Packet Centroids of the Riemann Zeta Function

Packet Centroids of the Riemann Zeta Function: A Smoothing Identity and a Displacement Sum Rule

Ondřej Dvořák¹

¹Independent researcher, Děčín, Czech Republic. on.dvorak@email.cz

AI models used: Anthropic Claude, OpenAI ChatGPT, Google Gemini, xAI Grok.

AI assistance: Large language models were used for computation, proof drafting, proof checking, literature consultation, cross-verification, editing, and manuscript preparation. The mathematical arguments were drafted and checked by these models, including repeated blind refereeing by independent model instances; the author has not independently verified every proof. The author originated and directed the research programme, made the methodological and editorial decisions, reviewed the manuscript, and accepts responsibility for presenting this material. The work is written so that every claim can be checked from what is printed and deposited, without trust in either the author or the models.

Record of work: These files are a record of work, not a record of results. They include measurements that were later corrected, conjectures that were refuted, and observations that have never been checked against the literature. Every claim is marked with which of those it is.

§1. Discovery and Motivation

1.1 Partial-sum spirals and the packet structure

For s = σ + it in the critical strip 0 < σ < 1, the partial Dirichlet sums S_N(s) = ∑_n=1^N n^−s trace a spiral on the Argand plane as N increases. The spiral is not uniform: when N crosses the saddle abscissa x_ν = t/2πν, the ν-th term of the Fourier expansion of the sawtooth ψ(x) = {x} − 12 appearing in the exact truncation formula (made explicit in §2.2–2.3) passes through a stationary point, and the running sum swings direction. Successive saddle crossings divide the integers into packets: the k-th packet is the interval of integers N between two consecutive crossings at x_k+1 < x_k.

Within each packet the running sum describes a coherent arc — a partial turn of a Cornu-type spiral driven by the single dominant frequency ν = k (Figure 1). The arc terminates and a new arc begins at each saddle. The packet centroid of the k-th arc, and its offset from ζ, are the natural summary statistics: ⟨S⟩_k(s) := 1|G_k ∩ ℤ|∑_{N ∈ Gk} S_N(s),    off_k(s) := ζ(s) − ⟨S⟩_k(s), where G_k = (x_k+1, x_k) and the sum is the uniform average over all integer truncation points in the gap. (For the proof in §2 we use a smooth weight supported in the middle third of G_k; Lemma 2.2 in §2 reconciles the two.)

A computation at any s in the strip reveals a striking fact: off_k(s) is χ-proportional with a k-term leading expression, and its deviation from that expression decays like t^−1/2 (relative to χ) as t → ∞, with oscillating phase. More precisely:

Theorem 1 (§2): For fixed k ≥ 1 and compact [σ₀,σ₁] ⊂ (0,1), off_k(s) = χ(s)P_k(s) + χ(s) O_k,ε(t^−1/2+ε), where P_k(s) = ∑_{m ≤ k} m^s−1 and χ is the functional-equation factor (defined explicitly in §2.1).

The centroid offset is not zero; it equals χ(s)P_k(s) to leading order — a sum of k terms with χ-proportional weight. Equivalently, the centroid itself satisfies ⟨S⟩_k(s) = ζ(s) − χ(s)P_k(s) + χ(s) O(t^−1/2+ε). The level-2 zeros — the zeros of the centroid ⟨S⟩_k itself — are thus the zeros of ζ(s) − χ(s)P_k(s), which by the functional equation are the points where ζ(w) = S_k(w) in the reflected variable w = 1−s = (1−σ) + it.

1.2 From spirals to value distribution

The map s ↦ w = (1−σ)+it sends the critical strip to itself and converts the geometric question — where does the centroid nearly vanish? — into a question in the analytic theory of the Riemann zeta function: for which w in the strip does ζ take the value S_k(w)?

For k=1, S₁(w) = 1: the roots are the 1-points of ζ, studied since Levinson [1] and carrying a known mean density (1/2π)log(t/4π) per unit height (the classical a-point density, distinct from ζ's own zero density (1/2π)log(t/2π); §3.4 uses the former to rescale the k-family gaps and the latter for the Riemann-zero control). For k ≥ 2, S_k(w) = 1 + 2^−w + ⋯ + k^−w varies with position, and the root family is new. Throughout, write F_k(w) := ζ(w) − S_k(w) for the comparison function; the roots in question are the zeros of F_k.

Two features of the root distribution emerge from a high-precision census:

(a) Displacement rate. Each root w_ρ = u_ρ + iv_ρ lies at signed displacement dσ_ρ := σ^*_ρ − 12 = 12 − u_ρ from the critical line — the sign convention of the census, positive on the side σ^* > 12, i.e. u_ρ < 12; note that u = 1 − σ^* makes u_ρ − 12 and dσ_ρ opposite in sign. The product |dσ_ρ| · log v_ρ has a distribution that collapses (KS p ≥ 0.45) to a common shape across heights from t = 200 to t = 100,040, with mean c = 1.023 ± 0.011. This constant is a finite-T analogue of Levinson's clustering rate, here measured directly from the 1-point locations.

(b) Displacement asymmetry. The mean displacement from 12, signed by u_ρ − 12, is positive and grows like log(k+1): D_k(T₁,T₂) := ∑_{Fk(wρ)=0 , vρ ∈ (T1,T2]} (u_ρ − 12) = T₂−T₁4πlog(k+1) + O_k(log T₂). (The sum rule's natural variable is u_ρ − 12 = −dσ_ρ; the positive budget D_k > 0 thus means that the net displacement lies on the side dσ < 0.)

Proposition 2 (§4) proves this law from Littlewood's lemma. The proof reveals that the comparison function H(w) = χ(w)F_k(1−w) appearing on the left of the Littlewood integral equals ζ(w) − χ(w)P_k(w) — exactly the centroid's leading expression, i.e. ζ minus Theorem 1's main term. The smoothing lemma and the sum rule thus share a single object; their derivations are two views of the same asymptotic identity.

1.3 Figures

The paper's five figures are collected in the companion volume Packet Centroids: Visuals [16], with the data behind each. They are cited in the text as Figure 1 ... Figure 5.

§2. The Smoothing Lemma

Theorem 1. Fix k ≥ 1, a compact [σ₀,σ₁] ⊂ (0,1), and ε > 0. Then, uniformly for σ ∈ [σ₀,σ₁], off_k(s) = χ(s)P_k(s) + χ(s) O_k,σ0,σ1,ε(t^−1/2+ε), where the error, written as χ(s) t^−1/2 Φ_k, carries an oscillating factor Φ_k ≪_k,ε t^ε whose dominant part has phase depending on {t/2π k} and {t/2π(k+1)} (§2.5).

2.1 Setup and notation

Write x_ν:=t/2πν for the ν-th saddle abscissa and let G_k:=(x_k+1, x_k),   |G_k|=t2π k(k+1), be the k-th packet gap. Fix a weight ω∈ C_c^∞(G_k), suppω contained in the middle third of G_k, ∫ω=1, ω^(j)≪_j|G_k|^−j, and set the discrete normalization ω(N)=ω(N)/∑_M∈ℤω(M). The smooth-weight centroid offset of §2 is off_k(s) = ζ(s) − ∑_Nω(N) S_N(s) = ∑_Nω(N) T_N(s),   T_N(s):=∑_n>Nn^−s, the tail T_N being defined for 0<σ<1 by analytic continuation from σ>1 (equivalently Abel regularization); all identities below are exact identities of the continuations. Here χ(s) = 2^sπ^s−1sin(π s/2) Γ(1−s) is the functional-equation factor, with the Stirling asymptotic χ(s) = (2π/t)^s−1/2e^i(t+π/4)(1+O(1/t)) as t → ∞, uniformly on compact σ-sets; its zeros and poles all lie on the real axis.

2.2 Step 1 — exact truncation formula

For integer N≥1 and σ>0, Stieltjes integration of x^−s against d⌊x⌋=dx−dψ(x), ψ(x):={x}−12, gives the exact formula T_N(s) = N^1−ss−1_{(a) smooth term} − 12 N^−s_{(b) local term} − s∫_N^∞ ψ(x) x^−s−1 dx_{(c) ψ−integral} . Terms (a) and (b) are error-sized relative to χ: with N≍ t/k, |N^1−s/(s−1)||χ(s)|≍ k^σ−1t^−1/2,   |N^−s||χ(s)|≍ k^σ t^−1/2, both uniform on [σ₀,σ₁]. Every main term therefore comes from (c).

