Prime Mystery Engine: cycles 1-4 — certified desert, refuted race conjecture, twice-confirmed correlation scaling law to 10^13
- Verification-first pipeline (4-role protocol), 37-check validation gate - Cycle 1: C2 race conjecture REFUTED (minimal counterexample 80966861/80966933); certified 260-gap desert at a 25-digit prime (hybrid covering system) - Cycles 2-4: scaling law rho(x)·ln x = c + d/ln x, c = -0.4835(2), d = -1.777; out-of-sample predictions confirmed at 10^12 and 10^13 (lag-1 and lag-2) - Distributed mergeable-statistics scans (exact integer accumulators, bit-identical cross-machine); custom Metal GPU sieve kernel validated byte-identical vs CPU - All 53 published maximal gaps below 10^13 reproduced (OEIS A005250/A002386) - LaTeX paper (13 pp.) with robustness checks and threats-to-validity Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
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| 1 | +# Prime Mystery Engine — Simon-Pierre Boucher — contact@spboucher.ai | |
| 2 | +__pycache__/ | |
| 3 | +*.pyc | |
| 4 | +.DS_Store | |
| 5 | +.claude/ | |
| 6 | +paper/*.aux | |
| 7 | +paper/*.log | |
| 8 | +paper/*.out | |
| 9 | +paper/*.toc | |
| 10 | +paper/*.fls | |
| 11 | +paper/*.fdb_latexmk | |
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CLAUDE.md
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| 1 | +# CLAUDE.md — Prime Mystery Engine | |
| 2 | + | |
| 3 | +## Mission | |
| 4 | + | |
| 5 | +You are the engine of an autonomous prime-number research laboratory. Your goal: produce **verifiable results** (record constructions, massively tested conjectures, counterexamples, primality certificates, formal or semi-formal proofs). You must NEVER assert a mathematical result without computational verification or proof. | |
| 6 | + | |
| 7 | +## Golden Rule | |
| 8 | + | |
| 9 | +**No claim without a certificate.** Every number declared prime must pass a deterministic primality test (or a probabilistic one with the number of rounds documented). Every conjecture must explicitly state the tested range. Every "record" must be checked against the literature (OEIS, primes.utm.edu, recent papers) before being called new. | |
| 10 | + | |
| 11 | +--- | |
| 12 | + | |
| 13 | +## Architecture: 4 roles you cycle through | |
| 14 | + | |
| 15 | +In every work cycle, you pass through the 4 roles **in this order**, skipping none: | |
| 16 | + | |
| 17 | +1. **EXPLORER** — computes, generates data, detects patterns. Output: data files + quantified observations. | |
| 18 | +2. **CONJECTURER** — turns observations into precise mathematical statements (explicit quantifiers, bounds, conditions). Output: numbered list of conjectures C1, C2, … | |
| 19 | +3. **ADVERSARY** — attacks each conjecture: modular obstructions, density heuristics (Hardy–Littlewood, Cramér), brute-force counterexample search over an extended range. Output: status of each conjecture (REFUTED with minimal counterexample / SURVIVOR with tested range / TRIVIAL). | |
| 20 | +4. **PROVER** — for survivors: attempt a proof (modular covering, sieve, elementary argument), or a partial proof, or a reduction to a known conjecture. Output: proof, honest sketch, or an explicit admission of failure. | |
| 21 | + | |
| 22 | +Document every cycle in `journal.md`: cycle N, role, actions, results, decisions. | |
| 23 | + | |
| 24 | +--- | |
| 25 | + | |
| 26 | +## Methodical Protocol — the 8 phases | |
| 27 | + | |
| 28 | +### Phase 0 — Setup (mandatory before any computation) | |
| 29 | +- Create the structure: `/src` (code), `/data` (raw results), `/certs` (certificates), `/journal.md`, `/conjectures.md`, `/records.md`. | |
| 30 | +- Write and TEST the core building blocks: segmented Sieve of Eratosthenes, deterministic Miller–Rabin (< 3.3×10^24 with the right bases), BPSW test, small-factor trial division / sieving. | |
| 31 | +- Validate each block against known values (π(10^6)=78498, π(10^8)=5761455, known record gaps, etc.). Do not proceed while any test fails. | |
| 32 | + | |
| 33 | +### Phase 1 — Problem selection | |
| 34 | +Select ONE main axis among the 10 in the source document (recommended: #3 constellations, #7 conjecture factory, or #10 prime deserts — best verifiability/originality ratios). Justify the choice in 5 lines: feasibility, measurable success criterion, state of the art. | |
| 35 | + | |
| 36 | +### Phase 2 — State of the art | |
| 37 | +- Look up known records and results (OEIS, literature, k-tuple databases). | |
| 38 | +- Write in `records.md`: what is known, what would count as a new result, the exact threshold to beat. | |
| 39 | + | |
| 40 | +### Phase 3 — Exploration (EXPLORER role) | |
| 41 | +- Generate data at small scale first (n ≤ 10^6), check consistency, THEN scale up (10^8, 10^9…). | |
| 42 | +- Always log: range, compute time, method, seed if randomized. | |
| 43 | +- Hunt for patterns: modular residues, densities, symmetries, statistical anomalies (compare against Hardy–Littlewood predictions). | |
| 44 | + | |
| 45 | +### Phase 4 — Conjecture (CONJECTURER role) | |
| 46 | +- State every conjecture in strict mathematical language: "For all n ≥ N₀, …" or "There exist infinitely many …". | |
| 47 | +- Classify: (a) probably known, (b) easy consequence of a known result, (c) potentially new. | |
| 48 | +- Immediately eliminate conjectures with an obvious modular obstruction (systematically check admissibility mod 2, 3, 5, 7). | |
| 49 | + | |
| 50 | +### Phase 5 — Attack (ADVERSARY role) | |
| 51 | +- For each surviving conjecture: test over a range 10× to 100× larger than the discovery range. | |
| 52 | +- Actively hunt for the minimal counterexample — don't just "verify". | |
| 53 | +- Apply the Cramér/Maier test: would the conjecture survive in a random model of the primes? If yes, it may be merely statistical, not structural — note it. | |
| 54 | + | |
| 55 | +### Phase 6 — Proof or certificate (PROVER role) | |
| 56 | +- Constructions (constellations, deserts): produce a complete certificate — list of numbers, primality test used, modular covering for the composites (N+i ≡ 0 mod qᵢ), independent re-verification script. | |
| 57 | +- Conjectures: attempt an elementary proof; otherwise reduce to Dickson/Hardy–Littlewood/Bunyakovsky; otherwise honestly document "open, verified up to X". | |
| 58 | +- If Lean or Sage is available, formalize the provable results. | |
| 59 | + | |
| 60 | +### Phase 7 — Synthesis | |
| 61 | +- Write a report: results, status of each conjecture, any records with certificates, instructive failures, next directions. | |
| 62 | +- Every announced result must be independently re-verifiable via a standalone script provided in `/certs`. | |
| 63 | + | |
| 64 | +--- | |
| 65 | + | |
| 66 | +## Rigor constraints (non-negotiable) | |
| 67 | + | |
| 68 | +- **Small scale first**: never launch massive computation before validation on known cases. | |
| 69 | +- **Reproducibility**: every script must run end-to-end without intervention; fix all seeds. | |
| 70 | +- **Epistemic honesty**: clearly distinguish PROVEN / VERIFIED UP TO X / CONJECTURED / SPECULATIVE in every output. | |
| 71 | +- **No hidden tables**: generators (axis #8) must contain no hardcoded list of primes. | |
| 72 | +- **Novelty**: before announcing a record or discovery, check OEIS and the literature. When in doubt, write "possibly known". | |
| 73 | +- **Budget**: estimate compute cost before every scale-up; prefer a better algorithm over more brute force (segmented sieve > individual tests, mod 30/210 wheels, etc.). | |
| 74 | + | |
| 75 | +## Cycle stopping criteria | |
| 76 | + | |
| 77 | +A cycle ends when: (a) a conjecture is refuted or proven, (b) a record construction is certified, or (c) 3 attempts in the same role fail — in that case, return to Phase 3 with a different angle and note it in the journal. | |
| 78 | + | |
| 79 | +## Output format for every session | |
| 80 | + | |
| 81 | +1. Cycle summary (5 lines max). | |
| 82 | +2. Table of conjectures with status. | |
| 83 | +3. New files created and how to re-verify them. | |
| 84 | +4. Next precise action (one only, concrete). | |
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README.md
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| 1 | +<!-- | |
| 2 | +README.md — Prime Mystery Engine | |
| 3 | +Author: Simon-Pierre Boucher — contact@spboucher.ai | |
| 4 | +--> | |
| 5 | + | |
| 6 | +# 🔬 Prime Mystery Engine | |
| 7 | + | |
| 8 | +**A verification-first computational laboratory for prime gap statistics.** | |
| 9 | + | |
| 10 | + | |
| 11 | + | |
| 12 | + | |
| 13 | + | |
| 14 | + | |
| 15 | + | |
| 16 | + | |
| 17 | + | |
| 18 | + | |
| 19 | + | |
| 20 | + | |
| 21 | +An autonomous research pipeline that studies gaps between consecutive primes under a strict | |
| 22 | +epistemic protocol: **no claim without a certificate**. Every statement is labelled | |
| 23 | +`PROVEN` / `CERTIFIED` / `VERIFIED UP TO X` / `CONJECTURED` / `REFUTED`, every scan is | |
| 24 | +deterministic and bit-reproducible across machines, and every headline object ships with a | |
| 25 | +standalone re-verification script. | |
| 26 | + | |
| 27 | +--- | |
| 28 | + | |
| 29 | +## 🏆 Headline results | |
| 30 | + | |
| 31 | +| Result | Status | | |
| 32 | +|---|---| | |
| 33 | +| **Race conjecture refuted** — "N(2,x) > N(4,x) for x ≥ 10⁶" is false: first tie at end-prime **80 966 861**, first strict overtake at **80 966 933**; the lead changes forever up to 10¹³ | `REFUTED` (minimal counterexample) | | |
| 34 | +| **Certified prime desert** — gap of exactly **260** after the 25-digit prime `1116336781708038449369693` (merit 4.70), built from a hybrid covering system mod primes ≤ 59 | `CERTIFIED` | | |
| 35 | +| **Scaling law for consecutive-gap correlation** — ρ(x)·ln x = c + d/ln x with **c = −0.4835 ± 0.002**, d = −1.777; out-of-sample predictions confirmed at 10¹² (−0.549 ± 0.005 → −0.54756) **and** 10¹³ (−0.5432 ± 0.004 → −0.54264) | `CONJECTURED`, 2/2 hits | | |
| 36 | +| **Lag-2 law** — ρ₂(x)·ln x → c₂ ≈ −0.278, prediction confirmed at 10¹³ (−0.2545 ± 0.004 → −0.25247) | `CONJECTURED`, 1/1 hit | | |
| 37 | +| **First-order Hardy–Littlewood triple model rejected** — right sign and drift form, but only ~32 % of the observed magnitude | `MODEL REJECTED` | | |
| 38 | +| **All published tables reproduced** — 53/53 maximal gaps below 10¹³ (OEIS A005250/A002386), twin counts, π(10ᵏ), CSG maxima | `VERIFIED` | | |
| 39 | + | |
| 40 | +**Predictions on record (falsifiable):** ρ·ln x(10¹⁴) = −0.5387 ± 0.004 · ρ·ln x(10¹⁵) = −0.5350 ± 0.004 · ρ₂·ln x(10¹⁴) = −0.2553 ± 0.004 | |
| 41 | + | |
| 42 | +--- | |
| 43 | + | |
| 44 | +## ⚡ Verify the headline claims yourself (stock Python 3) | |
| 45 | + | |
| 46 | +```bash | |
| 47 | +# the certified 260-gap desert (independent Miller–Rabin implementation) | |
| 48 | +python3 certs/verify_desert.py | |
| 49 | + | |
| 50 | +# the race-conjecture refutation (stdlib-only sieve, ~30 s) | |
| 51 | +python3 certs/verify_c2_refutation.py | |
| 52 | + | |
| 53 | +# the core-primitives validation gate (37 checks, needs numpy) | |
| 54 | +python3 src/test_core.py | |
| 55 | +``` | |
| 56 | + | |
| 57 | +## 🗂 Repository layout | |
| 58 | + | |
| 59 | +``` | |
| 60 | +src/ core primitives, scans, distributed worker/merger, Metal GPU sieve, models | |
| 61 | +certs/ certificates + standalone verifiers (the "trust nothing" layer) | |
| 62 | +data/ raw checkpoint JSONs for every campaign (deterministic, re-mergeable) | |
| 63 | +paper/ LaTeX manuscript (main.tex → main.pdf, 13 pp., full math detail) | |
| 64 | +journal.md complete research journal: every cycle, decision, failure | |
| 65 | +conjectures.md the conjecture ledger with epistemic statuses | |
| 66 | +records.md state of the art (what would count as new; thresholds to beat) | |
| 67 | +report_cycle{1..4}.md per-cycle syntheses | |
| 68 | +``` | |
| 69 | + | |
| 70 | +## 🖥 Compute | |
| 71 | + | |
| 72 | +| Campaign | Range | Fleet | Wall time | | |
| 73 | +|---|---|---|---| | |
| 74 | +| Cycle 1 | 4×10⁹ | 1 laptop core | 15 s | | |
| 75 | +| Cycle 2 | 4×10¹⁰ | laptop + M3 Ultra (1 core each) | ~2 min | | |
| 76 | +| Cycle 3 | 10¹² | 12 Apple-silicon nodes, 162 CPU cores | ~5 min | | |
| 77 | +| Cycle 4 | 10¹³ | 3 nodes: 58 CPU cores + **3 GPUs (custom Metal kernel)** | ~50 min | | |
| 78 | + | |
| 79 | +The distributed design uses *mergeable* streaming statistics (exact integers < 2⁵³ in | |
| 80 | +float64 → **zero rounding**, bit-identical across machines) with checkpoint-aligned chunking | |
| 81 | +and a junction buffer, so a 10 000-chunk merge equals the single-machine scan exactly. | |
| 82 | +The Metal sieve kernel was accepted only after producing byte-identical prime sets to the | |
| 83 | +CPU sieve; it processed 11.7 % of cycle 4. | |
| 84 | + | |
| 85 | +## 📄 Paper | |
| 86 | + | |
| 87 | +Full write-up with all equations, proofs of the infrastructure lemmas, robustness checks and | |
| 88 | +threats-to-validity: [`paper/main.tex`](paper/main.tex) → [`paper/main.pdf`](paper/main.pdf). | |
| 89 | + | |
| 90 | +## 👤 Author | |
| 91 | + | |
| 92 | +**Simon-Pierre Boucher** · [contact@spboucher.ai](mailto:contact@spboucher.ai) | |
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certs/desert_certificate.json
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| 1 | +{ | |
| 2 | + "title": "Certified prime desert via hybrid covering system", | |
| 3 | + "author": "Simon-Pierre Boucher \u2014 contact@spboucher.ai", | |
| 4 | + "date": "2026-08-06", | |
| 5 | + "claim": "N and N+260 are consecutive primes (gap exactly 260).", | |
| 6 | + "N": "1116336781708038449369693", | |
| 7 | + "gap": 260, | |
| 8 | + "merit": 4.69551, | |
| 9 | + "csg_ratio": 0.084799, | |
| 10 | + "construction": { | |
| 11 | + "prime_set_max": 59, | |
| 12 | + "primorial_P": "1922760350154212639070", | |
| 13 | + "crt_residue_x": "1135778618595118709093", | |
| 14 | + "shift_t": 580, | |
| 15 | + "classes_a_p": { | |
| 16 | + "2": 1, | |
| 17 | + "3": 1, | |
| 18 | + "5": 2, | |
| 19 | + "7": 6, | |
| 20 | + "11": 8, | |
| 21 | + "13": 1, | |
| 22 | + "17": 16, | |
| 23 | + "19": 6, | |
| 24 | + "23": 3, | |
| 25 | + "29": 2, | |
| 26 | + "31": 18, | |
| 27 | + "37": 24, | |
| 28 | + "41": 4, | |
| 29 | + "43": 9, | |
| 30 | + "47": 9, | |
| 31 | + "53": 28, | |
| 32 | + "59": 38 | |
| 33 | + }, | |
| 34 | + "holes": [ | |
| 35 | + 36, | |
| 36 | + 54, | |
| 37 | + 68, | |
| 38 | + 78, | |
| 39 | + 108, | |
| 40 | + 110, | |
| 41 | + 114, | |
| 42 | + 116, | |
| 43 | + 126, | |
| 44 | + 128, | |
| 45 | + 180, | |
| 46 | + 194, | |
| 47 | + 198, | |
| 48 | + 200, | |
| 49 | + 218, | |
| 50 | + 236 | |
| 51 | + ], | |
| 52 | + "note": "positions with a covering factor are composite for EVERY shift t (proven); hole positions are certified composite for THIS t by explicit factor or strong MR witness" | |
| 53 | + }, | |
| 54 | + "interior_certificates": { | |
| 55 | + "1": { | |
| 56 | + "type": "covering_factor", | |
| 57 | + "q": 2 | |
| 58 | + }, | |
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| 919 | + "217": { | |
| 920 | + "type": "covering_factor", | |
| 921 | + "q": 2 | |
| 922 | + }, | |
| 923 | + "218": { | |
| 924 | + "type": "trial_factor", | |
| 925 | + "q": 3637 | |
| 926 | + }, | |
| 927 | + "219": { | |
| 928 | + "type": "covering_factor", | |
| 929 | + "q": 2 | |
| 930 | + }, | |
| 931 | + "220": { | |
| 932 | + "type": "covering_factor", | |
| 933 | + "q": 3 | |
| 934 | + }, | |
| 935 | + "221": { | |
| 936 | + "type": "covering_factor", | |
| 937 | + "q": 2 | |
| 938 | + }, | |
| 939 | + "222": { | |
| 940 | + "type": "covering_factor", | |
| 941 | + "q": 5 | |
| 942 | + }, | |
| 943 | + "223": { | |
| 944 | + "type": "covering_factor", | |
| 945 | + "q": 2 | |
| 946 | + }, | |
| 947 | + "224": { | |
| 948 | + "type": "covering_factor", | |
| 949 | + "q": 43 | |
| 950 | + }, | |
| 951 | + "225": { | |
| 952 | + "type": "covering_factor", | |
| 953 | + "q": 2 | |
| 954 | + }, | |
| 955 | + "226": { | |
| 956 | + "type": "covering_factor", | |
| 957 | + "q": 3 | |
| 958 | + }, | |
| 959 | + "227": { | |
| 960 | + "type": "covering_factor", | |
| 961 | + "q": 2 | |
| 962 | + }, | |
| 963 | + "228": { | |
| 964 | + "type": "covering_factor", | |
| 965 | + "q": 11 | |
| 966 | + }, | |
| 967 | + "229": { | |
| 968 | + "type": "covering_factor", | |
| 969 | + "q": 2 | |
| 970 | + }, | |
| 971 | + "230": { | |
| 972 | + "type": "covering_factor", | |
| 973 | + "q": 7 | |
| 974 | + }, | |
| 975 | + "231": { | |
| 976 | + "type": "covering_factor", | |
| 977 | + "q": 2 | |
| 978 | + }, | |
| 979 | + "232": { | |
| 980 | + "type": "covering_factor", | |
| 981 | + "q": 3 | |
| 982 | + }, | |
| 983 | + "233": { | |
| 984 | + "type": "covering_factor", | |
| 985 | + "q": 2 | |
| 986 | + }, | |
| 987 | + "234": { | |
| 988 | + "type": "covering_factor", | |
| 989 | + "q": 19 | |
| 990 | + }, | |
| 991 | + "235": { | |
| 992 | + "type": "covering_factor", | |
| 993 | + "q": 2 | |
| 994 | + }, | |
| 995 | + "236": { | |
| 996 | + "type": "trial_factor", | |
| 997 | + "q": 349 | |
| 998 | + }, | |
| 999 | + "237": { | |
| 1000 | + "type": "covering_factor", | |
| 1001 | + "q": 2 | |
| 1002 | + }, | |
| 1003 | + "238": { | |
| 1004 | + "type": "covering_factor", | |
| 1005 | + "q": 3 | |
| 1006 | + }, | |
| 1007 | + "239": { | |
| 1008 | + "type": "covering_factor", | |
| 1009 | + "q": 2 | |
| 1010 | + }, | |
| 1011 | + "240": { | |
| 1012 | + "type": "covering_factor", | |
| 1013 | + "q": 53 | |
| 1014 | + }, | |
| 1015 | + "241": { | |
| 1016 | + "type": "covering_factor", | |
| 1017 | + "q": 2 | |
| 1018 | + }, | |
| 1019 | + "242": { | |
| 1020 | + "type": "covering_factor", | |
| 1021 | + "q": 5 | |
| 1022 | + }, | |
| 1023 | + "243": { | |
| 1024 | + "type": "covering_factor", | |
| 1025 | + "q": 2 | |
| 1026 | + }, | |
| 1027 | + "244": { | |
| 1028 | + "type": "covering_factor", | |
| 1029 | + "q": 3 | |
| 1030 | + }, | |
| 1031 | + "245": { | |
| 1032 | + "type": "covering_factor", | |
| 1033 | + "q": 2 | |
| 1034 | + }, | |
| 1035 | + "246": { | |
| 1036 | + "type": "covering_factor", | |
| 1037 | + "q": 37 | |
| 1038 | + }, | |
| 1039 | + "247": { | |
| 1040 | + "type": "covering_factor", | |
| 1041 | + "q": 2 | |
| 1042 | + }, | |
| 1043 | + "248": { | |
| 1044 | + "type": "covering_factor", | |
| 1045 | + "q": 13 | |
| 1046 | + }, | |
| 1047 | + "249": { | |
| 1048 | + "type": "covering_factor", | |
| 1049 | + "q": 2 | |
| 1050 | + }, | |
| 1051 | + "250": { | |
| 1052 | + "type": "covering_factor", | |
| 1053 | + "q": 3 | |
| 1054 | + }, | |
| 1055 | + "251": { | |
| 1056 | + "type": "covering_factor", | |
| 1057 | + "q": 2 | |
| 1058 | + }, | |
| 1059 | + "252": { | |
| 1060 | + "type": "covering_factor", | |
| 1061 | + "q": 5 | |
| 1062 | + }, | |
| 1063 | + "253": { | |
| 1064 | + "type": "covering_factor", | |
| 1065 | + "q": 2 | |
| 1066 | + }, | |
| 1067 | + "254": { | |
| 1068 | + "type": "covering_factor", | |
| 1069 | + "q": 17 | |
| 1070 | + }, | |
| 1071 | + "255": { | |
| 1072 | + "type": "covering_factor", | |
| 1073 | + "q": 2 | |
| 1074 | + }, | |
| 1075 | + "256": { | |
| 1076 | + "type": "covering_factor", | |
| 1077 | + "q": 3 | |
| 1078 | + }, | |
| 1079 | + "257": { | |
| 1080 | + "type": "covering_factor", | |
| 1081 | + "q": 2 | |
| 1082 | + }, | |
| 1083 | + "258": { | |
| 1084 | + "type": "covering_factor", | |
| 1085 | + "q": 7 | |
| 1086 | + }, | |
| 1087 | + "259": { | |
| 1088 | + "type": "covering_factor", | |
| 1089 | + "q": 2 | |
| 1090 | + } | |
| 1091 | + }, | |
| 1092 | + "endpoint_primality": { | |
| 1093 | + "method": "deterministic Miller-Rabin, bases {2..37} (12 bases), valid for n < 3.317e24 (Sorenson-Webster 2015)", | |
| 1094 | + "validity_bound": "3317044064679887385961981", | |
| 1095 | + "both_endpoints_below_bound": true | |
| 1096 | + }, | |
| 1097 | + "epistemic_status": "CERTIFIED construction; NOT a record (see records.md: merit record 41.94, largest maximal gap 1854). Baseline with same primes: classic primorial gives gap 61; pure covering (attempt 1) gave gap 90." | |
| 1098 | +} | |
| \ No newline at end of file | ||
added
certs/verify_c2_refutation.py
+71 −0
@@ -0,0 +1,71 @@ | ||
| 1 | +#!/usr/bin/env python3 | |
| 2 | +# ============================================================================= | |
| 3 | +# verify_c2_refutation.py — INDEPENDENT re-verification of the C2 counterexample | |
| 4 | +# Author: Simon-Pierre Boucher — contact@spboucher.ai | |
| 5 | +# ============================================================================= | |
| 6 | +# Claim refuted: "N(2,x) > N(4,x) for all x >= 10^6" (C2, cycle 1) | |
| 7 | +# where N(g,x) counts consecutive prime pairs (p, p+g) with p+g <= x. | |
| 8 | +# This script recomputes, stdlib-only (bytearray sieve, no numpy, no /src): | |
| 9 | +# - D(p) = N2 - N4 after each prime p (gaps attributed to their END prime), | |
| 10 | +# - checks D(p) > 0 for every end-prime p in (10^6, 80966861), | |
| 11 | +# - checks D = 0 at end-prime 80966861 (first tie), | |
| 12 | +# - checks D = -1 at end-prime 80966933 (first strict overtake by gap 4). | |
| 13 | +# Runtime ~ 30 s pure Python. Exit 0 + "VERIFICATION PASSED" iff all hold. | |
| 14 | +# ============================================================================= | |
| 15 | + | |
| 16 | +import sys | |
| 17 | + | |
| 18 | +LIMIT = 81_000_000 | |
| 19 | +FIRST_TIE = 80_966_861 | |
| 20 | +FIRST_NEG = 80_966_933 | |
| 21 | + | |
| 22 | + | |
| 23 | +def main(): | |
| 24 | + sieve = bytearray([1]) * (LIMIT + 1) | |
| 25 | + sieve[0] = sieve[1] = 0 | |
| 26 | + for i in range(2, int(LIMIT ** 0.5) + 1): | |
| 27 | + if sieve[i]: | |
| 28 | + sieve[i * i:: i] = bytearray(len(sieve[i * i:: i])) | |
| 29 | + D = 0 | |
| 30 | + prev = 2 | |
| 31 | + ok_positive = True | |
| 32 | + tie_val = neg_val = None | |
| 33 | + min_after_1e6 = None | |
| 34 | + for p in range(3, LIMIT + 1, 2): | |
| 35 | + if not sieve[p]: | |
| 36 | + continue | |
| 37 | + g = p - prev | |
| 38 | + if g == 2: | |
| 39 | + D += 1 | |
| 40 | + elif g == 4: | |
| 41 | + D -= 1 | |
| 42 | + prev = p | |
| 43 | + if 10**6 < p < FIRST_TIE and D <= 0: | |
| 44 | + ok_positive = False | |
| 45 | + print(f"unexpected D<=0 at {p}: D={D}") | |
| 46 | + if p == FIRST_TIE: | |
| 47 | + tie_val = D | |
| 48 | + if p == FIRST_NEG: | |
| 49 | + neg_val = D | |
| 50 | + failures = [] | |
| 51 | + if not ok_positive: | |
| 52 | + failures.append("D not strictly positive on (10^6, 80966861)") | |
| 53 | + if tie_val != 0: | |
| 54 | + failures.append(f"D at {FIRST_TIE} is {tie_val}, expected 0") | |
| 55 | + if neg_val != -1: | |
| 56 | + failures.append(f"D at {FIRST_NEG} is {neg_val}, expected -1") | |
| 57 | + if failures: | |
| 58 | + print("VERIFICATION FAILED:") | |
| 59 | + for f in failures: | |
| 60 | + print(" -", f) | |
| 61 | + sys.exit(1) | |
| 62 | + print("VERIFICATION PASSED:") | |
| 63 | + print(f" D(p) > 0 for every end-prime p in (10^6, {FIRST_TIE})") | |
| 64 | + print(f" D({FIRST_TIE}) = 0 (first tie after 10^6)") | |
| 65 | + print(f" D({FIRST_NEG}) = -1 (first strict N4 > N2 after 10^6)") | |
| 66 | + print(" => C2 is REFUTED; minimal counterexample x = 80966861 (tie), " | |
| 67 | + "80966933 (strict).") | |
| 68 | + | |
| 69 | + | |
| 70 | +if __name__ == "__main__": | |
| 71 | + main() | |
added
certs/verify_desert.py
+105 −0
@@ -0,0 +1,105 @@ | ||
| 1 | +#!/usr/bin/env python3 | |
| 2 | +# ============================================================================= | |
| 3 | +# verify_desert.py — INDEPENDENT re-verification of desert_certificate.json | |
| 4 | +# Author: Simon-Pierre Boucher — contact@spboucher.ai | |
| 5 | +# ============================================================================= | |
| 6 | +# Standalone: stdlib only, NO imports from /src (independent implementation). | |
| 7 | +# Verifies the claim "N and N+gap are consecutive primes": | |
| 8 | +# (1) every interior position 1..gap-1 has a compositeness certificate and | |
| 9 | +# the certificate checks out (divisor divides / MR witness is a witness); | |
| 10 | +# (2) both endpoints pass deterministic Miller-Rabin (12 bases, valid | |
| 11 | +# for n < 3.317e24) and lie below that bound; | |
| 12 | +# (3) the covering classes are consistent with N (N+i = 0 mod q claimed). | |
| 13 | +# Exit code 0 + "VERIFICATION PASSED" iff everything holds. | |
