ThetaWarpedHardyZStationaryPullback - crowlogic/arb4j GitHub Wiki

Θ-Warped Hardy Z Stationary Pullback and the J₀ Autocovariance

Notes on the wide-sense stationary structure of the Θ-warped Hardy Z function, the chirp-derivation of the $J_0$ autocovariance kernel, the closed-form two-parameter Bessel-style regularizer that makes the limit converge, and the conjectural Cramér orthogonal measure on the spectral interval $[-1,1]$.

1. Definitions and Conventions

1.1 The phase and the warp

Let $\vartheta$ be the RiemannSiegelThetaFunction:

$$\vartheta(t) = \text{Im} \log\Gamma\left(\tfrac{1}{4} + \tfrac{it}{2}\right) - \tfrac{t}{2}\log\pi.$$

It is odd, real-analytic on $\mathbb{R}$, with $\vartheta'(0) = -2.6860917\ldots$ (the minimum of $\vartheta'$ on $\mathbb{R}$) and $\vartheta'(t) \to +\infty$ as $|t| \to \infty$, with the asymptotic expansion

$$\vartheta'(t) = \tfrac{1}{2}\log\left(\tfrac{t}{2\pi}\right) + \tfrac{1}{8t} + O(t^{-3}). \qquad (1.1)$$

Fix $C > 2.687$ so that $\Theta(t) := \vartheta(t) + Ct$ satisfies $\Theta'(t) > 0$ for all $t \in \mathbb{R}$. Then $\Theta : \mathbb{R} \to \mathbb{R}$ is a real-analytic odd diffeomorphism. Its inverse is $t = \Theta^{-1}(u)$.

1.2 The two pullback processes

Define

$$Y(u) = Z\left(\Theta^{-1}(u)\right), \qquad \zeta_\Theta(u) = \zeta\left(\tfrac{1}{2} + i\Theta^{-1}(u)\right). \qquad (1.2)$$

Using $Z(t) = e^{i\vartheta(t)}\zeta(\tfrac12 + it)$ and $\Theta(t) = \vartheta(t) + Ct$ so $\vartheta(\Theta^{-1}(u)) = u - C\Theta^{-1}(u)$:

$$\zeta_\Theta(u) = e^{-i\vartheta(\Theta^{-1}(u))} Y(u) = e^{-i(u - C\Theta^{-1}(u))} Y(u). \qquad (1.3)$$

When $C = 0$ on the half-line where $\vartheta$ is monotone (i.e., $|t| > 6.29$), this collapses to $\zeta_\Theta(u) = e^{-iu} Y(u)$. The two processes carry the same information; whether one computes from $Z$ or from $\zeta$ amounts to multiplication by the unimodular carrier $e^{-iu}$ on the warped axis.

1.3 Time-averaged autocovariance

For a locally bounded measurable function $X : [t_0, \infty) \to \mathbb{C}$, define

$$R_X(s) := \lim_{T \to \infty} \frac{1}{T - t_0}\int_{t_0}^{T} X(t + s) \overline{X(t)} dt \qquad (1.4)$$

whenever the limit exists for every $s \in \mathbb{R}$. The function $X$ is wide-sense stationary in the Cesàro sense if (1.4) exists and is continuous in $s$. The lower limit $t_0$ is fixed (say $t_0 = 14$, beyond the first Riemann zero), and $R_X$ is independent of $t_0$ when the limit exists.

This is a deterministic Hilbert-space definition — no probability measure required. The Cramér representation theorem then asserts the existence of an orthogonal complex Borel measure $\Phi_X$ on $\mathbb{R}$ with $X(u) = \int e^{i\xi u} d\Phi_X(\xi)$ and $\mathbb{E}|d\Phi_X(\xi)|^2 = dF_X(\xi)$ where $R_X(s) = \int e^{i\xi s} dF_X(\xi)$.

2. The Empirical Observation

The input to the original computation (S. Crowley, c. 2023) was nothing more than the Hardy $Z$ function itself — no warp, no preconditioning, no auxiliary process. On a compact window $[a, b] \subset (0, \infty)$ of finite length, form the windowed autocovariance

$$\widehat R_{[a,b]}(s) := \frac{1}{b - a}\int_a^{b - s} Z(t + s) Z(t) dt, \qquad s \in [0, b - a]. \qquad (2.1)$$

Normalize by dividing through by $\widehat R_{[a,b]}(0)$. Two empirical facts came out, and both are crucial.

2.1 Finite-window translation-stability of the normalized $Z$ kernel

The input was $Z$ itself — not $Y$, not any pullback. The numerics computed $\widehat R_{[a,b]}(s)/\widehat R_{[a,b]}(0)$ on a sequence of windows whose centers ranged across $[10^2, 10^4]$ (e.g.\ $[100, 200]$, $[1000, 2000]$, $[5000, 6000]$, $[9000, 10,000]$, and similar). Across this range, the normalized kernel came out to the same curve, with the same constants, up to a finite-window discrepancy that decreased as the window length increased.

This is not exact translation-invariance. Exact translation-invariance of the windowed normalized kernel — i.e.\

$$\frac{\widehat R_{[a, b]}(s)}{\widehat R_{[a, b]}(0)} = \frac{\widehat R_{[a + h, b + h]}(s)}{\widehat R_{[a + h, b + h]}(0)} \quad\text{for all } h \qquad (\text{NOT THIS}) \qquad (2.2)$$

is impossible at finite window length: it would require an infinitely wide window. What the numerics show is the weaker, but more important, statement

$$\frac{\widehat R_{[a, b]}(s)}{\widehat R_{[a, b]}(0)} = \rho(s) + \varepsilon([a,b], s), \qquad \sup_{|s| \leq S} |\varepsilon([a,b], s)| \xrightarrow[(b - a) \to \infty]{} 0, \qquad (2.3)$$

where $\varepsilon([a, b], s)$ is the finite-window discrepancy and $\rho$ is a fixed function of $s$ alone. Empirically, $\varepsilon$ was small enough on windows of length $\sim 100$ in $[10^2, 10^4]$ that the curves visually coincided to high precision.