2.3 Step 2 — Poisson expansion of the ψ-integral

Insert ψ(x)=−∑_ν≥1sin 2πν xπν into (c): (c) = ∑_ν≥1s2π iν∫_N^∞ (e(ν x)−e(−ν x)) x^−s−1 dx = ∑_ν≥1(I_ν(N)−I_−ν(N)).

Justification of the interchange (dominated convergence). The termwise integration behind this display holds for each fixed N; no smoothing in N is needed at this step — the ω-average enters only later, in the Fresnel-defect bounds of §2.4–2.5. Let ψ_V(x):=−∑_{ν≤ V}sin2πν xπν be the partial sums of the sawtooth series. (i) ψ_V(x)→ψ(x) at every non-integer x (Dirichlet's convergence theorem for the piecewise-C¹ function ψ); the exceptional integers are a null set for dx. (ii) |ψ_V(x)|≤4 uniformly in V and x: at integer x every term vanishes; otherwise, with \|x\| the distance from x to the nearest integer and ν₀:=⌈1/\|x\|⌉, each term with ν≤ν₀ is at most 2\|x\| in modulus (from |sin2πν x|≤2πν\|x\|), so these contribute at most 2\|x\|ν₀≤2+2\|x\|≤3; for the terms ν₀<ν≤ V, the geometric-sum bound |∑_{ν≤ M}e(ν x)|≤1/(2\|x\|) gives |∑_{ν0<ν≤ M}sin2πν x|≤1/\|x\|, and partial summation against the decreasing weights 1/(πν) bounds their contribution by (1/π)·(1/\|x\|)·(ν₀+1)⁻¹≤1/π. The dominating function is therefore 4x^−σ−1, integrable on (N,∞) because σ≥σ₀>0. Dominated convergence gives ∫_N^∞ψ(x) x^−s−1dx=lim_V∫_N^∞ψ_V(x) x^−s−1dx, and each ∫_N^∞ψ_V x^−s−1dx is the finite sum ∑_{ν≤ V} of absolutely convergent integrals. The ν-series in the display is thus summed as the limit of its symmetric partial sums ∑_{ν≤ V}(I_ν−I_−ν); the estimates below show that the negative-frequency sum and the class-(iii) sum converge absolutely, so no rearrangement issue arises downstream.

The phase of I_±ν is φ_±ν(x)=±2πν x−tlog x, with φ_±ν'(x)=±2πν−t/x. For the negative frequencies φ_−ν'≤ −2πν on x>0: no stationary point, and repeated integration by parts yields ∑_ν|I_−ν(N)|=|χ(s)| O_k(t^−1/2) (this bound is input (ii) of the status note at the end of §2.6). The positive frequency ν has the unique saddle x_ν=t/2πν, φ_ν''(x_ν)=t/x_ν²>0.

2.4 Step 3 — saddle classification and endpoint weights

For N∈suppω (middle third of G_k) the saddles split into three classes.

(i) Captured saddles ν≤ k: full weight, and the identity that creates P_k. Here x_ν≥ x_k>N, so the saddle lies inside the integration range at distance ≥|G_k|/3 from the endpoint. Stationary phase at x_ν with amplitude s2π iνx^−s−1 gives, using 2πν x_ν=t and s/it=1+O(1/t), I_ν(N) = x_ν^−σ· x_ν√2πt e^{i(t−tlog x_ν+π/4)} (1+F_ν(N)+O(t⁻¹)) = (2πt)^s−12ν^s−1e^i(t+π/4)(1+F_ν(N)+O(t⁻¹)), and by the standard asymptotic χ(s)=(2π/t)^s−1/2e^i(t+π/4)(1+O(1/t)),  I_ν(N) = χ(s) ν^s−1(1+F_ν(N))+χ(s) O(t⁻¹)   (1≤ν≤ k). F_ν(N) is the Fresnel-truncation defect: the deficit of the half-line Fresnel integral cut at distance d_ν=|x_ν−N| from its critical point, of size (κ_ν d_ν)⁻¹ with Fresnel scale κ_ν=√φ_ν''(x_ν)=√t/x_ν (the size follows from one integration by parts on the Fresnel tail, ∫_D^∞ e^iu2 du = O(1/D); the full expansion with its uniform O(t⁻¹) error is the standard stationary-phase lemma — see the status note at the end of §2.6 and [18], ch. IV). For ν≤ k and N in the middle third, d_ν≥|G_k|/3, whence |F_ν(N)| ≪ x_ν√t d_ν ≪ k√t.

(ii) Ejected saddle ν=k+1: weight zero. Now x_k+1<N, the saddle lies outside [N,∞) at distance d_k+1≥|G_k|/3; only the Fresnel tail survives: I_k+1(N) = χ(s) (k+1)^s−1 F_k+1(N) + χ(s) O(t⁻¹),   |F_k+1(N)|≪ k t^−1/2.

(iii) Distant frequencies ν≥ k+2: |φ_ν'|≥ 2π(ν−(k+1))·(1+o(1)) on [N,∞); non-stationary integration by parts and summation over ν give a total χ(s) O_k(t^−1/2) (again input (ii) of the status note, §2.6).

Summing (i)–(iii) and averaging over N with ω: off_k(s) = χ(s)∑_{ν≤ k}ν^s−1 + χ(s) t^−1/2 Φ_k = χ(s) P_k(s) + χ(s) t^−1/2 Φ_k, with Φ_k ≪_k,ε t^ε — the t^ε absorbs the unexhibited ν-summation losses, the next-order stationary-phase corrections, and (under the uniform weight) the logarithm of Lemma 2.2; see the status note at the end of §2.6.

Remark 2.1 (endpoint weight w_k=1; recovery of the sharp-cutoff "½ + ½" bookkeeping). In the sharp-cutoff picture with truncation placed exactly at the saddle, N=x_k, the frequency-k term carries only half the Fresnel mass, F_k→−12, i.e. weight 12. Sweeping the truncation point across G_k restores the missing half over one Fresnel transition width κ_k⁻¹≍ √t/k, and under the centroid average the endpoint frequencies emerge with weights exactly 1 (ν=k) and 0 (ν=k+1): this is the unit endpoint weight w_k=1 of the sharp-cutoff bookkeeping, with the two halves recombined by the average. The geometric verification — Newton on the off₂ centroid returning w₂=1 with finite-t error 0.067/0.035/0.016 at t≈500/1250/5000 — measures precisely this factor, and its measured decay is the theorem's t^−1/2 (fitted exponent 0.504 over 24 heights, t∈[300,60000]).

Lemma 2.2 (Uniform-weight extension). Theorem 1 holds, with the same error order t^−1/2+ε, for the literal uniform average over all integers N ∈ G_k — the object measured in §3. Proof (given the Fresnel-defect input of §2.4). The proof above uses the smooth weight ω supported in the middle third of G_k, placing every N at distance ≥ |G_k|/3 from both saddle abscissae and bounding every Fresnel defect by O(k/√t). Under the uniform weight, the endpoint frequencies ν ∈ {k, k+1} see integers at all distances d from the saddle. Integers within the Fresnel transition width κ_k⁻¹ ≍ √t/k of either endpoint contribute defects F_ν(N) = O(1); there are at most O(√t/k) such integers near each endpoint, while the total count in G_k is |G_k| ≍ t/k², so their share of the average is O(k/√t) · O(1) · χ(s) = χ(s) · O(t^−1/2) — the same order as the middle-third estimate. An integer at larger distance d from the saddle carries a defect of size (κ_k d)⁻¹ = x_k/(√t d); averaging over d from κ_k⁻¹ up to |G_k| costs 1|G_k|∑_{κk⁻¹ ≤ d ≤ |Gk|} x_k√t d  ≪  x_k√t |G_k| log(κ_k|G_k|)  ≪  k√t log t, which is absorbed by the t^ε of the theorem (and is where that ε is spent under the uniform weight). The interior frequencies are unchanged. ∎ The oscillating prefactor Φ_k is, under this reading, the t-varying phase of the endpoint integers under the uniform average; the fitted decay exponent 0.504 of Remark 2.1, with its residual ± 0.3–0.4 wobble visible and uncollapsed, measures exactly this uniform-weight object.

2.5 Step 4 — error assembly and the fractional-part wobble

The dominant error is the pair of endpoint Fresnel defects F_k,F_k+1. Their phases are φ_ν(N)−φ_ν(x_ν)≈12φ_ν''(x_ν) (N−x_ν)² (mod 2π), evaluated at integers N; the offset of the integer lattice from the saddles is exactly {x_k}={t/2π k} and {x_k+1}={t/2π(k+1)}. Hence E_k(s) = χ(s) t^−1/2 Φ_k(t;{t/2π k},{t/2π(k+1)}),   Φ_k ≪_k,ε t^ε , oscillating. Φ_k must be carried as an oscillating factor and not absorbed into a smaller power of t: the measured residual wobble of ±0.3–0.4 in log scale is this factor, and any fit forcing a pure power law through it misestimates the exponent.

