Paper v4: figures (scaling law, race, histogram, certificate map), Gallagher/Brent/Rankin/Lemke Oliver-Soundararajan math and references; cycle 5 launch infra (SEG env, idempotent workers)

Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>

simon-pierre boucher committed 1 mo ago (Aug 6, 2026) parent 9b7916b

9 changed files +377 −12

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		`@@ -13,6 +13,7 @@`
13	13	\usepackage{algpseudocode}
14	14	\usepackage[hidelinks]{hyperref}
15	15	\usepackage{xcolor}
	16	+\usepackage{graphicx}
16	17
17	18	\newtheorem{theorem}{Theorem}
18	19	\newtheorem{lemma}{Lemma}
		`@@ -33,7 +34,7 @@ Certified Deserts, a Refuted Race Conjecture, and an Empirical Scaling Law\\`
33	34	for Consecutive-Gap Correlations}
34	35	\author{Simon-Pierre Boucher\\
35	36	\texttt{contact@spboucher.ai}}
36		−\date{August 6, 2026 --- working draft v3}
	37	+\date{August 6, 2026 --- working draft v4}
37	38
38	39	\begin{document}
39	40	\maketitle
		`@@ -141,13 +142,47 @@ Cram\'er's heuristic suggests $\limsup \mathrm{CSG} = 1$; Granville's correction`
141	142	$\limsup \ge 2e^{-\gamma} \approx 1.1229$. The largest CSG value ever observed is
142	143	$0.9206\ldots$ (Nyman's gap 1132 after 1693182318746371).
143	144
144		−\paragraph{Cram\'er random model.}
145		−In the Cram\'er model each integer $n$ is independently ``prime'' with probability
146		−$1/\ln n$; consecutive gaps near $x$ are then asymptotically i.i.d.\
147		−$\mathrm{Exponential}(\lambda)$ up to discretization,
148		−so \emph{any} statistic that vanishes for independent gaps (such as the correlations of
149		−Section~\ref{sec:c4}) measures genuinely arithmetic structure. We use this as a
150		−significance filter (the ``Cram\'er/Maier test'').
	145	+\paragraph{Cram\'er random model and Gallagher's theorem.}
	146	+In the Cram\'er model \cite{Cramer} each integer $n$ is independently ``prime'' with
	147	+probability $1/\ln n$; consecutive gaps near $x$ are then asymptotically i.i.d.\
	148	+$\mathrm{Exponential}(\lambda)$ up to discretization. Remarkably, this exponential law
	149	+is not merely a toy: Gallagher \cite{Gallagher} proved that the uniform Hardy--Littlewood
	150	+conjecture implies that the number of primes in the short interval $(x,\, x + \eta \ln x]$
	151	+converges to a Poisson law,
	152	+\begin{equation}
	153	+\#\bigl\{ p \in (x,\, x + \eta \ln x] \bigr\} \;\xrightarrow{d}\; \mathrm{Poisson}(\eta),
	154	+\qquad\text{equivalently}\qquad
	155	+\Pr\!\Bigl( \frac{g}{\ln x} > t \Bigr) \;\to\; e^{-t},
	156	+\label{eq:gallagher}
	157	+\end{equation}
	158	+because the singular series averages to one:
	159	+$\sum_{\mathcal H \subset [1,h],\, \|\mathcal H\| = k} \Sing(\mathcal H) \sim h^k / k!$.
	160	+Thus the \emph{marginal} exponential shape of gaps carries no arithmetic information at
	161	+leading order --- but joint statistics of \emph{consecutive} gaps do, since Gallagher's
	162	+averaging does not force independence at the $1/\lambda$ correction level. Any statistic
	163	+that vanishes for independent gaps (such as the correlations of Section~\ref{sec:c4})
	164	+therefore measures genuinely arithmetic structure. We use this as a significance filter
	165	+(the ``Cram\'er/Maier test'', cf.\ Maier's matrix method \cite{Maier} which shows the
	166	+Cram\'er model's predictions fail in short intervals).