| 14 | +# Usage: python3 verify_desert.py [certificate.json] | |
| 15 | +# ============================================================================= | |
| 16 | + | |
| 17 | +import json | |
| 18 | +import sys | |
| 19 | +from pathlib import Path | |
| 20 | + | |
| 21 | +MR_LIMIT = 3_317_044_064_679_887_385_961_981 | |
| 22 | +BASES = (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37) | |
| 23 | + | |
| 24 | + | |
| 25 | +def mr_decompose(n): | |
| 26 | + d, r = n - 1, 0 | |
| 27 | + while d % 2 == 0: | |
| 28 | + d //= 2 | |
| 29 | + r += 1 | |
| 30 | + return d, r | |
| 31 | + | |
| 32 | + | |
| 33 | +def is_strong_witness(n, a): | |
| 34 | + """True iff a proves n composite (strong Miller-Rabin witness).""" | |
| 35 | + d, r = mr_decompose(n) | |
| 36 | + x = pow(a, d, n) | |
| 37 | + if x == 1 or x == n - 1: | |
| 38 | + return False | |
| 39 | + for _ in range(r - 1): | |
| 40 | + x = x * x % n | |
| 41 | + if x == n - 1: | |
| 42 | + return False | |
| 43 | + return True | |
| 44 | + | |
| 45 | + | |
| 46 | +def is_prime_det(n): | |
| 47 | + """Deterministic MR for n < 3.317e24.""" | |
| 48 | + assert n < MR_LIMIT, "outside deterministic validity range" | |
| 49 | + if n < 2: | |
| 50 | + return False | |
| 51 | + for p in BASES: | |
| 52 | + if n % p == 0: | |
| 53 | + return n == p | |
| 54 | + return not any(is_strong_witness(n, a) for a in BASES) | |
| 55 | + | |
| 56 | + | |
| 57 | +def main(): | |
| 58 | + path = sys.argv[1] if len(sys.argv) > 1 else Path(__file__).parent / "desert_certificate.json" | |
| 59 | + cert = json.load(open(path)) | |
| 60 | + N = int(cert["N"]) | |
| 61 | + gap = int(cert["gap"]) | |
| 62 | + interior = cert["interior_certificates"] | |
| 63 | + failures = [] | |
| 64 | + | |
| 65 | + # (1) completeness: every interior position certified | |
| 66 | + missing = [i for i in range(1, gap) if str(i) not in interior] | |
| 67 | + if missing: | |
| 68 | + failures.append(f"positions without certificate: {missing}") | |
| 69 | + | |
| 70 | + # (2) each interior certificate is valid | |
| 71 | + for i_str, c in interior.items(): | |
| 72 | + i = int(i_str) | |
| 73 | + v = N + i | |
| 74 | + if c["type"] in ("covering_factor", "trial_factor"): | |
| 75 | + q = int(c["q"]) | |
| 76 | + if v % q != 0 or v <= q: | |
| 77 | + failures.append(f"position {i}: claimed factor {q} invalid") | |
| 78 | + elif c["type"] == "mr_witness": | |
| 79 | + if not is_strong_witness(v, int(c["a"])): | |
| 80 | + failures.append(f"position {i}: base {c['a']} is NOT a witness") | |
| 81 | + else: | |
| 82 | + failures.append(f"position {i}: unknown certificate type {c['type']}") | |
| 83 | + | |
| 84 | + # (3) endpoints prime, within deterministic range | |
| 85 | + if N + gap >= MR_LIMIT: | |
| 86 | + failures.append("endpoint exceeds deterministic MR bound") | |
| 87 | + else: | |
| 88 | + if not is_prime_det(N): | |
| 89 | + failures.append("N is not prime") | |
| 90 | + if not is_prime_det(N + gap): | |
| 91 | + failures.append(f"N+{gap} is not prime") | |
| 92 | + | |
| 93 | + if failures: | |
| 94 | + print("VERIFICATION FAILED:") | |
| 95 | + for f in failures: | |
| 96 | + print(" -", f) | |
| 97 | + sys.exit(1) | |
| 98 | + print(f"VERIFICATION PASSED: {N} and {N}+{gap} are consecutive primes;") | |
| 99 | + print(f"all {gap-1} interior numbers certified composite " | |
| 100 | + f"(covering/trial factors + MR witnesses).") | |
| 101 | + print(f"merit = {cert['merit']}, csg = {cert['csg_ratio']}") | |
| 102 | + | |
| 103 | + | |
| 104 | +if __name__ == "__main__": | |
| 105 | + main() | |
added
conjectures.md
+121 −0
@@ -0,0 +1,121 @@ | ||
| 1 | +<!-- | |
| 2 | +conjectures.md — Prime Mystery Engine: conjecture ledger | |
| 3 | +Author: Simon-Pierre Boucher — contact@spboucher.ai | |
| 4 | +--> | |
| 5 | + | |
| 6 | +# Conjecture Ledger | |
| 7 | + | |
| 8 | +Status vocabulary: PROVEN / VERIFIED UP TO X / CONJECTURED / REFUTED (minimal counterexample) / SPECULATIVE. | |
| 9 | +Notation: N(g,x) = #{consecutive prime pairs (p, p+g), p+g < x}. ρ(x) = Pearson correlation of | |
| 10 | +(g_n, g_{n+1}) over consecutive gaps below x. CSG(p) = g(p)/ln²p. G6(x) = largest G such that every | |
| 11 | +multiple of 6 ≤ G satisfies N(g,x) > N(g−2,x) and N(g,x) > N(g+2,x). | |
| 12 | + | |
| 13 | +## Cycle 1 (2026-08-06) — discovery range 10^8, deterministic scans (no seed) | |
| 14 | + | |
| 15 | +### C1 — Jumping champion | |
| 16 | +**Statement.** For all x ∈ [10^5, 4×10^9], argmax_g N(g,x) = 6. | |
| 17 | +**Classification.** (a) probably known — Odlyzko–Rubinstein–Wolf conjecture territory (champion 6 from ≈947 to ≈1.7×10^35). | |
| 18 | +**Modular admissibility.** Consistent (6 = 2·3 maximizes the singular series among small gaps). | |
| 19 | +**Status at discovery.** VERIFIED at checkpoints 10^6, 10^7, 10^8. | |
| 20 | +**FINAL STATUS (adversary, 4×10^9).** SURVIVOR — VERIFIED UP TO 4×10^9 (champion = 6 at every checkpoint). | |
| 21 | +Cramér/Maier note: NOT merely statistical — a Cramér random model has no mod-6 structure at all; this is | |
| 22 | +structural (singular series). Known conjecture (ORW); we add nothing but an independent check. | |
| 23 | + | |
| 24 | +### C2 — The 2-vs-4 race never crosses again | |
| 25 | +**Statement.** For all x ≥ 10^6, N(2,x) > N(4,x). | |
| 26 | +**Classification.** (c) potentially interesting — asymptotically N(2,x) ~ N(4,x) under Hardy–Littlewood | |
| 27 | +(identical singular series), so the sign of the difference is a second-order effect analogous to | |
| 28 | +Chebyshev bias; margins observed are tiny (+26 at 10^6, +359 at 10^7, +55 at 10^8, i.e. ~10^-4 relative) | |
| 29 | +and non-monotone, so a crossing is plausible. Possibly known in race literature. | |
| 30 | +**Modular admissibility.** No obstruction (both gaps admissible). | |
| 31 | +**Status at discovery.** VERIFIED at checkpoints 10^6, 10^7, 10^8. ⚠ Checkpoint-only: interior crossings not yet excluded. | |
| 32 | +**FINAL STATUS (adversary, full resolution to 4×10^9).** **REFUTED.** | |
| 33 | +Minimal counterexample above 10^6: first tie D = N2−N4 = 0 at end-prime **80966861**; first strict | |
| 34 | +overtake (N4 > N2) at end-prime **80966933** (D = −1). The checkpoint verification was misleading: | |
| 35 | +the race in fact changes leader forever in the tested range — D(10^9) = −173, D(2×10^9) = +1074, | |
| 36 | +D(4×10^9) = −2270, with 137,574,763 gap-events where D ≤ 0 and the last one at 3999999979 (= the | |
| 37 | +last prime scanned). Cramér/Maier note: consistent with a random-walk model of the difference | |
| 38 | +(N(2) and N(4) have identical singular series) — the refutation is exactly what the random model | |
| 39 | +predicts; the interesting open follow-up is whether the *logarithmic density* of the lead is biased | |
| 40 | +(Chebyshev-style). Re-verify: `python3 certs/verify_c2_refutation.py` (stdlib-only, ~30 s). | |
| 41 | + | |
| 42 | +### C3 — Local dominance of multiples of 6 | |
| 43 | +**Statement.** G6(x) ≥ 66 for all x ≥ 10^6, and G6(x) → ∞. | |
| 44 | +**Classification.** (b) easy consequence of Hardy–Littlewood heuristics (singular series of 6k beats | |
| 45 | +neighbors); the finite claim is the testable part — the threshold 66 at 10^8 is limited by sample noise | |
| 46 | +in the histogram tail, so G6 should grow with x. | |
| 47 | +**Status at discovery.** G6 = 66 exactly at 10^6, 10^7 and 10^8 (fails at 72 each time — noise or structure? adversary must decide). | |
| 48 | +**FINAL STATUS (adversary, 4×10^9).** SURVIVOR, strengthened — G6(10^9) = G6(4×10^9) = **216** ≥ 66. | |
| 49 | +The failure at 72 below 10^8 was histogram-tail noise, not structure: G6 grows with x as predicted. | |
| 50 | +VERIFIED UP TO 4×10^9; the G6(x) → ∞ part remains CONJECTURED (would follow from Hardy–Littlewood). | |
| 51 | + | |
| 52 | +### C4 — Scaled anticorrelation of consecutive gaps | |
| 53 | +**Statement.** ρ(x) < 0 for all x ≥ 10^6, and ρ(x)·ln x → c with c ≈ −0.6; sharply: ρ(x)·ln x ∈ [−0.65, −0.55] | |
| 54 | +for all x ∈ [10^6, 4×10^9]. | |
| 55 | +**Data.** ρ·ln x = −0.596 (10^6), −0.614 (10^7), −0.582 (10^8). | |
| 56 | +**Classification.** (a/c) anticorrelation itself is known empirically; the precise scaling constant ≈ −0.6 | |
| 57 | +is a sharper claim, possibly known (gap correlation literature), possibly a clean new datum. | |
| 58 | +**Status at discovery.** VERIFIED at 3 checkpoints. | |
| 59 | +**FINAL STATUS (adversary, 4×10^9).** SURVIVOR with **revision**. ρ(x) < 0 and ρ·ln x ∈ [−0.65, −0.55] | |
| 60 | +both VERIFIED UP TO 4×10^9 — but the sequence ρ·ln x = −0.596, −0.614, −0.582, −0.571, −0.567, −0.565 | |
| 61 | +(x = 10^6 … 4×10^9) drifts monotonically upward after 10^7, so the "limit ≈ −0.6" clause is retracted; | |
| 62 | +revised claim: ρ(x)·ln x converges to some c ∈ [−0.60, −0.50] — CONJECTURED, next test at 10^10+. | |
| 63 | +Cramér/Maier note: structural, NOT statistical — a Cramér model with independent gaps gives ρ = 0; | |
| 64 | +the anticorrelation is sieve-induced. This is the most promising quantitative lead of cycle 1. | |
| 65 | +**CYCLE 2 STATUS (laptop to 10^10 + M3U96a to 4×10^10, bit-for-bit identical on overlap).** | |
| 66 | +Window [−0.65, −0.55]: VERIFIED UP TO 4×10^10 (ρ·ln x = −0.55569 there) — but now expected to fail: | |
| 67 | +the 9-checkpoint fit (see C4′) predicts exit of the window near x ≈ 5×10^11. Superseded by C4′. | |
| 68 | + | |
| 69 | +### C4′ — refined scaling law for the gap anticorrelation (NEW, cycle 2) | |
| 70 | +**Statement.** ρ(x)·ln x = c + d/ln x + o(1/ln x) with c = −0.486 ± 0.010 and d ≈ −1.72. | |
| 71 | +Falsifiable near-term prediction: ρ(x)·ln x = −0.549 ± 0.005 at x = 10^12. | |
| 72 | +**Evidence.** Least-squares on 9 checkpoints 10^8…4×10^10: c = −0.48618, d = −1.7226, | |
| 73 | +max |residual| = 0.0025, leave-one-out c ∈ [−0.494, −0.482]. Data: `data/fit_c4.json`. | |
| 74 | +**Model support (PROVER, cycle 2).** The first-order HL triple-correlation model | |
| 75 | +(`src/model_c4.py`: P(g1,g2) ∝ S({0,g1,g1+g2})·exp(−(g1+g2)/λ)) predicts the correct SIGN and the | |
| 76 | +same c + d/λ drift FORM, but magnitude ≈ −0.154 vs observed ≈ −0.49 (factor ~3.2 too small): | |
| 77 | +first-order model quantitatively REJECTED — interior-compositeness (inclusion–exclusion) terms must | |
| 78 | +carry most of the effect. Open problem for a future cycle. | |
| 79 | +**Classification.** (c) potentially new as a precise constant; the phenomenon is known qualitatively. | |
| 80 | +**Status.** CONJECTURED; VERIFIED-consistent up to 4×10^10. | |
| 81 | +**CYCLE 3 STATUS (distributed scan to 10^12, 12 nodes / 1000 chunks / 162 cores).** | |
| 82 | +**PREDICTION CONFIRMED**: observed ρ·ln x(10^12) = −0.54756, inside the predicted band −0.549 ± 0.005. | |
| 83 | +The cycle-1 window [−0.65, −0.55] was exited between 4×10^11 (−0.55027) and 10^12 (−0.54756), matching | |
| 84 | +the predicted exit ≈ 5×10^11 — old C4 now REFUTED-as-predicted. Combined 13-point fit (10^8…10^12): | |
| 85 | +**c = −0.4845 ± 0.003 (LOO), d = −1.757**, max residual 0.0025. New falsifiable predictions: | |
| 86 | +ρ·ln x = −0.5432 ± 0.004 at 10^13; −0.5390 ± 0.004 at 10^14. | |
| 87 | +Cross-validation: distributed merge reproduces cycle-2 single-machine checkpoints exactly at | |
| 88 | +10^10/2×10^10/4×10^10. Status: CONJECTURED and now twice-tested; VERIFIED-consistent up to 10^12. | |
| 89 | +**CYCLE 4 STATUS (3 nodes, 58 CPU cores + 3 Metal GPUs, 10,000 chunks to 10^13).** | |
| 90 | +**SECOND PREDICTION CONFIRMED**: observed ρ·ln x(10^13) = −0.54264, predicted −0.5432 ± 0.004. | |
| 91 | +All 10 overlapping checkpoints identical to cycles 2/3 (exact). 16-point refit (10^8…10^13): | |
| 92 | +**c = −0.48354 (LOO [−0.48570, −0.48165]), d = −1.7772**, max residual 0.00255. | |
| 93 | +Next predictions: ρ·ln x(10^14) = −0.5387 ± 0.004; ρ·ln x(10^15) = −0.5350 ± 0.004. | |
| 94 | +Status: CONJECTURED, twice-confirmed out of sample; VERIFIED-consistent up to 10^13. | |
| 95 | + | |
| 96 | +### C6 — lag-2 anticorrelation (NEW, cycle 2) | |
| 97 | +**Statement.** ρ₂(x) < 0 (correlation of (g_n, g_{n+2})) with ρ₂(x)·ln x → c₂, c₂ ≈ −0.29 ± 0.03 | |
| 98 | +(2-point fit of the c + d/λ form: c₂ = −0.292, d₂ ≈ +1.05 — note the drift has OPPOSITE sign to C4′). | |
| 99 | +**Data.** ρ₂·ln x: −0.235 (10^8) → −0.248 (10^10) → −0.249 (4×10^10), monotone decreasing. | |
| 100 | +**Classification.** (c) same family as C4′; less tested (fit uses only endpoints — firm up in cycle 3). | |
| 101 | +**Status.** CONJECTURED; ρ₂ < 0 VERIFIED UP TO 4×10^10. | |
| 102 | +**CYCLE 3 STATUS.** ρ₂·ln x continues its monotone drift down: −0.25141 at 10^12. Proper 13-point fit: | |
| 103 | +**c₂ = −0.2806, d₂ = +0.782** (opposite-sign drift vs lag-1 confirmed). ρ₂ < 0 VERIFIED UP TO 10^12; | |
| 104 | +c₂ CONJECTURED ≈ −0.28 ± 0.02. Prediction: ρ₂·ln x(10^13) = −0.2545 ± 0.004. | |
| 105 | +**CYCLE 4 STATUS. PREDICTION CONFIRMED**: observed ρ₂·ln x(10^13) = −0.25247 ∈ −0.2545 ± 0.004. | |
| 106 | +16-point refit: c₂ = −0.27762, d₂ = +0.7187. Next prediction: ρ₂·ln x(10^14) = −0.2553 ± 0.004. | |
| 107 | +ρ₂ < 0 VERIFIED UP TO 10^13. | |
| 108 | + | |
| 109 | +### C5 — Cramér–Shanks–Granville ratio below 4×10^9 | |
| 110 | +**Statement.** max_{p ≤ 4×10^9} CSG(p) = 210/ln²(20831323) ≈ 0.7395, attained at p = 20831323 (gap 210). | |
| 111 | +**Classification.** (a) known — the maximal-gap table is exhaustively verified far beyond 4×10^9; | |
| 112 | +this is an independent re-verification, not a discovery. | |
| 113 | +**Status at discovery.** VERIFIED to 10^8 (max CSG 0.7395); the claim to 4×10^9 requires that | |
| 114 | +no later maximal gap below 4×10^9 exceeds it (adversary run). | |
| 115 | +**FINAL STATUS (adversary, 4×10^9).** VERIFIED UP TO 4×10^9 — max CSG = 0.73947 at p = 20831323; | |
| 116 | +all 32 maximal gaps below 4×10^9 reproduced and equal to the published table (A005250/A002386), | |
| 117 | +ending with gap 336 after 3842610773. Independent re-verification of known results, as intended. | |
| 118 | + | |
| 119 | +### Discarded at birth (Phase 4 filter) | |
| 120 | +- "Odd gaps never occur beyond (2,3)" — trivial (parity), not logged as a conjecture. | |
| 121 | +- Any conjecture about gaps ≡ 3 mod 6 etc. — trivial modular obstruction (all gaps beyond (2,3) are even). | |
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data/adversary_4e9.json
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| 453 | + "csg": 0.65423 | |
| 454 | + }, | |
| 455 | + { | |
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| 458 | + "merit": 13.8408, | |
| 459 | + "csg": 0.65606 | |
| 460 | + }, | |
| 461 | + { | |
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| 467 | + { | |
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| 470 | + "merit": 15.2247, | |
| 471 | + "csg": 0.68985 | |
| 472 | + } | |
| 473 | + ], | |
| 474 | + "max_csg": 0.73947 | |
| 475 | +} | |
| \ No newline at end of file | ||
added
data/cycle2_c4_1e10_laptop.json
+315 −0
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| 1 | +{ | |
| 2 | + "limit": 10000000000, | |
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| \ No newline at end of file | ||
added
data/cycle2_c4_1e8_sanity.json
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| \ No newline at end of file | ||
added
data/cycle2_c4_4e10_M3U96a.json
+355 −0
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| \ No newline at end of file | ||
added
data/cycle3_c4_1e12.json
+350 −0
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| 305 | + { | |
| 306 | + "gap": 500, | |
| 307 | + "after": 303371455241, | |
| 308 | + "csg": 0.71533 | |
| 309 | + }, | |
| 310 | + { | |
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| 312 | + "after": 304599508537, | |
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| 315 | + { | |
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| 317 | + "after": 416608695821, | |
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| 321 | + "gap": 532, | |
| 322 | + "after": 461690510011, | |
| 323 | + "csg": 0.7375 | |
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| 325 | + { | |
| 326 | + "gap": 534, | |
| 327 | + "after": 614487453523, | |
| 328 | + "csg": 0.72476 | |
| 329 | + }, | |
| 330 | + { | |
| 331 | + "gap": 540, | |
| 332 | + "after": 738832927927, | |
| 333 | + "csg": 0.72305 | |
| 334 | + } | |
| 335 | + ], | |
| 336 | + "cross_validation_vs_cycle2": { | |
| 337 | + "10000000000": { | |
| 338 | + "rho1_lnx_diff": 0.0, | |
| 339 | + "exact_D_match": true | |
| 340 | + }, | |
| 341 | + "20000000000": { | |
| 342 | + "rho1_lnx_diff": 0.0, | |
| 343 | + "exact_D_match": true | |
| 344 | + }, | |
| 345 | + "40000000000": { | |
| 346 | + "rho1_lnx_diff": 0.0, | |
| 347 | + "exact_D_match": true | |
| 348 | + } | |
| 349 | + } | |
| 350 | +} | |
| \ No newline at end of file | ||
added
data/cycle4_c4_1e13.json
+419 −0
@@ -0,0 +1,419 @@ | ||
| 1 | +{ | |
| 2 | + "limit": 10000000000000, | |
| 3 | + "n_chunks": 10000, | |
| 4 | + "hosts": [ | |
| 5 | + "MacStudnpierre2", | |
| 6 | + "MacStudnpierre2/gpu", | |
| 7 | + "MacStudnpierre4", | |
| 8 | + "MacStudnpierre4/gpu", | |
| 9 | + "MacStudnpierre6", | |
| 10 | + "MacStudnpierre6/gpu" | |
| 11 | + ], | |
| 12 | + "total_core_seconds": 147737.5, | |
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| 93 | + "D_N2_minus_N4": -37509 | |
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| 101 | + "G6": 576, | |
| 102 | + "D_N2_minus_N4": 8870 | |
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| 104 | + }, | |
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| 371 | + { | |
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| 380 | + } | |
| 381 | + ], | |
| 382 | + "cross_validation_vs_cycle2": { | |
| 383 | + "10000000000": { | |
| 384 | + "ref": "cycle3_c4_1e12.json", | |
| 385 | + "rho1_lnx_diff": 0.0, | |
| 386 | + "exact_D_match": true | |
| 387 | + }, | |
| 388 | + "20000000000": { | |
| 389 | + "ref": "cycle3_c4_1e12.json", | |
| 390 | + "rho1_lnx_diff": 0.0, | |
| 391 | + "exact_D_match": true | |
| 392 | + }, | |
| 393 | + "40000000000": { | |
| 394 | + "ref": "cycle3_c4_1e12.json", | |
| 395 | + "rho1_lnx_diff": 0.0, | |
| 396 | + "exact_D_match": true | |
| 397 | + }, | |
| 398 | + "100000000000": { | |
| 399 | + "ref": "cycle3_c4_1e12.json", | |
| 400 | + "rho1_lnx_diff": 0.0, | |
| 401 | + "exact_D_match": true | |
| 402 | + }, | |
| 403 | + "200000000000": { | |
| 404 | + "ref": "cycle3_c4_1e12.json", | |
| 405 | + "rho1_lnx_diff": 0.0, | |
| 406 | + "exact_D_match": true | |
| 407 | + }, | |
| 408 | + "400000000000": { | |
| 409 | + "ref": "cycle3_c4_1e12.json", | |
| 410 | + "rho1_lnx_diff": 0.0, | |
| 411 | + "exact_D_match": true | |
| 412 | + }, | |
| 413 | + "1000000000000": { | |
| 414 | + "ref": "cycle3_c4_1e12.json", | |
| 415 | + "rho1_lnx_diff": 0.0, | |
| 416 | + "exact_D_match": true | |
| 417 | + } | |
| 418 | + } | |
| 419 | +} | |
| \ No newline at end of file | ||
added
data/fit_c4.json
+11 −0
@@ -0,0 +1,11 @@ | ||
| 1 | +{ | |
| 2 | + "source": "cycle2_c4_4e10_M3U96a.json", | |
| 3 | + "n_points": 9, | |
| 4 | + "c_extrapolated": -0.48618, | |
| 5 | + "d": -1.7226, | |
| 6 | + "c_loo_range": [ | |
| 7 | + -0.49422, | |
| 8 | + -0.48193 | |
| 9 | + ], | |
| 10 | + "max_abs_residual": 0.00252 | |
| 11 | +} | |
| \ No newline at end of file | ||
added
data/gap_histogram_1e8.csv
+98 −0
@@ -0,0 +1,98 @@ | ||
| 1 | +gap,count | |
| 2 | +1,1 | |
| 3 | +2,440312 | |
| 4 | +4,440257 | |
| 5 | +6,768752 | |
| 6 | +8,334180 | |
| 7 | +10,430016 | |
| 8 | +12,538382 | |
| 9 | +14,293201 | |
| 10 | +16,215804 | |
| 11 | +18,384738 | |
| 12 | +20,202922 | |
| 13 | +22,175945 | |
| 14 | +24,257548 | |
| 15 | +26,119465 | |
| 16 | +28,129567 | |
| 17 | +30,222847 | |
| 18 | +32,68291 | |
| 19 | +34,71248 | |
| 20 | +36,114028 | |
| 21 | +38,51756 | |
| 22 | +40,60761 | |
| 23 | +42,86637 | |
| 24 | +44,34881 | |
| 25 | +46,29327 | |
| 26 | +48,49824 | |
| 27 | +50,27522 | |
| 28 | +52,20595 | |
| 29 | +54,33593 | |
| 30 | +56,16595 | |
| 31 | +58,14611 | |
| 32 | +60,28439 | |
| 33 | +62,8496 | |
| 34 | +64,8823 | |
| 35 | +66,15579 | |
| 36 | +68,6200 | |
| 37 | +70,8813 | |
| 38 | +72,8453 | |
| 39 | +74,4316 | |
| 40 | +76,3580 | |
| 41 | +78,6790 | |
| 42 | +80,3281 | |
| 43 | +82,2362 | |
| 44 | +84,4668 | |
| 45 | +86,1597 | |
| 46 | +88,1637 | |
| 47 | +90,3337 | |
| 48 | +92,1083 | |
| 49 | +94,971 | |
| 50 | +96,1641 | |
| 51 | +98,851 | |
| 52 | +100,878 | |
| 53 | +102,1059 | |
| 54 | +104,494 | |
| 55 | +106,404 | |
| 56 | +108,711 | |
| 57 | +110,454 | |
| 58 | +112,330 | |
| 59 | +114,487 | |
| 60 | +116,191 | |
| 61 | +118,181 | |
| 62 | +120,433 | |
| 63 | +122,131 | |
| 64 | +124,145 | |
| 65 | +126,204 | |
| 66 | +128,76 | |
| 67 | +130,78 | |
| 68 | +132,132 | |
| 69 | +134,50 | |
| 70 | +136,40 | |
| 71 | +138,93 | |
| 72 | +140,57 | |
| 73 | +142,30 | |
| 74 | +144,51 | |
| 75 | +146,22 | |
| 76 | +148,34 | |
| 77 | +150,37 | |
| 78 | +152,20 | |
| 79 | +154,13 | |
| 80 | +156,23 | |
| 81 | +158,10 | |
| 82 | +160,11 | |
| 83 | +162,8 | |
| 84 | +164,5 | |
| 85 | +166,1 | |
| 86 | +168,8 | |
| 87 | +170,6 | |
| 88 | +172,1 | |
| 89 | +174,3 | |
| 90 | +176,5 | |
| 91 | +178,4 | |
| 92 | +180,4 | |
| 93 | +182,1 | |
| 94 | +184,1 | |
| 95 | +196,1 | |
| 96 | +198,1 | |
| 97 | +210,2 | |
| 98 | +220,1 | |
added
data/gap_mod6_start_1e8.csv
+7 −0
@@ -0,0 +1,7 @@ | ||
| 1 | +p_mod6,gap_mod6,count | |
| 2 | +1,0,1264047 | |
| 3 | +1,4,1616470 | |
| 4 | +2,1,1 | |
| 5 | +3,2,1 | |
| 6 | +5,0,1264465 | |
| 7 | +5,2,1616470 | |
added
data/gapstats_1e8.json
+210 −0
@@ -0,0 +1,210 @@ | ||
| 1 | +{ | |
| 2 | + "limit": 100000000, | |
| 3 | + "scan_seconds": 0.3, | |
| 4 | + "checkpoints": { | |
| 5 | + "1000000": { | |
| 6 | + "jumping_champion": 6, | |
| 7 | + "N2": 8169, | |
| 8 | + "N4": 8143, | |
| 9 | + "N6": 13549, | |
| 10 | + "N2_gt_N4": true, | |
| 11 | + "mult6_local_max_up_to": 66, | |
| 12 | + "pearson_consecutive_gaps": -0.04314, | |
| 13 | + "HL_ratio_obs_over_pred": { | |
| 14 | + "2": 1.1638, | |
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| 62 | + "100": 0.2879, | |
| 63 | + "112": 0.5587, | |
| 64 | + "114": 0.4069 | |
| 65 | + } | |
| 66 | + }, | |
| 67 | + "10000000": { | |
| 68 | + "jumping_champion": 6, | |
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| 70 | + "N4": 58621, | |
| 71 | + "N6": 99987, | |
| 72 | + "N2_gt_N4": true, | |
| 73 | + "mult6_local_max_up_to": 66, | |
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| 135 | + "120": 0.1492 | |
| 136 | + } | |
| 137 | + }, | |
| 138 | + "100000000": { | |
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| 141 | + "N4": 440257, | |
| 142 | + "N6": 768752, | |
| 143 | + "N2_gt_N4": true, | |
| 144 | + "mult6_local_max_up_to": 66, | |
| 145 | + "pearson_consecutive_gaps": -0.03161, | |
| 146 | + "HL_ratio_obs_over_pred": { | |
| 147 | + "2": 1.1233, | |
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| 158 | + "24": 1.0072, | |
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| 162 | + "32": 0.9567, | |
| 163 | + "34": 1.1223, | |
| 164 | + "36": 0.6928, | |
| 165 | + "38": 1.0298, | |
| 166 | + "40": 0.8962, | |
| 167 | + "42": 1.0292, | |
| 168 | + "44": 0.9837, | |
| 169 | + "46": 0.9288, | |
| 170 | + "48": 0.8448, | |
| 171 | + "50": 0.8412, | |
| 172 | + "52": 0.9229, | |
| 173 | + "54": 0.861, | |
| 174 | + "56": 0.6814, | |
| 175 | + "58": 0.9254, | |
| 176 | + "60": 0.9739, | |
| 177 | + "62": 0.6775, | |
| 178 | + "64": 0.7893, | |
| 179 | + "66": 0.7561, | |
| 180 | + "68": 0.6979, | |
| 181 | + "70": 0.7817, | |
| 182 | + "72": 0.5203, | |
| 183 | + "74": 0.6855, | |
| 184 | + "76": 0.6376, | |
| 185 | + "78": 0.6595, | |
| 186 | + "80": 0.5209, | |
| 187 | + "82": 0.5931, | |
| 188 | + "84": 0.6405, | |
| 189 | + "86": 0.5041, | |
| 190 | + "88": 0.5793, | |
| 191 | + "90": 0.595, | |
| 192 | + "92": 0.4816, | |
| 193 | + "94": 0.484, | |
| 194 | + "96": 0.4483, | |
| 195 | + "98": 0.4487, | |
| 196 | + "100": 0.3112, | |
| 197 | + "102": 0.4079, | |
| 198 | + "104": 0.4355, | |
| 199 | + "106": 0.3991, | |
| 200 | + "108": 0.2648, | |
| 201 | + "110": 0.406, | |
| 202 | + "112": 0.36, | |
| 203 | + "114": 0.3723, | |
| 204 | + "116": 0.3333, | |
| 205 | + "118": 0.3539, | |
| 206 | + "120": 0.3075 | |
| 207 | + } | |
| 208 | + } | |
| 209 | + } | |
| 210 | +} | |
| \ No newline at end of file | ||
added
data/gapstats_1e8.npz
+0 −0
Binary file not shown.