Forward reference (companion paper). The functional form of Φ_k — left open here — is the subject of the companion paper [13]: the factor is measured directly, models in the two fractional parts alone are refuted (the factor carries explicit t through the Fresnel scale κ_ν = √t/x_ν), and a parameter-free Fresnel-endpoint prediction reproduces the measurement with median R² = 0.993 over k = 1..8, the residual being attributed to instrument artifacts of the prediction. Theorem 1's error term is thereby mechanism-identified, not merely bounded ([13], ch. 3–4). None of this is used in the present paper, and the claim in this paragraph is the companion's, verifiable only once [13] is available.

2.6 Step 5 — uniformity in σ

Every σ-dependence above enters through the bounded continuous factors k^σ−1, k^σ, the exact ratios x_ν^−σ(t/2π)^σ=ν^σ (in which the σ-dependence cancels against the χ-normalization), and the O(1/t) correction to s/it; on the compact [σ₀,σ₁] all are uniformly bounded, and the leading stationary-phase constant is σ-independent. Thus the implied constant in O(t^−1/2+ε) is uniform on the compact, and the error is χ-proportional with σ-independent ratio. ∎

Numerically, the error/χ ratio varied by ≤5.7% between σ=0.3 and σ=0.5; the specification of this check (window, sample size) is not carried in this paper, and the check is a consistency observation, not part of the proof.

Status of the proof. Two classes of estimates above are used in their standard form rather than proved from first principles: (i) the truncated stationary-phase expansion of §2.4(i)–(ii) — the main-term coefficient with its uniform O(t⁻¹) error, and the Fresnel-tail size (κ_ν d_ν)⁻¹ — and (ii) the repeated-integration-by-parts bounds, with summation over ν, for the negative frequencies (§2.3) and the distant frequencies (§2.4(iii)), on the infinite range [N,∞). Both are standard applications of the exponential-integral (van der Corput / stationary-phase) lemmas of [18], ch. IV; the hypotheses required there are elementary for the amplitudes s2π iνx^−s−1, but the verifications are not written out here, and the t^ε in the theorem absorbs the unexhibited summation losses (and the logarithm of Lemma 2.2). A reader requiring a fully self-contained proof should treat Theorem 1's error term as resting on these two standard inputs; the main term χ P_k, the error exponent (fitted 0.504 over 24 heights, Remark 2.1), the χ-proportionality, and the oscillating prefactor are each confirmed independently by the census of §3.

Corollary 2.3 (Zero correspondence). Since F_k has real Dirichlet coefficients, F_k(w) = 0 ⇔ F_k(w) = 0. The census records roots w_ρ = (1−σ^*) + it^* with t^* > 0, i.e. w_ρ = 1−s for s = σ^* + it^*. Dividing by χ(s) and applying the functional equation, the zeros of the leading expression ζ(s) − χ(s)P_k(s) at s = σ + it, t > 0, correspond exactly to roots w = (1−σ) + it of F_k(w) = 0. This exact correspondence is what identifies the census population.

The zeros of the centroid ⟨S⟩_k itself differ from these by the error term: at a centroid zero, the reflected equation reads F_k(w) = δ(w) with δ := E_k/χ, so |δ| ≪_k,ε t^−1/2+ε by Theorem 1. If w₀ is a simple root of F_k and r > 0 is such that the disk |w − w₀| ≤ r contains no other root of F_k and min_|w−w0|=r|F_k| > sup_{|w−w0|≤ r}|δ|, then Rouché's theorem gives exactly one zero of the perturbed equation in that disk, and to first order its displacement from w₀ is δ(w₀)/F_k'(w₀). These hypotheses — simplicity, separation, and the lower bound on the circle — are not verified here for the census populations, and they can fail exactly where |F_k'| is small, the regime that drives the displacement statistics of §3.2. The first-order reading |dσ| ≈ |δ|/|F_k'| is therefore a working approximation supported by the measured correlation Corr(|dσ|, 1/|F₂'|) = 0.58, monotone from Q1 to Q3 of |F₂'| (§3.2) — not a proved bound.

§3. The k-family Census

3.1 Instrument

Roots of F_k(w) = ζ(w) − S_k(w) = 0 are located by Newton's method on a dense grid. The evaluator uses the Euler–Maclaurin formula with N = max(100, ⌈t ⌉) Dirichlet terms and 12 Bernoulli correction pairs, with Kahan-compensated summation; this gives float64 accuracy for t ≤ 60,000 and approaches the double-precision floor (∼ 10⁻¹⁰) near t = 10⁵. The Newton grid has step sizes Δ t = 0.5, Δ u = 0.2 in the primary box u ∈ [−0.5, 1.5]; duplicate roots within tolerance 10⁻⁴ in both coordinates are merged. Before each census window, the evaluator is validated against a reference table (15/15 value matches, 12/12 Newton convergence on the low-range table; 12/12 and 12/12 on the high-range table; both tables and their coverage are recorded in the Supplementary Materials [15], §S3). The census dataset was frozen on 2026-07-08.

The strip family consists of roots with u ∈ [0,1] (equivalently σ^* ∈ [0,1]); roots with u > 1 (σ^* < 0) form the ROS family (Reflected-Outer-Strip), which extends to u ≤ u^*(k) as established in Lemma 4.1.1. The initial census box reached u ≤ 2.0 and therefore truncated the high-u ROS tail; the repaired window u ∈ [1.4, k+2.1] is proved complete on the right by Lemma 4.1.1(1). See §4.2 for the census window and its completeness; the record of the repair is in the Supplementary Materials [15], §S1.

3.2 Windows and the Levinson displacement constant

Seven strip windows were surveyed for k = 1 (F₁(w) = ζ(w) − 1 = 0, the 1-points of ζ):

The normalized displacement x_ρ = |dσ_ρ| · log t_ρ^* has a distribution that collapses across all windows (Figure 3; Kolmogorov–Smirnov p ≥ 0.45). Fitting a constant model gives c = 1.023 ± 0.011    (χ²/dof = 6.46/6), consistent with c independent of t (the highest-t window sits nearest the evaluator's stated precision floor, §3.1, and carries this fit's largest single value; that proximity has not been separately ruled out as a contributor). A free-exponent fit yields α = 0.954 ± 0.090, consistent with α = 1 at 0.5σ. The loglog-growth model c ∝ loglog t is rejected (χ²/dof = 28.05/6 against 6.46/6 for the constant model — the model comparison is these two fits). Within the measured range above, the displacement rate is constant; §6.2(a) records its asymptotic status.

For k = 2 and k = 3, the constant increases mildly: at t ∈ [1000,1500], c(k=2) = 1.079 ± 0.046 and c(k=3) = 1.197 ± 0.049. The increase is mechanically explained by the distribution of |F_k'(w)| across the strip: the displacement |dσ| is proportional to 1/|F_k'(w)| at first order (Corollary 2.3), and a regression of |dσ| on 1/|F₂'| over the k=2 census confirms Corr(|dσ|, 1/|F₂'|) = 0.58 with a monotone decrease from Q1 to Q3 of |F₂'|. The variation of c with k is not a new free parameter but a consequence of the varying derivative landscape.

3.3 Displacement split and the sum rule

The signed split of roots across the critical line u = 12 is diagnostic. For k = 1, the strip-family fraction with u > 12 is 49.3% (z = −1.01, p = 0.31), consistent with 50/50 at all windows individually and pooled. For k = 2, the fraction with u > 12 is 54.9% (z = 6.3, p = 3.7 × 10⁻¹⁰) — equivalently, 45.1% of the σ^*-values exceed 12. This is a highly significant departure from symmetry.

Proposition 2 (§4) predicts and explains both. The total signed displacement D_k = (Δ T/4π)log(k+1) grows with k while the ROS family's capacity to absorb the budget — the ROS points each contribute u_ρ − 12 ≈ 1.2–2.0 — is limited by the ROS vertical spacing 2π/logk+2k+1 (one root per ≈15.5 units of height at k=1; per ≈21.8 at k=2) — a heuristic two-term scale, not a derived density; its origin and unproved status are recorded in Remark 4.1.5, and the ROS displacements entering the closure accounting of §4.2 are measured roots, independent of it. At k=1, the ROS family absorbs ∼71\% of D₁, leaving the strip nearly balanced. At k=2, the ROS capacity is sparser relative to the larger budget, and the strip inherits a positive remainder — the observed right-bias.