This finite-window translation-stability of the normalized $Z$-kernel is the empirical fingerprint of an underlying exact stationarity that lives, not on the $t$-axis, but on the warped axis. The central hypothesis of these notes is the following.

Hypothesis (WSS of the pullback). The pullback

$$Y(u) := Z(\Theta^{-1}(u)) \qquad (2.4)$$

along the shifted Riemann–Siegel theta function $\Theta(t) = \vartheta(t) + Ct$ is wide-sense stationary in the Cesaro sense of §1.3. That is: the limit

$$R_Y(s) = \lim_{U \to \infty} \frac{1}{U}\int_0^U Y(u + s) Y(u) du \qquad (2.5)$$

exists for every $s \in \mathbb{R}$ and is continuous in $s$.

Wide-sense stationarity is a binary property: either (2.5) exists and defines a continuous positive-definite $R_Y$, or it does not. There is no "approximately." The hypothesis asserts that the yes case holds. Section 3 supplies a heuristic derivation via the chirp–Mellin–band-edge calculation; section 6 states the resulting Cramér representation. A rigorous proof is a separate matter — it is open — but the proposition itself is sharp.

The connection to the empirical observation on $Z$. If the hypothesis holds, the inverse warp $u \mapsto t = \Theta^{-1}(u)$ is real-analytic, monotone, and bilipschitz on any compact subinterval of $(0, \infty)$, with Jacobian $\Theta'(t) = \vartheta'(t) + C$ that varies slowly in $t$ — specifically $\Theta'(t) = \tfrac{1}{2}\log(t/2\pi) + C + O(1/t)$. Pushing $R_Y$ back to the $t$-axis through this warp produces precisely the finite-window phenomenon recorded in (2.3): a normalized kernel that is the same curve up to a discrepancy controlled by the relative variation of $\Theta'$ across $[a, b]$, which tends to zero as $(b-a) \to \infty$ at fixed scale (because $\Theta'(b)/\Theta'(a) \to 1$ when $a, b$ both grow with bounded ratio).

So:

Hypothesized: $Y$ is WSS on the $u$-axis. Binary; no fudge factor.
Empirically observed (c. 2023, $Z$-axis numerics): the normalized windowed autocovariance of $Z$ is translation-stable across windows in $[10^2, 10^4]$ up to a discrepancy $\varepsilon$ that shrinks with window length. This is not a statement that $Z$ is WSS — $Z$ is not, since its variance grows like $\log T$ — but it is exactly the fingerprint that the WSS hypothesis on $Y$ predicts after pulling back through the warp.
What is open: a proof of (2.5), and a proof that the resulting $R_Y$ is the kernel identified in §2.2.

2.2 The kernel shape is $J_0$

The limit kernel $\rho(s)$ was carefully normalized and compared to the Bessel function $J_0$. Its zero set matched the rescaled zero set of $J_0$ to six to seven decimal places, simultaneously across multiple zeros. The agreement was not a single-zero coincidence: the entire low-lying zero pattern of $\rho(s)$ tracked the rescaled zeros of $J_0$ to that precision. The recorded features are:

(i) First zero of $\rho$ at $s_* = j_{0,1}/2 \approx 1.2024$, where $j_{0,1} \approx 2.4048256$ is the first positive zero of $J_0$.

(ii) Subsequent zeros at $j_{0,k}/2$ for the first several $k$, matching $J_0$'s zero pattern to six to seven decimals.

(iii) Magnitude scale of the unnormalized peak amplitude approximately $\gamma_1/2 \approx 7.067$, where $\gamma_1 = 14.1347\ldots$ is the imaginary part of the first nontrivial zero of $\zeta$.

(iv) Logarithmic divergence at zero of the unnormalized covariance, $\widehat R_T(0) \sim \log(T/2\pi)$, consistent with the Hardy–Littlewood mean-square theorem (4.1) below. The growth is in the amplitude only; it does not destroy the kernel shape after normalization.

2.3 Why this is decisive

The combination is what gives the observation its weight:

Translation stability across $[10^2, 10^4]$ is the empirical signature of stationarity: the kernel is a property of the function, not of the window. A spurious or transient feature would not survive translation by two orders of magnitude.
Six-to-seven-digit zero-set agreement with $J_0$ across multiple zeros is the empirical signature of an identification, not a shape match. Random damped oscillations do not land on $j_{0,k}/2$ to that precision for several $k$ simultaneously.

Taken together, these two facts make $\rho(s) = J_0(2s)$ (after the chirp-slope correction explained in §3) the working hypothesis with very high confidence. The kernel form is therefore taken as

$$\boxed{; \rho(s) = J_0(2s) ;}$$

with the only outstanding items being:

Independent re-derivation of those numerics on the current arb4j framework (the original computation, performed roughly three years ago, is being re-run; the symbolic infrastructure for it is currently being dusted off).
A first-principles proof that $\rho(s) = J_0(2s)$ exactly (up to constants and arithmetic factors), rather than merely matching it to seven digits across the sampled $s$-range. The chirp–Mellin–band-edge mechanism in §3 produces this form heuristically; converting it to a rigorous theorem requires controlling the $Z_{\geq 2}$ contribution from (3.2) and the boundary subtraction in (3.8).

The warped process $Y(u)$ on the $u$-axis is the natural setting in which the same $\rho(s)$ appears without the $\log T$ amplitude growth: the chirp slope $\vartheta'(t) \sim \tfrac12\log(t/2\pi)$ becomes the constant $1$ on the $u$-axis after warping by $\Theta$, so the warp absorbs the slowly-varying instantaneous frequency, leaving an honestly stationary process whose autocovariance is $\rho$.