added
data/maximal_gaps_1e8.csv
+26 −0
@@ -0,0 +1,26 @@ | ||
| 1 | +gap,start_prime,merit,csg_ratio | |
| 2 | +1,2,1.442695,2.081369 | |
| 3 | +2,3,1.820478,1.657071 | |
| 4 | +4,7,2.055593,1.056366 | |
| 5 | +6,23,1.913574,0.610294 | |
| 6 | +8,89,1.782278,0.397065 | |
| 7 | +14,113,2.961466,0.626449 | |
| 8 | +18,523,2.875592,0.459390 | |
| 9 | +20,887,2.946443,0.434076 | |
| 10 | +22,1129,3.129851,0.445271 | |
| 11 | +34,1327,4.728345,0.657566 | |
| 12 | +36,9551,3.928244,0.428642 | |
| 13 | +44,15683,4.554709,0.471486 | |
| 14 | +52,19609,5.261164,0.532305 | |
| 15 | +72,31397,6.953520,0.671548 | |
| 16 | +86,155921,7.192377,0.601515 | |
| 17 | +96,360653,7.502537,0.586334 | |
| 18 | +112,370261,8.735012,0.681254 | |
| 19 | +114,492113,8.697998,0.663642 | |
| 20 | +118,1349533,8.359741,0.592248 | |
| 21 | +132,1357201,9.347823,0.661983 | |
| 22 | +148,2010733,10.197044,0.702566 | |
| 23 | +154,4652353,10.030689,0.653342 | |
| 24 | +180,17051707,10.809668,0.649161 | |
| 25 | +210,20831323,12.461452,0.739466 | |
| 26 | +220,47326693,12.448660,0.704405 | |
added
data/model_c4.json
+50 −0
@@ -0,0 +1,50 @@ | ||
| 1 | +{ | |
| 2 | + "model": "P(g1,g2) ~ W_HL(0,g1,g1+g2) * exp(-(g1+g2)/lambda)", | |
| 3 | + "predictions": { | |
| 4 | + "1e8": { | |
| 5 | + "lambda": 18.4207, | |
| 6 | + "rho": -0.008638, | |
| 7 | + "rho_times_lambda": -0.15911 | |
| 8 | + }, | |
| 9 | + "1e9": { | |
| 10 | + "lambda": 20.7233, | |
| 11 | + "rho": -0.007659, | |
| 12 | + "rho_times_lambda": -0.15872 | |
| 13 | + }, | |
| 14 | + "1e10": { | |
| 15 | + "lambda": 23.0259, | |
| 16 | + "rho": -0.006868, | |
| 17 | + "rho_times_lambda": -0.15814 | |
| 18 | + }, | |
| 19 | + "4e10": { | |
| 20 | + "lambda": 24.4121, | |
| 21 | + "rho": -0.006461, | |
| 22 | + "rho_times_lambda": -0.15774 | |
| 23 | + }, | |
| 24 | + "1e12": { | |
| 25 | + "lambda": 27.631, | |
| 26 | + "rho": -0.005673, | |
| 27 | + "rho_times_lambda": -0.15676 | |
| 28 | + }, | |
| 29 | + "1e15": { | |
| 30 | + "lambda": 34.5388, | |
| 31 | + "rho": -0.004487, | |
| 32 | + "rho_times_lambda": -0.15498 | |
| 33 | + }, | |
| 34 | + "1e20": { | |
| 35 | + "lambda": 46.0517, | |
| 36 | + "rho": -0.003335, | |
| 37 | + "rho_times_lambda": -0.15358 | |
| 38 | + }, | |
| 39 | + "1e30": { | |
| 40 | + "lambda": 69.0776, | |
| 41 | + "rho": -0.002218, | |
| 42 | + "rho_times_lambda": -0.1532 | |
| 43 | + }, | |
| 44 | + "1e40": { | |
| 45 | + "lambda": 92.1034, | |
| 46 | + "rho": -0.001671, | |
| 47 | + "rho_times_lambda": -0.15388 | |
| 48 | + } | |
| 49 | + } | |
| 50 | +} | |
| \ No newline at end of file | ||
added
data/summary_1e8.json
+30 −0
@@ -0,0 +1,30 @@ | ||
| 1 | +{ | |
| 2 | + "limit": 100000000, | |
| 3 | + "n_gaps": 5761454, | |
| 4 | + "scan_seconds": 0.6, | |
| 5 | + "largest_gap": { | |
| 6 | + "gap": 220, | |
| 7 | + "after_prime": 47326693 | |
| 8 | + }, | |
| 9 | + "n_maximal_gaps": 25, | |
| 10 | + "jumping_champion": 6, | |
| 11 | + "gap_fraction_mod6": { | |
| 12 | + "0": 0.4388669943385819, | |
| 13 | + "1": 1.7356729742179665e-07, | |
| 14 | + "2": 0.28056650283070905, | |
| 15 | + "3": 0.0, | |
| 16 | + "4": 0.28056632926341163, | |
| 17 | + "5": 0.0 | |
| 18 | + }, | |
| 19 | + "best_merit": { | |
| 20 | + "merit": 12.46145, | |
| 21 | + "gap": 210, | |
| 22 | + "after_prime": 20831323 | |
| 23 | + }, | |
| 24 | + "best_csg": { | |
| 25 | + "csg": 0.73947, | |
| 26 | + "gap": 210, | |
| 27 | + "after_prime": 20831323 | |
| 28 | + }, | |
| 29 | + "method": "segmented sieve of Eratosthenes, numpy, deterministic (no seed)" | |
| 30 | +} | |
| \ No newline at end of file | ||
added
journal.md
+203 −0
@@ -0,0 +1,203 @@ | ||
| 1 | +<!-- | |
| 2 | +journal.md — Prime Mystery Engine: research journal | |
| 3 | +Author: Simon-Pierre Boucher — contact@spboucher.ai | |
| 4 | +--> | |
| 5 | + | |
| 6 | +# Research Journal | |
| 7 | + | |
| 8 | +## 2026-08-06 — Phase 0: Setup | |
| 9 | + | |
| 10 | +- Created structure: `/src`, `/data`, `/certs`, `journal.md`, `conjectures.md`, `records.md`. | |
| 11 | +- Implemented `src/core.py`: odd-only numpy sieve, segmented sieve, deterministic Miller–Rabin | |
| 12 | + (12 bases, valid for n < 3.317×10^24, Sorenson–Webster), BPSW (strong Lucas, Selfridge params). | |
| 13 | +- Validation `src/test_core.py`: **37/37 PASS** in 0.3 s — | |
| 14 | + π(10^2..10^8) exact (incl. π(10^8)=5761455 via segmented sieve), segment-vs-full sieve equality, | |
| 15 | + full maximal-gap table below 10^6 reproduced, strong-pseudoprime & Carmichael traps, | |
| 16 | + MR≡sieve on [0,40000), BPSW≡det-MR on 300 random 62-bit integers (seed 42). | |
| 17 | +- Gate passed → allowed to proceed. | |
| 18 | + | |
| 19 | +## 2026-08-06 — Phase 1: Axis selection | |
| 20 | + | |
| 21 | +**Chosen axis: #10 — Prime deserts (large gaps).** Justification (5 lines): | |
| 22 | +1. Fully verifiable: a gap is certified by two primality certificates + compositeness of the interior (sieve or covering) — zero ambiguity. | |
| 23 | +2. Measurable success criteria exist at every scale: merit g/ln p, CSG ratio g/ln²p, reproduction of the known maximal-gap table below our compute bound. | |
| 24 | +3. Rich conjecture surface: Cramér/Granville corrections, residues of gap endpoints, jumping champions — testable to 10^9+ locally. | |
| 25 | +4. State of the art is precisely documented (records.md) so novelty checks are cheap and honest. | |
| 26 | +5. The PROVER role has a concrete deliverable even without records: covering-system desert constructions with independent re-verification scripts. | |
| 27 | + | |
| 28 | +## 2026-08-06 — Phase 2: State of the art | |
| 29 | + | |
| 30 | +- Web-checked current records (see `records.md`): largest maximal gap 1854 (2026), merit record 41.94 (2017), | |
| 31 | + CSG record 0.9206 (Nyman 1999), exhaustive bound ≈ 2×10^19. | |
| 32 | +- Realistic targets: (i) independent verification of the maximal-gap table to ~4×10^9, (ii) massively tested | |
| 33 | + statistical conjectures on gaps, (iii) certified covering-system desert + verification pipeline. | |
| 34 | + | |
| 35 | +## Cycle 1 — 2026-08-06 — COMPLETE (stopping criterion (a): a conjecture refuted + criterion (b): a construction certified) | |
| 36 | + | |
| 37 | +### EXPLORER (Phase 3) | |
| 38 | +- `src/explore_gaps.py 1e8` (0.6 s): all 25 maximal gaps below 10^8 reproduced = published table; | |
| 39 | + jumping champion 6; gaps ≡ 0 mod 6 are 43.9% of all gaps; best merit 12.46 (gap 210 after 20831323). | |
| 40 | +- `src/analyze_gaps.py 1e8` with checkpoints 10^6/10^7/10^8 (bug found & fixed: segment boundaries | |
| 41 | + must align on checkpoints — symptom: twin counts wrong at sub-segment checkpoints; after fix, | |
| 42 | + N(2,x) = 8169 / 58980 / 440312 = literature values exactly). | |
| 43 | +- Key quantified observations: (O1) N2−N4 margins +26/+359/+55 — tiny and non-monotone; | |
| 44 | + (O2) multiples of 6 strict local maxima up to g = 66 at all three scales; (O3) ρ(x)·ln x ≈ −0.6 stable; | |
| 45 | + (O4) naive HL gap model exp(−g/ln t) shows structured deviations (deficit at g = 36, 72, 100, 108) — | |
| 46 | + parked for a future cycle (needs inclusion–exclusion model before conjecturing). | |
| 47 | + | |
| 48 | +### CONJECTURER (Phase 4) | |
| 49 | +- C1 champion=6; C2 race N2>N4 for x ≥ 10^6; C3 G6(x) ≥ 66; C4 ρ·ln x ∈ [−0.65,−0.55] → c ≈ −0.6; | |
| 50 | + C5 max CSG below 4×10^9 = 0.7395. Two trivial/parity statements discarded at birth. See conjectures.md. | |
| 51 | + | |
| 52 | +### ADVERSARY (Phase 5) | |
| 53 | +- `src/adversary_race.py 4e9` (15 s, deterministic): range = 40× discovery. | |
| 54 | +- **C2 REFUTED** — full-resolution scan (every gap event): first tie after 10^6 at end-prime 80966861, | |
| 55 | + first strict overtake at 80966933; the race then changes leader forever in range (137.6M events with | |
| 56 | + D ≤ 0; D(4×10^9) = −2270). Checkpoint-only verification had been misleading — lesson recorded. | |
| 57 | +- C1, C3, C5 survive (C3 strengthened: G6 jumps to 216 at 10^9). C4 survives with revision (drift of | |
| 58 | + ρ·ln x toward −0.565; "≈ −0.6 limit" retracted). | |
| 59 | +- Cramér/Maier test applied: C2's refutation is exactly random-model behavior (fair random walk); | |
| 60 | + C1/C3/C4 are structural (absent from a Cramér model). | |
| 61 | +- All 32 maximal gaps below 4×10^9 = published table (independent verification, `data/adversary_4e9.json`). | |
| 62 | + | |
| 63 | +### PROVER (Phase 6) | |
| 64 | +- Attempt 1 (`src/prover_desert.py`): pure covering system, primes ≤ 59, greedy residues + CRT → | |
| 65 | + certified gap 90 at a 22-digit N, merit 1.78. Correct but weak. | |
| 66 | +- Attempt 2 (`src/prover_desert2.py`): **hybrid covering** — greedy covers 243/259 positions for every | |
| 67 | + shift t (PROVEN); 16 holes certified composite for the chosen t by explicit trial factors (15) and one | |
| 68 | + strong MR witness (1); endpoints deterministic-MR prime (< 3.317×10^24 bound). Result: | |
| 69 | + **certified gap 260 after N = 1116336781708038449369693 (25 digits), merit 4.6955** — | |
| 70 | + ×2.9 vs attempt 1, ×4.3 vs the classic primorial baseline (61) with the same primes. NOT a record | |
| 71 | + (records.md: merit record 41.94) — the deliverable is the certified pipeline. | |
| 72 | +- Proven statement (elementary): for every t ≥ 0, the 243 covered positions of x + t·59# + i are | |
| 73 | + composite — an explicit infinite family of long composite runs. | |
| 74 | +- Independent verifications: `certs/verify_desert.py` PASSED; `certs/verify_c2_refutation.py` PASSED. | |
| 75 | + | |
| 76 | +### Decisions | |
| 77 | +- Cycle closed. Next cycle: C4 at 10^10 (does ρ·ln x stabilize?) — see report_cycle1.md. | |
| 78 | + | |
| 79 | +## Cycle 2 — 2026-08-06 — COMPLETE (focus: C4 at 10^10/4×10^10, dual-machine) | |
| 80 | + | |
| 81 | +### Setup / budget | |
| 82 | +- User directive: run C4 at 10^10 on this computer AND on M3U96a. Strategy: identical deterministic | |
| 83 | + scan on both — laptop to 10^10 (40 s), M3U96a (M3 Ultra) to 4×10^10 (128 s, single-core numpy). | |
| 84 | + Deploy via `cluster-gateway.sh stage` to `~/cluster-projects/conjoncture/`. | |
| 85 | +- Sanity gate: `src/cycle2_c4.py 1e8` reproduces cycle-1 values exactly (ρ·ln x = −0.59604 / −0.61428 / | |
| 86 | + −0.58221 at 10^6/10^7/10^8; D = 26/359/55). PASS. | |
| 87 | + | |
| 88 | +### EXPLORER/ADVERSARY (Phases 3+5 merged — C4 escalation) | |
| 89 | +- **Cross-machine validation:** every overlapping checkpoint (10^6…10^10) bit-for-bit identical | |
| 90 | + between laptop and M3U96a. Independent-hardware reproducibility ✓. | |
| 91 | +- ρ₁·ln x continues drifting up: −0.58221 (10^8) → −0.56116 (10^10) → −0.55569 (4×10^10). | |
| 92 | + Cycle-1 window [−0.65, −0.55] still holds at 4×10^10 — barely. | |
| 93 | +- New: lag-2 correlation ρ₂·ln x drifts DOWN: −0.235 (10^8) → −0.249 (4×10^10). | |
| 94 | +- Free continuations: champion = 6 up to 4×10^10 (C1 ✓); G6 ≥ 66 everywhere but NON-monotone | |
| 95 | + (312 at 10^10 → 282 at 2×10^10 → 276 at 4×10^10 — tail noise, C3 statement unaffected); | |
| 96 | + C2 race still swinging (D = +2681 at 10^10, +7160 at 2×10^10, −804 at 4×10^10); | |
| 97 | + all 37 maximal gaps below 4×10^10 = published table incl. 384@20678048297, 394@22367084959, | |
| 98 | + 456@25056082087 (web-checked OEIS A002386 / t5k.org after a wrong hand-written checklist | |
| 99 | + briefly suggested a mismatch — the SCAN was right, the from-memory list was wrong; lesson: | |
| 100 | + never hand-write "known" tables from memory, always fetch). | |
| 101 | +- Max CSG below 4×10^10 = 0.79535 (gap 456 after 25056082087) — new in-range CSG high, known. | |
| 102 | + | |
| 103 | +### CONJECTURER (Phase 4) | |
| 104 | +- `src/fit_c4.py`: ρ₁·ln x = c + d/ln x fits 9 checkpoints (10^8…4×10^10) with max residual 0.0025: | |
| 105 | + **c = −0.486 ± 0.010 (LOO), d = −1.72** → C4′ stated (see conjectures.md), with the sharp | |
| 106 | + falsifiable prediction ρ·ln x(10^12) = −0.549 ± 0.005, and predicted exit of the cycle-1 window | |
| 107 | + near x ≈ 5×10^11. C6 (lag-2, c₂ ≈ −0.29) stated with lower confidence. | |
| 108 | + | |
| 109 | +### PROVER (Phase 6) | |
| 110 | +- `src/model_c4.py`: first-order HL triple-correlation model P(g1,g2) ∝ S({0,g1,g1+g2})·e^{−(g1+g2)/λ}. | |
| 111 | + Result: correct sign, correct c + d/λ drift form, magnitude ρ·λ ≈ −0.154 vs observed −0.49: | |
| 112 | + **quantitatively REJECTED** (factor ~3.2). Honest conclusion: the anticorrelation is dominated by | |
| 113 | + interior-compositeness constraints (inclusion–exclusion), not by the bare triple singular series. | |
| 114 | + Building the full inclusion–exclusion model is the open PROVER problem for a later cycle. | |
| 115 | + | |
| 116 | +### Decisions | |
| 117 | +- Cycle closed (criterion (a)-adjacent: C4 superseded by refined C4′ with a falsifiable constant). | |
| 118 | +- Next precise action: test C4′ at 10^12 — predicted ρ·ln x = −0.549 ± 0.005. Needs ~25× more compute | |
| 119 | + than 4×10^10 (~1 h on M3U96a single-core, or minutes core-proportionally across the cluster). | |
| 120 | + | |
| 121 | +## Cycle 3 — 2026-08-06 — COMPLETE (C4′ tested at 10^12, distributed on the cluster) | |
| 122 | + | |
| 123 | +### Setup / budget | |
| 124 | +- User directive: distribute; exclude M3U96b and M2U64. Design: 1000 chunks of 10^9 (boundaries = | |
| 125 | + multiples of 10^9 so checkpoints fall on chunk edges), mergeable streaming sums, 5000-wide junction | |
| 126 | + buffer per chunk (max gap < 10^12 is 540, so the buffer always contains the 2 preceding gaps). | |
| 127 | +- Small-scale gate first: 10 chunks × 10^9 merged locally reproduce the cycle-2 single-machine values | |
| 128 | + at 10^10 EXACTLY (ρ, D, G6, champion). PASS. | |
| 129 | +- Fleet reality vs plan: 18 candidate nodes → m2m16 unreachable; numpy installed on 5 more nodes | |
| 130 | + (Apple CLT Python 3.9); 5 nodes had NO usable python3 (missing Xcode CLT: m4ma, m4mb, m4mc, | |
| 131 | + M2M32c, m2m8b) → dropped. Final fleet: 12 nodes / 162 cores, core-proportional quotas | |
| 132 | + (174/99/99/99/86/86/74/74/62/49/49/49 chunks). | |
| 133 | + | |
| 134 | +### Incident (instructive failure) | |
| 135 | +- First launch: 9/12 nodes crashed instantly — `np.ndarray | None` annotation in core.py is | |
| 136 | + Python ≥ 3.10 syntax; the 9 nodes run Apple's Python 3.9. Fix: `from __future__ import annotations` | |
| 137 | + (annotations become lazy). Local test suite re-run (PASS), core.py re-pushed, import-checked on all | |
| 138 | + nine, relaunched. Lesson: pin the LOWEST interpreter version in the fleet as the compatibility | |
| 139 | + target, and import-check on every node class before a fleet launch. | |
| 140 | +- zsh gotchas hit twice in orchestration (`$NODES` not word-split → `${=NODES}`; `$n:c…` eaten as a | |
| 141 | + history-style modifier in scp targets → quote remote paths). | |
| 142 | + | |
| 143 | +### ADVERSARY (Phase 5 — the run) | |
| 144 | +- 1000/1000 chunks collected; merge verified exact tiling of [2, 10^12). | |
| 145 | +- **Cross-validation:** merged distributed results equal the cycle-2 single-machine scan exactly at | |
| 146 | + 10^10, 2×10^10, 4×10^10 (|Δρ·ln x| = 0 at 5 decimals; D identical). Pipeline validated end-to-end. | |
| 147 | +- **C4′ VERDICT: CONFIRMED.** ρ·ln x(10^12) = −0.54756 ∈ predicted −0.549 ± 0.005. | |
| 148 | + Old C4 window [−0.65, −0.55] exited between 4×10^11 and 10^12 as predicted (≈ 5×10^11). | |
| 149 | +- Checkpoints: −0.55381 (10^11), −0.55203 (2×10^11), −0.55027 (4×10^11), −0.54756 (10^12). | |
| 150 | +- Free continuations: champion = 6 to 10^12 (C1); G6 → 480 at 10^12 (C3, growing); C2 race STILL | |
| 151 | + swinging at 10^12 (D = −238 out of ~1.5×10^10 gap events — astonishingly tight); | |
| 152 | + all 48 maximal gaps below 10^12 = published table (A005250/A002386; tail spot-checked online, | |
| 153 | + ends 540 @ 738832927927); max CSG below 10^12 unchanged in significance (known values). | |
| 154 | +- Compute: 17,719 core-seconds ≈ 4.9 core-hours; wall ≈ 5 min over two launches. | |
| 155 | + | |
| 156 | +### CONJECTURER (Phase 4 — refit) | |
| 157 | +- Combined 13-point fit (10^8…10^12): c = −0.48449 (LOO [−0.48793, −0.48196]), d = −1.7571, | |
| 158 | + max residual 0.00245. New predictions: ρ·ln x(10^13) = −0.5432 ± 0.004, ρ·ln x(10^14) = −0.5390 ± 0.004. | |
| 159 | +- C6 firmed up: c₂ = −0.2806, d₂ = +0.7815 (opposite drift sign vs lag-1 confirmed); | |
| 160 | + prediction ρ₂·ln x(10^13) = −0.2545 ± 0.004. | |
| 161 | + | |
| 162 | +### Decisions | |
| 163 | +- Cycle closed (criterion (b)-adjacent: prediction confirmed, law strengthened). | |
| 164 | +- **Fleet policy change (user directive, recorded):** future distributed runs use ONLY | |
| 165 | + M3U96a + M4M64a + M4M64b, saturated, adding a Metal/GPU path (MLX or Metal kernels) for the sieve. | |
| 166 | + Benchmark GPU vs numpy before committing a full run. | |
| 167 | +- Next precise action: cycle 4 — C4′ at 10^13 on the 3-node fleet with a Metal-accelerated sieve | |
| 168 | + (10× cycle 3's work: ~50 core-hours CPU-only, so the GPU path is the enabler). | |
| 169 | + | |
| 170 | +## Cycle 4 — 2026-08-06 — COMPLETE (C4′ and C6 tested at 10^13; CPU+Metal GPU on 3 nodes) | |
| 171 | + | |
| 172 | +### Setup / gates | |
| 173 | +- Fleet per user directive: M3U96a (28c) + M4M64a (16c) + M4M64b (16c), saturated, plus Metal GPU. | |
| 174 | +- MLX installed where missing (PEP 668 → `--user --break-system-packages`). | |
| 175 | +- **GPU sieve** (`src/gpu_sieve.py`, `mx.fast.metal_kernel`): grid (blocks × base primes), byte marking | |
| 176 | + from max(p², first multiple ≥ block), benign write races. Gates passed: byte-identical prime sets vs | |
| 177 | + CPU sieve on 4 windows up to 10^13; full worker partial identical field-for-field on a 10^9 chunk. | |
| 178 | + Bench: GPU 1.2–1.3 s vs CPU-core 2.7–3.6 s per 10^9 near 10^12 → GPU ≈ 2–3 CPU cores per node. | |
| 179 | +- 10,000 chunks of 10^9; measured-rate quotas 4000/3000/3000 with GPU sub-quotas 370/400/400 | |
| 180 | + (27/15/15 CPU workers, 1 core reserved per node for GPU extraction). Local 10-chunk merge gate | |
| 181 | + reproduced cycle-2 values exactly before launch. | |
| 182 | + | |
| 183 | +### The run | |
| 184 | +- Wall ≈ 50 min; 147,738 core-seconds (~41 core-hours) including ~4% duplicated tail work from a | |
| 185 | + late rebalance (M4 nodes finished their quotas and absorbed 500 tail chunks of M3U96a's queue; | |
| 186 | + duplicates are byte-identical, tiling asserted at merge). GPU processed 1170 chunks (11.7%). | |
| 187 | +- Merge: 10,000 unique chunks tile [2, 10^13) exactly; **all 10 overlapping checkpoints identical to | |
| 188 | + cycles 2/3** (|Δρ·ln x| = 0, D exact). | |
| 189 | + | |
| 190 | +### VERDICTS | |
| 191 | +- **C4′ CONFIRMED (2nd out-of-sample hit): ρ·ln x(10^13) = −0.54264 vs predicted −0.5432 ± 0.004.** | |
| 192 | +- **C6 CONFIRMED (1st hit): ρ₂·ln x(10^13) = −0.25247 vs predicted −0.2545 ± 0.004.** | |
| 193 | +- 16-point refits: c = −0.48354 (LOO ±0.002), d = −1.7772; c₂ = −0.27762, d₂ = +0.7187. | |
| 194 | + Next: ρ·ln x(10^14) = −0.5387 ± 0.004; ρ₂·ln x(10^14) = −0.2553 ± 0.004. | |
| 195 | +- Continuations: champion 6 to 10^13; G6 = 576 at 10^13; C2 race STILL swinging (D: −98,967 at 2×10^12, | |
| 196 | + +8,870 at 10^13); all 53 maximal gaps below 10^13 = published table (new in range: 582, 588, 602, | |
| 197 | + 652, 674 — tail web-checked t5k/Nicely); max CSG below 10^13 = 0.79754 (gap 652 after 2614941710599). | |
| 198 | + | |
| 199 | +### Decisions | |
| 200 | +- Cycle closed. Next precise action: cycle 5 — 10^14 (~10× work: ~400 CPU-core-hours equivalent; | |
| 201 | + with the 3-node fleet ≈ 8–9 h wall, GPU included — schedule as an overnight run) to test | |
| 202 | + ρ·ln x(10^14) = −0.5387 ± 0.004. Alternative next action if compute is deferred: the PROVER | |
| 203 | + inclusion–exclusion model for the constant c (main open problem). | |
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| 1 | +% ============================================================================= | |
| 2 | +% main.tex — Prime Mystery Engine: methods and results to date (detailed) | |
| 3 | +% Author: Simon-Pierre Boucher — contact@spboucher.ai | |
| 4 | +% Build: pdflatex main.tex (twice) or latexmk -pdf main.tex | |
| 5 | +% ============================================================================= | |
| 6 | +\documentclass[11pt]{article} | |
| 7 | + | |
| 8 | +\usepackage[margin=1.05in]{geometry} | |
| 9 | +\usepackage{amsmath,amssymb,amsthm} | |
| 10 | +\usepackage{booktabs} | |
| 11 | +\usepackage{microtype} | |
| 12 | +\usepackage{algorithm} | |
| 13 | +\usepackage{algpseudocode} | |
| 14 | +\usepackage[hidelinks]{hyperref} | |
| 15 | +\usepackage{xcolor} | |
| 16 | + | |
| 17 | +\newtheorem{theorem}{Theorem} | |
| 18 | +\newtheorem{lemma}{Lemma} | |
| 19 | +\newtheorem{proposition}{Proposition} | |
| 20 | +\newtheorem{conjecture}{Conjecture} | |
| 21 | +\newtheorem{observation}{Observation} | |
| 22 | +\theoremstyle{definition} | |
| 23 | +\newtheorem{result}{Result} | |
| 24 | +\newtheorem{remark}{Remark} | |
| 25 | + | |
| 26 | +\newcommand{\lnx}{\ln x} | |
| 27 | +\newcommand{\statuslbl}[1]{\textsc{\small #1}} | |
| 28 | +\newcommand{\Sing}{\mathfrak{S}} | |
| 29 | +\DeclareMathOperator{\li}{li} | |
| 30 | + | |
| 31 | +\title{A Verification-First Computational Laboratory for Prime Gap Statistics:\\ | |
| 32 | +Certified Deserts, a Refuted Race Conjecture, and an Empirical Scaling Law\\ | |
| 33 | +for Consecutive-Gap Correlations} | |
| 34 | +\author{Simon-Pierre Boucher\\ | |
| 35 | +\texttt{contact@spboucher.ai}} | |
| 36 | +\date{August 6, 2026 --- working draft v3} | |
| 37 | + | |
| 38 | +\begin{document} | |
| 39 | +\maketitle | |
| 40 | + | |
| 41 | +\begin{abstract} | |
| 42 | +We describe an autonomous, verification-first research pipeline for the statistics of gaps | |
| 43 | +between consecutive primes, and the results it has produced to date. The pipeline enforces a | |
| 44 | +strict epistemic protocol (every claim is \emph{proven}, \emph{certified}, \emph{verified up | |
| 45 | +to an explicit bound}, or \emph{explicitly conjectural}) and cycles through four adversarial | |
| 46 | +roles: exploration, conjecture, attack, and proof. Main outcomes so far: (i) the refutation, | |
| 47 | +with minimal counterexample, of the conjecture that gaps of size 2 outnumber gaps of size 4 | |
| 48 | +from $10^6$ onward --- the two counts in fact exchange the lead indefinitely throughout the | |
| 49 | +tested range, with the difference at $10^{12}$ being $-238$ out of $\approx 3.76\times | |
| 50 | +10^{10}$ gap events; (ii) a fully certified prime desert of length 260 at a 25-digit | |
| 51 | +integer, constructed by a hybrid covering system with independently re-verifiable | |
| 52 | +certificates; (iii) an empirical scaling law for the Pearson correlation of consecutive | |
| 53 | +gaps, $\rho(x)\lnx = c + d/\lnx + o(1/\lnx)$ with $c = -0.4835 \pm 0.002$ and $d = -1.777$, | |
| 54 | +whose out-of-sample predictions at $x = 10^{12}$ ($-0.549 \pm 0.005$, measured $-0.54756$) | |
| 55 | +and $x = 10^{13}$ ($-0.5432 \pm 0.004$, measured $-0.54264$) were both subsequently | |
| 56 | +confirmed; and (iv) the quantitative rejection of the | |
| 57 | +first-order Hardy--Littlewood triple-correlation model for this constant, which predicts | |
| 58 | +the correct sign and drift form but only about $30\%$ of the observed magnitude. All scans | |
| 59 | +are deterministic; their accumulators are exact integers below $2^{53}$, which yields | |
| 60 | +bit-identical results across heterogeneous machines and exact chunked/merged decomposition. | |
| 61 | +All published gap tables below $10^{13}$ (53 maximal gaps) were reproduced exactly. | |
| 62 | +Computations ran on a small cluster of Apple-silicon machines, latterly using a custom | |
| 63 | +Metal GPU sieve kernel (validated byte-identical against the CPU implementation) alongside | |
| 64 | +saturated CPU cores; the correlation scan now extends to $10^{13}$, where the law's predictions for both the | |
| 65 | +lag-1 and lag-2 statistics were confirmed out of sample. | |
| 66 | +\end{abstract} | |
| 67 | + | |
| 68 | +\tableofcontents | |
| 69 | + | |
| 70 | +\section{Introduction} | |
| 71 | + | |
| 72 | +Let $p_1 < p_2 < \cdots$ denote the primes and | |
| 73 | +\begin{equation} | |
| 74 | +g_n \;=\; p_{n+1} - p_n | |
| 75 | +\end{equation} | |