	167	+
	168	+\paragraph{Refined single-gap model.}
	169	+The standard first-order heuristic for the count of gaps of size exactly $g$ is
	170	+\begin{equation}
	171	+N(g, x) \;\approx\; C(g) \int_2^x e^{-g/\ln t}\, \frac{dt}{(\ln t)^2},
	172	+\label{eq:single-gap}
	173	+\end{equation}
	174	+(interior compositeness absorbed into the exponential); Brent \cite{Brent} showed that
	175	+accurate gap counts require the full inclusion--exclusion over interior prime patterns,
	176	+\begin{equation}
	177	+N(g, x) \;\approx\; \sum_{k \ge 0} (-1)^k
	178	+\sum_{\substack{0 < h_1 < \cdots < h_k < g}}
	179	+\Sing\bigl(\{0, h_1, \dots, h_k, g\}\bigr) \int_2^x \frac{dt}{(\ln t)^{k+2}},
	180	+\label{eq:brent}
	181	+\end{equation}
	182	+truncated at moderate $k$. Our own measurements confirm the necessity: the naive
	183	+\eqref{eq:single-gap} exhibits structured residuals (Section~\ref{sec:lessons}), and its
	184	+pairwise analogue underpredicts the gap correlation threefold
	185	+(Section~\ref{sec:model}).
151	186
152	187	\section{Methodological protocol}
153	188	\label{sec:protocol}
		`@@ -383,6 +418,17 @@ $D = +26, +359, +55$, suggesting:`
383	418	$N(2, x) > N(4, x)$ for all $x \ge 10^6$.
384	419	\end{conjecture}
385	420
	421	+\begin{figure}[t]
	422	+\centering
	423	+\includegraphics[width=\textwidth]{figs/fig_race.pdf}
	424	+\caption{The 2-vs-4 race $D(x) = N(2,x) - N(4,x)$ on a symlog scale. Blue: exact
	425	+$D$ sampled at $\sim$60 log-spaced points per decade up to $10^{10}$ (recomputed for
	426	+this figure); orange diamonds: campaign checkpoints to $10^{13}$. The lead changes
	427	+indefinitely; the walk stays within its diffusive envelope
	428	+$O(\sqrt{N(2,x)+N(4,x)})$.}
	429	+\label{fig:race}
	430	+\end{figure}
	431	+
386	432	\begin{result}[\statuslbl{refuted with minimal counterexample}]
387	433	\label{res:race}
388	434	A full-resolution scan (every gap event inspected via cumulative sums) shows
		`@@ -423,6 +469,21 @@ Rubinstein--Sarnak analysis for $\pi(x; 4, 1)$ vs $\pi(x; 4, 3)$).`
423	469	\section{Result II: a certified prime desert from a hybrid covering system}
424	470	\label{sec:desert}
425	471
	472	+Explicit long runs of composites go back to Westzynthius \cite{West}, Erd\H{o}s
	473	+\cite{Erdos35} and Rankin \cite{Rankin}, whose residue-shifting refinements of the
	474	+primorial construction give
	475	+\begin{equation}
	476	+G(x) \;\ge\; (c + o(1))\,
	477	+\frac{\ln x \,\ln\ln x \,\ln\ln\ln\ln x}{(\ln\ln\ln x)^2}
	478	+\end{equation}
	479	+for the maximal gap $G(x)$ below $x$ (Rankin's form; the FGKMT/Maynard breakthroughs
	480	+\cite{FGKMT} improved the exponent structure of the triple-log factor and settled the
	481	+Erd\H{o}s \$10{,}000 problem). Our construction is a computational descendant of exactly
	482	+this residue-shifting idea, with two modern twists: the greedy choice of residues is
	483	+\emph{certified per position}, and a controlled number of uncovered positions
	484	+(``holes'') is permitted and certified a posteriori --- which is what lifts the achieved
	485	+length from the classical regime to $260$ at 25 digits.
	486	+
426	487	\begin{result}[\statuslbl{certified}]
427	488	\label{res:desert}
428	489	Let $N = 1{,}116{,}336{,}781{,}708{,}038{,}449{,}369{,}693$ (25 digits). Then $N$ and
		`@@ -469,6 +530,18 @@ endpoints, search over the shift $t$: at $t = 580$,`
469	530	Hence every $N + i$, $1 \le i \le 259$, carries an explicit compositeness certificate
470	531	and both endpoints an unconditional primality certificate: the gap is exactly 260.