3. The Chirp Calculation (Detailed)

3.1 The Riemann–Siegel main term

The Riemann–Siegel formula gives, for $t > 0$,

$$Z(t) = 2\sum_{n=1}^{N(t)} \frac{\cos(\vartheta(t) - t\log n)}{\sqrt{n}} + R(t), \qquad N(t) = \lfloor\sqrt{t/(2\pi)}\rfloor, \qquad (3.1)$$

with remainder $|R(t)| = O(t^{-1/4})$. In particular, the leading $n=1$ term dominates pointwise:

$$Z(t) = 2\cos\vartheta(t) + Z_{\geq 2}(t), \qquad Z_{\geq 2}(t) := 2\sum_{n=2}^{N(t)} \frac{\cos(\vartheta(t) - t\log n)}{\sqrt{n}} + R(t). \qquad (3.2)$$

The sum $Z_{\geq 2}$ is bounded in mean square by $\sum_{n=2}^{N(t)} 1/n \sim \tfrac12\log\log(t/2\pi)$ — a $\log\log$ correction to the leading $\log$.

3.2 The leading autocovariance

Set $X_1(t) := 2\cos\vartheta(t)$. Then

$$X_1(t+s) X_1(t) = 2\cos(\vartheta(t+s) - \vartheta(t)) + 2\cos(\vartheta(t+s) + \vartheta(t)). \qquad (3.3)$$

The second term has phase $\vartheta(t+s) + \vartheta(t) = 2\vartheta(t) + s\vartheta'(t) + O(s^2/t)$, growing like $t\log t$. By the standard non-stationary-phase estimate,

$$\frac{1}{T}\int_{t_0}^T \cos(2\vartheta(t) + s\vartheta'(t)) dt = O\left(\frac{1}{(\log T)^{1/2}}\right) \to 0 \qquad (3.4)$$

uniformly for $s$ in any compact interval. (The phase derivative is $2\vartheta'(t) + s\vartheta''(t) \sim \log(t/2\pi)$, bounded below by a positive constant for $t$ large, giving the $1/\log T$ decay after one integration by parts.)

The first term has phase $\Delta(t,s) := \vartheta(t+s) - \vartheta(t)$. Taylor-expanding in $s$:

$$\Delta(t,s) = s\vartheta'(t) + \tfrac{s^2}{2}\vartheta''(t) + O\left(\tfrac{s^3}{t^2}\right). \qquad (3.5)$$

Since $\vartheta''(t) = O(1/t)$, the second-order correction is $O(s^2/t)$, which vanishes in the time average for fixed $s$ as $T \to \infty$. So to leading order,

$$\frac{1}{T}\int_{t_0}^T 2\cos\Delta(t, s) dt = \frac{2}{T}\int_{t_0}^T \cos(s\vartheta'(t)) dt + O(s^2/\log T). \qquad (3.6)$$

3.3 The Mellin substitution

Substitute $v = \vartheta'(t)$. From (1.1), $v = \tfrac12\log(t/(2\pi)) + O(1/t)$, so $t = 2\pi e^{2v}(1 + O(e^{-2v}))$ and $dt/dv = 4\pi e^{2v}(1 + O(e^{-2v}))$. The bounds $t_0 \leq t \leq T$ become $v_0 \leq v \leq V$ with $v_0 = \vartheta'(t_0)$ and $V = \tfrac12\log(T/2\pi) + O(1/T)$. Thus

$$\frac{1}{T}\int_{t_0}^T \cos(s\vartheta'(t)) dt = \frac{1}{T}\int_{v_0}^V \cos(sv)\cdot 4\pi e^{2v} dv + O(1/T). \qquad (3.7)$$

The inner integral evaluates exactly:

$$\int_{v_0}^V \cos(sv)\cdot 4\pi e^{2v} dv = 4\pi\cdot\frac{e^{2V}(2\cos(sV) + s\sin(sV)) - e^{2v_0}(2\cos(sv_0) + s\sin(sv_0))}{4 + s^2}. \qquad (3.8)$$

Using $4\pi e^{2V} = T(1 + O(1/T))$ and dividing by $T$:

$$\widehat R_T^{(1)}(s) := \frac{1}{T}\int_{t_0}^T 2\cos(s\vartheta'(t)) dt = \frac{2(2\cos(sV) + s\sin(sV))}{4 + s^2} + O\left(\tfrac{1}{T}\right). \qquad (3.9)$$

This is the leading-order autocovariance from the $n=1$ Riemann–Siegel term, unrenormalized. The $s = 0$ value of the closed-form expression equals $1$, which is inconsistent with the direct mean $\overline{X_1^2} = \overline{4\cos^2\vartheta} = 2$. The discrepancy is a boundary-contribution artifact: the substitution $v = \vartheta'(t)$ is not uniformly valid near $t = t_0$, where $\vartheta'$ is small or negative, and the $e^{2v_0}$ subtraction in (3.8) is not negligible relative to the leading $e^{2V}$ term once one looks at $s = 0$. A careful match of the boundary terms restores $R_T^{(1)}(0) \to 2$. For the present purpose only the $s$-dependence of the kernel matters; the boundary discrepancy enters as an additive constant absorbed into renormalization.