| 76 | +the sequence of prime gaps. This project treats the empirical study of $(g_n)$ as an | |
| 77 | +engineering discipline: no numerical claim is admitted without a certificate, a | |
| 78 | +re-verification script, or an explicit statement of the verified range. The work reported | |
| 79 | +here was carried out by an autonomous research loop (the ``Prime Mystery Engine'') | |
| 80 | +operating under the fixed protocol of Section~\ref{sec:protocol} on the hardware of | |
| 81 | +Section~\ref{sec:compute}. | |
| 82 | + | |
| 83 | +Three kinds of outputs are reported. First, \emph{negative results}: a natural conjecture | |
| 84 | +about the race between twin gaps ($g=2$) and quadruple gaps ($g=4$) is refuted with a | |
| 85 | +minimal counterexample (Section~\ref{sec:race}). Second, \emph{constructions}: a certified | |
| 86 | +prime desert obtained from a hybrid covering system (Section~\ref{sec:desert}). Third, | |
| 87 | +\emph{quantitative empirical laws}: a two-parameter scaling law for the correlation of | |
| 88 | +consecutive gaps that has survived two rounds of genuine out-of-sample prediction | |
| 89 | +(Section~\ref{sec:c4}), together with the failure of the natural first-order model to | |
| 90 | +explain its constant (Section~\ref{sec:model}). | |
| 91 | + | |
| 92 | +Throughout, every statement carries one of the labels \statuslbl{proven}, | |
| 93 | +\statuslbl{certified}, \statuslbl{verified up to $X$}, \statuslbl{conjectured}, or | |
| 94 | +\statuslbl{refuted}. Nothing below is claimed to be new to the literature unless | |
| 95 | +explicitly discussed; where we suspect a phenomenon is known, we say so. | |
| 96 | + | |
| 97 | +\section{Notation and background} | |
| 98 | +\label{sec:background} | |
| 99 | + | |
| 100 | +$\pi(x)$ is the prime-counting function; the prime number theorem gives | |
| 101 | +$\pi(x) \sim \li(x) = \int_2^x dt/\ln t$, so the \emph{local scale} of gaps near $x$ is | |
| 102 | +\begin{equation} | |
| 103 | +\lambda \;=\; \lnx, \qquad \mathbb{E}[\,g \mid p \approx x\,] \sim \lambda . | |
| 104 | +\end{equation} | |
| 105 | +For an even $g$ we write | |
| 106 | +\begin{equation} | |
| 107 | +N(g, x) \;=\; \#\{\, n : g_n = g,\; p_{n+1} \le x \,\}, | |
| 108 | +\end{equation} | |
| 109 | +the number of gaps of size exactly $g$ ending below $x$ (the ``ending-prime'' attribution | |
| 110 | +is the convention used by all our accumulators; see Section~\ref{sec:merge}). | |
| 111 | + | |
| 112 | +\paragraph{Hardy--Littlewood singular series.} | |
| 113 | +For a finite set $\mathcal{H} = \{h_1, \dots, h_k\}$ of integers let | |
| 114 | +$\nu_p(\mathcal{H})$ be the number of distinct residues of $\mathcal{H}$ modulo $p$. The | |
| 115 | +Hardy--Littlewood $k$-tuple conjecture predicts | |
| 116 | +\begin{equation} | |
| 117 | +\#\{\, n \le x : n + h_1, \dots, n + h_k \ \text{all prime} \,\} | |
| 118 | +\;\sim\; \Sing(\mathcal{H}) \int_2^x \frac{dt}{(\ln t)^k}, | |
| 119 | +\qquad | |
| 120 | +\Sing(\mathcal{H}) \;=\; \prod_{p} \frac{1 - \nu_p(\mathcal{H})/p}{(1 - 1/p)^k}. | |
| 121 | +\label{eq:HL} | |
| 122 | +\end{equation} | |
| 123 | +For a pair $\mathcal{H} = \{0, g\}$ this specializes to | |
| 124 | +\begin{equation} | |
| 125 | +\Sing(\{0,g\}) \;=\; C(g) \;=\; 2 C_2 \prod_{\substack{p \mid g \\ p > 2}} \frac{p-1}{p-2}, | |
| 126 | +\qquad | |
| 127 | +C_2 = \prod_{p > 2}\Bigl(1 - \frac{1}{(p-1)^2}\Bigr) = 0.6601618158\ldots | |
| 128 | +\label{eq:singular-pair} | |
| 129 | +\end{equation} | |
| 130 | +Note $C(2) = C(4) = 2C_2$: the twin and quadruple constants coincide, which is what makes | |
| 131 | +the race of Section~\ref{sec:race} a second-order question. | |
| 132 | + | |
| 133 | +\paragraph{Merit and the Cram\'er--Shanks--Granville ratio.} | |
| 134 | +For a gap $g$ after $p$, | |
| 135 | +\begin{equation} | |
| 136 | +\text{merit}(g, p) = \frac{g}{\ln p}, | |
| 137 | +\qquad | |
| 138 | +\mathrm{CSG}(g, p) = \frac{g}{\ln^2 p}. | |
| 139 | +\end{equation} | |
| 140 | +Cram\'er's heuristic suggests $\limsup \mathrm{CSG} = 1$; Granville's correction argues | |
| 141 | +$\limsup \ge 2e^{-\gamma} \approx 1.1229$. The largest CSG value ever observed is | |
| 142 | +$0.9206\ldots$ (Nyman's gap 1132 after 1693182318746371). | |
| 143 | + | |
| 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''). | |
| 151 | + | |
| 152 | +\section{Methodological protocol} | |
| 153 | +\label{sec:protocol} | |
| 154 | + | |
| 155 | +Work proceeds in cycles. Each cycle passes through four roles in order: | |
| 156 | +\begin{enumerate} | |
| 157 | + \item \textbf{Explorer} --- generates data at small scale, checks it against known | |
| 158 | + values, then scales up; produces quantified observations only. | |
| 159 | + \item \textbf{Conjecturer} --- converts observations into precise statements with | |
| 160 | + explicit quantifiers and bounds; discards at birth any statement with a modular | |
| 161 | + obstruction (admissibility is checked mod $2,3,5,7$ systematically) or that is | |
| 162 | + trivially known. | |
| 163 | + \item \textbf{Adversary} --- attacks each conjecture on a range $10\times$--$100\times$ | |
| 164 | + the discovery range, hunting for the \emph{minimal} counterexample rather than | |
| 165 | + merely re-verifying; applies the Cram\'er/Maier test. | |
| 166 | + \item \textbf{Prover} --- for survivors, attempts a proof, a reduction to a standard | |
| 167 | + conjecture, or an explicit model; for constructions, produces a complete | |
| 168 | + certificate and an independent verification script. | |
| 169 | +\end{enumerate} | |
| 170 | +Non-negotiable constraints: small scale before large scale; determinism; every announced | |
| 171 | +object re-verifiable by a standalone script; every ``known value'' fetched from a source | |
| 172 | +(OEIS, t5k.org, the Nicely tables) rather than transcribed from memory --- a rule adopted | |
| 173 | +after an instructive incident (Section~\ref{sec:lessons}). | |
| 174 | + | |
| 175 | +\section{Computational infrastructure} | |
| 176 | +\label{sec:compute} | |
| 177 | + | |
| 178 | +\subsection{Hardware} | |
| 179 | + | |
| 180 | +All computation ran on Apple-silicon machines from a private cluster: a Mac Studio | |
| 181 | +M3~Ultra (28 CPU cores, 96\,GB RAM, 80-core GPU; node \texttt{M3U96a}), two Mac Studios | |
| 182 | +M4~Max (16 CPU cores, 64\,GB each; nodes \texttt{M4M64a/b}), and, in one earlier | |
| 183 | +campaign, nine additional small nodes (M1--M4 class, 8--16 cores) for a 162-core fleet. | |
| 184 | +Orchestration ran from a laptop over SSH; file distribution used an rsync gateway | |
| 185 | +through \texttt{M3U96a}. | |
| 186 | + | |
| 187 | +\subsection{Primality primitives} | |
| 188 | +\label{sec:primitives} | |
| 189 | + | |
| 190 | +\paragraph{Deterministic Miller--Rabin.} | |
| 191 | +Write $n - 1 = 2^s d$ with $d$ odd. A base $a$ with $a \not\equiv 0 \pmod n$ is a | |
| 192 | +\emph{strong witness} for the compositeness of $n$ if | |
| 193 | +\begin{equation} | |
| 194 | +a^{d} \not\equiv 1 \pmod n | |
| 195 | +\quad\text{and}\quad | |
| 196 | +a^{2^r d} \not\equiv -1 \pmod n \ \ \text{for all } 0 \le r < s. | |
| 197 | +\label{eq:mr} | |
| 198 | +\end{equation} | |
| 199 | +If no $a$ in the 12-element base set $\{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37\}$ | |
| 200 | +is a strong witness, then $n$ is prime for every | |
| 201 | +\begin{equation} | |
| 202 | +n \;<\; 3.317\times 10^{24} \;=\; 3{,}317{,}044{,}064{,}679{,}887{,}385{,}961{,}981 | |
| 203 | +\end{equation} | |
| 204 | +(Sorenson--Webster \cite{SW}). All primality assertions in our certificates stay below | |
| 205 | +this bound, so they are unconditional. | |
| 206 | + | |
| 207 | +\paragraph{Baillie--PSW.} | |
| 208 | +Beyond the bound (used only as a cross-check, never in a certificate) we apply BPSW: a | |
| 209 | +strong base-2 test plus a strong Lucas test with Selfridge parameters --- the first | |
| 210 | +$D \in \{5, -7, 9, -11, \dots\}$ with Jacobi symbol $\bigl(\tfrac{D}{n}\bigr) = -1$, | |
| 211 | +$P = 1$, $Q = (1-D)/4$. With $n + 1 = 2^s d$, $n$ is a strong Lucas probable prime if | |
| 212 | +\begin{equation} | |
| 213 | +U_d \equiv 0 \ \ \text{or} \ \ V_{2^r d} \equiv 0 \pmod n \ \ \text{for some } 0 \le r < s, | |
| 214 | +\end{equation} | |
| 215 | +where the Lucas sequences are computed by the binary ladder | |
| 216 | +\begin{equation} | |
| 217 | +U_{2k} = U_k V_k, \qquad | |
| 218 | +V_{2k} = V_k^2 - 2Q^k, \qquad | |
| 219 | +U_{k+1} = \tfrac{1}{2}(P U_k + V_k), \qquad | |
| 220 | +V_{k+1} = \tfrac{1}{2}(D U_k + P V_k) \pmod n . | |
| 221 | +\end{equation} | |
| 222 | +No BPSW pseudoprime is known; the test is deterministic below $2^{64}$. | |
| 223 | + | |
| 224 | +\paragraph{Validation gate.} | |
| 225 | +No computation is permitted until a 37-check gate passes ($0.3$\,s): $\pi(10^k)$ for | |
| 226 | +$k \le 8$ (e.g.\ $\pi(10^8) = 5{,}761{,}455$ via the segmented path), equality of | |
| 227 | +segmented and monolithic sieves on randomized windows, the complete table of maximal | |
| 228 | +gaps below $10^6$, strong-pseudoprime traps ($2047, 3277, 4033, \dots$), Carmichael | |
| 229 | +traps ($561, 1105, 41041, 321197185, \dots$), and cross-agreement of the three tests on | |
| 230 | +$[0, 4\times 10^4)$ and on 300 random 62-bit integers (fixed seed 42). | |
| 231 | + | |
| 232 | +\subsection{Segmented sieve and cost model} | |
| 233 | +\label{sec:sieve} | |
| 234 | + | |
| 235 | +To enumerate primes in $[\ell, h)$ we precompute base primes up to $\sqrt{h}$ and, per | |
| 236 | +segment of length $S = 5\times 10^7$, mark for every base prime $p$ the arithmetic | |
| 237 | +progression starting at $\max(p^2, \lceil \ell/p \rceil p)$. The marking work per | |
| 238 | +segment is | |
| 239 | +\begin{equation} | |
| 240 | +W(S, h) \;=\; \sum_{p \le \sqrt{h}} \Bigl( \frac{S}{p} + O(1) \Bigr) | |
| 241 | +\;=\; S \bigl( \ln\ln\sqrt{h} + M \bigr) + O\bigl(\pi(\sqrt h)\bigr), | |
| 242 | +\label{eq:mertens} | |
| 243 | +\end{equation} | |
| 244 | +by Mertens' second theorem ($M = 0.2614\ldots$), i.e.\ $O(\ln\ln h)$ byte-writes per | |
| 245 | +integer --- the procedure is memory-bandwidth-bound, which is why both the NumPy | |
| 246 | +implementation and the GPU kernel of Section~\ref{sec:gpu} are effective. Measured | |
| 247 | +throughput: $\approx 3\times 10^8$ integers per second per CPU core near $10^{10}$, | |
| 248 | +falling mildly with $h$ through the $\pi(\sqrt h)$ term ($\pi(\sqrt{10^{13}}) = | |
| 249 | +216{,}816$ base primes). | |
| 250 | + | |
| 251 | +\subsection{Mergeable streaming statistics; exact distribution} | |
| 252 | +\label{sec:merge} | |
| 253 | + | |
| 254 | +All statistics of a scan are accumulated in objects that form a commutative monoid | |
| 255 | +under merging, so a scan of $[2, X)$ decomposes into independent chunks: | |
| 256 | + | |
| 257 | +\begin{itemize} | |
| 258 | +\item \textbf{Histogram} $H[g] = N(g, \cdot)$ and the race counter | |
| 259 | + $D = H[2] - H[4]$ (integer vectors; merge $=$ addition). | |
| 260 | +\item \textbf{Record lists}: chunk-local running maxima $(g, p)$; the global merge | |
| 261 | + keeps increasing records in range order (any global record is a chunk-local | |
| 262 | + record, so no information is lost). | |
| 263 | +\item \textbf{Pearson sums}: for lag $j \in \{1, 2\}$, | |
| 264 | +\begin{equation} | |
| 265 | +T^{(j)} \;=\; \Bigl( n,\ \textstyle\sum a,\ \sum b,\ \sum a^2,\ \sum b^2,\ \sum ab \Bigr), | |
| 266 | +\qquad (a, b) = (g_{n-j},\, g_n), | |
| 267 | +\label{eq:sums} | |
| 268 | +\end{equation} | |
| 269 | + summed over all counted pairs; merge $=$ componentwise addition. The Pearson | |
| 270 | + coefficient is reconstructed at any checkpoint as | |
| 271 | +\begin{equation} | |
| 272 | +\rho \;=\; | |
| 273 | +\frac{\frac{1}{n}\sum ab - \frac{1}{n}\sum a \cdot \frac{1}{n}\sum b} | |
| 274 | +{\sqrt{\bigl(\frac{1}{n}\sum a^2 - (\frac{1}{n}\sum a)^2\bigr) | |
| 275 | + \bigl(\frac{1}{n}\sum b^2 - (\frac{1}{n}\sum b)^2\bigr)}} . | |
| 276 | +\label{eq:pearson} | |
| 277 | +\end{equation} | |
| 278 | +\end{itemize} | |
| 279 | + | |
| 280 | +\begin{proposition}[attribution and tiling] | |
| 281 | +Attribute each gap $g_n$ (and each pair $(g_{n-j}, g_n)$) to the chunk containing the | |
| 282 | +\emph{ending} prime $p_{n+1}$. Then chunks $[\ell_i, \ell_{i+1})$ tiling $[2, X)$ induce a | |
| 283 | +partition of all gap events, and the ordered cumulative merge reproduces the exact prefix | |
| 284 | +statistics at every chunk boundary. | |
| 285 | +\end{proposition} | |
| 286 | + | |
| 287 | +\begin{lemma}[junction buffer] | |
| 288 | +\label{lem:buffer} | |
| 289 | +Let $G^* $ be the maximal prime gap below $X$. If each worker sieves from $\ell - B$ | |
| 290 | +with $B \ge 3G^* + 1$, it observes the two gaps preceding its first counted gap, so the | |
| 291 | +lag-1 and lag-2 pair sums \eqref{eq:sums} are exact across chunk boundaries. | |
| 292 | +\end{lemma} | |
| 293 | +\begin{proof}[Proof sketch] | |
| 294 | +The first counted gap ends at some $p_{n+1} \ge \ell$, hence starts at | |
| 295 | +$p_n \ge \ell - G^*$; its two predecessors start at $\ge \ell - 3G^*$. Any window of | |
| 296 | +length $G^*$ below $X$ contains a prime by definition of $G^*$, so sieving from | |
| 297 | +$\ell - B$ recovers a full prime stream from below $\ell - 3G^*$. | |
| 298 | +\end{proof} | |
| 299 | +We use $B = 5000$; the maximal gap below $10^{13}$ is well under $800$ | |
| 300 | +($G^* = 540$ below $10^{12}$), so the margin is ample. | |
| 301 | + | |
| 302 | +\begin{proposition}[exactness of the accumulators] | |
| 303 | +\label{prop:exact} | |
| 304 | +All components of $H$, $D$, and $T^{(j)}$ are integers, and every partial sum arising in | |
| 305 | +their accumulation is bounded by $\sum g_n \le X \le 10^{13} < 2^{53}$, resp.\ | |
| 306 | +$\sum g_n^2 \approx \lambda^2 \pi(X) \lesssim 10^{15}$ for the ranges used here. Hence | |
| 307 | +the IEEE-754 double-precision accumulation is \emph{exact} (every intermediate value is | |
| 308 | +an integer representable in the 53-bit significand), the results are independent of | |
| 309 | +summation order and of the chunk decomposition, and identical bit-for-bit across | |
| 310 | +machines; the only rounding occurs in the final evaluation of \eqref{eq:pearson}. | |
| 311 | +\end{proposition} | |
| 312 | + | |
| 313 | +This proposition is not academic: it is what makes the cross-machine and | |
| 314 | +chunked-vs-monolithic validations come out \emph{exactly} equal (laptop vs.\ M3~Ultra | |
| 315 | +up to $4\times 10^{10}$; 10-chunk merge vs.\ single scan at $10^{10}$; 1000-chunk | |
| 316 | +distributed merge vs.\ prior scans at $10^{10}, 2\times 10^{10}, 4\times 10^{10}$ --- | |
| 317 | +race counters exactly equal, correlations equal to all published decimals). | |
| 318 | + | |
| 319 | +\subsection{A Metal GPU sieve kernel} | |
| 320 | +\label{sec:gpu} | |
| 321 | + | |
| 322 | +For the $10^{13}$ campaign the marking phase moved to the GPU via a custom Metal kernel | |
| 323 | +(MLX \texttt{fast.metal\_kernel}). The segment is a byte array initialized to 1; the | |
| 324 | +dispatch grid is $(\text{blocks}) \times (\text{base primes})$ with block width | |
| 325 | +$10^6$; the thread for block $b$ and prime $p$ executes | |
| 326 | +\begin{equation} | |
| 327 | +\text{for } m = \max\bigl(p^2, \lceil \mathrm{blo}/p\rceil\, p\bigr),\, | |
| 328 | +m {+} p,\, m {+} 2p, \dots < \mathrm{bhi}: \quad \texttt{out}[m - \ell] \leftarrow 0, | |
| 329 | +\end{equation} | |
| 330 | +in 64-bit integer arithmetic. Write races are benign (all threads store the same value). | |
| 331 | +Prime extraction (\texttt{nonzero} + differences) remains on the CPU; one CPU core per | |
| 332 | +node is reserved for it. Acceptance criteria (all met): byte-identical prime sets to the | |
| 333 | +CPU sieve on windows spanning $[2, 10^6)$ to $[10^{13} - 5\times 10^7,\, 10^{13})$, and | |
| 334 | +a worker partial identical field-for-field to the CPU worker's on a full $10^9$ chunk. | |
| 335 | +Measured single-chunk times near $10^{12}$ ($10^9$ integers): GPU $1.2$--$1.3$\,s versus | |
| 336 | +$2.7$--$3.6$\,s per CPU core, on all three nodes; the GPU path is thus worth | |
| 337 | +$\approx 2$--$3$ CPU cores per node; in the $10^{13}$ campaign it processed 1170 of the | |
| 338 | +10{,}000 chunks ($11.7\%$). | |
| 339 | + | |
| 340 | +\begin{algorithm}[t] | |
| 341 | +\caption{Distributed correlation scan of $[2, X)$ (one worker chunk)} | |
| 342 | +\begin{algorithmic}[1] | |
| 343 | +\Require chunk $[\ell, h)$ with $\ell, h$ multiples of $10^9$ (or $\ell = 2$); buffer $B = 5000$ | |
| 344 | +\State sieve primes in $[\ell - B, h)$ segment by segment (NumPy bytes or Metal kernel) | |
| 345 | +\State form gaps between consecutive primes; \emph{count} those whose end prime is $\ge \ell$ | |
| 346 | +\State accumulate $H$, $D$, records, and the lag-1/lag-2 sums $T^{(1)}, T^{(2)}$ of | |
| 347 | + \eqref{eq:sums}, using the buffered predecessor gaps at the boundary | |
| 348 | + (Lemma~\ref{lem:buffer}) | |
| 349 | +\State emit all accumulators as JSON \Comment{ordered cumulative merge afterwards} | |
| 350 | +\end{algorithmic} | |
| 351 | +\end{algorithm} | |
| 352 | + | |
| 353 | +\subsection{Campaigns} | |
| 354 | + | |
| 355 | +\begin{center} | |
| 356 | +\begin{tabular}{@{}llllll@{}} | |
| 357 | +\toprule | |
| 358 | +Campaign & Range & Fleet & Chunks & Wall time & Cost \\ | |
| 359 | +\midrule | |
| 360 | +Cycle 1 & $4\times 10^9$ & 1 laptop core & --- & 15\,s & --- \\ | |
| 361 | +Cycle 2 & $10^{10}$ / $4\times 10^{10}$ & laptop / M3 Ultra (1 core) & --- & 40 / 128\,s & --- \\ | |
| 362 | +Cycle 3 & $10^{12}$ & 12 nodes, 162 CPU cores & 1000 & $\approx 5$\,min & 17{,}719 core-s \\ | |
| 363 | +Cycle 4 & $10^{13}$ & 3 nodes, 58 CPU cores $+$ 3 GPUs & 10{,}000 & $\approx 50$\,min & 147{,}738 core-s \\ | |
| 364 | +\bottomrule | |
| 365 | +\end{tabular} | |
| 366 | +\end{center} | |
| 367 | +Chunk quotas are proportional to measured per-node rates (e.g.\ cycle 4: | |
| 368 | +4000/3000/3000 with per-node GPU shares 370/400/400). | |
| 369 | + | |
| 370 | +\section{Result I: the 2-vs-4 race conjecture is false} | |
| 371 | +\label{sec:race} | |
| 372 | + | |
| 373 | +Define | |
| 374 | +\begin{equation} | |
| 375 | +D(x) \;=\; N(2, x) - N(4, x). | |
| 376 | +\end{equation} | |
| 377 | +By \eqref{eq:singular-pair}, $C(2) = C(4)$, so under Hardy--Littlewood | |
| 378 | +$N(2,x) \sim N(4,x)$ and $\operatorname{sign} D$ is a second-order phenomenon, | |
| 379 | +analogous to a Chebyshev bias. Checkpoints at $10^6, 10^7, 10^8$ gave | |
| 380 | +$D = +26, +359, +55$, suggesting: | |
| 381 | + | |
| 382 | +\begin{conjecture}[C2; \statuslbl{refuted}] | |
| 383 | +$N(2, x) > N(4, x)$ for all $x \ge 10^6$. | |
| 384 | +\end{conjecture} | |
| 385 | + | |
| 386 | +\begin{result}[\statuslbl{refuted with minimal counterexample}] | |
| 387 | +\label{res:race} | |
| 388 | +A full-resolution scan (every gap event inspected via cumulative sums) shows | |
| 389 | +\[ | |
| 390 | +D = 0 \ \text{first at end-prime } 80{,}966{,}861, \qquad | |
| 391 | +D = -1 \ \text{first at end-prime } 80{,}966{,}933 . | |
| 392 | +\] | |
| 393 | +The race then changes leader indefinitely throughout the tested range. Up to | |
| 394 | +$4\times 10^9$ there are $137{,}574{,}763$ gap events with $D \le 0$; sampled values: | |
| 395 | +\begin{center} | |
| 396 | +\begin{tabular}{@{}lcccccccc@{}} | |
| 397 | +\toprule | |
| 398 | +$x$ & $10^9$ & $4{\cdot}10^9$ & $10^{10}$ & $2{\cdot}10^{10}$ & $4{\cdot}10^{10}$ & | |
| 399 | +$10^{11}$ & $4{\cdot}10^{11}$ & $10^{12}$ \\ | |
| 400 | +\midrule | |
| 401 | +$D(x)$ & $-173$ & $-2270$ & $+2681$ & $+7160$ & $-804$ & $+2888$ & $+22767$ & $-238$ \\ | |
| 402 | +\bottomrule | |
| 403 | +\end{tabular} | |
| 404 | +\end{center} | |
| 405 | +\end{result} | |
| 406 | + | |
| 407 | +\paragraph{Null model.} Restricting to steps where $g \in \{2, 4\}$, $D$ performs a | |
| 408 | +$\pm 1$ walk with $\approx$ balanced steps; with $N_{\pm}(x) = N(2,x) + N(4,x)$ steps, | |
| 409 | +the diffusive scale is $\sqrt{N_\pm(x)}$ --- about $6\times 10^4$ at $x = 10^{12}$ | |
| 410 | +(where $N_\pm \approx 3.7\times 10^9$). The observed $|D|$ stays well within this | |
| 411 | +envelope (max sampled $|D| \approx 2.3\times 10^4$), so the data are consistent with a | |
| 412 | +recurrent, unbiased race; the original conjecture was precisely the anti-random-model | |
| 413 | +claim, and the Cram\'er/Maier test correctly flagged it. The refutation is | |
| 414 | +re-verifiable by a standalone, stdlib-only script | |
| 415 | +(\texttt{certs/verify\_c2\_refutation.py}, $\approx 30$\,s). | |
| 416 | + | |
| 417 | +\paragraph{Methodological point.} The conjecture \emph{passed} at every decade | |
| 418 | +checkpoint while failing $\sim 1.4\times 10^8$ times in between: race-type statements | |
| 419 | +must be attacked at full resolution. Open follow-up: whether the \emph{logarithmic | |
| 420 | +density} of the $\{D > 0\}$ region exists and is biased (the analogue of the | |
| 421 | +Rubinstein--Sarnak analysis for $\pi(x; 4, 1)$ vs $\pi(x; 4, 3)$). | |
| 422 | + | |
| 423 | +\section{Result II: a certified prime desert from a hybrid covering system} | |
| 424 | +\label{sec:desert} | |
| 425 | + | |
| 426 | +\begin{result}[\statuslbl{certified}] | |
| 427 | +\label{res:desert} | |
| 428 | +Let $N = 1{,}116{,}336{,}781{,}708{,}038{,}449{,}369{,}693$ (25 digits). Then $N$ and | |
| 429 | +$N + 260$ are consecutive primes: the gap after $N$ is exactly $260$, with | |
| 430 | +$\mathrm{merit} = 260/\ln N = 4.6955$. | |
| 431 | +\end{result} | |
| 432 | + | |
| 433 | +\paragraph{Construction.} | |
| 434 | +Let $Q = \{2, 3, \dots, 59\}$ be the 17 primes up to $m = 59$ and $P = 59\# = | |
| 435 | +\prod_{p \in Q} p = 1{,}922{,}760{,}350{,}154{,}212{,}639{,}070$. Choose one residue | |
| 436 | +class $a_p \bmod p$ per $p \in Q$, greedily maximizing coverage of positions | |
| 437 | +$\{1, \dots, L\}$, $L = 259$, under the constraint that no class contains $0$ or | |
| 438 | +$L + 1$: | |
| 439 | +\begin{equation} | |
| 440 | +\mathcal{C} \;=\; \bigl\{ i \in \{1,\dots,L\} : \exists\, p \in Q,\ i \equiv a_p | |
| 441 | +\!\!\pmod p \bigr\}, \qquad |\mathcal{C}| = 243 . | |
| 442 | +\end{equation} | |
| 443 | +By the Chinese Remainder Theorem pick $x$ with | |
| 444 | +\begin{equation} | |
| 445 | +x \;\equiv\; -a_p \pmod{p} \quad \text{for all } p \in Q, \qquad 0 \le x < P . | |
| 446 | +\label{eq:crt} | |
| 447 | +\end{equation} | |
| 448 | + | |
| 449 | +\begin{lemma}[\statuslbl{proven}] | |
| 450 | +For every integer $t \ge 0$ and every $i \in \mathcal{C}$, the number $x + tP + i$ is | |
| 451 | +composite. | |
| 452 | +\end{lemma} | |
| 453 | +\begin{proof} | |
| 454 | +If $i \equiv a_p \pmod p$ then $x + tP + i \equiv -a_p + 0 + a_p \equiv 0 \pmod p$ by | |
| 455 | +\eqref{eq:crt}, and $x + tP + i > P > p$, so $p$ is a proper divisor. | |
| 456 | +\end{proof} | |
| 457 | + | |
| 458 | +This yields an explicit infinite family of runs of $\ge 243$ composites in arithmetic | |
| 459 | +progression of modulus $P$. For the $16$ uncovered positions (``holes'') and the two | |
| 460 | +endpoints, search over the shift $t$: at $t = 580$, | |
| 461 | +\begin{itemize} | |
| 462 | +\item $N = x + 580 P$ and $N + 260$ both pass deterministic Miller--Rabin | |
| 463 | + (Section~\ref{sec:primitives}; both $< 3.317\times 10^{24}$, so unconditionally | |
| 464 | + prime); | |
| 465 | +\item each hole $N + i$ is composite, certified by an explicit prime factor found by | |
| 466 | + trial division (15 holes) or by an explicit strong witness $a$ satisfying | |
| 467 | + \eqref{eq:mr} (1 hole) --- a strong witness \emph{is} a compositeness proof. | |
| 468 | +\end{itemize} | |
| 469 | +Hence every $N + i$, $1 \le i \le 259$, carries an explicit compositeness certificate | |
| 470 | +and both endpoints an unconditional primality certificate: the gap is exactly 260. | |
| 471 | + | |
| 472 | +\paragraph{Endpoint-search heuristic.} | |
| 473 | +$N$ and $N + L + 1$ are constructed coprime to all $p \le 59$, so by the standard | |
| 474 | +sieve heuristic each is prime with probability | |
| 475 | +$\approx e^{\gamma} \ln(59) / \ln N \approx 0.148$ per shift, both endpoints with | |
| 476 | +probability $\approx 0.022$; including hole compositeness | |
| 477 | +($\approx 0.856^{16} \approx 0.083$) the expected number of shifts is a few hundred | |
| 478 | +($t = 580$ observed), against | |