471	532
	533	+\begin{figure}[t]
	534	+\centering
	535	+\includegraphics[width=\textwidth]{figs/fig_certificate.pdf}
	536	+\caption{Certificate map of the 260-desert. Each interior position $i$ carries an
	537	+explicit compositeness certificate for $N+i$: blue dots — smallest covering prime
	538	+(valid for \emph{every} CRT shift $t$, Lemma 2); orange diamonds — explicit trial
	539	+factor at the 15 holes; aqua square — the one hole certified by a strong Miller--Rabin
	540	+witness (plotted at an arbitrary height: a witness proves compositeness without
	541	+exhibiting a factor).}
	542	+\label{fig:certificate}
	543	+\end{figure}
	544	+
472	545	\paragraph{Endpoint-search heuristic.}
473	546	$N$ and $N + L + 1$ are constructed coprime to all $p \le 59$, so by the standard
474	547	sieve heuristic each is prime with probability
		`@@ -520,6 +593,19 @@ $10^{13}$ & $-0.54264$ & $-0.25247$ \\`
520	593	\end{tabular}
521	594	\end{center}
522	595
	596	+\begin{figure}[t]
	597	+\centering
	598	+\includegraphics[width=\textwidth]{figs/fig_scaling.pdf}
	599	+\caption{Left: the scaling law. Circles: the 16 measured checkpoints for lag-1 (blue)
	600	+and lag-2 (aqua) with their two-parameter fits $c + d/\lambda$; orange diamonds: the
	601	+three out-of-sample predictions later confirmed by measurement ($10^{12}$, $10^{13}$
	602	+lag-1; $10^{13}$ lag-2); gray squares: the standing predictions at $10^{14}$ and
	603	+$10^{15}$. Right: the same lag-1 data against the first-order Hardy--Littlewood
	604	+triple-correlation model \eqref{eq:triple-model}, which has the right sign and drift
	605	+form but $\approx 32\%$ of the magnitude.}
	606	+\label{fig:scaling}
	607	+\end{figure}
	608	+
523	609	\begin{conjecture}[C4$'$; \statuslbl{conjectured}, one prediction confirmed]
524	610	\label{conj:c4p}
525	611	With $\lambda = \lnx$,
		`@@ -611,8 +697,13 @@ out-of-sample ones in (i).`
611	697
612	698	We have not located these constants in the literature. The qualitative anticorrelation
613	699	of consecutive gaps is known empirically, and $c, c_2$ may be derivable from existing
614		−conditional results (e.g.\ pair-correlation heuristics); we make no novelty claim
615		−pending a proper literature pass.
	700	+conditional results; we make no novelty claim pending a proper literature pass. The
	701	+closest well-studied phenomenon is the Lemke Oliver--Soundararajan bias
	702	+\cite{LOS}: consecutive primes are measurably biased against repeating their residue
	703	+class mod $q$, with a $1/\ln x$-order correction governed by the same triple singular
	704	+series that appears in \eqref{eq:triple-model} --- our $\rho(x)$ can be viewed as the
	705	+gap-space aggregate of those pattern biases, and their conditional analysis is the
	706	+natural route to a proof of Conjecture~\ref{conj:c4p}.
616	707
617	708	\section{Result IV: the first-order Hardy--Littlewood model fails quantitatively}
618	709	\label{sec:model}
		`@@ -666,6 +757,16 @@ $\operatorname{argmax}_g N(g, x) = 6$ at every checkpoint from $10^6$ to $10^{12`
666	757	(\statuslbl{verified}), consistent with the Odlyzko--Rubinstein--Wolf conjecture that 6
667	758	is the champion from $\approx 947$ to $\approx 1.7\times 10^{35}$.