3.4 Spacing statistics

Nearest-neighbor gaps Δ t_ρ between consecutive strip-family roots (sorted by v_ρ), rescaled by the local mean density of the respective family — (1/2π)log(t/2π(k+1)), i.e. (1/2π)log(t/4π) at k=1 and (1/2π)log(t/6π) at k=2, and (1/2π)log(t/2π) for the Riemann-zero control — are strikingly sub-Poissonian and sub-GUE. All rows of the table below are computed on the single window t ∈ [3000,8000] — the same window used for the control, so that the comparison is like-for-like — with the density evaluated at the earlier endpoint of each gap and the first gap of the window dropped (n = 4559 gaps at k=1, 4173 at k=2, 5359 for the control):

The control uses Riemann zeros in t ∈ [3000,8000] processed on the identical pipeline (same normalisation, same boundary trim). The control is not GUE-consistent, and that is an expected finite-height effect rather than a defect of the pipeline: against the exact row above it carries a small-gap deficit of 20.8% (0.0896 against 0.1131) and an upper-tail deficit of 22.1% (0.0138 against 0.0177), with a standard deviation 7.0% narrower (0.395 against 0.4243). Deficits of the same size at both ends together with a narrower spread are the signature of a distribution compressed relative to GUE, not of a small-gap-specific anomaly; at t ∼ 5×10³ the zeros of ζ are not expected to have reached the GUE limit. The control is therefore an empirical reference at this height, not a stand-in for GUE, and the two comparisons are reported separately below and never merged. The 1-points at k=1 are stiffer than the control by a factor of 6 in small-gap suppression, and at k=2 by a factor of 4.5 — both factors are measurement against measurement and are unaffected by the correction to the reference row. The large-gap tail (frac>2) is indistinguishable between the k=1 family and the control (0.0140 against 0.0138); at k=2 it is 0.0098, below both the control and the exact GUE value (0.0177), from which it is a 45% deficit. The small-gap deficit is therefore the dominant anomaly at both indices, and it is not accompanied by a compensating excess in the upper tail. The pooled seven-window value of frac<0.5 at k=1 is 0.0165; Figure 4 (in the companion Visuals volume) is drawn on that pooled set (n = 5688) and is therefore a different population from this table — both are correct. A diagnostic on the normalisation: the mean rescaled gap, which a correct density constant should place near 1, reads 1.074 at k=2 on this table's constant. (This subsection was corrected on 2026-07-31; the revision record, with the superseded values, is in the Supplementary Materials [15], §S1. It was corrected again on 2026-08-04: the reference row printed 0.090 / 0.42 / 0.010, which are not GUE values — 0.090 was this table's own empirical control restated. The exact Gaudin values are computed by Nyström evaluation of the sine-kernel Fredholm determinant, stable to ten digits over quadrature orders 40–240: F(0.5) = 0.11305539, 1−F(2) = 0.01771044, std = 0.42425685. Every sentence calibrated against the wrong row is rewritten above. No suppression factor and no conclusion of this paper changes, all of them being measured against the control.)

Two artifact checks were passed: (i) refining the seed grid from Δ t = 0.5 to 0.25 at t ∈ [1000,1500] left frac<0.5 unchanged (Δ = 0.0000) and found zero extra strip points; (ii) the Riemann-zero control, computed on the same pipeline, reproduces the published empirical spacing statistics for ζ's zeros at this height. It does not reproduce the exact GUE values, and is not claimed to (see the paragraph above).

Whether this stiffness is known for a-point families with |a| > 1 is an open literature question (see §6).

3.5 Distributional moments and autocorrelation

The distribution of x = dσ · log t (signed displacement times log t, with the sign convention of §1.2: dσ = σ^* − 12 = 12 − u) has negative skew throughout: pooled skew −0.713, with ∼40\% attributable to boundary points near u → 1 (σ^* → 0, where dσ → −12). After trimming u > 0.9, the residual pooled skew is −0.443, statistically significant in the large windows and growing at high t (caveat: small n at t > 15000). The left tail of dσ is heavier: more roots lie with σ^* far below 12 — equivalently u_ρ far above 12 — than the reverse. This is the same side as the sum rule's budget: D_k > 0 means net displacement at u > 12, i.e. dσ < 0 (§1.2).

The pooled weighted autocorrelation function of dσ (ordered by t_ρ^*) at lag ℓ=5 is −0.077 (3σ outside the white-noise band ±0.026); lags 1–4 are within the band. Consistent with the spacing stiffness: level repulsion produces negative serial correlation at longer lags. A multiple-comparisons correction is required before interpreting the lag-5 value as individually significant.

§4. The Littlewood Displacement Sum Rule

4.1 Proposition 2 (Littlewood sum rule) — full derivation

Standing assumptions. k≥1 fixed; 2π(k+2)≤ T₁<T₂=:T. Roots are the zeros ρ=u_ρ+iv_ρ of F_k(w)=ζ(w)−S_k(w), strip family and ROS family together. Define D_k(T₁,T₂):=∑_{Fk(ρ)=0, vρ∈(T1,T2]}(u_ρ−12).

Proposition 2. D_k(T₁,T₂) = T₂−T₁4π log(k+1) + O_k(log T₂).

4.1.1 Right normalization and analytic confinement

Lemma 4.1.1. Let G(w):=(k+1)^wF_k(w). For u>1, G(w) = 1 + ∑_{m≥ k+2}(k+1m)^w,   |G(w)−1| ≤ Σ_k(u):=∑_{m≥ k+2}(k+1m)^u. Σ_k is continuous and strictly decreasing on (1,∞) with Σ_k(1+)=∞, Σ_k(∞)=0; let u^*(k) be the unique solution of Σ_k(u)=1. Then:

1. (Confinement.) F_k has no zeros with u>u^*(k). Numerically u^*(1)=2.424, u^*(2)=3.118, u^*(3)=3.811 (computed by bisection on Σ_k(u)=1 to six significant figures; the displayed values are truncated).
2. (Right anchor.) ∫_T1^T₂|log|G(B+it)|| dt≤ 2(T₂−T₁) Σ_k(B)→0 as B→∞, uniformly in T₁,T₂; and |G(B+it)−1|≤12 for all B≥ u^*(k)+c_k, where c_k is the least c ≥ 0 with Σ_k(u^*(k)+c)≤12 (finite, since Σ_k is continuous and strictly decreasing to 0).

Proof. The series identity is the Dirichlet expansion of F_k for u>1 (valid by absolute convergence of the Dirichlet series for Re(w) > 1); the bound is the triangle inequality; F_k(w)=0 forces |G−1|≥1, i.e. Σ_k(u)≥1, i.e. u≤ u^*(k); (2) is |log(1+z)|≤2|z| for |z|≤12. ∎

Remark 4.1.1-bis (asymptotics of u^*(k)). Numerically, u^*(k)−k decreases by a near-constant increment per unit k (−0.3061 to −0.3068 over k=1,…,12, converging to ln2−1=−0.306853… by k≈30). This matches the heuristic asymptotic u^*(k)∼(k+1)ln2: writing K=k+1 and u=cK, the geometric limit of Σ_k(cK) as K→∞ is 1/(e^c−1), so Σ_k=1 forces c=ln2 at leading order (the O(1/K) correction is not bounded rigorously). The sharp numerical confirmation is in the per-unit-k increment just quoted: −0.3061 at k=1 is already within 0.3% of the limit ln2−1, reaching 4–5 significant figures by k≈12–30; the bare comparison of u^*(k) with (k+1)ln2 is only the leading-order heuristic (1.386 against the true 2.424 at k=1), the relative gap narrowing like 1/k. The confinement bound crosses u=k between k=5 and k=6 (directly computed: offsets +0.198 and −0.109; interpolated crossing ≈5.64). None of this affects the proof, which uses u^*(k) exactly; numerics in §4.2 use k≤3.

4.1.2 Left normalization: the reflected comparison function

Lemma 4.1.2. Let H(w):=χ(w) F_k(1−w). Then:

1. H(w)=ζ(w)−χ(w)P_k(w) (the centroid's leading expression: the reflected comparison function of the sum rule is ζ minus the smoothing-lemma main term, i.e. ⟨S⟩_k to leading order).
2. H is analytic on {u≥12, T₁<v≤ T₂} (the pole of ζ at w=1 and all zeros/poles of χ lie on the real axis), and χ is non-vanishing there; hence the zeros of H in that half-strip are exactly w=1−ρ for roots ρ with u_ρ<12 (the reflection ρ↦1−ρ preserves the ordinate v_ρ; since F_k has real Dirichlet coefficients, ρ is a root whenever ρ is, so conjugation is an involution on the root set and the multiset of real parts is unchanged), with Rew−12=12−u_ρ.
3. (Line identity — exact, not asymptotic.) |H(12+it)|=|F_k(12+it)| for all t, since |χ(12+it)|=1 and F_k(w)=F_k(w) (real Dirichlet coefficients), so |F_k(12−it)|=|F_k(12+it)|.
4. (Right anchor.) For u=B and t∈(T₁,T₂]: |H(B+it)−1|≤ 2^1−B+(t/2π)^1/2(2π k/t)^B(1+o(1)), which is ≤12 once B≥ B₀(k,T₁) — a threshold independent of T₂, and nonincreasing in T₁, so that B₀(k) := B₀(k,2π(k+2)) serves for every admissible T₁ — and ∫_T1^T₂|log|H(B+it)||dt→0 as B→∞. In particular every root satisfies u_ρ≥1−B₀, so the left family is finite and fully captured for B≥ B₀.