| 479 | +$t_{\max} = \lfloor (3.317\times 10^{24} - x - L - 1)/P \rfloor \approx 1725$ available | |
| 480 | +below the deterministic bound. | |
| 481 | + | |
| 482 | +\paragraph{Calibration.} | |
| 483 | +Same prime set $Q$: the classic primorial construction ($a_p \equiv 0$) yields a gap of | |
| 484 | +only $61$ (next prime after 59); pure covering with no holes yields $90$; the hybrid | |
| 485 | +yields $260$. This is \emph{not} a record of any kind (Section~\ref{sec:sota}); the | |
| 486 | +deliverable is the certified, independently re-verifiable pipeline | |
| 487 | +(\texttt{certs/desert\_certificate.json} $+$ \texttt{certs/verify\_desert.py}, the | |
| 488 | +verifier carrying its own independent Miller--Rabin implementation). | |
| 489 | + | |
| 490 | +\section{Result III: a scaling law for consecutive-gap correlations} | |
| 491 | +\label{sec:c4} | |
| 492 | + | |
| 493 | +Let $\rho(x)$ (resp.\ $\rho_2(x)$) be the Pearson correlation \eqref{eq:pearson} of the | |
| 494 | +pairs $(g_n, g_{n+1})$ (resp.\ $(g_n, g_{n+2})$) over all gaps ending below $x$. In the | |
| 495 | +Cram\'er model $\rho \equiv \rho_2 \equiv 0$; the observed anticorrelation is | |
| 496 | +structural. All 16 checkpoints: | |
| 497 | + | |
| 498 | +\begin{center} | |
| 499 | +\begin{tabular}{@{}lcc@{}} | |
| 500 | +\toprule | |
| 501 | +$x$ & $\rho(x)\lnx$ & $\rho_2(x)\lnx$ \\ | |
| 502 | +\midrule | |
| 503 | +$10^{8}$ & $-0.58221$ & $-0.23535$ \\ | |
| 504 | +$2\times 10^{8}$ & $-0.57397$ & $-0.23980$ \\ | |
| 505 | +$4\times 10^{8}$ & $-0.57089$ & $-0.24457$ \\ | |
| 506 | +$10^{9}$ & $-0.57055$ & $-0.24252$ \\ | |
| 507 | +$2\times 10^{9}$ & $-0.56652$ & $-0.24340$ \\ | |
| 508 | +$4\times 10^{9}$ & $-0.56513$ & $-0.24524$ \\ | |
| 509 | +$10^{10}$ & $-0.56116$ & $-0.24783$ \\ | |
| 510 | +$2\times 10^{10}$ & $-0.55954$ & $-0.24814$ \\ | |
| 511 | +$4\times 10^{10}$ & $-0.55569$ & $-0.24929$ \\ | |
| 512 | +$10^{11}$ & $-0.55381$ & $-0.24949$ \\ | |
| 513 | +$2\times 10^{11}$ & $-0.55203$ & $-0.25035$ \\ | |
| 514 | +$4\times 10^{11}$ & $-0.55027$ & $-0.25076$ \\ | |
| 515 | +$10^{12}$ & $-0.54756$ & $-0.25141$ \\ | |
| 516 | +$2\times 10^{12}$ & $-0.54601$ & $-0.25162$ \\ | |
| 517 | +$4\times 10^{12}$ & $-0.54448$ & $-0.25205$ \\ | |
| 518 | +$10^{13}$ & $-0.54264$ & $-0.25247$ \\ | |
| 519 | +\bottomrule | |
| 520 | +\end{tabular} | |
| 521 | +\end{center} | |
| 522 | + | |
| 523 | +\begin{conjecture}[C4$'$; \statuslbl{conjectured}, one prediction confirmed] | |
| 524 | +\label{conj:c4p} | |
| 525 | +With $\lambda = \lnx$, | |
| 526 | +\begin{equation} | |
| 527 | +\rho(x)\,\lambda \;=\; c + \frac{d}{\lambda} + o\!\Bigl(\frac{1}{\lambda}\Bigr), | |
| 528 | +\qquad c = -0.4835 \pm 0.002, \quad d = -1.777 . | |
| 529 | +\label{eq:c4law} | |
| 530 | +\end{equation} | |
| 531 | +\end{conjecture} | |
| 532 | + | |
| 533 | +The $1/\lambda$ form is the natural first correction: the mean gap is $\lambda$, all | |
| 534 | +discrete/arithmetic effects (the parity of gaps, the mod-6 structure, the singular | |
| 535 | +series) enter through moments whose relative size is $O(1/\lambda)$, and the model of | |
| 536 | +Section~\ref{sec:model} produces exactly this drift shape. | |
| 537 | + | |
| 538 | +\paragraph{Fit.} | |
| 539 | +Least squares on the 16 points $(\lambda_i, v_i)$, $v_i = \rho(x_i)\lambda_i$: | |
| 540 | +\begin{equation} | |
| 541 | +(\hat c, \hat d) \;=\; \arg\min_{c, d} \sum_{i=1}^{13} | |
| 542 | +\Bigl( v_i - c - \frac{d}{\lambda_i} \Bigr)^{\!2} | |
| 543 | +\;=\; (-0.48354,\ -1.7772), | |
| 544 | +\end{equation} | |
| 545 | +with maximum residual $0.00255$ and leave-one-out range | |
| 546 | +$\hat c \in [-0.48570, -0.48165]$ (16 refits, each dropping one point). The quoted | |
| 547 | +uncertainty on $c$ is this leave-one-out spread, not a formal confidence interval. | |
| 548 | + | |
| 549 | +\paragraph{Out-of-sample confirmation.} | |
| 550 | +The law was first fitted on the 9 checkpoints up to $4\times 10^{10}$ | |
| 551 | +($\hat c = -0.48618$, $\hat d = -1.7226$) \emph{before} the $10^{12}$ campaign, giving | |
| 552 | +the prediction | |
| 553 | +\[ | |
| 554 | +\rho\lnx(10^{12}) \;=\; \hat c + \hat d / \ln 10^{12} \;=\; -0.549 \pm 0.005 . | |
| 555 | +\] | |
| 556 | +The subsequent measurement returned $-0.54756$. The previously conjectured static | |
| 557 | +window $\rho\lnx \in [-0.65, -0.55]$ was exited between $4\times 10^{11}$ and | |
| 558 | +$10^{12}$, as the law predicted (exit at $\lambda = \hat d/(-0.55 - \hat c) | |
| 559 | +\Rightarrow x \approx 5\times 10^{11}$). | |
| 560 | + | |
| 561 | +\paragraph{Second confirmation at $10^{13}$.} | |
| 562 | +The 13-point refit after the $10^{12}$ campaign predicted | |
| 563 | +$\rho\lnx(10^{13}) = -0.5432 \pm 0.004$ and, for the lag-2 statistic, | |
| 564 | +$\rho_2\lnx(10^{13}) = -0.2545 \pm 0.004$. The subsequent $10^{13}$ campaign | |
| 565 | +(Section~\ref{sec:compute}) measured | |
| 566 | +\[ | |
| 567 | +\rho\lnx(10^{13}) = -0.54264, \qquad \rho_2\lnx(10^{13}) = -0.25247, | |
| 568 | +\] | |
| 569 | +confirming both. Current out-of-sample predictions from the 16-point fit: | |
| 570 | +\begin{equation} | |
| 571 | +\rho\lnx(10^{14}) = -0.5387 \pm 0.004, \qquad | |
| 572 | +\rho\lnx(10^{15}) = -0.5350 \pm 0.004, \qquad | |
| 573 | +\rho_2\lnx(10^{14}) = -0.2553 \pm 0.004 . | |
| 574 | +\end{equation} | |
| 575 | + | |
| 576 | +\paragraph{Lag 2.} | |
| 577 | +The same fit form on $\rho_2$ gives | |
| 578 | +\begin{equation} | |
| 579 | +c_2 = -0.2806, \qquad d_2 = +0.782, | |
| 580 | +\end{equation} | |
| 581 | +with the drift in the \emph{opposite} direction (lag-1 relaxes upward toward its | |
| 582 | +constant, lag-2 downward), and the prediction $\rho_2\lnx(10^{13}) = -0.2545 \pm | |
| 583 | +0.004$ (\statuslbl{conjectured}; $\rho_2 < 0$ \statuslbl{verified up to $10^{12}$}). | |
| 584 | + | |
| 585 | +\paragraph{Robustness checks.} | |
| 586 | +(i) \emph{Out-of-sample record}: the law's only two genuine forecasts to date both hit --- | |
| 587 | +\begin{center} | |
| 588 | +\begin{tabular}{@{}lccc@{}} | |
| 589 | +\toprule | |
| 590 | +target & fit data used & predicted & measured \\ | |
| 591 | +\midrule | |
| 592 | +$\rho\lnx(10^{12})$ & 9 pts $\le 4\times 10^{10}$ & $-0.5485 \pm 0.005$ & $-0.54756$ \\ | |
| 593 | +$\rho\lnx(10^{13})$ & 13 pts $\le 10^{12}$ & $-0.5432 \pm 0.004$ & $-0.54264$ \\ | |
| 594 | +$\rho_2\lnx(10^{13})$ & 13 pts $\le 10^{12}$ & $-0.2545 \pm 0.004$ & $-0.25247$ \\ | |
| 595 | +\bottomrule | |
| 596 | +\end{tabular} | |
| 597 | +\end{center} | |
| 598 | +(ii) \emph{Split-sample stability}: fitting \eqref{eq:c4law} on the 8 low checkpoints | |
| 599 | +($10^8$--$2\times 10^{10}$) gives $c = -0.4890$, on the 8 high checkpoints | |
| 600 | +($4\times 10^{10}$--$10^{13}$) gives $c = -0.4830$ --- consistent within the quoted | |
| 601 | +uncertainty. (iii) \emph{Functional-form sensitivity}: adding a $1/\lambda^2$ term | |
| 602 | +(three-parameter fit) leaves the in-range residuals unchanged (max $0.0025$) but moves | |
| 603 | +the extrapolated constant to $c = -0.4749$: with only $\lambda \in [18.4, 29.9]$ | |
| 604 | +observed, the \emph{asymptotic} constant is determined only at the $\pm 0.01$ level, | |
| 605 | +even though in-range and near-range predictions are tightly pinned. We therefore quote | |
| 606 | +$c = -0.4835 \pm 0.002$ (statistical, fixed form) with a systematic form-uncertainty | |
| 607 | +of order $0.01$. (iv) \emph{Dependence caveat}: the 16 checkpoints are nested prefixes | |
| 608 | +of one dataset, hence strongly positively correlated; ordinary least squares is used | |
| 609 | +only as a summarizing device, and the confirmations that matter are the genuinely | |
| 610 | +out-of-sample ones in (i). | |
| 611 | + | |
| 612 | +We have not located these constants in the literature. The qualitative anticorrelation | |
| 613 | +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. | |
| 616 | + | |
| 617 | +\section{Result IV: the first-order Hardy--Littlewood model fails quantitatively} | |
| 618 | +\label{sec:model} | |
| 619 | + | |
| 620 | +The natural first-order model for the joint law of two consecutive gaps at scale | |
| 621 | +$\lambda$ treats the triple $\{p,\, p + g_1,\, p + g_1 + g_2\}$ as an HL pattern and | |
| 622 | +ignores the requirement that the \emph{interior} of each gap be prime-free: | |
| 623 | +\begin{equation} | |
| 624 | +\Pr(g_1, g_2) \;\propto\; \Sing\bigl(\{0,\, g_1,\, g_1 + g_2\}\bigr)\, | |
| 625 | +e^{-(g_1 + g_2)/\lambda}, | |
| 626 | +\qquad g_1, g_2 \in 2\mathbb{N}. | |
| 627 | +\label{eq:triple-model} | |
| 628 | +\end{equation} | |
| 629 | +Using \eqref{eq:HL} with $k = 3$ and writing $\nu_p = \nu_p(\{0, g_1, g_1{+}g_2\})$: | |
| 630 | +$\nu_p = 1$ iff $p \mid g_1$ and $p \mid g_2$; $\nu_p = 2$ iff exactly one of | |
| 631 | +$p \mid g_1$, $p \mid g_2$, $p \mid g_1{+}g_2$ holds; else $\nu_p = 3$. For | |
| 632 | +$p > g_1 + g_2$ always $\nu_p = 3$, so the factor | |
| 633 | +$(1 - 3/p)/(1 - 1/p)^3 = 1 + O(1/p^2)$ is common to all pairs and cancels in the | |
| 634 | +normalization; the computable part is the finite product over $p \le g_1 + g_2$. | |
| 635 | +If $\nu_3 = 3$ the pattern is inadmissible and the weight vanishes --- e.g.\ | |
| 636 | +$(g_1, g_2) \equiv (1, 1) \pmod 3$, so the pair of consecutive gaps $(4, 4)$ is | |
| 637 | +\emph{forbidden}: this arithmetic coupling is the model's only source of correlation | |
| 638 | +(with $\Sing \equiv 1$ it factorizes and $\rho = 0$). | |
| 639 | + | |
| 640 | +\begin{result}[\statuslbl{model rejected}] | |
| 641 | +Numerical evaluation of \eqref{eq:triple-model} (grid $g_i \le 14\lambda$, exact finite | |
| 642 | +singular products) gives | |
| 643 | +\begin{center} | |
| 644 | +\begin{tabular}{@{}lccccccc@{}} | |
| 645 | +\toprule | |
| 646 | +$x$ & $10^8$ & $10^{10}$ & $10^{12}$ & $10^{15}$ & $10^{20}$ & $10^{30}$ & $10^{40}$\\ | |
| 647 | +\midrule | |
| 648 | +$\rho^{\mathrm{model}}\lambda$ & $-0.1591$ & $-0.1581$ & $-0.1568$ & $-0.1550$ & | |
| 649 | +$-0.1536$ & $-0.1532$ & $-0.1539$ \\ | |
| 650 | +\bottomrule | |
| 651 | +\end{tabular} | |
| 652 | +\end{center} | |
| 653 | +The model reproduces the sign and the $c + d/\lambda$ drift form, but its plateau | |
| 654 | +($\approx -0.154$) is only $\approx 32\%$ of the observed $c = -0.4845$. The dominant | |
| 655 | +share of the anticorrelation must therefore come from the interior-compositeness | |
| 656 | +(inclusion--exclusion) constraints that \eqref{eq:triple-model} ignores. Constructing | |
| 657 | +the corrected model --- in the spirit of the Odlyzko--Rubinstein--Wolf treatment of | |
| 658 | +jumping champions --- is the main open problem left by this cycle. | |
| 659 | +\end{result} | |
| 660 | + | |
| 661 | +\section{Secondary observations} | |
| 662 | +\label{sec:secondary} | |
| 663 | + | |
| 664 | +\paragraph{Jumping champion.} | |
| 665 | +$\operatorname{argmax}_g N(g, x) = 6$ at every checkpoint from $10^6$ to $10^{12}$ | |
| 666 | +(\statuslbl{verified}), consistent with the Odlyzko--Rubinstein--Wolf conjecture that 6 | |
| 667 | +is the champion from $\approx 947$ to $\approx 1.7\times 10^{35}$. | |
| 668 | + | |
| 669 | +\paragraph{Local dominance of multiples of 6.} | |
| 670 | +By \eqref{eq:singular-pair}, $3 \mid g$ multiplies $C(g)$ by | |
| 671 | +$(3-1)/(3-2) = 2$, so multiples of 6 should be strict local maxima of | |
| 672 | +$g \mapsto N(g, x)$: | |
| 673 | +\begin{equation} | |
| 674 | +G_6(x) \;=\; \max\bigl\{ G : \forall\, 6 \mid g \le G,\ | |
| 675 | +N(g,x) > \max(N(g{-}2,x),\, N(g{+}2,x)) \bigr\} | |
| 676 | +\end{equation} | |
| 677 | +grows through $66\ (10^8)$, $216\ (10^9)$, $312\ (10^{10})$, $480\ (10^{12})$ | |
| 678 | +(\statuslbl{verified}; the growth is sample-size-limited, as HL predicts). | |
| 679 | + | |
| 680 | +\paragraph{CSG maxima.} | |
| 681 | +$\max_{p \le 4\times 10^9} \mathrm{CSG} = 0.73947$ (gap 210 after 20{,}831{,}323); | |
| 682 | +$\max_{p \le 10^{13}} \mathrm{CSG} = 0.79754$ (gap 652 after 2{,}614{,}941{,}710{,}599) --- | |
| 683 | +all agree with the published record tables. | |
| 684 | + | |
| 685 | +\paragraph{Reproduction of published tables.} | |
| 686 | +Every campaign reproduces the known tables in its range (\statuslbl{verified}): all 53 | |
| 687 | +maximal gaps below $10^{13}$ (OEIS A005250/A002386; 53 below $10^{13}$, ending with gap 674 after | |
| 688 | +7{,}177{,}162{,}611{,}713), twin-pair counts $N(2, 10^{6,7,8}) = 8169,\ 58980,\ 440312$, | |
| 689 | +and $\pi(10^k)$, $k \le 8$. | |
| 690 | + | |
| 691 | +\section{Threats to validity} | |
| 692 | +\label{sec:threats} | |
| 693 | + | |
| 694 | +We list what could invalidate each headline claim, and why we believe it does not. | |
| 695 | +\begin{enumerate} | |
| 696 | +\item \emph{Software error in the scan.} Mitigations: the 37-check validation gate; | |
| 697 | + exact reproduction of independent published tables in every campaign (53 maximal | |
| 698 | + gaps, twin counts, $\pi(10^k)$, CSG maxima); bit-identical cross-machine and | |
| 699 | + chunked-vs-monolithic agreement (Proposition~\ref{prop:exact}); two independent | |
| 700 | + sieve implementations (NumPy CPU and Metal GPU) forced to byte-identical outputs. | |
| 701 | + A bug would have to affect all of these coherently. | |
| 702 | +\item \emph{Floating-point artefacts in $\rho$.} Excluded by | |
| 703 | + Proposition~\ref{prop:exact}: all accumulators are exact integers; rounding | |
| 704 | + enters only in the final division, at relative level $\sim 10^{-16}$. | |
| 705 | +\item \emph{Selection effects / overfitting in C4$'$.} The law has two free parameters | |
| 706 | + and sixteen highly correlated points; the defense is not the fit quality but the | |
| 707 | + two pre-registered out-of-sample confirmations (Section~\ref{sec:c4}, | |
| 708 | + robustness item (i)) and the split-sample stability. The asymptotic constant | |
| 709 | + carries an explicit form-systematic of $\pm 0.01$. | |
| 710 | +\item \emph{Certificate validity for the desert.} Endpoints rely on the | |
| 711 | + Sorenson--Webster bound for the 12-base Miller--Rabin test; both endpoints are | |
| 712 | + $\approx 10^{24} < 3.317\times 10^{24}$. Interior certificates are elementary | |
| 713 | + (explicit divisors) or strong witnesses \eqref{eq:mr}, checked by an independent | |
| 714 | + implementation. No probabilistic step remains. | |
| 715 | +\item \emph{Literature novelty.} We claim novelty for nothing; phenomena analogous to | |
| 716 | + C4$'$/C6 may exist in the pair-correlation literature. All ``known value'' | |
| 717 | + comparisons were fetched from OEIS/t5k/Nicely at run time. | |
| 718 | +\end{enumerate} | |
| 719 | + | |
| 720 | +\section{Instructive failures} | |
| 721 | +\label{sec:lessons} | |
| 722 | + | |
| 723 | +Recorded because they shaped the protocol: | |
| 724 | +\begin{enumerate} | |
| 725 | + \item \emph{Checkpoint-only verification is a trap}: C2 held at every decade | |
| 726 | + checkpoint while failing $\sim 1.4\times 10^8$ times in between | |
| 727 | + (Result~\ref{res:race}). | |
| 728 | + \item \emph{Never transcribe known values from memory}: a hand-written ``known | |
| 729 | + table'' briefly contradicted a correct scan; fetching OEIS resolved it in the | |
| 730 | + scan's favour. | |
| 731 | + \item \emph{Fleet heterogeneity}: a Python-3.10-only type annotation crashed 9 of 12 | |
| 732 | + nodes running Python 3.9 (fixed by lazy annotation evaluation); five further | |
| 733 | + nodes had no usable interpreter at all. Target the lowest interpreter in the | |
| 734 | + fleet and import-check per node class before launching. | |
| 735 | + \item \emph{Naive models fail beyond their remit}: the single-gap model | |
| 736 | + $N(g,x) \approx C(g) \int_2^x e^{-g/\ln t} (\ln t)^{-2} dt$ shows structured | |
| 737 | + residuals (deficits at $g = 36, 72, 100, 108$ relative to neighbours) --- do | |
| 738 | + not conjecture on top of it without the inclusion--exclusion correction. | |
| 739 | +\end{enumerate} | |
| 740 | + | |
| 741 | +\section{State of the art (for honesty of any claim)} | |
| 742 | +\label{sec:sota} | |
| 743 | + | |
| 744 | +Published records against which any of our outputs must be measured: largest known | |
| 745 | +maximal gap $1854$ after $p = 101412319996363309069$ (2026), with exhaustive | |
| 746 | +maximal-gap verification to $\approx 2\times 10^{19}$; largest known gap between | |
| 747 | +probable primes $16{,}045{,}848$ (2024); record merit $41.938\ldots$ (Gapcoin, 2017); | |
| 748 | +record CSG ratio $0.9206\ldots$ (Nyman, 1999). Ford--Green--Konyagin--Maynard--Tao | |
| 749 | +\cite{FGKMT} give the best unconditional lower bound | |
| 750 | +\begin{equation} | |
| 751 | +\max_{p_{n+1} \le x} g_n \;\gg\; | |
| 752 | +\frac{\ln x \,\ln\ln x \,\ln\ln\ln\ln x}{\ln\ln\ln x}, | |
| 753 | +\end{equation} | |
| 754 | +while Baker--Harman--Pintz give $g_n \ll p_n^{0.525}$ unconditionally. None of our | |
| 755 | +constructions approaches the computational records; none is claimed to. | |
| 756 | + | |
| 757 | +\section{Reproducibility} | |
| 758 | + | |
| 759 | +Everything is re-runnable end to end: \texttt{src/test\_core.py} (validation gate, 37 | |
| 760 | +checks), \texttt{src/cycle2\_c4.py} (single-machine scans), | |
| 761 | +\texttt{src/worker\_c4.py} $+$ \texttt{src/merge\_c4.py} (distributed scans with | |
| 762 | +built-in cross-validation against prior campaigns), \texttt{src/gpu\_sieve.py} (Metal | |
| 763 | +kernel with its correctness gate), \texttt{src/model\_c4.py} (the HL triple model), | |
| 764 | +\texttt{certs/verify\_desert.py} and \texttt{certs/verify\_c2\_refutation.py} | |
| 765 | +(standalone verifiers, stdlib-only where feasible). All scans are deterministic and | |
| 766 | +seed-free; all raw checkpoint data are kept as JSON under \texttt{/data}; the research | |
| 767 | +journal (\texttt{journal.md}) records every cycle, decision, and failure. | |
| 768 | + | |
| 769 | +\paragraph{Data and code availability.} | |
| 770 | +The complete repository (source, certificates, raw checkpoint JSONs, journal, and this | |
| 771 | +manuscript) is available at \url{https://github.com/spboucher-ai/prime-mystery-engine}. | |
| 772 | +Re-verification of the two headline certificates requires only a stock Python~3: | |
| 773 | +\texttt{python3 certs/verify\_desert.py} and | |
| 774 | +\texttt{python3 certs/verify\_c2\_refutation.py}. | |
| 775 | + | |
| 776 | +\begin{thebibliography}{9} | |
| 777 | +\bibitem{FGKMT} K.~Ford, B.~Green, S.~Konyagin, J.~Maynard, T.~Tao, | |
| 778 | +\emph{Long gaps between primes}, J.~Amer.~Math.~Soc.~31 (2018), 65--105. | |
| 779 | +\bibitem{ORW} A.~Odlyzko, M.~Rubinstein, M.~Wolf, | |
| 780 | +\emph{Jumping champions}, Experiment.~Math.~8 (1999), 107--118. | |
| 781 | +\bibitem{SW} J.~Sorenson, J.~Webster, | |
| 782 | +\emph{Strong pseudoprimes to twelve prime bases}, Math.~Comp.~86 (2017), 985--1003. | |
| 783 | +\bibitem{BPSW} R.~Baillie, S.~S.~Wagstaff, | |
| 784 | +\emph{Lucas pseudoprimes}, Math.~Comp.~35 (1980), 1391--1417. | |
| 785 | +\bibitem{RS} M.~Rubinstein, P.~Sarnak, \emph{Chebyshev's bias}, | |
| 786 | +Experiment.~Math.~3 (1994), 173--197. | |
| 787 | +\bibitem{Granville} A.~Granville, \emph{Harald Cram\'er and the distribution of prime | |
| 788 | +numbers}, Scand.~Actuar.~J.~1 (1995), 12--28. | |
| 789 | +\bibitem{A005250} OEIS Foundation, \emph{Sequences A005250, A002386 (record gaps | |
| 790 | +between primes)}, \url{https://oeis.org}. | |
| 791 | +\bibitem{t5k} \emph{Table of known maximal gaps between primes}, | |
| 792 | +\url{https://t5k.org/notes/GapsTable.html}. | |
| 793 | +\bibitem{Nicely} T.~R.~Nicely, \emph{First occurrence prime gaps}, | |
| 794 | +\url{https://faculty.lynchburg.edu/~nicely/gaps/gaplist.html}. | |
| 795 | +\end{thebibliography} | |
| 796 | + | |
| 797 | +\end{document} | |
added
records.md
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| 1 | +<!-- | |
| 2 | +records.md — Prime Mystery Engine: state of the art (Phase 2) | |
| 3 | +Author: Simon-Pierre Boucher — contact@spboucher.ai | |
| 4 | +Last updated: 2026-08-06 | |
| 5 | +--> | |
| 6 | + | |
| 7 | +# State of the Art — Axis #10: Prime Deserts (large prime gaps) | |
| 8 | + | |
| 9 | +## What is known (checked against the literature, 2026-08-06) | |
| 10 | + | |
| 11 | +| Quantity | Known record | Source | | |
| 12 | +|---|---|---| | |
| 13 | +| Largest **maximal** prime gap | 1854, after p = 101412319996363309069 (85th maximal gap, Robert Smith / Brian Kehrig code, 2026) | primerecords.dk, t5k.org GapsTable | | |
| 14 | +| Exhaustive verification bound for maximal gaps | ≈ 2×10^19 (beyond 2^64) | t5k.org/notes/GapsTable.html | | |
| 15 | +| Largest known gap (any size, PRP endpoints) | 16,045,848 near a 385,713-digit number (A. Höglund, 2024) | primerecords.dk/primegaps/megagap3.htm | | |
| 16 | +| Highest known **merit** g/ln(p) | 41.93878… (gap 8350 after an 87-digit prime, Gapcoin network, 2017) | primegap-list-project | | |
| 17 | +| Highest Cramér–Shanks–Granville ratio g/ln²(p) | 0.9206… (Nyman's gap 1132 after 1693182318746371, 1999) | MathWorld Prime Gaps | | |
| 18 | +| Theory (lower bound) | Ford–Green–Konyagin–Maynard–Tao 2016: max gap ≥ c·(ln x · lnln x · lnlnlnln x)/lnlnln x | published | | |
| 19 | +| Theory (upper bound) | Baker–Harman–Pintz: gap = O(x^0.525); RH gives O(√x ln x) | published | | |
| 20 | +| Heuristic | Cramér: limsup g/ln²p = 1; Granville correction: ≥ 2e^{-γ} ≈ 1.1229 | published | | |
| 21 | +| Jumping champion | 6 is the most common gap from ≈ 947 up to ≈ 1.7×10^35 (conjectured, verified in ranges); then 30 | published (Odlyzko–Rubinstein–Wolf) | | |
| 22 | + | |
| 23 | +## What would count as NEW here | |
| 24 | + | |
| 25 | +1. **A new maximal gap** — requires exhaustive sieving beyond 2×10^19: **out of reach** locally; not the goal. | |
| 26 | +2. **A gap with merit > 41.94** — requires massive targeted search (Gapcoin-scale): out of reach in one cycle; a merit > 20 gap found by our own sieve pipeline would still only be "personal best" territory (thousands of merit-20+ gaps are catalogued). Any gap we find must be checked against the primegap-list-project tables before any claim. | |
| 27 | +3. **Structural conjectures about gap statistics** (residues of maximal-gap primes, corrections to Cramér in arithmetic progressions, gap-pair correlations) — realistic: findings are likely "(a) probably known", but a precise, massively tested statement with clean data is a legitimate output. | |
| 28 | +4. **Certified desert constructions** (covering systems): any construction is reproducible and certifiable, but primorial/covering constructions have low merit (~2–4) — value is the **certificate pipeline**, not records. | |
| 29 | + | |
| 30 | +## Exact thresholds to beat (for honesty in any claim) | |
| 31 | + | |
| 32 | +- Maximal gap: must exceed 1854 AND be proven maximal (impossible without exhaustive sieve > 2×10^19) — or simply reproduce/verify the known table below our compute bound (verification, not record). | |
| 33 | +- Merit record: > 41.93878. | |
| 34 | +- CSG ratio record: > 0.92064. | |
| 35 | +- First-occurrence gaps: the Nicely / primegap-list-project tables catalogue first known occurrences for every even gap; before claiming a "first occurrence", check those tables. | |
| 36 | + | |
| 37 | +Sources: [t5k.org GapsTable](https://t5k.org/notes/GapsTable.html), [primerecords.dk top-20](http://primerecords.dk/primegaps/gaps20.htm), [primerecords.dk megagap](http://primerecords.dk/primegaps/megagap3.htm), [primegap-list-project merit record](https://primegap-list-project.github.io/record/2017/12/31/new-prime-gap-of-maximum-merit/), [MathWorld Prime Gaps](https://mathworld.wolfram.com/PrimeGaps.html), [Wikipedia Prime gap](https://en.wikipedia.org/wiki/Prime_gap), [Nicely first-occurrence tables](https://faculty.lynchburg.edu/~nicely/gaps/gaplist.html) | |
added
report_cycle1.md
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| 1 | +<!-- | |
| 2 | +report_cycle1.md — Prime Mystery Engine: Cycle 1 synthesis (Phase 7) | |
| 3 | +Author: Simon-Pierre Boucher — contact@spboucher.ai | |
| 4 | +Date: 2026-08-06 | |
| 5 | +--> | |
| 6 | + | |
| 7 | +# Cycle 1 Report — Prime Deserts (axis #10) | |
| 8 | + | |
| 9 | +## Results | |
| 10 | + | |
| 11 | +1. **REFUTATION (main result).** Conjecture C2 — "N(2,x) > N(4,x) for all x ≥ 10^6" — is **false**. | |
| 12 | + Minimal counterexample: the count of gap-4 pairs ties the count of twin pairs at end-prime | |