3.4 The Mellin-renormalized autocovariance

The natural normalization treats $\widehat R_T(s)$ as growing like $\log T$ (from the $s = 0$ value $\sim \log(T/2\pi)$) and extracts the kernel as

$$\rho(s) := \lim_{T \to \infty} \frac{\widehat R_T(s)}{\widehat R_T(0)}. \qquad (3.10)$$

For $X_1(t) = 2\cos\vartheta(t)$, $\widehat R_T^{(1)}(0) = 2 + O(1/\log T)$ (since $\overline{\cos^2\vartheta} = 1/2$ on average and the prefactor is $4$). So $\rho^{(1)}(s)$ is the limit of (3.9) divided by $2$:

$$\rho^{(1)}(s) = \lim_{T\to\infty}\frac{2\cos(sV) + s\sin(sV)}{4 + s^2}. \qquad (3.11)$$

But $V = \tfrac12\log(T/2\pi) \to \infty$, so $\cos(sV)$ and $\sin(sV)$ oscillate without limit in the ordinary sense. The limit (3.11) does not exist as a pointwise limit; it exists only in the distributional sense, and its value is read off via Cesàro averaging in $V$ or via Riemann–Lebesgue:

$$\frac{1}{V_2 - V_1}\int_{V_1}^{V_2}\frac{2\cos(sV) + s\sin(sV)}{4+s^2} dV \to 0 \quad\text{as } V_1, V_2 \to \infty\text{ for } s \neq 0. \qquad (3.12)$$

So the unregularized $X_1$-autocovariance limit is $\rho^{(1)}(s) = \delta_{s,0}$ in the Cesàro-distributional sense — pure white noise on the $u$-axis. This is not what the empirical $J_0$ kernel looks like.

3.5 Where the $J_0$ comes from: the warped axis

The fix is to do the autocovariance on the warped $u$-axis, not the $t$-axis. Define the warped chirp

$$\widetilde X_1(u) := X_1(\Theta^{-1}(u)) = 2\cos\vartheta(\Theta^{-1}(u)). \qquad (3.13)$$

The autocovariance on the $u$-axis is

$$\widehat R^Y_U(s) = \frac{1}{U}\int_0^U \widetilde X_1(u + s)\widetilde X_1(u) du. \qquad (3.14)$$

Pulling back via $u = \Theta(t)$, $du = \Theta'(t) dt$:

$$\widehat R^Y_U(s) = \frac{1}{U}\int_{t_0(U)}^{T(U)} 2\cos\left(\vartheta(\Theta^{-1}(\Theta(t)+s)) - \vartheta(t)\right)\Theta'(t) dt + O(1/\log U) \qquad (3.15)$$

where $T(U) = \Theta^{-1}(U)$ and the $O(1/\log U)$ swallows the rapidly-oscillating second cosine.

Now $\Theta^{-1}(\Theta(t) + s) = t + s/\Theta'(t) + O(s^2/(\Theta'(t)^2 t))$, so

$$\vartheta(\Theta^{-1}(\Theta(t)+s)) - \vartheta(t) = \vartheta'(t)\cdot \frac{s}{\Theta'(t)} + O\left(\tfrac{s^2}{t(\log t)^2}\right) = \frac{s\vartheta'(t)}{\vartheta'(t)+C} + O\left(\tfrac{s^2}{t(\log t)^2}\right). \qquad (3.16)$$

As $t \to \infty$, $\vartheta'(t)/(\vartheta'(t)+C) \to 1$, so the phase tends to $s$, independent of $t$. This is the stationarity miracle of the warp: the chirp's logarithmically-varying instantaneous frequency $\vartheta'(t)$ becomes the constant unit frequency on the $u$-axis after warping.

3.6 Spectral support and conjectural kernel form

To identify the kernel one substitutes the band-edge variable $\xi := \vartheta'(t)/(\vartheta'(t)+C) \in (0, 1)$. From $\vartheta'(t) = C\xi/(1-\xi)$ one gets $d\xi/dt = C\vartheta''(t)/(\vartheta'(t)+C)^2$. Using $\vartheta''(t) = 1/(2t) + O(1/t^3)$ and $\vartheta'(t) \sim \tfrac12\log(t/2\pi)$, one obtains a measure $d\nu(\xi)$ on $(0,1)$ with density

$$\frac{d\nu}{d\xi}(\xi) = \frac{(\vartheta'(t)+C)^2}{C\vartheta''(t)}\Bigg|_{\vartheta'(t) = C\xi/(1-\xi)} \sim \frac{2C^2 t}{(1-\xi)^2}\cdot\frac{1}{C}\cdot 2t \sim \frac{4C t^2}{(1-\xi)^2}, \qquad (3.17)$$

which when integrated against $\cos(s\xi)$ on $(0, 1)$ produces (after the relation between $V$ and $T$) a kernel of the form

$$K(s) = \int_0^1 \cos(s\xi) w(\xi) d\xi \qquad (3.18)$$

for some weight $w$ supported on $[0,1]$. The integral representation

$$J_0(s) = \frac{2}{\pi}\int_0^1 \frac{\cos(s\xi)}{\sqrt{1 - \xi^2}} d\xi \qquad (3.19)$$

is precisely of this form with $w(\xi) = (2/\pi)/\sqrt{1-\xi^2}$. The empirical observation that the autocovariance kernel matches $J_0$ is consistent with $w(\xi) \sim c/\sqrt{1-\xi^2}$ near $\xi = 1$, i.e., a square-root singularity at the band edge $\xi = 1$. This is the universality class of BesselFunctionOfTheFirstKind kernels arising from Hankel-transform spectral decompositions on $(0, 1)$ — see J0CovarianceIntegralOperator.

3.7 The chirp-slope factor and the $1/2$

The argument scaling factor inside the warped kernel is not $s$ but $s\cdot\vartheta'(t)/(\vartheta'(t)+C)$ at the location $t$ that contributes the spectral mass at edge frequency $\xi$. As $t \to \infty$, $\vartheta'(t) \to \infty$ and $\vartheta'(t)/(\vartheta'(t)+C) \to 1$, so the outer edge of the spectral support sits at $\xi = 1$.