668	759
	760	+\begin{figure}[t]
	761	+\centering
	762	+\includegraphics[width=\textwidth]{figs/fig_hist.pdf}
	763	+\caption{The gap histogram $N(g, 10^8)$ (log scale). Multiples of 6 (blue) are strict
	764	+local maxima --- the factor $(p-1)/(p-2) = 2$ at $p = 3$ in the singular series
	765	+\eqref{eq:singular-pair}; the jumping champion is $g = 6$ at every checkpoint from
	766	+$10^6$ to $10^{13}$.}
	767	+\label{fig:hist}
	768	+\end{figure}
	769	+
669	770	\paragraph{Local dominance of multiples of 6.}
670	771	By \eqref{eq:singular-pair}, $3 \mid g$ multiplies $C(g)$ by
671	772	$(3-1)/(3-2) = 2$, so multiples of 6 should be strict local maxima of
		`@@ -784,6 +885,25 @@ Re-verification of the two headline certificates requires only a stock Python~3:`
784	885	\emph{Lucas pseudoprimes}, Math.~Comp.~35 (1980), 1391--1417.
785	886	\bibitem{RS} M.~Rubinstein, P.~Sarnak, \emph{Chebyshev's bias},
786	887	Experiment.~Math.~3 (1994), 173--197.
	888	+\bibitem{Cramer} H.~Cram\'er, \emph{On the order of magnitude of the difference between
	889	+consecutive prime numbers}, Acta Arith.~2 (1936), 23--46.
	890	+\bibitem{Gallagher} P.~X.~Gallagher, \emph{On the distribution of primes in short
	891	+intervals}, Mathematika 23 (1976), 4--9.
	892	+\bibitem{Maier} H.~Maier, \emph{Primes in short intervals}, Michigan Math.~J.~32
	893	+(1985), 221--225.
	894	+\bibitem{Brent} R.~P.~Brent, \emph{The distribution of small gaps between successive
	895	+primes}, Math.~Comp.~28 (1974), 315--324.
	896	+\bibitem{LOS} R.~J.~Lemke Oliver, K.~Soundararajan, \emph{Unexpected biases in the
	897	+distribution of consecutive primes}, Proc.~Natl.~Acad.~Sci.~113 (2016), E4446--E4454.
	898	+\bibitem{West} E.~Westzynthius, \emph{\"Uber die Verteilung der Zahlen, die zu den $n$
	899	+ersten Primzahlen teilerfremd sind}, Comm.~Phys.-Math.~Soc.~Sci.~Fenn.~5 (1931), 1--37.
	900	+\bibitem{Erdos35} P.~Erd\H{o}s, \emph{On the difference of consecutive primes},
	901	+Q.~J.~Math.~6 (1935), 124--128.
	902	+\bibitem{Rankin} R.~A.~Rankin, \emph{The difference between consecutive prime numbers},
	903	+J.~London Math.~Soc.~13 (1938), 242--247.
	904	+\bibitem{OeSHP} T.~Oliveira e Silva, S.~Herzog, S.~Pardi, \emph{Empirical verification
	905	+of the even Goldbach conjecture and computation of prime gaps up to $4\times 10^{18}$},
	906	+Math.~Comp.~83 (2014), 2033--2060.
787	907	\bibitem{Granville} A.~Granville, \emph{Harald Cram\'er and the distribution of prime
788	908	numbers}, Scand.~Actuar.~J.~1 (1995), 12--28.
789	909	\bibitem{A005250} OEIS Foundation, \emph{Sequences A005250, A002386 (record gaps

added src/figures.py +239 −0

		`@@ -0,0 +1,239 @@`
	1	+#!/usr/bin/env python3
	2	+# =============================================================================
	3	+# figures.py — paper figures from existing campaign data
	4	+# Author: Simon-Pierre Boucher — contact@spboucher.ai
	5	+# =============================================================================
	6	+# Generates (into paper/figs/):
	7	+# fig_scaling.pdf rho·lnx vs lnx: data, fit, confirmed predictions, HL model
	8	+# fig_race.pdf D(x) = N2-N4 lead changes (fine trace to 1e10 + checkpoints)
	9	+# fig_hist.pdf N(g, 1e8) with the mod-6 comb structure
	10	+# fig_certificate.pdf the 260-gap desert certificate map
	11	+# Palette (validated, dataviz method): blue #2a78d6, orange #eb6834, aqua #1baf7a
	12	+# Deterministic; the race trace is recomputed locally (scan to 1e10, ~40 s).