Proof. (1): F_k(1−w)=χ(1−w)ζ(w)−∑_{m≤ k}m^w−1; multiply by χ(w) and use χ(w)χ(1−w)=1. (2): χ(w)=2^wπ^w−1sin(π w/2)Γ(1−w) has zeros at even integers and poles at positive integers, all real. (4): Write H−1=(ζ(w)−1)−χ(w)P_k(w) and bound the two pieces on u≥3, t∈(T₁,T₂]. Dirichlet tail: |ζ(B+it)−1|≤∑_n≥2n^−B≤2^−B+∫₂^∞ x^−Bdx=2^−B(1+2B−1)≤2^1−B for B≥3. Second piece: |P_k(B+it)|≤∑_{m≤ k}m^B−1≤ k^B, and the Stirling asymptotic |χ(B+it)|=(t/2π)^1/2−B(1+O(1/t)) — valid as t→∞ at fixed B, which is the regime of the displayed (1+o(1)) — gives the stated two-term bound by the triangle inequality. The two claims that need uniformity in B (the threshold B₀ and the vanishing of the anchor integral as B→∞) we derive instead from an unconditional bound. The reflection formula rewrites the definition of χ as χ(w)=2^w−1π^w/(cos(π w/2) Γ(w)); for u≥3 and t≥ T₁≥2π(k+2) we have |cos(π w/2)|≥sinh(π t/2)≥0.49 e^{π t/2}, while writing u=x+m with x∈[2,3), m=⌊u⌋−2, the recursion |Γ(u+it)|=|Γ(x+it)|∏_j=0^m−1|x+j+it|≥|Γ(x+it)| (⌊u⌋−1)! and Stirling on the fixed strip x∈[2,3] (|Γ(x+it)|≥ c₀ t^3/2e^{−π t/2}, c₀ absolute, t≥6π) give, with an absolute constant C, |χ(u+it) P_k(u+it)| ≤ C (2π k)^⌊u⌋+1t^3/2 (⌊u⌋−1)!  (u≥3, t≥ T₁), a bound that decreases geometrically at successive integers once ⌊u⌋≥4π k and tends to 0 super-exponentially, uniformly in t≥ T₁ and in T₂. Choose B₀=B₀(k,T₁)≥3 so that this bound is ≤14 for all u≥ B₀ (the displayed bound is decreasing in t, so the choice made at T₁ = 2π(k+2) serves for every larger T₁: B₀ may be taken to depend on k alone); then |H(u+it)−1|≤2^1−u+14≤12 on {u≥ B₀, T₁<t≤ T₂}, so H≠0 there. Since F_k has real coefficients, a root ρ with u_ρ<12 and v_ρ∈(T₁,T₂] yields the zero 1−ρ of H at ordinate v_ρ with Re(1−ρ)=1−u_ρ; hence 1−u_ρ≤ B₀, i.e. u_ρ≥1−B₀, and the left family is finite and fully captured for B≥ B₀. Finally, on |H−1|≤12 we have |log|H||≤2|H−1| (as in Lemma 4.1.1(2)), so ∫_T1^T₂|log|H(B+it)|| dt≤2(T₂−T₁)(2^1−B+C(2π k)^⌊B⌋+1/(T₁^3/2(⌊B⌋−1)!))→0 as B→∞. ∎

4.1.3 Horizontal edges and zero-counting

Lemma 4.1.3. Let Ψ∈{G,H} and B≥ B₀.

1. |Ψ(σ+it)|≪_k t^A on [12,B]×[T₁,T₂] for an absolute constant A (A=1 suffices), uniformly in B; and |Ψ(B+it)|≥12.
2. (Edge bound.) Along either horizontal side t=T_j: argΨ(σ+iT_j)≪_klog T for σ∈[12,u^*+1] (resp. [12,B₀] for H), while for larger σ, |argΨ|≪Σ_k(σ) resp. |H−1|, which is integrable in σ uniformly in B. Hence the horizontal contributions to Littlewood's lemma are O_k(log T), uniformly in B.
3. (Counting.) N_Fk(T+1)−N_Fk(T)≪_klog T, where N_Fk(T) counts the roots with 0 < v_ρ ≤ T.
4. (Contour hygiene.) If a zero of F_k lies on an edge of the rectangle, translate T_j by at most 1 to a zero-free ordinate; by (3) and by confinement (12<u_ρ≤ u^*(k) on the right of the line by Lemma 4.1.1(1), and 1−B₀ ≤ u_ρ<12 on the left by Lemma 4.1.2(4), with B₀ = O_k(1)), each root crossed by the moved edge contributes O_k(1) to the displacement sum, so the induced change in D_k is O_k(log T).

Proof sketch and status. (1): for u≥ u^*(k) (resp. u≥ B₀ for H), |Ψ−1|≤1 by Lemmas 4.1.1–4.1.2, uniformly in B; on the remaining strip 12≤ u≤ u^*(k)+1 (resp. B₀), the Dirichlet-polynomial factors are O_k(1), |χ(u+it)|≤1+O(1/t) for u≥12 by Stirling, and |ζ(u+it)|≪ t^1/2log t by the classical convexity bound; so A=1 suffices. (2) and (3) are the standard Jensen–Borel–Carathéodory apparatus for functions of polynomial growth in a strip, run from anchor disks centered at u=B+1 (where |Ψ−1|≤12) resp. u=u^*+2 (where |G−1|<1): the number of zeros of ReΨ on a horizontal segment of length ℓ on which |Ψ|≤ M is ≪ ℓlog M+1; here M≪_k t^A by (1) and ℓ≪ u^*(k)≪ k, giving the O_k(log T) bounds. This is the argument behind the classical estimates argζ(σ+iT)≪log T and N(T+1)−N(T)≪log T ([18], ch. IX), applied to G and H in place of ζ; we use it in its standard form and do not evaluate the constants — which is why the operating tolerance ± O(log T) of §4.2.4 is a working figure rather than a derived one. (4) is immediate from (3) and the two confinement bounds.

4.1.4 Proof of Proposition 2

Apply Littlewood's lemma to G on R_B=[12,B]×[T₁,T₂] (edges zero-free by Lemma 4.1.3(4)). By Lemma 4.1.1(1) all roots with u_ρ>12 lie in R_B: 2π∑_uρ>1/2(u_ρ−12) = ∫_T1^T₂log|G(12+it)| dt − ∫_T1^T₂log|G(B+it)| dt + O_k(log T), and on the line log|G(12+it)| = log|F_k(12+it)| + 12log(k+1). Apply Littlewood's lemma to H on the same rectangle; by Lemma 4.1.2(2,4) its zeros there are in bijection with all roots u_ρ<12, with distance to the left edge 12−u_ρ: 2π∑_uρ<1/2(12−u_ρ) = ∫_T1^T₂log|H(12+it)| dt − ∫_T1^T₂log|H(B+it)| dt + O_k(log T), and by the exact line identity, log|H(12+it)|=log|F_k(12+it)| pointwise.

Subtract the two displays and let B→∞ (both right-anchor integrals vanish by Lemmas 4.1.1(2), 4.1.2(4); the horizontal bounds are uniform in B): 2π D_k(T₁,T₂) = T₂−T₁2 log(k+1) + O_k(log T₂).   ∎

Remark 4.1.4 (why the rule is "free"). The two line integrals ∫log|F_k(12+it)| dt cancel identically — no estimate of the line integral, no information about the value distribution of F_k on the line, is used. The entire displacement budget is the normalization asymmetry between the two ends of the strip: (k+1)^−w on the right versus χ on the left. Roots exactly on u=12 contribute 0 to both sides (formally: apply Littlewood's lemma on [12+δ,B] and on its reflected counterpart, and let δ→0⁺; roots on the line contribute 0 in the limit).