The critical observation is that the chirp slope is $1/2$, not $1$:

$$\vartheta'(t) = \tfrac{1}{2}\log(t/2\pi) + O(1/t).$$

The Mellin substitution of §3.3 maps $t \mapsto v = \vartheta'(t)$ with Jacobian $dt/dv = 4\pi e^{2v}$ — the factor $2$ in the exponent is exactly the inverse of the chirp slope $1/2$. When this Jacobian is rolled through the band-edge change of variables (3.17)–(3.18), the kernel acquires argument $2s$ rather than $s$:

$$K(s) = c_1 \int_0^1 \frac{\cos(2 s\xi)}{\sqrt{1 - \xi^2}} d\xi = \tfrac{\pi c_1}{2} J_0(2 s). \qquad (3.20)$$

The first zero of $K$ then sits at $2 s_* = j_{0,1}$, i.e., $s_* = j_{0,1}/2 \approx 1.2024$ — matching the empirical location.

Status of the kernel form.

Item	Status
Empirical match $K(s) \propto J_0(2s)$ to $\sim 6$–$7$ decimal digits across multiple zeros	Numerical observation (Crowley c. 2023), reproduction in progress
Chirp/Mellin/band-edge mechanism producing $J_0$ at argument $2s$	Heuristic derivation (this section)
Rigorous determination of $c_1$ from arithmetic factor and $W_{\alpha_1,\alpha_2}$ renormalization	Open; requires matching (3.17) to (3.19) with full constants
Rigorous identification of exact kernel $K(s) = c_1 J_0(2s)$ (not just leading-order)	Open; requires bounding the $Z_{\geq 2}$ contribution from (3.2) and showing it does not perturb the kernel shape

The combination of the seven-digit empirical match and the chirp-Mellin mechanism producing exactly the right argument scaling makes $K(s) = c_1 J_0(2s)$ the natural working hypothesis. Calling it "merely conjectural" understates the evidence. Calling it "theorem" overstates: the constants and the higher Riemann–Siegel terms are not yet pinned down rigorously.

4. The $L^2$-Divergence Problem

The autocovariance $\widehat R_T(0) = T^{-1}\int_0^T Z(t)^2 dt$ satisfies (Hardy–Littlewood, 1918)

$$\frac{1}{T}\int_0^T Z(t)^2 dt = \log\left(\tfrac{T}{2\pi}\right) + 2\gamma - 1 + O(T^{-1/2 + \epsilon}). \qquad (4.1)$$

This grows like $\log T$, so $Z \notin L^2(\mathbb{R})$ and no warped-axis stationarity statement holds without renormalization. Three regularization candidates:

(a) Polynomial damping $w_\varepsilon(t) = (1+\Theta(t)^2)^{-\varepsilon}$: kills $L^1$ tail for $\varepsilon > 0$ but does not give Hardy-space analyticity in any strip, so no boundary-value theorem applies. Insufficient.

(b) Exponential damping $w_\alpha(t) = e^{-\alpha|\Theta(t)|}$: kills tails at exponential rate, gives Hardy-space analyticity in $|\text{Im} \xi| < \alpha$, but is too aggressive — it kills the chirp itself. Insufficient.

(c) Logarithmic-power damping: $w_\beta(t) = (\log(2+|t|))^{-1/2 - \beta}$. This balances (4.1) exactly: $w_\beta(t)^2 \cdot \log(t/2\pi) \sim (\log t)^{-2\beta}$, integrable at infinity for $\beta > 1/2$ and giving a finite stationary variance. This is the right family.

5. The Two-Parameter Bessel-Style Regularizer

5.1 Definition and closed form

For $0 < \alpha_1 < \alpha_2 < \infty$ define

$$W_{\alpha_1, \alpha_2}(t) := \int_{\alpha_1}^{\alpha_2} \bigl(\log(2 + |t|)\bigr)^{-1/2 - s} ds. \qquad (5.1)$$

Setting $L = \log(2+|t|)$ and using $L^{-1/2-s} = L^{-1/2} e^{-s\log L}$:

$$\boxed{\quad W_{\alpha_1, \alpha_2}(t) = \frac{\bigl(\log(2+|t|)\bigr)^{-1/2 - \alpha_1} - \bigl(\log(2+|t|)\bigr)^{-1/2 - \alpha_2}}{\log\log(2+|t|)} \quad}$$

a closed elementary expression. The denominator $\log\log$ is the Hankel-frequency variable.

5.2 Boundary values

(i) $\alpha_2 \to \infty$ at fixed $\alpha_1 > 0$: the second term vanishes (since $L \geq \log 2 > 1$ gives $L^{-1/2-\alpha_2} \to 0$), leaving

$$W_{\alpha_1, \infty}(t) = \frac{(\log(2+|t|))^{-1/2 - \alpha_1}}{\log\log(2+|t|)}. \qquad (5.2)$$

(ii) Then $\alpha_1 \to 0^+$:

$$W_{0, \infty}(t) = \frac{(\log(2+|t|))^{-1/2}}{\log\log(2+|t|)}. \qquad (5.3)$$

5.3 $L^2$-integrability of the regularized process

Lemma 5.1. For $\beta > 0$ and $W_\beta(t) := (\log(2+|t|))^{-1/2-\beta}/\log\log(2+|t|)$,

$$\int_{t_0}^\infty Z(t)^2 W_\beta(t)^2 dt < \infty. \qquad (5.4)$$

Proof. By (4.1), $\int_{t_0}^T Z(t)^2 dt = T(\log(T/2\pi) + 2\gamma - 1) + O(T^{1/2+\epsilon})$. Integration by parts:

$$\int_{t_0}^T Z(t)^2 W_\beta(t)^2 dt = \left[\int_{t_0}^t Z(s)^2 ds\right] W_\beta(t)^2 \Big|{t_0}^T - \int{t_0}^T \left[\int_{t_0}^t Z(s)^2 ds\right] (W_\beta^2)'(t) dt. \qquad (5.5)$$

The boundary term at $T$ is $T\log(T/2\pi)\cdot (\log T)^{-1-2\beta}/(\log\log T)^2 \sim T/((\log T)^{2\beta} (\log\log T)^2)$, which diverges as $T \to \infty$ for any $\beta < \infty$. So the lemma as stated is false: the regularization $W_\beta$ does not make $Z\cdot W_\beta$ pointwise $L^2$ on $\mathbb{R}$.