| 13 | + **80966861** and strictly overtakes it at **80966933**. Beyond that, the race changes leader | |
| 14 | + endlessly up to 4×10^9 (last lead-change at 3999999979, i.e. still swinging at the scan boundary; | |
| 15 | + difference −2270 at 4×10^9). Epistemic status: REFUTED, re-verifiable in ~30 s with a | |
| 16 | + stdlib-only script. Consistent with the random-walk heuristic (equal singular series) — the | |
| 17 | + right follow-up question is the logarithmic density of each leader (Chebyshev-bias analogue). | |
| 18 | + | |
| 19 | +2. **CERTIFIED CONSTRUCTION.** A prime desert of length **260** at 25 digits: | |
| 20 | + N = 1116336781708038449369693 and N+260 are consecutive primes (merit 4.6955). | |
| 21 | + Method: hybrid covering system mod primes ≤ 59 (243 positions composite for *every* CRT shift — | |
| 22 | + proven; 16 holes certified by explicit factors/MR witnesses; endpoints by deterministic | |
| 23 | + Miller–Rabin, below the 3.317×10^24 validity bound). ×4.3 over the classic primorial baseline. | |
| 24 | + **Not a record** (records.md) — the certified, independently re-verifiable pipeline is the point. | |
| 25 | + | |
| 26 | +3. **INDEPENDENT VERIFICATIONS of known tables** (validation of the whole engine): | |
| 27 | + all 32 maximal gaps below 4×10^9, twin counts at 10^6/10^7/10^8, π(10^k) k ≤ 8, | |
| 28 | + max CSG ratio 0.73947 below 4×10^9 — all equal to published values. | |
| 29 | + | |
| 30 | +## Conjecture status table | |
| 31 | + | |
| 32 | +| # | Statement (short) | Status | | |
| 33 | +|---|---|---| | |
| 34 | +| C1 | Jumping champion = 6 | SURVIVOR — VERIFIED UP TO 4×10^9 (known, ORW) | | |
| 35 | +| C2 | N(2,x) > N(4,x) for x ≥ 10^6 | **REFUTED** — min. counterexample 80966861/80966933 | | |
| 36 | +| C3 | Multiples of 6 local maxima; G6(x) ≥ 66, → ∞ | SURVIVOR, strengthened (G6 = 216 at 10^9..4×10^9) | | |
| 37 | +| C4 | ρ(x) < 0, ρ·ln x ∈ [−0.65, −0.55] | SURVIVOR with revision (drift → −0.565; limit clause retracted) | | |
| 38 | +| C5 | max CSG below 4×10^9 = 0.7395 | VERIFIED (matches published table; not new) | | |
| 39 | + | |
| 40 | +## Instructive failures | |
| 41 | + | |
| 42 | +- **Checkpoint-only verification is a trap**: C2 "held" at every decade checkpoint while failing | |
| 43 | + ~137 million times in between. Full-resolution (every-event) adversary scans are now mandatory | |
| 44 | + for race-type conjectures. | |
| 45 | +- A first analysis run had segment boundaries misaligned with checkpoints (caught because twin | |
| 46 | + counts disagreed with literature values — the validate-against-known-values rule paid off). | |
| 47 | +- The naive HL gap model exp(−g/ln t) is inadequate beyond g ≈ 40 (structured residuals at | |
| 48 | + g = 36, 72, 100, 108): do not conjecture on it before building the inclusion–exclusion model. | |
| 49 | + | |
| 50 | +## Files and re-verification | |
| 51 | + | |
| 52 | +- `src/core.py` + `src/test_core.py` — primitives; run `python3 src/test_core.py` (37/37 PASS, 0.3 s). | |
| 53 | +- `src/explore_gaps.py`, `src/analyze_gaps.py`, `src/adversary_race.py` — deterministic scans; | |
| 54 | + outputs in `/data` (`adversary_4e9.json`, `gapstats_1e8.*`, `maximal_gaps_1e8.csv`, …). | |
| 55 | +- `certs/desert_certificate.json` + `certs/verify_desert.py` — desert; run `python3 certs/verify_desert.py`. | |
| 56 | +- `certs/verify_c2_refutation.py` — refutation; run `python3 certs/verify_c2_refutation.py`. | |
| 57 | +- `src/prover_desert.py` (attempt 1), `src/prover_desert2.py` (attempt 2, current certificate). | |
| 58 | + | |
| 59 | +## Next directions (ranked) | |
| 60 | + | |
| 61 | +1. **C4 at 10^10** (cluster job): does ρ·ln x stabilize? Best quantitative lead — structural | |
| 62 | + (absent from Cramér model), cheap to test, precise falsifiable target c ∈ [−0.60, −0.50]. | |
| 63 | +2. C2 follow-up: logarithmic density of the gap-2 lead (Chebyshev-bias analogue) — needs the | |
| 64 | + full crossing record, one scan with lead-time accounting. | |
| 65 | +3. Proper inclusion–exclusion HL model for N(g,x), then revisit the g = 36/72/100/108 deficits. | |
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report_cycle2.md
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| 1 | +<!-- | |
| 2 | +report_cycle2.md — Prime Mystery Engine: Cycle 2 synthesis (Phase 7) | |
| 3 | +Author: Simon-Pierre Boucher — contact@spboucher.ai | |
| 4 | +Date: 2026-08-06 | |
| 5 | +--> | |
| 6 | + | |
| 7 | +# Cycle 2 Report — C4 escalation to 4×10^10 (dual-machine) | |
| 8 | + | |
| 9 | +## Results | |
| 10 | + | |
| 11 | +1. **C4 refined into C4′ (main result).** The consecutive-gap anticorrelation obeys, empirically, | |
| 12 | + **ρ(x)·ln x = c + d/ln x** with **c = −0.486 ± 0.010, d = −1.72** (9 checkpoints 10^8…4×10^10, | |
| 13 | + max residual 0.0025, leave-one-out spread on c: [−0.494, −0.482]). Falsifiable prediction for | |
| 14 | + cycle 3: **ρ·ln x = −0.549 ± 0.005 at x = 10^12**; the cycle-1 window [−0.65, −0.55] should be | |
| 15 | + exited near x ≈ 5×10^11. Epistemic status: CONJECTURED (consistent up to 4×10^10). | |
| 16 | + | |
| 17 | +2. **First-order HL model rejected (PROVER).** The triple-correlation model | |
| 18 | + P(g₁,g₂) ∝ S({0,g₁,g₁+g₂})·e^{−(g₁+g₂)/λ} reproduces the sign and the c + d/λ drift form but | |
| 19 | + gives ρ·λ ≈ −0.154, a factor ~3.2 too small. The anticorrelation is therefore dominated by | |
| 20 | + interior-compositeness (inclusion–exclusion) effects, not the bare singular series. Open problem. | |
| 21 | + | |
| 22 | +3. **Dual-machine reproducibility.** Laptop (10^10) and M3U96a (4×10^10) produced bit-for-bit | |
| 23 | + identical values at every shared checkpoint — deterministic pipeline validated on independent | |
| 24 | + hardware. | |
| 25 | + | |
| 26 | +4. **New conjecture C6.** Lag-2 gaps also anticorrelate: ρ₂·ln x → c₂ ≈ −0.29 ± 0.03, drifting in the | |
| 27 | + opposite direction to lag-1. Low-confidence fit; to be firmed up. | |
| 28 | + | |
| 29 | +5. **Free continuations.** Champion = 6 (C1) and G6 ≥ 66 (C3) hold to 4×10^10 (G6 is non-monotone: | |
| 30 | + 312 → 282 → 276 — tail noise). C2 race keeps changing leader (D = −804 at 4×10^10). All 37 | |
| 31 | + maximal gaps below 4×10^10 match OEIS A002386 (web-checked); max CSG = 0.79535 (456 after | |
| 32 | + 25056082087, known). | |
| 33 | + | |
| 34 | +## Instructive failure | |
| 35 | + | |
| 36 | +A hand-written "known table" check briefly claimed our maximal gaps disagreed with the literature; | |
| 37 | +fetching OEIS A002386 showed the scan was right and the from-memory list wrong. Rule reinforced: | |
| 38 | +**never transcribe "known" values from memory — always fetch the source.** | |
| 39 | + | |
| 40 | +## Conjecture table (cumulative) | |
| 41 | + | |
| 42 | +| # | Statement (short) | Status | | |
| 43 | +|---|---|---| | |
| 44 | +| C1 | Jumping champion = 6 | VERIFIED UP TO 4×10^10 | | |
| 45 | +| C2 | N(2,x) > N(4,x) for x ≥ 10^6 | REFUTED (cycle 1); leader still swapping at 4×10^10 | | |
| 46 | +| C3 | G6(x) ≥ 66 | VERIFIED UP TO 4×10^10 | | |
| 47 | +| C4 | ρ·ln x ∈ [−0.65, −0.55] | VERIFIED UP TO 4×10^10, predicted to FAIL ≈ 5×10^11 — superseded | | |
| 48 | +| C4′ | ρ·ln x = c + d/ln x, c = −0.486(10), d ≈ −1.72 | CONJECTURED; falsifiable at 10^12 | | |
| 49 | +| C5 | Maximal-gap table reproduction | VERIFIED UP TO 4×10^10 (37/37 = A002386) | | |
| 50 | +| C6 | ρ₂·ln x → c₂ ≈ −0.29 | CONJECTURED (new, low confidence) | | |
| 51 | + | |
| 52 | +## Files and re-verification | |
| 53 | + | |
| 54 | +- `src/cycle2_c4.py` — deterministic scan; sanity: `python3 src/cycle2_c4.py 1e8 sanity` | |
| 55 | + must reproduce cycle-1 checkpoints exactly. | |
| 56 | +- `data/cycle2_c4_1e10_laptop.json`, `data/cycle2_c4_4e10_M3U96a.json` — raw results (identical on | |
| 57 | + shared checkpoints; diff them to re-verify cross-machine agreement). | |
| 58 | +- `src/fit_c4.py` → `data/fit_c4.json` — the C4′ fit; `src/model_c4.py` → `data/model_c4.json` — the | |
| 59 | + rejected first-order model. | |
| 60 | + | |
| 61 | +## Next precise action (one) | |
| 62 | + | |
| 63 | +Run the C4′ test at 10^12 distributed core-proportionally across the MacLustr cluster | |
| 64 | +(mergeable streaming sums make the scan embarrassingly parallel; ~284 cores ⇒ minutes), | |
| 65 | +to accept/refute **ρ·ln x(10^12) = −0.549 ± 0.005**. | |
| 66 | + | |
| 67 | +Sources: [OEIS A002386](https://oeis.org/A002386), [t5k.org GapsTable](https://t5k.org/notes/GapsTable.html) | |
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report_cycle3.md
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| 1 | +<!-- | |
| 2 | +report_cycle3.md — Prime Mystery Engine: Cycle 3 synthesis (Phase 7) | |
| 3 | +Author: Simon-Pierre Boucher — contact@spboucher.ai | |
| 4 | +Date: 2026-08-06 | |
| 5 | +--> | |
| 6 | + | |
| 7 | +# Cycle 3 Report — C4′ tested at 10^12 (distributed, 12 nodes / 162 cores) | |
| 8 | + | |
| 9 | +## Headline | |
| 10 | + | |
| 11 | +**C4′'s prediction was CONFIRMED.** Cycle 2 predicted ρ(x)·ln x = −0.549 ± 0.005 at x = 10^12; | |
| 12 | +the distributed scan measured **−0.54756**. The superseded cycle-1 window [−0.65, −0.55] was exited | |
| 13 | +between 4×10^11 and 10^12, matching the predicted exit ≈ 5×10^11. This is the engine's first | |
| 14 | +confirmed quantitative prediction. | |
| 15 | + | |
| 16 | +## Results | |
| 17 | + | |
| 18 | +1. **Refined law.** 13-point fit (10^8…10^12): ρ·ln x = c + d/ln x with | |
| 19 | + **c = −0.4845 ± 0.003 (LOO), d = −1.757**, max residual 0.0025. | |
| 20 | + New falsifiable predictions: **ρ·ln x(10^13) = −0.5432 ± 0.004**, ρ·ln x(10^14) = −0.5390 ± 0.004. | |
| 21 | +2. **C6 firmed up.** Lag-2: c₂ = −0.2806, d₂ = +0.782 (drift sign opposite to lag-1); | |
| 22 | + prediction ρ₂·ln x(10^13) = −0.2545 ± 0.004. | |
| 23 | +3. **Pipeline validation.** The 1000-chunk distributed merge reproduces the cycle-2 single-machine | |
| 24 | + checkpoints exactly (10^10, 2×10^10, 4×10^10) and all 48 maximal gaps below 10^12 equal the | |
| 25 | + published table (tail spot-checked: 540 after 738832927927). | |
| 26 | +4. **Free continuations.** Champion = 6 up to 10^12 (C1). G6 grows to 480 (C3). The C2 twin-vs-quad | |
| 27 | + race is still changing leader at 10^12: D = −238 out of ~1.5×10^10 gap events. | |
| 28 | +5. **Cost.** 17,719 core-seconds (≈ 4.9 core-hours), ≈ 5 minutes wall across the fleet. | |
| 29 | + | |
| 30 | +## Instructive failures (logged for future cycles) | |
| 31 | + | |
| 32 | +- 9/12 nodes crashed on launch: a Python-3.10-only annotation (`np.ndarray | None`) met Apple's | |
| 33 | + Python 3.9. Fixed with `from __future__ import annotations`; rule: target the fleet's LOWEST | |
| 34 | + interpreter and import-check per node class before launching. | |
| 35 | +- 5 nodes had no usable python3 at all (missing Xcode CLT), 1 was unreachable — the theoretical | |
| 36 | + 228-core fleet was really 162 cores. | |
| 37 | +- zsh word-splitting and modifier gotchas in orchestration (documented in journal). | |
| 38 | + | |
| 39 | +## Conjecture table (cumulative) | |
| 40 | + | |
| 41 | +| # | Statement (short) | Status | | |
| 42 | +|---|---|---| | |
| 43 | +| C1 | Jumping champion = 6 | VERIFIED UP TO 10^12 | | |
| 44 | +| C2 | N(2,x) > N(4,x), x ≥ 10^6 | REFUTED (cycle 1); race still swinging at 10^12 (D = −238) | | |
| 45 | +| C3 | G6(x) ≥ 66 | VERIFIED UP TO 10^12 (G6 = 480 and growing) | | |
| 46 | +| C4 | ρ·ln x ∈ [−0.65, −0.55] | REFUTED-as-predicted (exited between 4×10^11 and 10^12) | | |
| 47 | +| C4′ | ρ·ln x = c + d/ln x, c = −0.4845(30) | **PREDICTION CONFIRMED at 10^12**; next test 10^13 | | |
| 48 | +| C5 | Maximal-gap table reproduction | VERIFIED UP TO 10^12 (48/48 = published table) | | |
| 49 | +| C6 | ρ₂·ln x → c₂ ≈ −0.28 | CONJECTURED, firmed (13-point fit); prediction at 10^13 stated | | |
| 50 | + | |
| 51 | +## Files and re-verification | |
| 52 | + | |
| 53 | +- `src/worker_c4.py`, `src/merge_c4.py` — distributed chunk worker + merger. Re-verify the pipeline: | |
| 54 | + run any subset of chunks locally and merge; the built-in cross-validation block compares against | |
| 55 | + `data/cycle2_c4_4e10_M3U96a.json`. | |
| 56 | +- `data/cycle3_c4_1e12.json` — merged result (checkpoints, maximal gaps, cross-val report). | |
| 57 | +- Fits reproducible from the two checkpoint JSONs (script in journal / `src/fit_c4.py` pattern). | |
| 58 | + | |
| 59 | +## Next precise action (one) | |
| 60 | + | |
| 61 | +Cycle 4: test ρ·ln x(10^13) = −0.5432 ± 0.004 on the **3-node fleet only (M3U96a, M4M64a, M4M64b — | |
| 62 | +user directive), saturated, with a Metal/GPU sieve path** (benchmark MLX/Metal kernel vs numpy first; | |
| 63 | +10^13 is ~10× cycle 3's work, ~50 CPU core-hours, so GPU acceleration is the enabler). | |
| 64 | + | |
| 65 | +Sources: [OEIS A005250](https://oeis.org/A005250), [OEIS A002386](https://oeis.org/A002386), | |
| 66 | +[t5k.org GapsTable](https://t5k.org/notes/GapsTable.html) | |
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report_cycle4.md
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| 1 | +<!-- | |
| 2 | +report_cycle4.md — Prime Mystery Engine: Cycle 4 synthesis (Phase 7) | |
| 3 | +Author: Simon-Pierre Boucher — contact@spboucher.ai | |
| 4 | +Date: 2026-08-06 | |
| 5 | +--> | |
| 6 | + | |
| 7 | +# Cycle 4 Report — C4′/C6 at 10^13, CPU + Metal GPU on 3 nodes | |
| 8 | + | |
| 9 | +## Headline | |
| 10 | + | |
| 11 | +**Both out-of-sample predictions confirmed at 10^13.** | |
| 12 | +ρ·ln x(10^13) = **−0.54264** (predicted −0.5432 ± 0.004 — C4′'s second consecutive hit); | |
| 13 | +ρ₂·ln x(10^13) = **−0.25247** (predicted −0.2545 ± 0.004 — C6's first hit). | |
| 14 | +The scaling law ρ(x)·ln x = c + d/ln x now stands twice-tested; 16-point refit: | |
| 15 | +**c = −0.48354 (LOO ± 0.002), d = −1.7772** (max residual 0.0026). | |
| 16 | + | |
| 17 | +## The compute | |
| 18 | + | |
| 19 | +- Fleet (user directive): M3U96a + M4M64a + M4M64b only — 58 CPU cores saturated + **3 GPUs via a | |
| 20 | + custom Metal kernel** (MLX `fast.metal_kernel`), which passed byte-identity gates against the CPU | |
| 21 | + sieve before use and processed **1170 of 10,000 chunks (11.7%)**. | |
| 22 | +- 10,000 chunks of 10^9, measured-rate quotas, ~50 min wall, 147,738 core-seconds (~41 core-hours, | |
| 23 | + incl. ~4% duplicated tail work from a late rebalance; duplicates byte-identical, tiling asserted). | |
| 24 | +- **Validation**: 10,000 unique chunks tile [2, 10^13) exactly; all 10 overlapping checkpoints | |
| 25 | + bit-identical to cycles 2/3; all 53 maximal gaps below 10^13 equal the published table | |
| 26 | + (new in range: 582, 588, 602, 652, 674 — web-checked against t5k/Nicely). | |
| 27 | + | |
| 28 | +## Conjecture table (cumulative) | |
| 29 | + | |
| 30 | +| # | Statement | Status | | |
| 31 | +|---|---|---| | |
| 32 | +| C1 | Champion = 6 | VERIFIED UP TO 10^13 | | |
| 33 | +| C2 | N(2,x) > N(4,x) | REFUTED (cycle 1); race still swinging at 10^13 (D = +8870) | | |
| 34 | +| C3 | G6 ≥ 66 | VERIFIED UP TO 10^13 (G6 = 576) | | |
| 35 | +| C4′ | ρ·ln x = c + d/ln x | **CONFIRMED at 10^12 AND 10^13**; c = −0.48354 ± 0.002 | | |
| 36 | +| C5 | Maximal-gap table | VERIFIED UP TO 10^13 (53/53); max CSG 0.79754 (652 after 2614941710599) | | |
| 37 | +| C6 | ρ₂·ln x → c₂ | **CONFIRMED at 10^13**; c₂ = −0.27762, d₂ = +0.719 | | |
| 38 | + | |
| 39 | +## Predictions on record (falsifiable) | |
| 40 | + | |
| 41 | +- ρ·ln x(10^14) = **−0.5387 ± 0.004**; ρ·ln x(10^15) = −0.5350 ± 0.004. | |
| 42 | +- ρ₂·ln x(10^14) = **−0.2553 ± 0.004**. | |
| 43 | + | |
| 44 | +## Files and re-verification | |
| 45 | + | |
| 46 | +- `src/gpu_sieve.py` — Metal kernel + its correctness/benchmark gate (`python3 src/gpu_sieve.py`). | |
| 47 | +- `src/gpu_runner.py`, refactored `src/worker_c4.py` (CPU/GPU modes, identical outputs). | |
| 48 | +- `data/cycle4_c4_1e13.json` — merged result with built-in cross-validation records. | |
| 49 | +- Paper updated: `paper/main.tex` (12 pp., compiles clean). | |
| 50 | + | |
| 51 | +## Next precise action (one) | |
| 52 | + | |
| 53 | +Cycle 5: 10^14 (~10× the work; ≈ 8–9 h wall on the 3-node fleet — overnight run) to test | |
| 54 | +ρ·ln x(10^14) = −0.5387 ± 0.004. If compute is deferred: attack the main open problem instead — | |
| 55 | +the inclusion–exclusion correction to the HL triple model that should explain c ≈ −0.484. | |
| 56 | + | |
| 57 | +Sources: [t5k.org GapsTable](https://t5k.org/notes/GapsTable.html), | |
| 58 | +[Nicely maximal gaps](https://faculty.lynchburg.edu/~nicely/gaps/gaps.html) | |
added
src/adversary_race.py
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| 1 | +#!/usr/bin/env python3 | |
| 2 | +# ============================================================================= | |
| 3 | +# adversary_race.py — Cycle 1 ADVERSARY: attack C1..C5 up to 4e9 | |
| 4 | +# Author: Simon-Pierre Boucher — contact@spboucher.ai | |
| 5 | +# ============================================================================= | |
| 6 | +# Single deterministic segmented scan of [2, LIMIT). Attacks: | |
| 7 | +# C2: FULL-RESOLUTION hunt for crossings of D(x) = N(2,x) - N(4,x): | |
| 8 | +# every gap event is inspected (vectorized cumsum), every x where | |
| 9 | +# D <= 0 after x >= START_TRACK is recorded (minimal counterexample). | |
| 10 | +# Also records ALL sign-change epochs from the very start (context). | |
| 11 | +# C1/C3/C4: histogram + correlation snapshots at decade checkpoints. | |
| 12 | +# C5: complete maximal-gap record list with CSG ratios. | |
| 13 | +# Output: data/adversary_<L>.json (+ console digest). | |
| 14 | +# Usage: python3 adversary_race.py 4e9 | |
| 15 | +# ============================================================================= | |
| 16 | + | |
| 17 | +import json | |
| 18 | +import sys | |
| 19 | +import time | |
| 20 | +from math import log | |
| 21 | +from pathlib import Path | |
| 22 | + | |
| 23 | +import numpy as np | |
| 24 | + | |
| 25 | +from core import primes_upto, sieve_segment | |
| 26 | + | |
| 27 | +DATA = Path(__file__).resolve().parent.parent / "data" | |
| 28 | +DATA.mkdir(exist_ok=True) | |
| 29 | + | |
| 30 | + | |
| 31 | +def pearson(S): | |
| 32 | + n = S["n"] | |
| 33 | + cov = S["xy"] / n - (S["x"] / n) * (S["y"] / n) | |
| 34 | + vx = S["xx"] / n - (S["x"] / n) ** 2 | |
| 35 | + vy = S["yy"] / n - (S["y"] / n) ** 2 | |
| 36 | + return cov / (vx * vy) ** 0.5 | |
| 37 | + | |
| 38 | + | |
| 39 | +def main(): | |
| 40 | + limit = int(float(sys.argv[1])) if len(sys.argv) > 1 else 4 * 10**9 | |
| 41 | + seg_size = 50_000_000 | |
| 42 | + cps = [c for c in (10**6, 10**7, 10**8, 10**9, 2 * 10**9, 4 * 10**9) if c <= limit] | |
| 43 | + bounds = sorted({limit, *cps, *range(2, limit, seg_size)} - {2}) | |
| 44 | + base = primes_upto(int(limit ** 0.5) + 1) | |
| 45 | + | |
| 46 | + hist = np.zeros(5000, dtype=np.int64) | |
| 47 | + S = dict(n=0, x=0.0, y=0.0, xx=0.0, yy=0.0, xy=0.0) | |
| 48 | + D = 0 # running N2 - N4 | |
| 49 | + last_nonpos_prime = None # last prime p (end of gap) where D <= 0 | |
| 50 | + n_nonpos_events = 0 | |
| 51 | + zero_epochs = [] # (prime, D) samples where D==0 (first 200) | |
| 52 | + maximal = [] | |
| 53 | + best = 0 | |
| 54 | + checkpoints_out = {} | |
| 55 | + prev = None | |
| 56 | + prev_gap = None | |
| 57 | + t0 = time.time() | |
| 58 | + lo = 2 | |
| 59 | + cps_left = list(cps) | |
| 60 | + for hi in bounds: | |
| 61 | + primes = sieve_segment(lo, hi, base) | |
| 62 | + if len(primes): | |
| 63 | + if prev is not None: | |
| 64 | + primes = np.concatenate(([prev], primes)) | |
| 65 | + if len(primes) >= 2: | |
| 66 | + gaps = np.diff(primes) | |
| 67 | + ends = primes[1:] | |
| 68 | + hist += np.bincount(gaps, minlength=len(hist))[: len(hist)] | |
| 69 | + # --- C2 full-resolution race --- | |
| 70 | + delta = (gaps == 2).astype(np.int64) - (gaps == 4).astype(np.int64) | |
| 71 | + run = D + np.cumsum(delta) | |
| 72 | + nonpos = np.nonzero(run <= 0)[0] | |
| 73 | + if len(nonpos): | |
| 74 | + n_nonpos_events += len(nonpos) | |
| 75 | + last_nonpos_prime = int(ends[nonpos[-1]]) | |
| 76 | + zeros = nonpos[run[nonpos] == 0] | |
| 77 | + for i in zeros[: max(0, 200 - len(zero_epochs))]: | |
| 78 | + zero_epochs.append((int(ends[i]), 0)) | |
| 79 | + D = int(run[-1]) | |
| 80 | + # --- correlations --- | |
| 81 | + gl = np.concatenate(([prev_gap], gaps)) if prev_gap is not None else gaps | |
| 82 | + a, b = gl[:-1].astype(np.float64), gl[1:].astype(np.float64) | |
| 83 | + S["n"] += len(a) | |
| 84 | + S["x"] += a.sum(); S["y"] += b.sum() | |
| 85 | + S["xx"] += (a * a).sum(); S["yy"] += (b * b).sum() | |
| 86 | + S["xy"] += (a * b).sum() | |
| 87 | + prev_gap = int(gaps[-1]) | |
| 88 | + # --- C5 records --- | |
| 89 | + if int(gaps.max()) > best: | |
| 90 | + starts = primes[:-1] | |
| 91 | + for i in np.nonzero(gaps > best)[0]: | |
| 92 | + g, p = int(gaps[i]), int(starts[i]) | |
| 93 | + if g > best: | |
| 94 | + best = g | |
| 95 | + maximal.append((g, p)) | |
| 96 | + prev = primes[-1] | |
| 97 | + while cps_left and hi >= cps_left[0]: | |
| 98 | + c = cps_left.pop(0) | |
| 99 | + nz = np.nonzero(hist)[0] | |
| 100 | + champ = int(nz[np.argmax(hist[nz])]) | |
| 101 | + g6 = 0 | |
| 102 | + for g in range(6, int(nz.max()) - 2, 6): | |
| 103 | + if hist[g] > hist[g - 2] and hist[g] > hist[g + 2]: | |
| 104 | + g6 = g | |
| 105 | + else: | |
| 106 | + break | |
| 107 | + rho = pearson(S) | |
| 108 | + checkpoints_out[str(c)] = { | |
| 109 | + "champion": champ, | |
| 110 | + "N2": int(hist[2]), "N4": int(hist[4]), | |
| 111 | + "D=N2-N4": int(hist[2] - hist[4]), | |
| 112 | + "G6": g6, | |
| 113 | + "rho": round(rho, 6), | |
| 114 | + "rho_times_lnx": round(rho * log(c), 4), | |
| 115 | + } | |
| 116 | + print(f" cp {c:.0e}: champ={champ} D={hist[2]-hist[4]} G6={g6} " | |
| 117 | + f"rho*lnx={rho*log(c):.4f} last_nonpos={last_nonpos_prime} " | |
| 118 | + f"[{time.time()-t0:.0f}s]", flush=True) | |
| 119 | + lo = hi | |
| 120 | + | |
| 121 | + out = { | |
| 122 | + "limit": limit, | |
| 123 | + "scan_seconds": round(time.time() - t0, 1), | |
| 124 | + "C2_race": { | |
| 125 | + "final_D": D, | |
| 126 | + "n_events_D_nonpositive": int(n_nonpos_events), | |
| 127 | + "last_prime_with_D_nonpositive": last_nonpos_prime, | |
| 128 | + "first_zero_epochs_sample": zero_epochs[:50], | |
| 129 | + "definition": "D(p) = N2 - N4 counting gaps whose END prime <= p", | |
| 130 | + }, | |
| 131 | + "checkpoints": checkpoints_out, | |
| 132 | + "maximal_gaps": [ | |
| 133 | + {"gap": g, "after": p, "merit": round(g / log(p), 4), | |
| 134 | + "csg": round(g / log(p) ** 2, 5)} for g, p in maximal | |
| 135 | + ], | |
| 136 | + "max_csg": max(round(g / log(p) ** 2, 5) for g, p in maximal if p > 100), | |
| 137 | + } | |
| 138 | + with open(DATA / f"adversary_{limit:.0e}.json".replace("+0", "").replace("+", ""), "w") as f: | |
| 139 | + json.dump(out, f, indent=2) | |
| 140 | + print(json.dumps(out["C2_race"], indent=2)) | |
| 141 | + print("max CSG:", out["max_csg"], "| largest gap:", maximal[-1]) | |
| 142 | + | |
| 143 | + | |
| 144 | +if __name__ == "__main__": | |
| 145 | + main() | |
added
src/analyze_gaps.py
+166 −0
@@ -0,0 +1,166 @@ | ||
| 1 | +#!/usr/bin/env python3 | |
| 2 | +# ============================================================================= | |
| 3 | +# analyze_gaps.py — Cycle 1 EXPLORER/ADVERSARY: gap statistics vs Hardy-Littlewood | |
| 4 | +# Author: Simon-Pierre Boucher — contact@spboucher.ai | |
| 5 | +# ============================================================================= | |
| 6 | +# One segmented scan up to LIMIT with checkpoints. Produces (in /data): | |
| 7 | +# gapstats_<L>.npz : per-checkpoint gap histograms, consecutive-gap pair | |
| 8 | +# matrix (g_n, g_{n+1}) for gaps < 400, correlation sums | |
| 9 | +# gapstats_<L>.json: HL-normalized ratios, champion tables, correlations | |
| 10 | +# HL model used (documented): | |
| 11 | +# N(g, x) ~ C(g) * INT_2^x exp(-g/ln t) / ln^2 t dt, | |
| 12 | +# C(g) = 2*C2 * prod_{p | g, p odd} (p-1)/(p-2), C2 = 0.6601618158468696. | |
| 13 | +# Deterministic; no randomness. | |
| 14 | +# Usage: python3 analyze_gaps.py 1e8 [checkpoints default 1e6,1e7,1e8,1e9,4e9 <= limit] | |
| 15 | +# ============================================================================= | |
| 16 | + | |
| 17 | +import json | |
| 18 | +import sys | |
| 19 | +import time | |
| 20 | +from math import exp, log | |
| 21 | +from pathlib import Path | |
| 22 | + | |
| 23 | +import numpy as np | |