	13	+# =============================================================================
	14	+
	15	+import json
	16	+from math import log
	17	+from pathlib import Path
	18	+
	19	+import matplotlib
	20	+matplotlib.use("Agg")
	21	+import matplotlib.pyplot as plt
	22	+import numpy as np
	23	+
	24	+from core import primes_upto, sieve_segment
	25	+
	26	+ROOT = Path(__file__).resolve().parent.parent
	27	+FIGS = ROOT / "paper" / "figs"
	28	+FIGS.mkdir(exist_ok=True)
	29	+DATA = ROOT / "data"
	30	+
	31	+BLUE, ORANGE, AQUA = "#2a78d6", "#eb6834", "#1baf7a"
	32	+INK, MUTED = "#0b0b0b", "#52514e"
	33	+
	34	+plt.rcParams.update({
	35	+ "font.size": 8.5, "axes.labelsize": 9, "axes.titlesize": 9.5,
	36	+ "axes.spines.top": False, "axes.spines.right": False,
	37	+ "axes.linewidth": 0.7, "xtick.major.width": 0.7, "ytick.major.width": 0.7,
	38	+ "grid.linewidth": 0.5, "grid.alpha": 0.25, "legend.frameon": False,
	39	+ "figure.dpi": 150, "text.color": INK, "axes.edgecolor": MUTED,
	40	+ "xtick.color": MUTED, "ytick.color": MUTED, "axes.labelcolor": INK,
	41	+})
	42	+
	43	+
	44	+def load_checkpoints():
	45	+ c2 = json.load(open(DATA / "cycle2_c4_4e10_M3U96a.json"))["checkpoints"]
	46	+ c4 = json.load(open(DATA / "cycle4_c4_1e13.json"))["checkpoints"]
	47	+ pts = [(float(x), v["rho1_lnx"], v["rho2_lnx"], v["D_N2_minus_N4"])
	48	+ for x, v in c2.items() if 1e8 <= float(x) < 1e10]
	49	+ pts += [(float(x), v["rho1_lnx"], v["rho2_lnx"], v["D_N2_minus_N4"])
	50	+ for x, v in c4.items()]
	51	+ return sorted(pts)
	52	+
	53	+
	54	+def fit_cd(lams, vals):
	55	+ A = np.vstack([np.ones_like(lams), 1 / lams]).T
	56	+ (c, d), *_ = np.linalg.lstsq(A, vals, rcond=None)
	57	+ return c, d
	58	+
	59	+
	60	+def fig_scaling(pts):
	61	+ lam = np.array([log(x) for x, *_ in pts])
	62	+ v1 = np.array([p[1] for p in pts])
	63	+ v2 = np.array([p[2] for p in pts])
	64	+ c1, d1 = fit_cd(lam, v1)
	65	+ c2_, d2_ = fit_cd(lam, v2)
	66	+ model = json.load(open(DATA / "model_c4.json"))["predictions"]
	67	+ ml = np.array([m["lambda"] for m in model.values()])
	68	+ mv = np.array([m["rho_times_lambda"] for m in model.values()])
	69	+
	70	+ fig, (ax, axb) = plt.subplots(1, 2, figsize=(6.4, 2.7),
	71	+ gridspec_kw={"width_ratios": [3, 2]})
	72	+ ll = np.linspace(18.0, 34.6, 200)
	73	+ # panel (a): data + fits + confirmed predictions
	74	+ ax.plot(ll, c1 + d1 / ll, color=BLUE, lw=1.4)
	75	+ ax.plot(ll, c2_ + d2_ / ll, color=AQUA, lw=1.4)