Correction. What $W_\beta$ does is make the Cesàro-time-averaged autocovariance converge:

$$\widetilde R_\beta(s) := \lim_{T \to \infty} \frac{1}{T}\int_{t_0}^T Z(t+s) Z(t) W_\beta(t+s) W_\beta(t) dt \qquad (5.6)$$

exists for each $s$, finite, when $\beta > 1/2$. This is the right convergence statement.

Proof of (5.6) for $\beta > 1/2$. Using $W_\beta(t+s)W_\beta(t) = W_\beta(t)^2(1 + O(s/(t\log t)))$ for $|s|$ bounded and $t \to \infty$:

$$\frac{1}{T}\int_{t_0}^T Z(t+s)Z(t) W_\beta(t)^2 dt + O\left(\frac{1}{T}\int_{t_0}^T \frac{|Z(t)|^2}{t\log t} dt\right). \qquad (5.7)$$

The error term is $O((\log T)/T \cdot \log T) = O((\log T)^2 / T) \to 0$. The main term: substitute $W_\beta(t)^2 = (\log t)^{-1-2\beta}/(\log\log t)^2$ and use the chirp identity from §3 with this weight inserted. The key estimate is

$$\frac{1}{T}\int_{t_0}^T |Z(t+s)Z(t)| (\log t)^{-1-2\beta} dt \leq \left(\frac{1}{T}\int_{t_0}^T Z(t)^2 (\log t)^{-1-2\beta} dt\right) \cdot \text{(Cauchy–Schwarz)} \qquad (5.8)$$

and $$\int_{t_0}^T Z(t)^2 (\log t)^{-1-2\beta} dt \sim \int_{t_0}^T \frac{\log(t/2\pi)}{(\log t)^{1+2\beta}} dt \sim \int_{t_0}^T \frac{dt}{(\log t)^{2\beta}}.$$

Substituting $u = \log t$ gives $\int e^u u^{-2\beta} du \sim T/(\log T)^{2\beta}$ as $T \to \infty$. Dividing by $T$: convergence to $1/(\log T)^{2\beta} \to 0$ if $\beta > 0$. For convergence to a positive limit one needs the $1/(\log\log t)^2$ factor in the denominator of $W_\beta^2$. With that included:

$$\int_{t_0}^T Z(t)^2 \cdot \frac{(\log t)^{-1-2\beta}}{(\log\log t)^2} dt \sim \int_{t_0}^T \frac{1}{(\log t)^{2\beta}(\log\log t)^2} dt \sim \frac{T}{(\log T)^{2\beta}(\log\log T)^2}, \qquad (5.9)$$

so dividing by $T$ gives $1/((\log T)^{2\beta}(\log\log T)^2) \to 0$ for $\beta > 0$.

So in fact $\widetilde R_\beta(s) \to 0$ as $T \to \infty$ for each $s$ — the regularization is too strong. The correct statement is:

Lemma 5.2 (Renormalized stationarity). Define

$$\widetilde R_\beta^{\text{ren}}(s) := \lim_{T \to \infty} \frac{(\log T)^{2\beta}(\log\log T)^2}{T}\int_{t_0}^T Z(t+s)Z(t) W_\beta(t)^2 dt. \qquad (5.10)$$

Then $\widetilde R_\beta^{\text{ren}}(s)$ exists, is finite, continuous in $s$, and positive-definite. It is the autocovariance of the renormalized stationary process associated with $W_\beta$.

Proof sketch. The factor $(\log T)^{2\beta}(\log\log T)^2/T$ exactly cancels the variance growth identified in (5.9). Substitute the chirp expansion (3.2), apply the Mellin substitution of §3.3 with weight $W_\beta^2$, and identify the resulting integral as a Hankel transform on $(0,1)$ (after the band-edge substitution of §3.6). The square-root weight emerges from the Jacobian of $\xi = \vartheta'(t)/(\vartheta'(t)+C)$ near the band edge $\xi = 1$. ∎

The two-parameter version $\widetilde R_{\alpha_1, \alpha_2}^{\text{ren}}$ is built the same way with $W_{\alpha_1,\alpha_2}^2$ and the corresponding renormalization factor depending on $\alpha_1$. The boundary values $\alpha_2 \to \infty$, $\alpha_1 \to 0^+$ recover the unregularized chirp-Mellin kernel $K(s)$ of §3.6.

5.4 Connection to incomplete-gamma kernels

A cognate family in the linear variable:

$$\int_{\alpha_1}^{\alpha_2} e^{-s|u|} s^{\nu - 1} ds = |u|^{-\nu}\bigl[\Gamma(\nu, \alpha_1|u|) - \Gamma(\nu, \alpha_2|u|)\bigr], \qquad (5.11)$$

a difference of upper incomplete gammas. For $\nu = 1/2$ this is a difference of error functions; for the Macdonald function $K_0$ via

$$K_0(au) = \int_0^\infty \frac{e^{-au\cosh\theta} + e^{au\cosh\theta}}{2} d\theta = \int_0^\infty \cos(au\sinh\theta) d\theta, \qquad (5.12)$$

the family lies in the Bessel/Macdonald class — see BesselFunctionOfTheFirstKind, BesselIntegral, Mellin.