Remark 4.1.5 (consequences, restated for §4.2). With Δ T=T₂−T₁: D_k=(Δ T/4π)log(k+1) gives D₁[3000,8000]=275.8,  D₁[1000,1500]=27.6,  D₂[3000,8000]=437.1,  D₂[1000,1500]=43.7,  D₃[1000,1500]=55.2. The k=1 50/50 strip split is a cancellation special to a=1 (the ROS family at u>1 absorbs essentially all of D₁, each contributing u_ρ − 12 ≈ 1.2–2.0, measured mean 1.244 at k=1 [3000,8000]); for k≥2 the ROS family is sparser (vertical spacing 2π/logk+2k+1: 15.5 at k=1, 21.8 at k=2) while the budget grows like log(k+1), so the strip inherits a positive remainder — the measured 54.9% of k=2 roots with u>12 (z=6.3), equivalently 45.1% with σ^*>12.

Status of the ROS spacing scale (heuristic, not proved). For u>1, F_k(w)=∑_{m≥ k+1}m^−w=(k+1)^−wG(w) with G as in Lemma 4.1.1, and after the constant term the slowest-rotating term of G is ((k+1)/(k+2))^w, whose phase completes one revolution over the vertical period 2π/logk+2k+1; one root per revolution of the slowest phase is the generic expectation for such a balance, and that is the sole origin of the quoted scale. It is not a derived density, and we do not prove it: at the relevant u the remaining tail of G is comparable to the (k+2)-term (by the definition of u^*(k), the full tail sums to 1 there), so no two-term reduction is available, and an actual asymptotic vertical density for the ROS family would require a mean-motion analysis of the almost periodic function G in the sense of Jessen–Tornehave [17], which we have not carried out. The scale enters this paper only in the mechanism-level reading of the strip/ROS partition, here and in §3.3; every quantitative ROS statement — the closure table of §4.2.4 and the strip/ROS anatomy there — sums measured roots and does not use the formula.

4.2 The census window: analytic confinement and closure

4.2.1 The census box and its cutoff

The original census box was u∈[−0.5,1.5], with an acceptance cutoff at u = 2.0: roots with u>2.0 were excluded. The k-th ROS family, governed at its outer edge by the balance of (k+1)^−w against the tail ∑_{m≥ k+2}m^−w, reaches further right as k grows, because (k+1)^−w decays more slowly in u. The cutoff therefore excluded exactly the high-u, high-displacement points — each carrying u−12≈1.2–2.0 — and excluded more of them at larger k. (The chronology of the defect and its repair is recorded in the Supplementary Materials [15], §S1.)

4.2.2 The analytic confinement window

Lemma 4.1.1(1) makes the repaired window an analytic statement rather than a guess: every root satisfies u≤ u^*(k), the unique solution of ∑_{m≥ k+2}((k+1)/m)^u=1, with u^*(1)=2.424,   u^*(2)=3.118,   u^*(3)=3.811, each strictly below the patch ceiling k+2.1 (=3.1, 4.1, 5.1). The patch window u∈[1.4, k+2.1], abutting the original box, is therefore provably complete on the right, with margin. The data agree: observed patch maxima u=1.78 (k=1), 2.47 (k=2), 2.93 (k=3) sit well below both u^*(k) and the ceiling — no roots near the ceiling for any k.

4.2.3 Exclusion of an in-box explanation

An alternative explanation of the k≥2 deficits — ∼362 missing in-box roots (near-critical-line points with small |F₂'|) — fails on the counting density: the expected in-box count per unit height is 12πlogt2π(k+1), not k-independent. (The k-dependence enters through the leading Dirichlet coefficient (k+1)^−w of F_k, exactly as the classical a-point density (1/2π)log(t/4π) [1] reflects the leading coefficient 2^−w at k=1; the formula is used here as a counting normalisation, and its asymptotic is not proved in this paper.) Correctly:

• k=2, [3000,8000]: ≈4480 expected vs. 4442 found → ∼38 missing in-box (not 362);
• k=3, [1000,1500]: ≈309 expected vs. 299 found → ∼10 missing in-box.

Near-line stragglers carry mean (u−12)≈0.2: forty of them explain ≈8 of the −74.7 deficit — off by an order of magnitude. The in-box hypothesis is refuted; the deficit had to sit outside the box, in few points of large displacement.

4.2.4 Pre-registration and closure

Proposition 2 fixed the deficits before the patch was run: the sum rule predicted per-window shortfalls of 7.9, 8.3, 74.7, 21.2. The patch census over u∈[1.4,k+2.1] then returned:

The fifth window, k=1 [1000,1500], closed without any patch at −7.3% of D₁=27.6 (residual ≈2.0, within an assumed operating tolerance of ± O(log T)≈±10 — the constant in Lemma 4.1.3's bound is not evaluated explicitly, so this is a working figure, not a derived one): consistent, since at k=1 the truncation bites least (u^*(1)≈2.4 barely exceeds the old 2.0 cutoff — six roots in 5000 units of height at [3000,8000], and none detectable above noise in the short window).

All five windows close within tolerance; the four patched windows close within 0.6% of D_k. The sum rule closes on all five tested window/index pairs (k = 1, 2, 3).

Strip/ROS anatomy at k=1 [3000,8000]: strip (u∈[0,1]) +79.41; ROS existing +188.52; ROS patch +7.47; total 275.40 vs. D₁=275.8. ROS carries 195.99=71.3% of the total. The strip net +79 is real, not ≈0: the sum rule constrains the total displacement, not the strip alone. Roughly 70% of D_k is carried by the ROS family at large u; the strip net displacement is the small positive remainder, consistent with the sum rule constraining the total rather than the strip alone.

Instrument note. Initial census box u∈[−0.5,1.5] (hard cutoff u≤2.0) missed the high-u ROS branch, which is analytically confined to u≤ u^*(k)<k+2.1; the patch census over u∈[1.4,k+2.1] recovered 6/5/45/11 roots per window with mean (u−12)≈1.2–2.0, closing all deficits to within 0.6% of D_k. The record of the repair is in the Supplementary Materials [15], §S1.

Summary. (1) Original census: k=1 closes (−2.9%), k≥2 shows 17–38% deficits. (2) Diagnosis: ROS truncation, not in-box incompleteness — the in-box hypothesis fails the counting density by an order of magnitude. (3) Patch census over the analytic confinement window: all deficits recovered, all rows within 1% of D_k, and the recovered u-ranges independently confirm the confinement bound.

The census reported here is final; no further windows were surveyed.

§5. Connection to the Riemann–Siegel Formula

At the Riemann–Siegel cutoff NRS = ⌊√t/2π⌋, the approximate functional equation takes the form ζ(s) = S_NRS(s) + χ(s) S_NRS(1−s) + R(s),    |R(s)| = (t2π)^−σ/2 F(p) (1+o(1)),   p = {√t/2π}, where F is a bounded, oscillating, σ-independent function of the fractional part p alone, and R is the classical Riemann–Siegel remainder. The σ-dependence of the remainder is the pure power (t/2π)^−σ/2, not a bounded prefactor; at σ = 12 this reduces to the t^−1/4 rate quoted below. This is the leading term of the classical Riemann–Siegel expansion at general s (Siegel [10]; explicit remainder bounds on the line Gabcke [11]; rigorous general-s formulation Arias de Reyna [12], whose leading coefficient C₀ is σ-independent), rediscovered and confirmed at finite height by the boomerang construction — see the Post Scriptum for the measured basis. The boomerang construction (Figures 2 and 5) asks: what happens if we compute the reversed-chirality extension, adding χ(s) S_NRS(1−s) to the partial sum S_NRS(s)?

Numerically (using Newton-iterated EM ζ, t ∈ [236, 3533]): the construction lands at ζ(s) + R(s), not at 0 or ζ(s). The landing magnitude |R| is ≈ 0.25 at t ≈ 237, 0.22 at t ≈ 1419, 0.10 at t ≈ 3533 — consistent with t^−1/4 decay (the σ=12 case of the general law above). The construction is path-independent: traversing the χ-extension top-down vs bottom-up (reversing the order of the NRS terms) reaches the same terminal point to ∼ 10⁻¹⁵ (commutativity of addition). The spiral picture does not "close" at Riemann zeros; the landing misses ζ(s) by |R| ∼ t^−1/4 regardless of whether ζ(s) = 0.

On the critical line, the landing has a geometric interpretation: by |χ(12 + it)| = 1 and the real-coefficient symmetry |S_NRS(12 − it)| = |S_NRS(12 + it)|, the landing vector is 2 proj_eθ(R) where e_θ = e^{i argχ(12+it)/2}. At a Riemann zero, ζ(s) = 0, so |landing|= |R| is the entire displacement from the origin. This is precisely the Riemann–Siegel formula for Z(t) = e^iϑ(t)ζ(12+it): the "boomerang" rediscovers it geometrically.

