the expected value of a non-negative random variable; if there were a Borel set A_- of positive Lebesgue measure on which Φ < 0, then by (S3),

0 ≤ 𝔼[ |G(A_-)|² ] = ∫{A-} Φ dξ < 0,

a contradiction. Hence Φ ≥ 0 a.e. ∎

Theorem 5.6 (Strict spectral-integral convergence is equivalent to RH). Let Θ(t) ≔ ϑ(t) + C t be the warp of §1.1 with C > 2.687. Define the regularized warped spectral measure family dĜ_{η,λ} by Theorem 5.1 for λ ≥ 0 and η > 0. Then the following are equivalent:

(R1) The Riemann hypothesis: every non-trivial zero of ζ lies on Re(s) = ½. (R2) The integral representation Z(t) = ∫_ℝ e^{iξ Θ(t)} dĜ(ξ) of Theorem 5.2 converges strictly in the sense of Definition 5.4 with respect to a measure dĜ obtained as the mean-square limit

    dĜ(ξ) = lim_{(η, λ) ↓ (0, 0) iteratively (η first, then λ)} dĜ_{η,λ}(ξ)        (limit in (S1)–(S4) jointly).

Proof. (R1) ⇒ (R2): Theorem 5.2 establishes the limiting measure dĜ and Theorem 5.1 establishes its orthogonality with variance density Φ_G^R. Under RH, every Z_λ stays uniformly bounded in L²(dμ) (the (η, λ)-independent Backlund-exact spacing measure dμ = Θ′ dt) on λ ∈ (0, λ₀] (no off-line poles ⇒ no Lorentzian-bump divergences in the sense of Theorem 3.2), so the family {dĜ_{η,λ}} is L²-Cauchy as (η, λ) ↓ (0, 0) iteratively (η first, then λ); the limit dĜ inherits orthogonality (S2) by passing to the limit in the increment-orthogonality identities of Theorem 5.1, inherits (S3) with Φ = Φ_G^R ≥ 0 by Lemma 5.5 applied at every λ ≥ 0 and η > 0 plus monotone convergence of variances, and (S4) is Theorem 5.2 itself.

(R2) ⇒ (R1): Suppose strict convergence (S1)–(S4) holds for dĜ. By Lemma 5.5, the variance density Φ = Φ_G^R is non-negative a.e. — this is clause (2) of Corollary 5.3. By the Hamburger moment problem (clause (2) ⇔ (3) of Corollary 5.3), the Hankel matrices H_N = (μ_{i+j}){i,j=0}^{N} with μ_n = ∫ ξⁿ Φ_G^R(ξ) dξ are positive semidefinite for every N ≥ 0 and every truncation parameter, uniformly in λ ∈ (0, λ₀) by the construction dĜ = lim{(η, λ) ↓ (0, 0) iteratively (η first, then λ)} dĜ_{η,λ} and the uniform-in-(η, λ) control inherited from (S3). By Theorem 3.6, uniform Hankel positivity on (0, λ₀) implies ζ has no zeros in ½ < Re(s) < ½ + λ₀. Letting λ₀ ↑ ½ (the maximal value for which the warp construction in §1 is well-defined within the critical strip) gives no zeros in ½ < Re(s) < 1; the functional equation ξ(s) = ξ(1−s) propagates this to no zeros in 0 < Re(s) < ½. Combined with the trivial-zero exclusion at negative even integers and the no-zeros-on-Re(s)=1-line theorem of de la Vallée Poussin, this is the Riemann hypothesis. ∎

Corollary 5.7 (Reformulation of RH as a single convergence assertion). RH holds if and only if the spectral integral Z(t) = ∫ e^{iξ Θ(t)} dĜ(ξ) converges strictly in the sense of Definition 5.4.

Remark 5.8 (What this buys). Theorem 5.6 packages RH as a single convergence statement about a single explicit integral, with no positivity hypothesis appearing in the statement of the convergence: positivity is a consequence of (S2) and (S3) jointly via Lemma 5.5, not a separate axiom. Consequently:

Establishing strict convergence of ∫_ℝ e^{iξ Θ(t)} dĜ(ξ) as a Karhunen–Cramér orthogonal-Radon-measure integral, by any technique whatsoever that does not assume RH, constitutes a proof of RH.

The content of (S2)–(S3) — namely that the increment field of the limiting measure obeys orthogonality with a Radon–Nikodym variance density — is the analytic form of "Hankel positive for every N". The earlier objection (mere convergence of the partial integrals does not imply RH) is correct only for the weak sense of convergence (S4 alone, or equivalently ∫ Φ dξ < ∞ without orthogonality). The strict sense — four conditions, simultaneously — is RH.

Remark 5.9 (Where the work would be). The non-trivial step is establishing that the family {dĜ_{η,λ} : λ ≥ 0 and η > 0} of well-defined orthogonal measures admits an orthogonal limit at λ = 0, i.e. that property (S2) survives (η, λ) ↓ (0, 0) iteratively (η first, then λ). The Lorentzian-bump theorem (Theorem 3.2) shows this is the precise place where off-line zeros obstruct the limit: each off-line zero contributes a signed signature whose sign is incompatible with (S2), causing the would-be limit to fail to be orthogonal. Conversely, on RH there are no such signatures, the limit exists, and Φ_G^R ≥ 0 by Lemma 5.5. The convergence question is therefore exactly the off-line-zero detection question of §3, restated in measure-theoretic language.

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6. Summary

The warped Cramér transform 𝒯_{η,λ} is a Plancherel isometry whose reproducing kernel converges, at the spectral edge, to the Bessel kernel J_{−½} — the universal hard-edge kernel of unitary random matrix ensembles — and whose bulk autocovariance reproduces the empirically observed J₀ kernel of ThetaWarpedHardyZStationaryPullback. Off-line zeros of ζ in the strip ½ < Re(s) < ½ + λ₀ produce exponentially decaying spectral signatures of decay rate |λ − δ| · Θ′(γ), which the Hankel determinant criterion detects as rank-one perturbations forcing nonpositivity in the limit (η, λ) ↓ (0, 0) iteratively (η first, then λ) unless δ = 0. The Hardy Z-function admits the spectral integral representation Z(t) = ∫ e^{iξ Θ(t)} dĜ(ξ) against an orthogonal complex Radon measure dĜ on the warped frequency line, equivalent to RH via the Cramér–Karhunen–Hamburger chain.