6. The Cramér Spectral Representation

6.1 Two coordinatizations of the same Hilbert space

The interpretation that makes the structure transparent is the following. $Z$ is a sample path of a non-stationary Gaussian process on $(0, \infty)$ (Gaussianity meant in the precise sense of Selberg's CLT / Bourgade–Kuan log-correlated structure on the critical line). Its pullback $Y(u) = Z(\Theta^{-1}(u))$, by the WSS hypothesis of §2.1, is a sample path of a stationary Gaussian process on $\mathbb{R}$ — Gaussianity is preserved under deterministic time-change, and stationarity is the content of the hypothesis.

For any second-order stationary process $Y$, Bochner's theorem gives a non-negative Borel measure $dF_Y$ on $\mathbb{R}$ with

$$R_Y(s) = \int_{\mathbb{R}} e^{i\xi s} dF_Y(\xi), \qquad (6.1)$$

and the Cramér representation theorem gives an orthogonal complex-valued random measure $\Phi_Y$ on $\mathbb{R}$ — the Cramér random Lebesgue–Stieltjes measure — such that

$$Y(u) = \int_{\mathbb{R}} e^{i\xi u} d\Phi_Y(\xi), \qquad \mathbb{E}\bigl[d\Phi_Y(\xi) \overline{d\Phi_Y(\xi')}\bigr] = \delta(\xi - \xi') dF_Y(\xi). \qquad (6.2)$$

The band-edge calculation of §3.6 (heuristically) localizes the support: $\mathrm{supp}(dF_Y) \subseteq [-1, 1]$, with density $\sim c/\sqrt{1-\xi^2}$ near $\xi = \pm 1$ — the universality class of $J_0$-kernels (§2.2, §3.6–§3.7).

6.2 The Cramér isomorphism

The Cramér representation is not just a formula; it is an isomorphism of Hilbert spaces. Define

$$\mathcal{H}Y := \overline{\mathrm{span}}{L^2(\Omega)}\bigl{Y(u) : u \in \mathbb{R}\bigr}, \qquad \mathcal{K}_Y := L^2(\mathbb{R}, dF_Y). \qquad (6.3)$$

The map $\mathcal{I}_Y : \mathcal{K}_Y \to \mathcal{H}_Y$ given by

$$\mathcal{I}Y(g) := \int{\mathbb{R}} g(\xi) d\Phi_Y(\xi) \qquad (6.4)$$

is a unitary isomorphism. The basis case $g(\xi) = e^{i\xi u}$ recovers $Y(u)$ itself. The inverse is the Cramér-measure inversion (§6.3).

Two coordinatizations of the same Hilbert space. $\mathcal{H}_Y$ is the time-domain coordinatization: random variables are linear functionals of the sample path $Y$. $\mathcal{K}_Y$, identified with random measures via $g \mapsto \int g d\Phi_Y$, is the spectral-domain coordinatization: random variables are stochastic spectral integrals. The sample path $Y$ and the random orthogonal measure $\Phi_Y$ are two presentations of the same object — every linear functional of $Y$ corresponds bijectively to a deterministic function $g \in L^2(dF_Y)$, and vice versa.

6.3 Cramér-measure inversion

By Fourier inversion at continuity points $\xi_1 < \xi_2$,

$$\Phi_Y(\xi_2) - \Phi_Y(\xi_1) = \lim_{U \to \infty}\frac{1}{2\pi}\int_{-U}^{U}\frac{e^{-i\xi_1 u} - e^{-i\xi_2 u}}{iu} Y(u) du. \qquad (6.5)$$

Pulling back via $u = \Theta(t)$, $du = \Theta'(t) dt$, $Y(u) = Z(t)$:

$$\Phi_Y(\xi_2) - \Phi_Y(\xi_1) = \lim_{T \to \infty}\frac{1}{2\pi}\int_{t_0}^T \frac{e^{-i\xi_1 \Theta(t)} - e^{-i\xi_2 \Theta(t)}}{i\Theta(t)} Z(t) \Theta'(t) dt. \qquad (6.6)$$

This is a deterministic Lebesgue–Stieltjes integral against the explicit sample path $Z(t) \Theta'(t) dt$, not a stochastic integral. The randomness sits inside $Z$ itself (as a sample of the Gaussian process); the integral against a fixed sample is ordinary Lebesgue–Stieltjes. The regularization $W_\beta$ from §5 is needed to control $L^1$-convergence of (6.6), but the form of the inversion is independent of regularization.

7. Higher Moments via the Cramér Isomorphism

7.1 Higher moments are determined by $\Phi_Y$ alone

Because $Y$ is hypothesized Gaussian and stationary, $\Phi_Y$ is a complex Gaussian orthogonal random measure. A Gaussian orthogonal random measure is completely determined as a probability law by its second-moment measure $dF_Y$. In particular:

The full (joint) law of $(\Phi_Y(B_1), \ldots, \Phi_Y(B_n))$ for any disjoint Borel sets $B_1, \ldots, B_n$ is a centered complex Gaussian with covariance matrix $\bigl(\delta_{ij} F_Y(B_i)\bigr)$.
The full law of $\Phi_Y$ as a random distribution is fixed by $dF_Y$ alone.
Through the Cramér isomorphism (6.4), the full law of $Y$ as a random function is fixed by $dF_Y$ alone.

In other words, $R_Y$ alone, equivalently $dF_Y$ alone, determines every joint moment of $Y$. The $J_0(2s)$ kernel of §2.2, taken with its boxed identity status, contains the entire higher-moment data of $Y$.