The construction illustrates the second scale at which the geometric picture of §1 encounters classical machinery: at the packet scale (N ∼ t/k) it produces P_k and the smoothing lemma; at the RS scale (N ∼ √t) it produces the RS remainder. The two scales are not in competition — the same truncation-plus-stationary-phase machinery used in §2, applied at the cutoff N ∼ √t/2π rather than N ∼ t/2π k, is the classical route to the approximate functional equation and, pushed to full asymptotic depth, to the RS formula ([10], [12]).

Forward reference (companion paper). A third regime of the same spiral is developed quantitatively in [13]: past the last saddle x₁ = t/2π the partial sum S_M closes into coils around ζ(s), and on the band M ∈ [2.5 x₁, 6 x₁] the Euler–Maclaurin tail converges, giving the coil radius in closed form, |S_M(s) − ζ(s)|  =  M^1−σ|s−1|·u/2sin(u/2)·(1+O(1/t)),    u = t/M, with derived slope-defect constant −0.250778 and winding number −1 per coil at smooth order ([13], ch. 7). The reversed-chirality extension used above belongs to the same instrument family as the winding-number classifiers and the pair-product reality criterion of [13], ch. 8–9.

§6. Scope, Framing, and Open Questions

6.1 What this paper does and does not claim

The paper identifies, proves, and numerically confirms two structural properties of the k-family of partial-sum roots F_k(w) = 0 in the critical strip:

1. Theorem 1 (the smoothing lemma): the centroid offset off_k equals χ P_k — equivalently, the packet centroid ⟨S⟩_k equals ζ − χ P_k — to within a χ-proportional O(t^−1/2+ε) error, uniform on compact σ-sets. The proof is via Poisson summation and stationary phase, with two standard exponential-integral estimates used as inputs (the status note of §2.6); the error exponent, the χ-proportionality, and the oscillating prefactor are each confirmed by the census data.

2. Proposition 2 (the Littlewood sum rule): the total signed displacement of all F_k-roots from the critical line grows as (Δ T/4π)log(k+1). The proof uses Littlewood's lemma twice, with the key cancellation being the pointwise equality |H(12+it)| = |F_k(12+it)| on the line (Lemma 4.1.2(3)); no estimate of log|F_k| on the line is needed.

The route — geometric/packet/centroid — is, to our knowledge, novel as a starting point for these results. The destinations are classical: the displacement rate c ≈ 1 is a finite-T avatar of Levinson's a-point clustering theorem [1]; the spacing statistics connect to GUE universality in the sense of Montgomery–Odlyzko [4,5]; the sum rule is an application of Littlewood's lemma [6].

The paper makes no claim about the Riemann Hypothesis. The roots of F_k(w) = 0 in the strip are objects distinct from the zeros of ζ(s); their distribution is governed by the sum rule and the smoothing lemma, both of which apply equally whether or not RH holds. The framing is: the geometry of partial sums is a high-precision microscope on the finite-T value distribution of ζ; the mechanisms it exposes are explained by classical theorems, not enforcing them.

Two papers, one finding. This paper and its companion [13] — published with a Supplementary Materials volume [14] — form one two-part, modular finding. The present paper is the framework and the census: the smoothing identity, the sum rule, and the frozen measurements. The companion paper identifies the mechanism of Theorem 1's error term, answers the literature questions of §6.2 (status notes added below), reports its instruments and closure result, and carries the corresponding negative results. The present paper's theorems, proofs, and frozen measurements stand independently: nothing in §1–§5 requires [13] or [14]; the companions are cited at exactly those points where their material is described, not as a dependency of the results themselves.

6.2 Open questions

(a) Is c ≈ 1.02 known? The constant c = ⟨|dσ| · log t ⟩ = 1.023 ± 0.011 governs the displacement rate. Levinson's theorem [1] gives the density of a-points; Selberg's value-distribution results [2] and Steuding's monograph [3, ch. 7] are the closest references. Whether c has an exact closed form or has been previously identified as a universal a-point constant is open. Status (answered in [14], §12.2): the mean displacement law is known and is not a constant — unconditionally ⟨|dσ|⟩·log t ∼ (1/√π)√loglog t (Selberg [2]; Tsang [19]), with no closed-form constant. The statement of record becomes: c = 1.023 ± 0.011 is a finite-height plateau of a quantity whose asymptotic law is 0.564√loglog t (1+o(1)); the census range cannot discriminate the two shapes. The census itself — apparently the first numerical census of a-point displacements at all — the plateau value, and everything at k ≥ 2 have no located counterpart.

(b) Is the sub-GUE stiffness known? The suppression of small gaps by a factor of 6 relative to the Riemann-zero distribution (and GUE) is the sharpest numerical signal in the census. Whether this stiffness is known for the a-point distribution of ζ at any |a| is an open literature question. Status (answered in [14], §12.4, by a failed-search record): no spacing statistics for a-points of ζ — any a, any height, theorem, conjecture, or numerics — were located, nor any a-point pair correlation. The measurement appears to be the first of its kind; the closest analog-in-spirit is the spacing rigidification of ξ'-zeros (Farmer–Gonek; Farmer–Rhoades [20]), whose mechanism does not transfer as stated and does not predict the observed tail identity.

(c) Is the negative skew known? The distribution of dσ · log t (dσ = σ^* − 12, the sign convention of §1.2 and §3.5) has residual skew −0.443 after boundary trimming, growing at high t; the measured heavy tail lies on the side σ^* < 12, i.e. roots at u_ρ > 12. This asymmetry is not predicted by the sum rule (which constrains the mean, not the shape) and may reflect the non-symmetric value distribution of ζ in the strip. Status (compared in [14], §12.3): the qualitative one-sidedness has a classical counterpart conditional on RH (Selberg [2]; Tsang [19]: a one-sided Gaussian tail at scale √loglog T/log T on one side of the line, excursions on the other side confined to a strictly smaller scale, with Selberg's conjectured asymptotic split of 3/4 to 1/4), which also anticipates the measured growth of the skew with height. Caution: the classical statements are phrased for the a-point abscissa u_ρ, while the measured skew is in dσ = 12 − u_ρ; the identification of the heavy sides under this sign reversal is carried in [14] and is not re-verified here. A qualitative literature comparison at most — not evidence for or against RH, since the present measurement is unconditional. The skewness coefficients themselves, the boundary-trim decomposition, and the lag-5 autocorrelation have no located counterpart.

(d) Partial-sum zero literature. The zeros of S_N(s) (partial sums, fixed N) have been studied by Spira [7], Montgomery [8], and Borwein et al. [9]. The objects of this paper are different — roots of ζ(w) = S_k(w), a comparison equation. We note the exact identity F_k(w) = ζ(w) − S_k(w) = ζ(w, k+1), the Hurwitz zeta function at integer parameter k+1: for k=1 the roots are the classical 1-points of ζ (Landau; Levinson [1]), while for k ≥ 2 the zero distribution of ζ(w,α) at integer α ≥ 3 appears to be unstudied — the Hurwitz-zero literature (Davenport–Heilbronn 1936 [21]; Cassels 1961 [22]; Spira 1976 [23]) treats 0 < α ≤ 1, where the phenomenology (zeros dense in σ, including σ > 1) is qualitatively opposite to the critical-line clustering measured here at k ≥ 2 and known at k = 1 by Levinson [1]. Connecting the two families (e.g., via the perturbation F_k(w) = 0 → S_N(w) = 0 as N → ∞) is a natural direction. Status (developed in [14], §12.1): the integer-parameter Hurwitz positioning is carried through the literature there; the F_k → S_N bridge has no located antecedent and remains open ([13], ch. 14).

6.3 Calibration note

The paper was developed by combining symbolic analysis (proof routes, sum-rule derivation) with a high-precision computational census (root-finding via Euler–Maclaurin Newton). We distinguish: (i) verified mathematical results — Theorem 1 and Proposition 2 with complete proofs; (ii) empirical measurements — the specific numerical constants confirmed at the indicated precision and windows; (iii) structural predictions — the sum rule's deficits were computed and stated before the patch census was run (§4.2.4). The match between those predictions and the census outcomes (≤ 0.6% residuals) is independent confirmation that the census recovers what the proved proposition predicts, rather than a calibration artifact — the proposition's correctness rests on the proof of §4.1, not on this match.