	76	+ ax.plot(lam, v1, "o", ms=3.4, color=BLUE, mfc="white", mew=1.1)
	77	+ ax.plot(lam, v2, "o", ms=3.4, color=AQUA, mfc="white", mew=1.1)
	78	+ # confirmed out-of-sample predictions (1e12 and 1e13 for lag1; 1e13 for lag2)
	79	+ for X, obs in ((1e12, -0.54756), (1e13, -0.54264)):
	80	+ ax.plot(log(X), obs, "D", ms=5.5, color=ORANGE, mfc="none", mew=1.4)
	81	+ ax.plot(log(1e13), -0.25247, "D", ms=5.5, color=ORANGE, mfc="none", mew=1.4)
	82	+ for X, txt in ((1e14, -0.5387), (1e15, -0.5350)):
	83	+ ax.plot(log(X), txt, "s", ms=4.5, color=MUTED, mfc="none", mew=1.0)
	84	+ ax.annotate("lag 1: $\\rho\\,\\ln x = c + d/\\ln x$", (18.4, -0.525),
	85	+ color=BLUE, fontsize=8)
	86	+ ax.annotate("lag 2", (19.8, -0.272), color=AQUA, fontsize=8)
	87	+ ax.annotate("confirmed predictions", (29.2, -0.572),
	88	+ color=ORANGE, fontsize=7.5, ha="center")
	89	+ ax.annotate("", xy=(log(1e12) + 0.1, -0.553), xytext=(28.2, -0.568),
	90	+ arrowprops=dict(arrowstyle="->", color=ORANGE, lw=0.9))
	91	+ ax.annotate("", xy=(log(1e13) - 0.1, -0.547), xytext=(30.0, -0.568),
	92	+ arrowprops=dict(arrowstyle="->", color=ORANGE, lw=0.9))
	93	+ ax.annotate("next targets", (31.6, -0.528), color=MUTED, fontsize=7.5)
	94	+ ax.set_xlabel("$\\ln x$")
	95	+ ax.set_ylabel("$\\rho(x)\\,\\ln x$")
	96	+ ax.grid(True)
	97	+ ax.set_xlim(17.8, 35.4)
	98	+ for X in (1e8, 1e10, 1e12, 1e14):
	99	+ ax.axvline(log(X), color=MUTED, lw=0.4, alpha=0.18)
	100	+ ax.text(log(X), ax.get_ylim()[1], f"$10^{{{int(round(log(X,10)))}}}$",
	101	+ fontsize=6.5, color=MUTED, ha="center", va="bottom")
	102	+ # panel (b): observed vs first-order HL model
	103	+ axb.plot(lam, v1, "o", ms=3.2, color=BLUE, mfc="white", mew=1.0, label="observed (lag 1)")
	104	+ axb.plot(ll, c1 + d1 / ll, color=BLUE, lw=1.2)
	105	+ axb.plot(ml, mv, "--", color=ORANGE, lw=1.4, label="first-order HL model")
	106	+ axb.axhline(0, color=MUTED, lw=0.6)
	107	+ axb.set_xlim(17.8, 95)
	108	+ axb.set_ylim(-0.65, 0.02)
	109	+ axb.set_xlabel("$\\lambda = \\ln x$")
	110	+ axb.grid(True)
	111	+ axb.annotate("observed", (23, -0.52), color=BLUE, fontsize=8)
	112	+ axb.annotate("first-order HL model\n($\\approx 32\\%$ of magnitude)",
	113	+ (40, -0.115), color=ORANGE, fontsize=8)
	114	+ fig.tight_layout()
	115	+ fig.savefig(FIGS / "fig_scaling.pdf")
	116	+ plt.close(fig)
	117	+ print("fig_scaling.pdf")
	118	+
	119	+
	120	+def race_trace(limit=10**10, per_decade=60):
	121	+ """D(x) sampled at ~per_decade log-spaced points per decade, exact."""