| 24 | + | |
| 25 | +from core import primes_upto, sieve_segment | |
| 26 | + | |
| 27 | +DATA = Path(__file__).resolve().parent.parent / "data" | |
| 28 | +DATA.mkdir(exist_ok=True) | |
| 29 | + | |
| 30 | +C2 = 0.6601618158468696 | |
| 31 | +GMAX = 400 # pair matrix covers gaps < GMAX | |
| 32 | + | |
| 33 | + | |
| 34 | +def singular(g: int) -> float: | |
| 35 | + """C(g) = 2*C2 * prod_{p|g, p>2} (p-1)/(p-2).""" | |
| 36 | + c = 2 * C2 | |
| 37 | + d, m = 3, g | |
| 38 | + while d * d <= m: | |
| 39 | + if m % d == 0: | |
| 40 | + if d > 2: | |
| 41 | + c *= (d - 1) / (d - 2) | |
| 42 | + while m % d == 0: | |
| 43 | + m //= d | |
| 44 | + d += 2 if d > 2 else 1 | |
| 45 | + if m > 2: | |
| 46 | + c *= (m - 1) / (m - 2) | |
| 47 | + return c | |
| 48 | + | |
| 49 | + | |
| 50 | +def hl_integral(g: int, x: int) -> float: | |
| 51 | + """INT_2^x exp(-g/ln t)/ln^2 t dt, trapezoid on a log grid.""" | |
| 52 | + ts = np.exp(np.linspace(log(3), log(x), 4000)) | |
| 53 | + ys = np.exp(-g / np.log(ts)) / np.log(ts) ** 2 | |
| 54 | + return float(np.trapezoid(ys, ts)) | |
| 55 | + | |
| 56 | + | |
| 57 | +def scan(limit: int, checkpoints, segment_size: int = 50_000_000): | |
| 58 | + base = primes_upto(int(limit ** 0.5) + 1) | |
| 59 | + hist = np.zeros(4000, dtype=np.int64) | |
| 60 | + pair = np.zeros((GMAX, GMAX), dtype=np.int64) | |
| 61 | + # streaming sums for Pearson correlation of (g_n, g_{n+1}) | |
| 62 | + S = dict(n=0, x=0.0, y=0.0, xx=0.0, yy=0.0, xy=0.0) | |
| 63 | + cp_hists, cp_corr = {}, {} | |
| 64 | + cps = sorted(checkpoints) | |
| 65 | + # segment boundaries aligned on checkpoints so each checkpoint histogram | |
| 66 | + # covers exactly [2, checkpoint) | |
| 67 | + bounds = sorted({limit, *cps, *range(2, limit, segment_size)} - {2}) | |
| 68 | + prev = None # last prime seen | |
| 69 | + prev_gap = None # gap ending at prev | |
| 70 | + t0 = time.time() | |
| 71 | + lo = 2 | |
| 72 | + for hi in bounds: | |
| 73 | + primes = sieve_segment(lo, hi, base) | |
| 74 | + if len(primes) == 0: | |
| 75 | + continue | |
| 76 | + if prev is not None: | |
| 77 | + primes = np.concatenate(([prev], primes)) | |
| 78 | + if len(primes) >= 2: | |
| 79 | + gaps = np.diff(primes) | |
| 80 | + hist += np.bincount(gaps, minlength=len(hist))[: len(hist)] | |
| 81 | + # consecutive pairs, including the junction pair across segments | |
| 82 | + if prev_gap is not None: | |
| 83 | + gl = np.concatenate(([prev_gap], gaps)) | |
| 84 | + else: | |
| 85 | + gl = gaps | |
| 86 | + a, b = gl[:-1].astype(np.float64), gl[1:].astype(np.float64) | |
| 87 | + am, bm = gl[:-1], gl[1:] | |
| 88 | + mask = (am < GMAX) & (bm < GMAX) | |
| 89 | + np.add.at(pair, (am[mask], bm[mask]), 1) | |
| 90 | + S["n"] += len(a) | |
| 91 | + S["x"] += a.sum(); S["y"] += b.sum() | |
| 92 | + S["xx"] += (a * a).sum(); S["yy"] += (b * b).sum() | |
| 93 | + S["xy"] += (a * b).sum() | |
| 94 | + prev_gap = int(gaps[-1]) | |
| 95 | + prev = primes[-1] | |
| 96 | + while cps and hi >= cps[0]: | |
| 97 | + c = cps.pop(0) | |
| 98 | + cp_hists[c] = hist.copy() | |
| 99 | + cp_corr[c] = dict(S) | |
| 100 | + print(f" checkpoint {c:.1e} reached at {time.time()-t0:.1f}s") | |
| 101 | + lo = hi | |
| 102 | + return hist, pair, cp_hists, cp_corr, time.time() - t0 | |
| 103 | + | |
| 104 | + | |
| 105 | +def pearson(S): | |
| 106 | + n = S["n"] | |
| 107 | + cov = S["xy"] / n - (S["x"] / n) * (S["y"] / n) | |
| 108 | + vx = S["xx"] / n - (S["x"] / n) ** 2 | |
| 109 | + vy = S["yy"] / n - (S["y"] / n) ** 2 | |
| 110 | + return cov / (vx * vy) ** 0.5 | |
| 111 | + | |
| 112 | + | |
| 113 | +def main(): | |
| 114 | + limit = int(float(sys.argv[1])) if len(sys.argv) > 1 else 10**8 | |
| 115 | + cps = [int(c) for c in (10**6, 10**7, 10**8, 10**9, 4 * 10**9) if c <= limit] | |
| 116 | + tag = f"{limit:.0e}".replace("+0", "").replace("+", "") | |
| 117 | + hist, pair, cp_hists, cp_corr, dt = scan(limit, cps) | |
| 118 | + | |
| 119 | + np.savez_compressed( | |
| 120 | + DATA / f"gapstats_{tag}.npz", | |
| 121 | + hist=hist, pair=pair, | |
| 122 | + checkpoints=np.array(sorted(cp_hists)), | |
| 123 | + **{f"hist_{c}": h for c, h in cp_hists.items()}, | |
| 124 | + ) | |
| 125 | + | |
| 126 | + out = {"limit": limit, "scan_seconds": round(dt, 1), "checkpoints": {}} | |
| 127 | + for c in sorted(cp_hists): | |
| 128 | + h = cp_hists[c] | |
| 129 | + nz = np.nonzero(h)[0] | |
| 130 | + even = nz[nz % 2 == 0] | |
| 131 | + # HL comparison for even gaps up to 120 | |
| 132 | + ratios = {} | |
| 133 | + for g in [int(v) for v in even if 2 <= v <= 120]: | |
| 134 | + pred = singular(g) * hl_integral(g, c) | |
| 135 | + ratios[g] = round(int(h[g]) / pred, 4) if pred > 0 else None | |
| 136 | + champ = int(nz[np.argmax(h[nz])]) | |
| 137 | + # largest G such that every multiple of 6 up to G is a strict local max | |
| 138 | + # of the gap histogram (h[g] > h[g-2] and h[g] > h[g+2]) | |
| 139 | + mult6_localmax_upto = 0 | |
| 140 | + for g in range(6, int(even.max()) - 2, 6): | |
| 141 | + if h[g] > h[g - 2] and h[g] > h[g + 2]: | |
| 142 | + mult6_localmax_upto = g | |
| 143 | + else: | |
| 144 | + break | |
| 145 | + out["checkpoints"][str(c)] = { | |
| 146 | + "jumping_champion": champ, | |
| 147 | + "N2": int(h[2]), "N4": int(h[4]), "N6": int(h[6]), | |
| 148 | + "N2_gt_N4": bool(h[2] > h[4]), | |
| 149 | + "mult6_local_max_up_to": int(mult6_localmax_upto), | |
| 150 | + "pearson_consecutive_gaps": round(pearson(cp_corr[c]), 5), | |
| 151 | + "HL_ratio_obs_over_pred": ratios, | |
| 152 | + } | |
| 153 | + with open(DATA / f"gapstats_{tag}.json", "w") as f: | |
| 154 | + json.dump(out, f, indent=2) | |
| 155 | + # console digest | |
| 156 | + for c in sorted(cp_hists): | |
| 157 | + o = out["checkpoints"][str(c)] | |
| 158 | + r = o["HL_ratio_obs_over_pred"] | |
| 159 | + rv = [v for v in r.values() if v] | |
| 160 | + print(f"x={c:.0e}: champ={o['jumping_champion']} N2={o['N2']} N4={o['N4']} " | |
| 161 | + f"N2>N4={o['N2_gt_N4']} 6|g-localmax-upto={o['mult6_local_max_up_to']} " | |
| 162 | + f"rho={o['pearson_consecutive_gaps']} HLratio[min,max]=[{min(rv):.3f},{max(rv):.3f}]") | |
| 163 | + | |
| 164 | + | |
| 165 | +if __name__ == "__main__": | |
| 166 | + main() | |
added
src/core.py
+207 −0
@@ -0,0 +1,207 @@ | ||
| 1 | +#!/usr/bin/env python3 | |
| 2 | +# ============================================================================= | |
| 3 | +# core.py — Prime Mystery Engine: core primitives | |
| 4 | +# Author: Simon-Pierre Boucher — contact@spboucher.ai | |
| 5 | +# ============================================================================= | |
| 6 | +# Building blocks (Phase 0 of the protocol): | |
| 7 | +# - primes_upto(n) : numpy Sieve of Eratosthenes (odd-only) | |
| 8 | +# - segmented_primes(lo,hi) : segmented sieve, constant memory per segment | |
| 9 | +# - is_prime(n) : deterministic Miller-Rabin for n < 3.317e24 | |
| 10 | +# - bpsw(n) : Baillie-PSW (MR base 2 + strong Lucas) | |
| 11 | +# - prime_count(n) : pi(n) via sieve (validation helper) | |
| 12 | +# All routines are pure-Python/numpy, no hidden prime tables beyond the | |
| 13 | +# 12 deterministic MR bases (which are witnesses, not a prime list). | |
| 14 | +# ============================================================================= | |
| 15 | + | |
| 16 | +from __future__ import annotations # keep annotations lazy (Python 3.9 nodes) | |
| 17 | + | |
| 18 | +import numpy as np | |
| 19 | + | |
| 20 | +# Deterministic Miller-Rabin witness set: correct for all n < 3.317e24 | |
| 21 | +# (Sorenson & Webster 2015). Documented per the Golden Rule. | |
| 22 | +_MR_BASES = (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37) | |
| 23 | +_MR_LIMIT = 3_317_044_064_679_887_385_961_981 | |
| 24 | + | |
| 25 | + | |
| 26 | +def primes_upto(n: int) -> np.ndarray: | |
| 27 | + """All primes <= n, via odd-only Sieve of Eratosthenes (numpy).""" | |
| 28 | + if n < 2: | |
| 29 | + return np.array([], dtype=np.int64) | |
| 30 | + if n < 3: | |
| 31 | + return np.array([2], dtype=np.int64) | |
| 32 | + # index i represents the odd number 2i+1; index 0 (=1) is not prime | |
| 33 | + size = (n + 1) // 2 | |
| 34 | + sieve = np.ones(size, dtype=bool) | |
| 35 | + sieve[0] = False | |
| 36 | + for i in range(1, (int(n ** 0.5) + 1) // 2 + 1): | |
| 37 | + if sieve[i]: | |
| 38 | + p = 2 * i + 1 | |
| 39 | + start = (p * p) // 2 | |
| 40 | + sieve[start::p] = False | |
| 41 | + odds = 2 * np.nonzero(sieve)[0].astype(np.int64) + 1 | |
| 42 | + return np.concatenate(([np.int64(2)], odds)) | |
| 43 | + | |
| 44 | + | |
| 45 | +def sieve_segment(lo: int, hi: int, base_primes: np.ndarray | None = None) -> np.ndarray: | |
| 46 | + """Primes in [lo, hi) via segmented sieve. base_primes must cover sqrt(hi).""" | |
| 47 | + if hi <= 2: | |
| 48 | + return np.array([], dtype=np.int64) | |
| 49 | + lo = max(lo, 2) | |
| 50 | + if base_primes is None: | |
| 51 | + base_primes = primes_upto(int(hi ** 0.5) + 1) | |
| 52 | + seg = np.ones(hi - lo, dtype=bool) | |
| 53 | + for p in base_primes: | |
| 54 | + p = int(p) | |
| 55 | + if p * p >= hi: | |
| 56 | + break | |
| 57 | + start = max(p * p, ((lo + p - 1) // p) * p) | |
| 58 | + seg[start - lo::p] = False | |
| 59 | + if lo <= 1: | |
| 60 | + seg[: 2 - lo] = False | |
| 61 | + return lo + np.nonzero(seg)[0].astype(np.int64) | |
| 62 | + | |
| 63 | + | |
| 64 | +def segmented_primes(lo: int, hi: int, segment_size: int = 10_000_000): | |
| 65 | + """Yield numpy arrays of primes covering [lo, hi) segment by segment.""" | |
| 66 | + base = primes_upto(int(hi ** 0.5) + 1) | |
| 67 | + for start in range(lo, hi, segment_size): | |
| 68 | + end = min(start + segment_size, hi) | |
| 69 | + yield sieve_segment(start, end, base) | |
| 70 | + | |
| 71 | + | |
| 72 | +def prime_count(n: int) -> int: | |
| 73 | + """pi(n), computed by segmented sieve (validation helper).""" | |
| 74 | + total = 0 | |
| 75 | + for chunk in segmented_primes(2, n + 1): | |
| 76 | + total += len(chunk) | |
| 77 | + return total | |
| 78 | + | |
| 79 | + | |
| 80 | +def _mr_witness(n: int, a: int, d: int, r: int) -> bool: | |
| 81 | + """True if a is a Miller-Rabin witness for compositeness of n.""" | |
| 82 | + x = pow(a, d, n) | |
| 83 | + if x == 1 or x == n - 1: | |
| 84 | + return False | |
| 85 | + for _ in range(r - 1): | |
| 86 | + x = x * x % n | |
| 87 | + if x == n - 1: | |
| 88 | + return False | |
| 89 | + return True | |
| 90 | + | |
| 91 | + | |
| 92 | +def is_prime(n: int) -> bool: | |
| 93 | + """Deterministic primality for n < 3.317e24 (Miller-Rabin, 12 bases). | |
| 94 | + For larger n, falls back to BPSW (documented: no known counterexample, | |
| 95 | + verified exhaustively below 2^64).""" | |
| 96 | + if n < 2: | |
| 97 | + return False | |
| 98 | + for p in (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37): | |
| 99 | + if n % p == 0: | |
| 100 | + return n == p | |
| 101 | + d, r = n - 1, 0 | |
| 102 | + while d % 2 == 0: | |
| 103 | + d //= 2 | |
| 104 | + r += 1 | |
| 105 | + if n < _MR_LIMIT: | |
| 106 | + return not any(_mr_witness(n, a, d, r) for a in _MR_BASES if a % n) | |
| 107 | + return bpsw(n) | |
| 108 | + | |
| 109 | + | |
| 110 | +def _jacobi(a: int, n: int) -> int: | |
| 111 | + """Jacobi symbol (a/n), n odd positive.""" | |
| 112 | + a %= n | |
| 113 | + result = 1 | |
| 114 | + while a: | |
| 115 | + while a % 2 == 0: | |
| 116 | + a //= 2 | |
| 117 | + if n % 8 in (3, 5): | |
| 118 | + result = -result | |
| 119 | + a, n = n, a | |
| 120 | + if a % 4 == 3 and n % 4 == 3: | |
| 121 | + result = -result | |
| 122 | + a %= n | |
| 123 | + return result if n == 1 else 0 | |
| 124 | + | |
| 125 | + | |
| 126 | +def _strong_lucas(n: int) -> bool: | |
| 127 | + """Strong Lucas probable-prime test (Selfridge parameters). n odd, >2, | |
| 128 | + not a perfect square, no small factors.""" | |
| 129 | + # Selfridge: first D in 5,-7,9,-11,... with Jacobi(D/n) = -1 | |
| 130 | + D = 5 | |
| 131 | + while True: | |
| 132 | + j = _jacobi(D % n, n) | |
| 133 | + if j == -1: | |
| 134 | + break | |
| 135 | + if j == 0 and abs(D) != n: | |
| 136 | + return False | |
| 137 | + D = -D - 2 if D > 0 else -D + 2 | |
| 138 | + if abs(D) > 1_000_000: # would indicate a square slipped through | |
| 139 | + raise ArithmeticError("no suitable D found; n likely a square") | |
| 140 | + Q = (1 - D) // 4 | |
| 141 | + # factor n+1 = d * 2^s | |
| 142 | + d, s = n + 1, 0 | |
| 143 | + while d % 2 == 0: | |
| 144 | + d //= 2 | |
| 145 | + s += 1 | |
| 146 | + # Lucas sequences U_d, V_d by binary ladder | |
| 147 | + U, V, k = 0, 2, 0 | |
| 148 | + Qk = 1 | |
| 149 | + inv2 = (n + 1) // 2 # inverse of 2 mod n (n odd) | |
| 150 | + for bit in bin(d)[2:]: | |
| 151 | + # double: U_{2k}=U_k V_k ; V_{2k}=V_k^2 - 2 Q^k | |
| 152 | + U, V = U * V % n, (V * V - 2 * Qk) % n | |
| 153 | + Qk = Qk * Qk % n | |
| 154 | + if bit == "1": | |
| 155 | + # increment: U_{k+1}=(P U + V)/2 ; V_{k+1}=(D U + P V)/2 ; P=1 | |
| 156 | + U, V = (U + V) * inv2 % n, (D * U + V) * inv2 % n | |
| 157 | + Qk = Qk * Q % n | |
| 158 | + if U == 0 or V == 0: | |
| 159 | + return True | |
| 160 | + for _ in range(s - 1): | |
| 161 | + V = (V * V - 2 * Qk) % n | |
| 162 | + if V == 0: | |
| 163 | + return True | |
| 164 | + Qk = Qk * Qk % n | |
| 165 | + return False | |
| 166 | + | |
| 167 | + | |
| 168 | +def _is_square(n: int) -> bool: | |
| 169 | + r = math_isqrt(n) | |
| 170 | + return r * r == n | |
| 171 | + | |
| 172 | + | |
| 173 | +from math import isqrt as math_isqrt # noqa: E402 | |
| 174 | + | |
| 175 | + | |
| 176 | +def bpsw(n: int) -> bool: | |
| 177 | + """Baillie-PSW: strong MR base 2 + strong Lucas (Selfridge). | |
| 178 | + No known counterexample; deterministic below 2^64.""" | |
| 179 | + if n < 2: | |
| 180 | + return False | |
| 181 | + for p in (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37): | |
| 182 | + if n % p == 0: | |
| 183 | + return n == p | |
| 184 | + d, r = n - 1, 0 | |
| 185 | + while d % 2 == 0: | |
| 186 | + d //= 2 | |
| 187 | + r += 1 | |
| 188 | + if _mr_witness(n, 2, d, r): | |
| 189 | + return False | |
| 190 | + if _is_square(n): | |
| 191 | + return False | |
| 192 | + return _strong_lucas(n) | |
| 193 | + | |
| 194 | + | |
| 195 | +def gaps_in_range(lo: int, hi: int, segment_size: int = 50_000_000): | |
| 196 | + """Iterate consecutive prime gaps over [lo, hi). Yields (p, gap) where | |
| 197 | + gap = next_prime(p) - p, for all consecutive pairs with p in [lo, hi).""" | |
| 198 | + prev = None | |
| 199 | + for chunk in segmented_primes(max(lo, 2), hi, segment_size): | |
| 200 | + if len(chunk) == 0: | |
| 201 | + continue | |
| 202 | + if prev is not None: | |
| 203 | + yield int(prev), int(chunk[0] - prev) | |
| 204 | + diffs = np.diff(chunk) | |
| 205 | + for p, g in zip(chunk[:-1], diffs): | |
| 206 | + yield int(p), int(g) | |
| 207 | + prev = chunk[-1] | |
added
src/cycle2_c4.py
+138 −0
@@ -0,0 +1,138 @@ | ||
| 1 | +#!/usr/bin/env python3 | |
| 2 | +# ============================================================================= | |
| 3 | +# cycle2_c4.py — Cycle 2 ADVERSARY: C4 (scaled gap anticorrelation) to 10^10+ | |
| 4 | +# Author: Simon-Pierre Boucher — contact@spboucher.ai | |
| 5 | +# ============================================================================= | |
| 6 | +# Attacks C4: rho(x) < 0 and rho(x)*ln x -> c in [-0.60, -0.50] ? | |
| 7 | +# Single deterministic segmented scan of [2, LIMIT). Collects: | |
| 8 | +# - lag-1 Pearson rho(x) and lag-2 Pearson rho2(x) of consecutive gaps, | |
| 9 | +# at checkpoints {1,2,4}x10^k (streaming sums; junction gaps handled) | |
| 10 | +# - C1/C3 continuation: jumping champion, G6(x) at checkpoints | |
| 11 | +# - C2 race D = N2 - N4 at checkpoints (lead-change census continuation) | |
| 12 | +# - C5 continuation: maximal gaps (validated against published table) | |
| 13 | +# Output: data/cycle2_c4_<L>.json — bit-for-bit reproducible on any machine. | |
| 14 | +# Usage: python3 cycle2_c4.py 1e10 [hosttag] | |
| 15 | +# ============================================================================= | |
| 16 | + | |
| 17 | +import json | |
| 18 | +import platform | |
| 19 | +import sys | |
| 20 | +import time | |
| 21 | +from math import log | |
| 22 | +from pathlib import Path | |
| 23 | + | |
| 24 | +import numpy as np | |
| 25 | + | |
| 26 | +from core import primes_upto, sieve_segment | |
| 27 | + | |
| 28 | +DATA = Path(__file__).resolve().parent.parent / "data" | |
| 29 | +DATA.mkdir(exist_ok=True) | |
| 30 | + | |
| 31 | + | |
| 32 | +def pearson(S): | |
| 33 | + n = S["n"] | |
| 34 | + if n < 2: | |
| 35 | + return 0.0 | |
| 36 | + cov = S["xy"] / n - (S["x"] / n) * (S["y"] / n) | |
| 37 | + vx = S["xx"] / n - (S["x"] / n) ** 2 | |
| 38 | + vy = S["yy"] / n - (S["y"] / n) ** 2 | |
| 39 | + return cov / (vx * vy) ** 0.5 | |
| 40 | + | |
| 41 | + | |
| 42 | +def acc(S, a, b): | |
| 43 | + S["n"] += len(a) | |
| 44 | + S["x"] += a.sum(); S["y"] += b.sum() | |
| 45 | + S["xx"] += (a * a).sum(); S["yy"] += (b * b).sum() | |
| 46 | + S["xy"] += (a * b).sum() | |
| 47 | + | |
| 48 | + | |
| 49 | +def main(): | |
| 50 | + limit = int(float(sys.argv[1])) if len(sys.argv) > 1 else 10**10 | |
| 51 | + tag = sys.argv[2] if len(sys.argv) > 2 else platform.node() | |
| 52 | + seg_size = 50_000_000 | |
| 53 | + cps = sorted(c for k in range(6, 12) for c in (10**k, 2 * 10**k, 4 * 10**k) | |
| 54 | + if c <= limit) | |
| 55 | + bounds = sorted({limit, *cps, *range(2, limit, seg_size)} - {2}) | |
| 56 | + base = primes_upto(int(limit ** 0.5) + 1) | |
| 57 | + | |
| 58 | + hist = np.zeros(6000, dtype=np.int64) | |
| 59 | + S1 = dict(n=0, x=0.0, y=0.0, xx=0.0, yy=0.0, xy=0.0) # lag-1 | |
| 60 | + S2 = dict(n=0, x=0.0, y=0.0, xx=0.0, yy=0.0, xy=0.0) # lag-2 | |
| 61 | + D = 0 | |
| 62 | + maximal = [] | |
| 63 | + best = 0 | |
| 64 | + out_cps = {} | |
| 65 | + prev = None | |
| 66 | + tail = [] # last up-to-2 gaps before current segment | |
| 67 | + t0 = time.time() | |
| 68 | + lo = 2 | |
| 69 | + cps_left = list(cps) | |
| 70 | + for hi in bounds: | |
| 71 | + primes = sieve_segment(lo, hi, base) | |
| 72 | + if len(primes): | |
| 73 | + if prev is not None: | |
| 74 | + primes = np.concatenate(([prev], primes)) | |
| 75 | + if len(primes) >= 2: | |
| 76 | + gaps = np.diff(primes) | |
| 77 | + hist += np.bincount(gaps, minlength=len(hist))[: len(hist)] | |
| 78 | + D += int((gaps == 2).sum() - (gaps == 4).sum()) | |
| 79 | + gl = np.concatenate((np.array(tail, dtype=gaps.dtype), gaps)) | |
| 80 | + k = len(tail) | |
| 81 | + if k >= 1: | |
| 82 | + acc(S1, gl[k - 1:-1].astype(np.float64), | |
| 83 | + gl[k:].astype(np.float64)) | |
| 84 | + else: # very first segment: all internal lag-1 pairs | |
| 85 | + acc(S1, gl[:-1].astype(np.float64), gl[1:].astype(np.float64)) | |
| 86 | + if k >= 2: | |
| 87 | + acc(S2, gl[k - 2:-2].astype(np.float64), | |
| 88 | + gl[k:].astype(np.float64)) | |
| 89 | + elif len(gl) > 2: # first segments: internal lag-2 pairs | |
| 90 | + acc(S2, gl[:-2].astype(np.float64), gl[2:].astype(np.float64)) | |
| 91 | + if int(gaps.max()) > best: | |
| 92 | + starts = primes[:-1] | |
| 93 | + for i in np.nonzero(gaps > best)[0]: | |
| 94 | + g, p = int(gaps[i]), int(starts[i]) | |
| 95 | + if g > best: | |
| 96 | + best = g | |
| 97 | + maximal.append((g, p)) | |
| 98 | + tail = [int(v) for v in gaps[-2:]] | |
| 99 | + prev = primes[-1] | |
| 100 | + while cps_left and hi >= cps_left[0]: | |
| 101 | + c = cps_left.pop(0) | |
| 102 | + nz = np.nonzero(hist)[0] | |
| 103 | + champ = int(nz[np.argmax(hist[nz])]) | |
| 104 | + g6 = 0 | |
| 105 | + for g in range(6, int(nz.max()) - 2, 6): | |
| 106 | + if hist[g] > hist[g - 2] and hist[g] > hist[g + 2]: | |
| 107 | + g6 = g | |
| 108 | + else: | |
| 109 | + break | |
| 110 | + r1, r2 = pearson(S1), pearson(S2) | |
| 111 | + out_cps[str(c)] = { | |
| 112 | + "rho1": round(r1, 7), "rho1_lnx": round(r1 * log(c), 5), | |
| 113 | + "rho2": round(r2, 7), "rho2_lnx": round(r2 * log(c), 5), | |
| 114 | + "champion": champ, "G6": g6, "D_N2_minus_N4": D, | |
| 115 | + "n_gaps": int(S1["n"] + 1), | |
| 116 | + } | |
| 117 | + print(f" cp {c:.0e}: rho1*lnx={r1*log(c):+.5f} rho2*lnx={r2*log(c):+.5f} " | |
| 118 | + f"champ={champ} G6={g6} D={D} [{time.time()-t0:.0f}s]", flush=True) | |
| 119 | + lo = hi | |
| 120 | + | |
| 121 | + out = { | |
| 122 | + "limit": limit, | |
| 123 | + "host": tag, | |
| 124 | + "scan_seconds": round(time.time() - t0, 1), | |
| 125 | + "checkpoints": out_cps, | |
| 126 | + "maximal_gaps": [{"gap": g, "after": p, "csg": round(g / log(p) ** 2, 5)} | |
| 127 | + for g, p in maximal], | |
| 128 | + "method": "deterministic segmented sieve; streaming Pearson sums; " | |
| 129 | + "gaps attributed to END prime; no randomness", | |
| 130 | + } | |
| 131 | + fname = DATA / f"cycle2_c4_{limit:.0e}_{tag}.json".replace("+", "").replace("e0", "e") | |
| 132 | + with open(fname, "w") as f: | |
| 133 | + json.dump(out, f, indent=2) | |
| 134 | + print("written:", fname) | |
| 135 | + | |
| 136 | + | |
| 137 | +if __name__ == "__main__": | |
| 138 | + main() | |
added
src/explore_gaps.py
+120 −0
@@ -0,0 +1,120 @@ | ||
| 1 | +#!/usr/bin/env python3 | |
| 2 | +# ============================================================================= | |
| 3 | +# explore_gaps.py — Cycle 1 EXPLORER: prime-gap statistics up to a bound | |
| 4 | +# Author: Simon-Pierre Boucher — contact@spboucher.ai | |
| 5 | +# ============================================================================= | |
| 6 | +# Fully vectorized (numpy) segmented scan of [2, LIMIT). Produces: | |
| 7 | +# data/maximal_gaps_<L>.csv : record (maximal) gaps with merit and CSG ratio | |
| 8 | +# data/gap_histogram_<L>.csv : count of each even gap g | |
| 9 | +# data/gap_mod6_start_<L>.csv : histogram of (start prime mod 6, gap mod 6) | |
| 10 | +# data/summary_<L>.json : headline numbers (deterministic, no seed used) | |
| 11 | +# Method: segmented Sieve of Eratosthenes (core.py), np.diff per segment, | |
| 12 | +# np.bincount accumulation. No per-prime Python loop. | |
| 13 | +# Usage: python3 explore_gaps.py 1e8 | |
| 14 | +# ============================================================================= | |
| 15 | + | |
| 16 | +import json | |
| 17 | +import sys | |
| 18 | +import time | |
| 19 | +from math import log | |
| 20 | +from pathlib import Path | |
| 21 | + | |
| 22 | +import numpy as np | |
| 23 | + | |
| 24 | +from core import primes_upto, sieve_segment | |
| 25 | + | |
| 26 | +DATA = Path(__file__).resolve().parent.parent / "data" | |
| 27 | +DATA.mkdir(exist_ok=True) | |
| 28 | + | |
| 29 | + | |
| 30 | +def scan(limit: int, segment_size: int = 50_000_000): | |
| 31 | + base = primes_upto(int(limit ** 0.5) + 1) | |
| 32 | + hist = np.zeros(4000, dtype=np.int64) # gap -> count | |
| 33 | + mod6 = np.zeros((6, 6), dtype=np.int64) # (p mod 6, gap mod 6) -> count | |
| 34 | + maximal = [] # (gap, start prime) records | |
| 35 | + best = 0 | |
| 36 | + prev = None # last prime of previous segment | |
| 37 | + n_gaps = 0 | |
| 38 | + t0 = time.time() | |
| 39 | + for lo in range(2, limit, segment_size): | |
| 40 | + hi = min(lo + segment_size, limit) | |
| 41 | + primes = sieve_segment(lo, hi, base) | |
| 42 | + if len(primes) == 0: | |
| 43 | + continue | |
| 44 | + if prev is not None: | |
| 45 | + primes = np.concatenate(([prev], primes)) | |
| 46 | + if len(primes) >= 2: | |
| 47 | + gaps = np.diff(primes) | |
| 48 | + starts = primes[:-1] | |
| 49 | + n_gaps += len(gaps) | |
| 50 | + hist += np.bincount(gaps, minlength=len(hist))[: len(hist)] | |
| 51 | + np.add.at(mod6, (starts % 6, gaps % 6), 1) | |
| 52 | + # record gaps within this segment | |
| 53 | + seg_best = int(gaps.max()) | |
| 54 | + if seg_best > best: | |
| 55 | + idx = np.nonzero(gaps > best)[0] | |
| 56 | + for i in idx: | |
| 57 | + g, p = int(gaps[i]), int(starts[i]) | |
| 58 | + if g > best: | |
| 59 | + best = g | |
| 60 | + maximal.append((g, p)) | |
| 61 | + prev = primes[-1] | |
| 62 | + dt = time.time() - t0 | |
| 63 | + return hist, mod6, maximal, n_gaps, dt | |
| 64 | + | |
| 65 | + | |
| 66 | +def main(): | |
| 67 | + limit = int(float(sys.argv[1])) if len(sys.argv) > 1 else 10**8 | |
| 68 | + tag = f"{limit:.0e}".replace("+0", "").replace("+", "") | |
| 69 | + hist, mod6, maximal, n_gaps, dt = scan(limit) | |
| 70 | + | |
| 71 | + # maximal gaps with merit and Cramér-Shanks-Granville ratio | |
| 72 | + with open(DATA / f"maximal_gaps_{tag}.csv", "w") as f: | |
| 73 | + f.write("gap,start_prime,merit,csg_ratio\n") | |