Post Scriptum: the general-σ remainder — measurement and classical identification

An earlier version of this paper stated the §5 remainder as |R(s)| ∼ C(σ) t^−1/4 with C(σ) bounded on the strip; the measurement below corrects this, and the revision is recorded in the Supplementary Materials [15], §S1. Over a 300-point band at t ≈ 73850 (t/2π ≈ 11746), the σ-ratio test returned |R|(0.3)|R|(0.5) = 2.55281  vs  (t2π)^0.1 = 2.5528,    |R|(0.7)|R|(0.5) = 0.39173  vs  (t2π)^−0.1 = 0.3917 (rel-std ≤ 4×10⁻⁵ across the band), and at the strip boundary |R|(0.0)/|R|(0.5) = 10.41227 vs (t/2π)^0.25 at band centre = 10.41227, with pointwise deviation ≤ 6.1×10⁻⁷ (n = 300): the exponent is −σ/2 to five significant figures, including at σ = 0. Pooled over all σ-legs and heights (n = 4858, t up to 2×10⁶), the rescaled remainder K := |R|·(t/2π)^σ/2 collapses to a function of the single variable p = {√t/2π} alone (Fourier-3 fit R² = 0.993; σ-uniformity CV 5×10⁻⁸ within t-cells), and arg R is σ-independent to ≤ 0.005 rad. Hence the corrected law displayed in §5: |R(s)| = (t2π)^−σ/2 F(p) (1+o(1)),    p = {√t/2π}, with F bounded, oscillating, and σ-independent. (The measurements of this Post Scriptum were made with dedicated high-precision probe instruments, separate from the float64 census evaluator of §3.1; the §3.1 precision range does not bound them.)

Only after the measurement was the law identified as classical: it is the leading term of the Riemann–Siegel expansion at general s (Siegel [10]; explicit remainder bounds on the line, Gabcke [11]; rigorous general-s formulation, Arias de Reyna [12]). In [12] the leading coefficient C₀(p) is σ-independent by construction — σ enters the coefficient recurrence only from C₁ on — so the measured σ-independence of F and of arg R is the classical structure, not a new phenomenon. We claim no priority for the law. What the geometric route contributes is the measurement itself: the −σ/2 power and the single-variable collapse were obtained from the boomerang construction of §5 before the classical expansion was consulted. All critical-line statements of §5 are the σ = 12 case and were unaffected by the correction.

References

[1] N. Levinson, Almost all roots of ζ(s) = a are arbitrarily close to σ = 1/2, Proc. Nat. Acad. Sci. USA 72 (1975), 1322–1324.

[2] A. Selberg, Old and new conjectures and results about a class of Dirichlet series, in Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989), Univ. Salerno (1992), 367–385.

[3] J. Steuding, Value-Distribution of L-Functions, Lecture Notes in Mathematics 1877, Springer, 2007.

[4] H. L. Montgomery, The pair correlation of zeros of the zeta function, Proc. Sympos. Pure Math. 24, Amer. Math. Soc. (1973), 181–193.

[5] A. M. Odlyzko, On the distribution of spacings between zeros of the zeta function, Math. Comp. 48 (1987), 273–308.

[6] J. E. Littlewood, On the zeros of the Riemann zeta-function, Proc. Cambridge Philos. Soc. 22 (1924), 295–318.

[7] R. Spira, Zeros of sections of the zeta function. I, Math. Comp. 20 (1966), 542–550; II, Math. Comp. 22 (1968), 163–173.

[8] H. L. Montgomery, Zeros of approximations to the zeta function, in Studies in Pure Mathematics: To the Memory of Paul Turán, Birkhäuser (1983), 497–506.

[9] P. Borwein, G. Fee, R. Ferguson, A. van der Waall, Zeros of partial sums of the Riemann zeta function, Experiment. Math. 16 (2007), 21–40.

[10] C. L. Siegel, Über Riemanns Nachlaß zur analytischen Zahlentheorie, Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik, Abt. B: Studien 2 (1932), 45–80. Reprinted in Gesammelte Abhandlungen, Vol. I, Springer, 1966. English translation: E. Barkan, D. Sclar, arXiv:1810.05198.

[11] W. Gabcke, Neue Herleitung und explizite Restabschätzung der Riemann-Siegel-Formel, Dissertation, Georg-August-Universität zu Göttingen, 1979. DOI 10.53846/goediss-5113.

[12] J. Arias de Reyna, High precision computation of Riemann's zeta function by the Riemann-Siegel formula, I, Math. Comp. 80 (2011), no. 274, 995–1009.

[13] O. Dvořák, Packet Centroids II: The Fresnel Mechanism, Coil Geometry, and Zero Conditions of the Partial-Sum Walk, companion paper, in preparation (2026).

[14] O. Dvořák, Packet Centroids II: Supplementary Materials — Literature Positioning, Negative Results, and the Identity Ledger, companion document (2026).

[15] O. Dvořák, Packet Centroids: Supplementary Materials, companion document (2026).

[16] O. Dvořák, Packet Centroids: Visuals, companion document (2026).

[17] B. Jessen, H. Tornehave, Mean motions and zeros of almost periodic functions, Acta Math. 77 (1945), 137–279.

[18] E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, 2nd ed., revised by D. R. Heath-Brown, Oxford University Press, 1986.

[19] K.-M. Tsang, The Distribution of the Values of the Riemann Zeta-Function, Ph.D. thesis, Princeton University, 1984.

[20] D. W. Farmer, R. C. Rhoades, Differentiation evens out zero spacings, Trans. Amer. Math. Soc. 357 (2005), 3789–3811.

[21] H. Davenport, H. Heilbronn, On the zeros of certain Dirichlet series, J. London Math. Soc. 11 (1936), 181–185.

[22] J. W. S. Cassels, Footnote to a note of Davenport and Heilbronn, J. London Math. Soc. 36 (1961), 177–184.

[23] R. Spira, Zeros of Hurwitz zeta functions, Math. Comp. 30 (1976), 863–866.

Project materials

The complete project — all papers with their supplementary and visual companions, and the data behind them — is available at zeta.pukapasoft.xyz.

This paper is one part of a series. Its own companion files are Packet Centroids: Supplementary Materials [15] (revision record and bibliographic verification) and Packet Centroids: Visuals [16] (the five figures, with the data behind each).

Nothing in this work decides the location of any zero of the Riemann zeta function, and no result here is progress toward a proof of the Riemann Hypothesis.

Figures

4 figures. Each opens with the commentary the paper wrote for it; click a thumbnail for the full-size render.

This file carries every figure of the paper named above, with its caption exactly as the paper states it and the data behind it. No figure computes a mathematical quantity. The paper's own text remains the sole document of record.

Figure 1

Figure 1 (spiral + packet anatomy). The partial sum S_N(12 + 5000i) traced on the Argand plane for N = 1 to ⌊x₁ ⌋. The k=1 and k=2 packets are shaded; the respective centroids (filled circles) lie at ⟨S⟩_k = ζ − off_k ≈ ζ − χ P_k.

Figure 2

Figure 2 (boomerang pair, the banked data table). Top-down (solid) and bottom-up (dashed) reversed-chirality extensions of S_NRS at Riemann zero ρ₁₀₀₀ (t ≈ 1419). Both traversal orders land at the same terminal point ζ(s) + R(s) (cross), with |R| ≈ 0.22, consistent with (t/2π)^−1/4 decay [erratum 2026-07-19, ratified — item 7]. At a non-zero (triangle), the landing shifts to ζ(s) + R(s); the offset from ζ is not small. See §5.

Figure 3

Figure 3 (displacement collapse). The empirical distribution of |dσ|·log t for all census windows, overlaid. Mean c = 1.023 (vertical line); loglog prediction (dashed, scaled to the same mean) has the wrong shape.

Figure 4

Figure 4 (spacing histogram). Rescaled gap density for 1-points (shaded, k=1, n=5688 gaps, the pooled seven-window census) vs Riemann zeros (open histogram, n=5359 gaps, the single window t ∈ [3000,8000]) and GUE curve. Small-gap suppression is the dominant signal; the large-gap tails are identical. (Both are normalised densities, so the shapes are comparable; the counts differ because the two populations are drawn from different height ranges. The table in §3.4 above is computed on the single window for both, and its k=1 small-gap cell therefore reads 0.0160 where this pooled set gives 0.0165 — E-P1D-2.)

Figure 5 (fig2_boomerang (same render as Figure 2))

Figure 5 (boomerang pair): see caption at Figure 2. The key visual point is that the two path shapes are distinct while the endpoint is identical — path-independence made visible.

Note on the figure set

Five captions, four renders: Figure 5 is the Figure 2 render read for a different point, and the paper says so in its own caption. Figure 2's data file is named in the caption itself.

Supplementary materials

The audit layer: how the numbers above were checked, what was corrected, and what is owed to whom.

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