	122	+ targets = np.unique(np.round(np.logspace(6, np.log10(limit),
	123	+ int(per_decade * (np.log10(limit) - 6)))).astype(np.int64))
	124	+ base = primes_upto(int(limit ** 0.5) + 1)
	125	+ D = 0
	126	+ out_x, out_d = [], []
	127	+ prev = None
	128	+ ti = 0
	129	+ for lo in range(2, limit, 50_000_000):
	130	+ hi = min(lo + 50_000_000, limit)
	131	+ primes = sieve_segment(lo, hi, base)
	132	+ if prev is not None:
	133	+ primes = np.concatenate(([prev], primes))
	134	+ gaps = np.diff(primes)
	135	+ ends = primes[1:]
	136	+ delta = (gaps == 2).astype(np.int64) - (gaps == 4).astype(np.int64)
	137	+ run = D + np.cumsum(delta)
	138	+ while ti < len(targets) and targets[ti] < hi:
	139	+ j = np.searchsorted(ends, targets[ti], side="right") - 1
	140	+ out_x.append(int(targets[ti]))
	141	+ out_d.append(int(run[j]) if j >= 0 else D)
	142	+ ti += 1
	143	+ D = int(run[-1])
	144	+ prev = primes[-1]
	145	+ return np.array(out_x, dtype=float), np.array(out_d, dtype=float)
	146	+
	147	+
	148	+def fig_race(pts):
	149	+ xs, ds = race_trace()
	150	+ fig, ax = plt.subplots(figsize=(6.4, 2.5))
	151	+ ax.axhline(0, color=MUTED, lw=0.7)
	152	+ ax.plot(xs, ds, color=BLUE, lw=1.1)
	153	+ cx = [p[0] for p in pts if p[0] >= 2e10]
	154	+ cd = [p[3] for p in pts if p[0] >= 2e10]
	155	+ ax.plot(cx, cd, "D", ms=4.5, color=ORANGE, mfc="none", mew=1.2)
	156	+ ax.set_xscale("log")
	157	+ ax.set_yscale("symlog", linthresh=200)
	158	+ ax.set_xlim(1e6, 2e13)
	159	+ ax.set_ylim(-2.5e5, 6e4)
	160	+ ax.set_xlabel("$x$")
	161	+ ax.set_ylabel("$D(x) = N(2,x) - N(4,x)$")
	162	+ ax.grid(True, which="major")
	163	+ ax.annotate("first tie after $10^6$\n$x = 80{,}966{,}861$",
	164	+ xy=(8.1e7, 0), xytext=(2.5e6, -2600), fontsize=7.5, color=INK,
	165	+ arrowprops=dict(arrowstyle="->", color=MUTED, lw=0.8))
	166	+ ax.annotate("fine trace (every gap, sampled)", (2.6e6, 6000), color=BLUE, fontsize=8)
	167	+ ax.annotate("campaign checkpoints", (2.2e11, -12000), color=ORANGE, fontsize=8)
	168	+ fig.tight_layout()
	169	+ fig.savefig(FIGS / "fig_race.pdf")
	170	+ plt.close(fig)
	171	+ print("fig_race.pdf")
	172	+
	173	+
	174	+def fig_hist():
	175	+ rows = np.loadtxt(DATA / "gap_histogram_1e8.csv", delimiter=",", skiprows=1)
	176	+ g, cnt = rows[:, 0].astype(int), rows[:, 1]
	177	+ m = (g >= 2) & (g <= 150) & (g % 2 == 0)
	178	+ g, cnt = g[m], cnt[m]
	179	+ six = g % 6 == 0
	180	+ fig, ax = plt.subplots(figsize=(6.4, 2.5))
	181	+ ax.vlines(g[~six], 1, cnt[~six], color=MUTED, lw=1.6, alpha=0.75)
	182	+ ax.vlines(g[six], 1, cnt[six], color=BLUE, lw=1.9)
	183	+ ax.set_yscale("log")
	184	+ ax.set_xlabel("gap $g$")
	185	+ ax.set_ylabel("$N(g, 10^8)$")
	186	+ ax.grid(True, axis="y")
	187	+ ax.annotate("multiples of 6", (66, 2.2e5), color=BLUE, fontsize=8.5)
	188	+ ax.annotate("other even gaps", (80, 1.1e4), color=MUTED, fontsize=8.5)
	189	+ ax.annotate("jumping champion $g = 6$", xy=(6, 8.8e5), xytext=(16, 1.5e6),
	190	+ fontsize=7.5, color=INK,
	191	+ arrowprops=dict(arrowstyle="->", color=MUTED, lw=0.8))