7.2 Wick / Isserlis as a consequence, not an extra hypothesis

Because the law of a centered Gaussian process is determined by its covariance, the Isserlis–Wick formula for $2n$-point moments

$$\mathbb{E}\bigl[Y(u_1) \cdots Y(u_{2n})\bigr] = \sum_{\pi \in P_2(2n)}\prod_{{j,k} \in \pi} R_Y(u_j - u_k) \qquad (7.1)$$

is a theorem — a consequence of Gaussianity — not an additional ansatz. Equivalently, the multi-fold spectral integral

$$Y(u_1) \cdots Y(u_n) = \int_{[-1,1]^n} e^{i(\xi_1 u_1 + \cdots + \xi_n u_n)} d\Phi_Y(\xi_1) \cdots d\Phi_Y(\xi_n) \qquad (7.2)$$

has expectation given by the Wick contraction sum, by orthogonality and Gaussianity of $\Phi_Y$.

7.3 Connection to Keating–Snaith

The Keating–Snaith moment conjecture:

$$\frac{1}{T}\int_0^T |\zeta(\tfrac{1}{2}+it)|^{2k} dt \sim a_k g_k (\log T)^{k^2}, \qquad (7.3)$$

with $g_k$ the CUE random-matrix factor and $a_k$ the arithmetic factor, becomes the prediction of the Gaussian-WSS framework: the $(\log T)^{k^2}$ growth is exactly Wick combinatorics for a log-correlated Gaussian field, and the $g_k a_k$ prefactor is the precise Wick contraction value once the renormalized kernel from §5 and the band-edge density from §6.1 are inserted.

Status. The Gaussian-WSS hypothesis on $Y$ is rigorously open beyond the second moment ($k = 1$ is Hardy–Littlewood; $k = 2$ is Ingham; $k = 3, 4$ have moment lower-bounds matching Keating–Snaith but full asymptotics are open). A proof that $\Phi_Y$ is Gaussian and orthogonal with the conjectured spectral measure would yield Keating–Snaith for all $k$ as a corollary, by (7.1)–(7.2).

7.4 Status table

Claim	Status
$\vartheta$ encodes mean zero spacing AND mean phase	Classical
Riemann–Siegel chirp expansion (3.1)	Classical, Riemann (1859), Siegel (1932)
Hardy–Littlewood mean-square (4.1)	Theorem (Hardy–Littlewood 1918)
Warp $u = \Theta(t)$ produces stationarity miracle (3.16)	Rigorous
Translation stability of the normalized $Z$-autocovariance kernel across windows in $[10^2, 10^4]$	High-precision numerical observation (Crowley c. 2023); empirical signature of WSS in the Cesaro sense; reproduction in progress
Identification $\widehat R(s)/\widehat R(0) = J_0(2s)$ via 6–7-decimal zero-set agreement across multiple zeros	High-precision numerical observation (Crowley c. 2023); reproduction in progress on current arb4j framework
Chirp–Mellin–band-edge derivation of $J_0(2s)$ form	Heuristic; rigorous version requires controlled treatment of (3.17)–(3.19) and the $Z_{\geq 2}$ tail
Closed form (5.1)–(5.3) for $W_{\alpha_1,\alpha_2}$	Elementary, proved
Renormalized stationarity Lemma 5.2	Proof sketch given; full proof requires careful Mellin analysis
Spectral support $\mathrm{supp}(dF_Y) \subseteq [-1, 1]$ with band-edge density $\sim 1/\sqrt{1-\xi^2}$	Conjectural; band edge $\xi_{\rm edge}(T) = \vartheta'(T)/(\vartheta'(T)+C) \to 1$
Selberg CLT for $\log\zeta(\tfrac12+it)$	Theorem (Selberg 1946)
Cramér isomorphism $\mathcal{H}_Y \cong L^2(dF_Y)$ for any 2nd-order WSS process	Theorem (Cramér 1940), assuming WSS-of-$Y$ hypothesis
Gaussian + orthogonal $\Rightarrow$ full law of $\Phi_Y$ determined by $dF_Y$	Theorem (standard Gaussian theory)
Isserlis–Wick (7.1) for the higher moments of $Y$	Theorem, conditional on Gaussian-WSS-of-$Y$ hypothesis
$Y$ Gaussian and WSS, with autocovariance $J_0(2s)$	Hypothesis; supported by §2.1, §2.2, §3.7; rigorous proof open
Keating–Snaith $(7.3)$ for all $k$	Corollary of the Gaussian-WSS hypothesis; rigorous unconditional proof open beyond $k = 2$

8. Computational Path

The closed form (5.1) makes the regularized autocovariance computable at finite $(\alpha_1, \alpha_2)$. The chirp-Mellin steps (3.6)–(3.18) with weight $W_{\alpha_1,\alpha_2}^2$ inserted produce an explicit Hankel-type integral. The boundary limit $(\alpha_2 \to \infty, \alpha_1 \to 0^+)$ should yield the empirical $J_0$ kernel quantitatively, fixing the constants $c_1, c_2$ in $K(s) = c_1 J_0(c_2 s)$.

Cross-check program:

(i) Sample $Z(t)$ on $[t_0, T]$ via arb4j at high precision. Compute $\widehat R_T(s)$ on a grid in $s$.

(ii) Fit $\widehat R_T(s) / \widehat R_T(0)$ to $J_0(c_2 s)$ for $c_2$ near $2$. Verify first node $s_* \cdot c_2 = j_{0,1}$.

(iii) Compute the regularized $\widetilde R_{\alpha_1,\alpha_2}^{\text{ren}}(s)$ for several $(\alpha_1,\alpha_2)$. Check stability of first-node location as $(\alpha_1, \alpha_2) \to (0, \infty)$.

(iv) Compute $|\widehat\zeta_T(\tfrac12+it)|^{2k}$ moments numerically and compare to the Wick prediction (7.3) using the kernel from (ii). Discrepancy detects non-Gaussian corrections.