	192	+ ax.set_xlim(0, 152)
	193	+ ax.set_ylim(1, 4e6)
	194	+ fig.tight_layout()
	195	+ fig.savefig(FIGS / "fig_hist.pdf")
	196	+ plt.close(fig)
	197	+ print("fig_hist.pdf")
	198	+
	199	+
	200	+def fig_certificate():
	201	+ cert = json.load(open(ROOT / "certs" / "desert_certificate.json"))
	202	+ interior = cert["interior_certificates"]
	203	+ cov_x, cov_y, tf_x, tf_y, mr_x = [], [], [], [], []
	204	+ for i_str, c in interior.items():
	205	+ i = int(i_str)
	206	+ if c["type"] == "covering_factor":
	207	+ cov_x.append(i); cov_y.append(c["q"])
	208	+ elif c["type"] == "trial_factor":
	209	+ tf_x.append(i); tf_y.append(c["q"])
	210	+ else:
	211	+ mr_x.append(i)
	212	+ fig, ax = plt.subplots(figsize=(6.4, 2.6))
	213	+ ax.plot(cov_x, cov_y, "o", ms=2.6, color=BLUE, mew=0)
	214	+ ax.plot(tf_x, tf_y, "D", ms=4.5, color=ORANGE, mfc="none", mew=1.2)
	215	+ top = max(tf_y) * 3
	216	+ ax.plot(mr_x, [top] * len(mr_x), "s", ms=5, color=AQUA, mfc="none", mew=1.3)
	217	+ ax.set_yscale("log")
	218	+ ax.set_xlabel("position $i$ in the desert ($N + i$, $1 \\leq i \\leq 259$)")
	219	+ ax.set_ylabel("certifying prime factor $q_i$")
	220	+ ax.grid(True, axis="y")
	221	+ ax.annotate("covering system (proven for every shift $t$)", (4, 230),
	222	+ color=BLUE, fontsize=8)
	223	+ ax.annotate("holes: explicit trial factor", (138, 2.6e4),
	224	+ color=ORANGE, fontsize=8)
	225	+ ax.annotate("hole: strong MR witness", (mr_x[0] + 8, top * 0.75),
	226	+ color=AQUA, fontsize=8)
	227	+ ax.set_xlim(0, 260)
	228	+ fig.tight_layout()
	229	+ fig.savefig(FIGS / "fig_certificate.pdf")
	230	+ plt.close(fig)
	231	+ print("fig_certificate.pdf")
	232	+
	233	+
	234	+if __name__ == "__main__":
	235	+ pts = load_checkpoints()
	236	+ fig_scaling(pts)
	237	+ fig_hist()
	238	+ fig_certificate()
	239	+ fig_race(pts) # last: recomputes a 1e10 scan (~40 s)

modified src/merge_c4.py +2 −1

		`@@ -21,7 +21,8 @@ import numpy as np`
21	21	DATA = Path(__file__).resolve().parent.parent / "data"
22	22	CHECKPOINTS = [10*10, 2 10*10, 4 10**10,
23	23	10*11, 2 10*11, 4 1011, 1012,
24		− 2 * 10*12, 4 1012, 1013]
	24	+ 2 * 10*12, 4 1012, 1013,
	25	+ 2 * 10*13, 4 1013, 1014]
25	26
26	27
27	28	def pearson(S):

modified src/worker_c4.py +6 −1

		`@@ -20,8 +20,9 @@ import numpy as np`
20	20
21	21	from core import primes_upto, sieve_segment
22	22
	23	+import os
23	24	BUFFER = 5000
24		−SEG = 50_000_000
	25	+SEG = int(os.environ.get("C4_SEG", 50_000_000))
25	26
26	27
27	28	def run(lo, hi, out_path, sieve_fn=None, tag=""):
		`@@ -101,6 +102,10 @@ def run(lo, hi, out_path, sieve_fn=None, tag=""):`
101	102	def main():
102	103	lo, hi = int(float(sys.argv[1])), int(float(sys.argv[2]))
103	104	out_path = sys.argv[3]
	105	+ from pathlib import Path as _P
	106	+ if _P(out_path).exists():
	107	+ print(f"skip existing {out_path}", flush=True)
	108	+ return
104	109	if len(sys.argv) > 4 and sys.argv[4] == "gpu":
105	110	from gpu_sieve import sieve_segment_gpu
106	111	run(lo, hi, out_path, sieve_fn=sieve_segment_gpu, tag="/gpu")
107	112