| 74 | + for g, p in maximal: | |
| 75 | + f.write(f"{g},{p},{g/log(p):.6f},{g/log(p)**2:.6f}\n") | |
| 76 | + | |
| 77 | + nz = np.nonzero(hist)[0] | |
| 78 | + with open(DATA / f"gap_histogram_{tag}.csv", "w") as f: | |
| 79 | + f.write("gap,count\n") | |
| 80 | + for g in nz: | |
| 81 | + f.write(f"{g},{hist[g]}\n") | |
| 82 | + | |
| 83 | + with open(DATA / f"gap_mod6_start_{tag}.csv", "w") as f: | |
| 84 | + f.write("p_mod6,gap_mod6,count\n") | |
| 85 | + for a in range(6): | |
| 86 | + for b in range(6): | |
| 87 | + if mod6[a, b]: | |
| 88 | + f.write(f"{a},{b},{mod6[a, b]}\n") | |
| 89 | + | |
| 90 | + # headline stats | |
| 91 | + champion = int(nz[np.argmax(hist[nz])]) | |
| 92 | + g_arr = nz.astype(float) | |
| 93 | + c_arr = hist[nz].astype(float) | |
| 94 | + total = c_arr.sum() | |
| 95 | + frac_mod6 = {r: float(c_arr[g_arr % 6 == r].sum() / total) for r in range(6)} | |
| 96 | + gmax, pmax = maximal[-1] | |
| 97 | + best_merit = max((g / log(p), g, p) for g, p in maximal if p > 100) | |
| 98 | + best_csg = max((g / log(p) ** 2, g, p) for g, p in maximal if p > 100) | |
| 99 | + summary = { | |
| 100 | + "limit": limit, | |
| 101 | + "n_gaps": int(n_gaps), | |
| 102 | + "scan_seconds": round(dt, 1), | |
| 103 | + "largest_gap": {"gap": gmax, "after_prime": pmax}, | |
| 104 | + "n_maximal_gaps": len(maximal), | |
| 105 | + "jumping_champion": champion, | |
| 106 | + "gap_fraction_mod6": frac_mod6, | |
| 107 | + "best_merit": {"merit": round(best_merit[0], 5), "gap": best_merit[1], "after_prime": best_merit[2]}, | |
| 108 | + "best_csg": {"csg": round(best_csg[0], 5), "gap": best_csg[1], "after_prime": best_csg[2]}, | |
| 109 | + "method": "segmented sieve of Eratosthenes, numpy, deterministic (no seed)", | |
| 110 | + } | |
| 111 | + with open(DATA / f"summary_{tag}.json", "w") as f: | |
| 112 | + json.dump(summary, f, indent=2) | |
| 113 | + print(json.dumps(summary, indent=2)) | |
| 114 | + print(f"\nmaximal gaps ({len(maximal)}):") | |
| 115 | + for g, p in maximal: | |
| 116 | + print(f" gap {g:4d} after {p:>15d} merit {g/log(p):7.4f} csg {g/log(p)**2:.4f}") | |
| 117 | + | |
| 118 | + | |
| 119 | +if __name__ == "__main__": | |
| 120 | + main() | |
added
src/fit_c4.py
+66 −0
@@ -0,0 +1,66 @@ | ||
| 1 | +#!/usr/bin/env python3 | |
| 2 | +# ============================================================================= | |
| 3 | +# fit_c4.py — Cycle 2 CONJECTURER: extrapolate rho1(x)*ln x from scan data | |
| 4 | +# Author: Simon-Pierre Boucher — contact@spboucher.ai | |
| 5 | +# ============================================================================= | |
| 6 | +# Fits rho1*lnx = c + d/lambda (lambda = ln x) by least squares on checkpoints | |
| 7 | +# x >= 1e8 (below that, transient), reports the extrapolated constant c with | |
| 8 | +# leave-one-out spread, and compares with the HL triple-correlation model | |
| 9 | +# (data/model_c4.json), which predicts the same 1/lambda-drift FORM. | |
| 10 | +# Usage: python3 fit_c4.py data/cycle2_c4_4e10_M3U96a.json | |
| 11 | +# ============================================================================= | |
| 12 | + | |
| 13 | +import json | |
| 14 | +import sys | |
| 15 | +from math import log | |
| 16 | +from pathlib import Path | |
| 17 | + | |
| 18 | +import numpy as np | |
| 19 | + | |
| 20 | +DATA = Path(__file__).resolve().parent.parent / "data" | |
| 21 | + | |
| 22 | + | |
| 23 | +def fit(lams, vals): | |
| 24 | + A = np.vstack([np.ones_like(lams), 1.0 / lams]).T | |
| 25 | + (c, d), res, *_ = np.linalg.lstsq(A, vals, rcond=None) | |
| 26 | + return c, d | |
| 27 | + | |
| 28 | + | |
| 29 | +def main(): | |
| 30 | + src = Path(sys.argv[1]) if len(sys.argv) > 1 else DATA / "cycle2_c4_4e10_M3U96a.json" | |
| 31 | + d = json.load(open(src)) | |
| 32 | + pts = [(float(x), v["rho1_lnx"]) for x, v in d["checkpoints"].items() | |
| 33 | + if float(x) >= 1e8] | |
| 34 | + pts.sort() | |
| 35 | + lams = np.array([log(x) for x, _ in pts]) | |
| 36 | + vals = np.array([v for _, v in pts]) | |
| 37 | + c, dd = fit(lams, vals) | |
| 38 | + # leave-one-out spread on c | |
| 39 | + cs = [] | |
| 40 | + for i in range(len(pts)): | |
| 41 | + m = np.ones(len(pts), dtype=bool); m[i] = False | |
| 42 | + cs.append(fit(lams[m], vals[m])[0]) | |
| 43 | + resid = vals - (c + dd / lams) | |
| 44 | + print(f"points used (x >= 1e8): {len(pts)}") | |
| 45 | + for (x, v), r in zip(pts, resid): | |
| 46 | + print(f" x={x:.0e} rho1*lnx={v:+.5f} fit-residual={r:+.5f}") | |
| 47 | + print(f"\nfit: rho1*lnx = c + d/lnx c = {c:+.5f} d = {dd:+.4f}") | |
| 48 | + print(f"leave-one-out c range: [{min(cs):+.5f}, {max(cs):+.5f}]") | |
| 49 | + print(f"max |residual| = {np.abs(resid).max():.5f}") | |
| 50 | + model = DATA / "model_c4.json" | |
| 51 | + if model.exists(): | |
| 52 | + m = json.load(open(model)) | |
| 53 | + big = m["predictions"].get("1e40", {}) | |
| 54 | + print(f"\nHL triple model for comparison: rho*lambda -> ~{big.get('rho_times_lambda')}" | |
| 55 | + f" (same 1/lambda-drift form, ~3.6x smaller magnitude)") | |
| 56 | + out = {"source": src.name, "n_points": len(pts), | |
| 57 | + "c_extrapolated": round(float(c), 5), "d": round(float(dd), 4), | |
| 58 | + "c_loo_range": [round(min(cs), 5), round(max(cs), 5)], | |
| 59 | + "max_abs_residual": round(float(np.abs(resid).max()), 6)} | |
| 60 | + with open(DATA / "fit_c4.json", "w") as f: | |
| 61 | + json.dump(out, f, indent=2) | |
| 62 | + print("written:", DATA / "fit_c4.json") | |
| 63 | + | |
| 64 | + | |
| 65 | +if __name__ == "__main__": | |
| 66 | + main() | |
added
src/gpu_runner.py
+28 −0
@@ -0,0 +1,28 @@ | ||
| 1 | +#!/usr/bin/env python3 | |
| 2 | +# ============================================================================= | |
| 3 | +# gpu_runner.py — Cycle 4: persistent GPU worker looping over a chunk list | |
| 4 | +# Author: Simon-Pierre Boucher — contact@spboucher.ai | |
| 5 | +# ============================================================================= | |
| 6 | +# Processes chunks sequentially in ONE process so the compiled Metal kernel | |
| 7 | +# and base-prime array are reused. Skips chunks whose output already exists | |
| 8 | +# (idempotent restart). Usage: python3 gpu_runner.py <chunks_file> | |
| 9 | +# ============================================================================= | |
| 10 | + | |
| 11 | +import sys | |
| 12 | +from pathlib import Path | |
| 13 | + | |
| 14 | +from gpu_sieve import sieve_segment_gpu | |
| 15 | +from worker_c4 import run | |
| 16 | + | |
| 17 | +def main(): | |
| 18 | + chunks = Path(sys.argv[1]).read_text().split() | |
| 19 | + for i in range(0, len(chunks), 3): | |
| 20 | + lo, hi, out = chunks[i], chunks[i + 1], chunks[i + 2] | |
| 21 | + if Path(out).exists(): | |
| 22 | + continue | |
| 23 | + run(int(float(lo)), int(float(hi)), out, | |
| 24 | + sieve_fn=sieve_segment_gpu, tag="/gpu") | |
| 25 | + | |
| 26 | + | |
| 27 | +if __name__ == "__main__": | |
| 28 | + main() | |
added
src/gpu_sieve.py
+100 −0
@@ -0,0 +1,100 @@ | ||
| 1 | +#!/usr/bin/env python3 | |
| 2 | +# ============================================================================= | |
| 3 | +# gpu_sieve.py — Cycle 4: segmented sieve on Apple GPU via MLX Metal kernel | |
| 4 | +# Author: Simon-Pierre Boucher — contact@spboucher.ai | |
| 5 | +# ============================================================================= | |
| 6 | +# One thread per (block, prime): marks multiples of base_primes[y] inside | |
| 7 | +# block x of the segment byte-array. Write races are benign (all store 0). | |
| 8 | +# Marking starts at max(p*p, first multiple >= block start) — identical | |
| 9 | +# semantics to core.sieve_segment (validated in __main__ before any use). | |
| 10 | +# ============================================================================= | |
| 11 | + | |
| 12 | +import numpy as np | |
| 13 | + | |
| 14 | +import mlx.core as mx | |
| 15 | + | |
| 16 | +_SRC = r""" | |
| 17 | + uint b = thread_position_in_grid.x; // block index | |
| 18 | + uint pi = thread_position_in_grid.y; // prime index | |
| 19 | + int64_t lo = params[0]; | |
| 20 | + int64_t hi = params[1]; | |
| 21 | + int64_t B = params[2]; | |
| 22 | + int64_t p = primes[pi]; | |
| 23 | + if (p * p >= hi) return; | |
| 24 | + int64_t blo = lo + (int64_t)b * B; | |
| 25 | + int64_t bhi = blo + B < hi ? blo + B : hi; | |
| 26 | + if (blo >= bhi) return; | |
| 27 | + int64_t start = ((blo + p - 1) / p) * p; | |
| 28 | + if (start < p * p) start = p * p; | |
| 29 | + for (int64_t m = start; m < bhi; m += p) { | |
| 30 | + out[m - lo] = 0; | |
| 31 | + } | |
| 32 | +""" | |
| 33 | + | |
| 34 | +_kernel = mx.fast.metal_kernel( | |
| 35 | + name="segmented_sieve", | |
| 36 | + input_names=["primes", "params"], | |
| 37 | + output_names=["out"], | |
| 38 | + source=_SRC, | |
| 39 | +) | |
| 40 | + | |
| 41 | +_BLOCK = 1_000_000 | |
| 42 | + | |
| 43 | + | |
| 44 | +def sieve_segment_gpu(lo: int, hi: int, base_primes) -> np.ndarray: | |
| 45 | + """Primes in [lo, hi) — GPU-marked composites, CPU extraction.""" | |
| 46 | + lo = max(lo, 2) | |
| 47 | + n = hi - lo | |
| 48 | + if n <= 0: | |
| 49 | + return np.array([], dtype=np.int64) | |
| 50 | + primes_mx = mx.array(np.asarray(base_primes, dtype=np.int64)) | |
| 51 | + params = mx.array(np.array([lo, hi, _BLOCK], dtype=np.int64)) | |
| 52 | + nblocks = (n + _BLOCK - 1) // _BLOCK | |
| 53 | + (out,) = _kernel( | |
| 54 | + inputs=[primes_mx, params], | |
| 55 | + grid=(nblocks, len(base_primes), 1), | |
| 56 | + threadgroup=(min(nblocks, 32), 8, 1), | |
| 57 | + output_shapes=[(n,)], | |
| 58 | + output_dtypes=[mx.uint8], | |
| 59 | + init_value=1, | |
| 60 | + ) | |
| 61 | + flags = np.array(out, copy=False) | |
| 62 | + if lo <= 2: | |
| 63 | + # positions of 0 and 1 if present (lo==2 means none) | |
| 64 | + pass | |
| 65 | + res = lo + np.nonzero(flags)[0].astype(np.int64) | |
| 66 | + # base primes < sqrt(hi) inside the segment survive (marking starts at p^2) | |
| 67 | + return res | |
| 68 | + | |
| 69 | + | |
| 70 | +if __name__ == "__main__": | |
| 71 | + import sys | |
| 72 | + import time | |
| 73 | + from core import primes_upto, sieve_segment | |
| 74 | + | |
| 75 | + print("device:", mx.default_device()) | |
| 76 | + # correctness gate: GPU == CPU on varied windows | |
| 77 | + ok = True | |
| 78 | + windows = [(2, 10**6), (999_000_000, 1_001_000_000), | |
| 79 | + (10**12, 10**12 + 5 * 10**7), (10**13 - 5 * 10**7, 10**13)] | |
| 80 | + for lo, hi in windows: | |
| 81 | + base = primes_upto(int(hi ** 0.5) + 1) | |
| 82 | + g = sieve_segment_gpu(lo, hi, base) | |
| 83 | + c = sieve_segment(lo, hi, base) | |
| 84 | + same = np.array_equal(g, c) | |
| 85 | + ok &= same | |
| 86 | + print(f" [{'PASS' if same else 'FAIL'}] [{lo:.2e},{hi:.2e}): " | |
| 87 | + f"{len(g)} primes (gpu) vs {len(c)} (cpu)") | |
| 88 | + if not ok: | |
| 89 | + sys.exit(1) | |
| 90 | + # benchmark: one 1e9 chunk near 1e12, GPU vs single-core CPU | |
| 91 | + lo, hi = 10**12, 10**12 + 10**9 | |
| 92 | + base = primes_upto(int(hi ** 0.5) + 1) | |
| 93 | + for name, fn in (("gpu", sieve_segment_gpu), ("cpu", sieve_segment)): | |
| 94 | + t0 = time.time() | |
| 95 | + tot = 0 | |
| 96 | + for s in range(lo, hi, 50_000_000): | |
| 97 | + tot += len(fn(s, min(s + 50_000_000, hi), base)) | |
| 98 | + print(f" bench {name}: 1e9-chunk near 1e12 in {time.time()-t0:.1f}s " | |
| 99 | + f"({tot} primes)") | |
| 100 | + print("GPU SIEVE VALIDATED") | |
added
src/merge_c4.py
+117 −0
@@ -0,0 +1,117 @@ | ||
| 1 | +#!/usr/bin/env python3 | |
| 2 | +# ============================================================================= | |
| 3 | +# merge_c4.py — Cycle 3: merge distributed worker partials into checkpoints | |
| 4 | +# Author: Simon-Pierre Boucher — contact@spboucher.ai | |
| 5 | +# ============================================================================= | |
| 6 | +# Reads all worker partial JSONs, verifies the chunks tile [2, LIMIT) exactly, | |
| 7 | +# merges cumulatively in range order, and emits checkpoint statistics at every | |
| 8 | +# requested boundary. Cross-validates overlapping checkpoints against the | |
| 9 | +# cycle-2 single-machine scan when available (5-decimal tolerance: summation | |
| 10 | +# order differs, float64 pairwise sums agree to ~1e-9). | |
| 11 | +# Usage: python3 merge_c4.py <partials_dir> <limit> [out.json] | |
| 12 | +# ============================================================================= | |
| 13 | + | |
| 14 | +import json | |
| 15 | +import sys | |
| 16 | +from math import log | |
| 17 | +from pathlib import Path | |
| 18 | + | |
| 19 | +import numpy as np | |
| 20 | + | |
| 21 | +DATA = Path(__file__).resolve().parent.parent / "data" | |
| 22 | +CHECKPOINTS = [10**10, 2 * 10**10, 4 * 10**10, | |
| 23 | + 10**11, 2 * 10**11, 4 * 10**11, 10**12, | |
| 24 | + 2 * 10**12, 4 * 10**12, 10**13] | |
| 25 | + | |
| 26 | + | |
| 27 | +def pearson(S): | |
| 28 | + n = S["n"] | |
| 29 | + cov = S["xy"] / n - (S["x"] / n) * (S["y"] / n) | |
| 30 | + vx = S["xx"] / n - (S["x"] / n) ** 2 | |
| 31 | + vy = S["yy"] / n - (S["y"] / n) ** 2 | |
| 32 | + return cov / (vx * vy) ** 0.5 | |
| 33 | + | |
| 34 | + | |
| 35 | +def main(): | |
| 36 | + pdir = Path(sys.argv[1]) | |
| 37 | + limit = int(float(sys.argv[2])) | |
| 38 | + out_path = Path(sys.argv[3]) if len(sys.argv) > 3 else DATA / f"cycle3_c4_{limit:.0e}.json".replace("+", "").replace("e0", "e") | |
| 39 | + parts = [json.load(open(p)) for p in sorted(pdir.glob("part_*.json"))] | |
| 40 | + parts.sort(key=lambda d: d["lo"]) | |
| 41 | + # tiling check | |
| 42 | + assert parts[0]["lo"] == 2, "first chunk must start at 2" | |
| 43 | + for a, b in zip(parts, parts[1:]): | |
| 44 | + assert a["hi"] == b["lo"], f"gap/overlap between chunks at {a['hi']} vs {b['lo']}" | |
| 45 | + assert parts[-1]["hi"] == limit, f"last chunk ends at {parts[-1]['hi']}, expected {limit}" | |
| 46 | + print(f"{len(parts)} chunks tile [2, {limit:.0e}) exactly; " | |
| 47 | + f"hosts: {sorted(set(p['host'] for p in parts))}") | |
| 48 | + | |
| 49 | + hist = np.zeros(6000, dtype=np.int64) | |
| 50 | + S1 = dict(n=0, x=0.0, y=0.0, xx=0.0, yy=0.0, xy=0.0) | |
| 51 | + S2 = dict(n=0, x=0.0, y=0.0, xx=0.0, yy=0.0, xy=0.0) | |
| 52 | + D = 0 | |
| 53 | + records = [] | |
| 54 | + best = 0 | |
| 55 | + cps = {} | |
| 56 | + want = [c for c in CHECKPOINTS if c <= limit] | |
| 57 | + for part in parts: | |
| 58 | + for g, c in part["hist"].items(): | |
| 59 | + hist[int(g)] += c | |
| 60 | + for k in S1: | |
| 61 | + S1[k] += part["S1"][k] | |
| 62 | + S2[k] += part["S2"][k] | |
| 63 | + D += part["D"] | |
| 64 | + for g, p in part["records"]: | |
| 65 | + if g > best: | |
| 66 | + best = g | |
| 67 | + records.append((g, p)) | |
| 68 | + if want and part["hi"] == want[0]: | |
| 69 | + c = want.pop(0) | |
| 70 | + nz = np.nonzero(hist)[0] | |
| 71 | + champ = int(nz[np.argmax(hist[nz])]) | |
| 72 | + g6 = 0 | |
| 73 | + for g in range(6, int(nz.max()) - 2, 6): | |
| 74 | + if hist[g] > hist[g - 2] and hist[g] > hist[g + 2]: | |
| 75 | + g6 = g | |
| 76 | + else: | |
| 77 | + break | |
| 78 | + r1, r2 = pearson(S1), pearson(S2) | |
| 79 | + cps[str(c)] = {"rho1": round(r1, 7), "rho1_lnx": round(r1 * log(c), 5), | |
| 80 | + "rho2": round(r2, 7), "rho2_lnx": round(r2 * log(c), 5), | |
| 81 | + "champion": champ, "G6": g6, "D_N2_minus_N4": D} | |
| 82 | + print(f"cp {c:.0e}: rho1*lnx={r1*log(c):+.5f} rho2*lnx={r2*log(c):+.5f} " | |
| 83 | + f"champ={champ} G6={g6} D={D}") | |
| 84 | + assert not want, f"missing chunk boundaries for checkpoints {want}" | |
| 85 | + | |
| 86 | + # cross-validation against previous cycles' scans | |
| 87 | + xval = {} | |
| 88 | + for ref_name in ("cycle2_c4_4e10_M3U96a.json", "cycle3_c4_1e12.json"): | |
| 89 | + ref_file = DATA / ref_name | |
| 90 | + if not ref_file.exists(): | |
| 91 | + continue | |
| 92 | + ref = json.load(open(ref_file))["checkpoints"] | |
| 93 | + for c in list(ref): | |
| 94 | + if c not in cps: | |
| 95 | + continue | |
| 96 | + d = abs(cps[c]["rho1_lnx"] - ref[c]["rho1_lnx"]) | |
| 97 | + ok = (d < 2e-5 and cps[c]["D_N2_minus_N4"] == ref[c]["D_N2_minus_N4"] | |
| 98 | + and cps[c]["champion"] == ref[c]["champion"]) | |
| 99 | + xval[c] = {"ref": ref_name, "rho1_lnx_diff": round(d, 7), "exact_D_match": ok} | |
| 100 | + print(f"x-val {float(c):.0e} vs {ref_name}: |drho1_lnx|={d:.2e} " | |
| 101 | + f"D {'==' if ok else '!='} -> {'OK' if ok else 'FAIL'}") | |
| 102 | + | |
| 103 | + total_s = sum(p["seconds"] for p in parts) | |
| 104 | + out = {"limit": limit, "n_chunks": len(parts), | |
| 105 | + "hosts": sorted(set(p["host"] for p in parts)), | |
| 106 | + "total_core_seconds": round(total_s, 1), | |
| 107 | + "checkpoints": cps, | |
| 108 | + "maximal_gaps": [{"gap": g, "after": p, "csg": round(g / log(p) ** 2, 5)} | |
| 109 | + for g, p in records], | |
| 110 | + "cross_validation_vs_cycle2": xval} | |
| 111 | + with open(out_path, "w") as f: | |
| 112 | + json.dump(out, f, indent=2) | |
| 113 | + print("written:", out_path, f"(total {total_s:.0f} core-seconds)") | |
| 114 | + | |
| 115 | + | |
| 116 | +if __name__ == "__main__": | |
| 117 | + main() | |
added
src/model_c4.py
+86 −0
@@ -0,0 +1,86 @@ | ||
| 1 | +#!/usr/bin/env python3 | |
| 2 | +# ============================================================================= | |
| 3 | +# model_c4.py — Cycle 2 PROVER: HL triple-correlation model for the C4 constant | |
| 4 | +# Author: Simon-Pierre Boucher — contact@spboucher.ai | |
| 5 | +# ============================================================================= | |
| 6 | +# Heuristic model for the joint law of consecutive gaps (g1, g2) at scale | |
| 7 | +# lambda = ln x: | |
| 8 | +# P(g1, g2) PROPORTIONAL TO W(g1, g2) * exp(-(g1+g2)/lambda) | |
| 9 | +# where W is the Hardy-Littlewood singular series of the triple {0, g1, g1+g2} | |
| 10 | +# restricted to primes p <= diameter (the tail p > g1+g2 is constant across | |
| 11 | +# pairs and cancels in the normalization): | |
| 12 | +# W = prod_{p odd, p <= G} [ (1 - nu_p/p) / (1 - 1/p)^3 ] * [p=2 term const] | |
| 13 | +# nu_p = #{0, g1, g1+g2 mod p} (in {1,2,3}; equals 3 unless p | g1, | |
| 14 | +# p | g2 or p | g1+g2) | |
| 15 | +# KNOWN LIMITATION (documented): no inclusion-exclusion for "no prime strictly | |
| 16 | +# inside the gaps" — this is the crude first-order model. If W == 1 (Cramer), | |
| 17 | +# rho == 0 identically; all predicted anticorrelation is singular-series | |
| 18 | +# coupling (the p | g1+g2 factor prevents factorization W = u(g1) v(g2)). | |
| 19 | +# Output: model rho(lambda) * lambda for lambdas of interest + large-lambda | |
| 20 | +# trend, printed and saved to data/model_c4.json. Deterministic. | |
| 21 | +# ============================================================================= | |
| 22 | + | |
| 23 | +import json | |
| 24 | +from math import exp, log | |
| 25 | +from pathlib import Path | |
| 26 | + | |
| 27 | +import numpy as np | |
| 28 | + | |
| 29 | +DATA = Path(__file__).resolve().parent.parent / "data" | |
| 30 | + | |
| 31 | + | |
| 32 | +def small_primes(n): | |
| 33 | + s = np.ones(n + 1, dtype=bool); s[:2] = False | |
| 34 | + for i in range(2, int(n ** 0.5) + 1): | |
| 35 | + if s[i]: | |
| 36 | + s[i * i:: i] = False | |
| 37 | + return np.nonzero(s)[0] | |
| 38 | + | |
| 39 | + | |
| 40 | +def model_rho(lam: float, gmax: int) -> float: | |
| 41 | + """Pearson correlation of (g1,g2) under the HL-coupled exponential model.""" | |
| 42 | + gs = np.arange(2, gmax + 1, 2, dtype=np.int64) | |
| 43 | + g1 = gs[:, None] | |
| 44 | + g2 = gs[None, :] | |
| 45 | + gsum = g1 + g2 | |
| 46 | + logW = np.zeros((len(gs), len(gs)), dtype=np.float64) | |
| 47 | + for p in small_primes(2 * gmax): | |
| 48 | + if p == 2: | |
| 49 | + continue # constant factor for even gaps (nu=1), cancels | |
| 50 | + # nu_p on the grid | |
| 51 | + d1 = (g1 % p == 0) | |
| 52 | + d2 = (g2 % p == 0) | |
| 53 | + ds = (gsum % p == 0) | |
| 54 | + nu = np.full(logW.shape, 3, dtype=np.int64) | |
| 55 | + nu[d1 | d2 | ds] = 2 | |
| 56 | + nu[d1 & d2] = 1 | |
| 57 | + logW += np.log(1 - nu / p) - 3 * log(1 - 1 / p) | |
| 58 | + P = np.exp(logW - gsum / lam) | |
| 59 | + P /= P.sum() | |
| 60 | + m1 = (P * g1).sum(); m2 = (P * g2).sum() | |
| 61 | + v1 = (P * (g1 - m1) ** 2).sum(); v2 = (P * (g2 - m2) ** 2).sum() | |
| 62 | + cov = (P * (g1 - m1) * (g2 - m2)).sum() | |
| 63 | + return float(cov / (v1 * v2) ** 0.5) | |
| 64 | + | |
| 65 | + | |
| 66 | +def main(): | |
| 67 | + out = {"model": "P(g1,g2) ~ W_HL(0,g1,g1+g2) * exp(-(g1+g2)/lambda)", | |
| 68 | + "predictions": {}} | |
| 69 | + print(f"{'lambda':>8} {'x=e^lam':>10} {'rho_model':>12} {'rho*lambda':>12}") | |
| 70 | + for lam, label in [(log(1e8), "1e8"), (log(1e9), "1e9"), (log(1e10), "1e10"), | |
| 71 | + (log(4e10), "4e10"), (log(1e12), "1e12"), | |
| 72 | + (log(1e15), "1e15"), (log(1e20), "1e20"), | |
| 73 | + (log(1e30), "1e30"), (log(1e40), "1e40")]: | |
| 74 | + gmax = int(14 * lam) # truncation: weight e^-14 ~ 8e-7 | |
| 75 | + r = model_rho(lam, gmax) | |
| 76 | + out["predictions"][label] = {"lambda": round(lam, 4), | |
| 77 | + "rho": round(r, 6), | |
| 78 | + "rho_times_lambda": round(r * lam, 5)} | |
| 79 | + print(f"{lam:8.3f} {label:>10} {r:12.6f} {r*lam:12.5f}") | |
| 80 | + with open(DATA / "model_c4.json", "w") as f: | |
| 81 | + json.dump(out, f, indent=2) | |
| 82 | + print("written:", DATA / "model_c4.json") | |
| 83 | + | |
| 84 | + | |
| 85 | +if __name__ == "__main__": | |
| 86 | + main() | |
added
src/prover_desert.py
+167 −0
@@ -0,0 +1,167 @@ | ||
| 1 | +#!/usr/bin/env python3 | |
| 2 | +# ============================================================================= | |
| 3 | +# prover_desert.py — Cycle 1 PROVER: certified prime desert via covering system | |
| 4 | +# Author: Simon-Pierre Boucher — contact@spboucher.ai | |
| 5 | +# ============================================================================= | |
| 6 | +# Construction: | |
| 7 | +# 1. Prime set Q = primes <= m. Greedily pick a residue class a_p mod p for | |
| 8 | +# each p in Q so that every position i in 1..L satisfies i = a_p (mod p) | |
| 9 | +# for some p, while NO class contains position 0 or L+1. | |
| 10 | +# 2. CRT: x = -a_p (mod p) for all p. Then for EVERY t >= 0, | |
| 11 | +# N = x + t*P (P = prod Q) has N+i = 0 (mod q_i) with N+i > q_i, | |
| 12 | +# hence N+1 .. N+L are ALL composite. <-- PROVEN by covering | |
| 13 | +# 3. Search the smallest t with N and N+L+1 both prime (deterministic | |
| 14 | +# Miller-Rabin, valid since N + L + 1 < 3.317e24). Then the gap after N | |
| 15 | +# is EXACTLY L+1. <-- CERTIFIED | |
| 16 | +# Baseline to beat: the classic primorial construction with the same Q gives | |
| 17 | +# only L = (next prime after m) - 1 - 1 interior composites (gap ~ next prime). | |
| 18 | +# Output: certs/desert_certificate.json (+ console report) | |
| 19 | +# Deterministic: greedy tie-break is lexicographic; t-search is sequential. | |
| 20 | +# ============================================================================= | |
| 21 | + | |
| 22 | +import json | |
| 23 | +import time | |
| 24 | +from math import log, prod | |
| 25 | +from pathlib import Path | |
| 26 | + | |
| 27 | +from core import is_prime, primes_upto | |
| 28 | + | |
| 29 | +CERTS = Path(__file__).resolve().parent.parent / "certs" | |
| 30 | +CERTS.mkdir(exist_ok=True) | |
| 31 | + | |
| 32 | +MR_LIMIT = 3_317_044_064_679_887_385_961_981 | |
| 33 | + | |
| 34 | + | |
| 35 | +def greedy_cover(qs, L, order): | |
| 36 | + """Try to cover positions 1..L with one class per prime (order given). | |
| 37 | + Classes containing position 0 or L+1 are forbidden. Returns {p: a_p} or None.""" | |
| 38 | + uncovered = set(range(1, L + 1)) | |
| 39 | + classes = {} | |
| 40 | + for p in order: | |
| 41 | + best_class, best_hits = None, -1 | |
| 42 | + for a in range(p): | |
| 43 | + if a == 0 % p or a == (L + 1) % p: | |
| 44 | + continue | |
| 45 | + hits = sum(1 for i in uncovered if i % p == a) | |
| 46 | + if hits > best_hits: | |
| 47 | + best_class, best_hits = a, hits | |
| 48 | + if best_class is None: | |
| 49 | + return None | |
| 50 | + classes[p] = best_class | |
| 51 | + uncovered = {i for i in uncovered if i % p != best_class} | |
| 52 | + return classes if not uncovered else None | |
| 53 | + | |
| 54 | + | |
| 55 | +def max_cover(qs): | |
| 56 | + """Largest L (with its classes) coverable by greedy, over a few orderings.""" | |
| 57 | + best = (0, None) | |
| 58 | + orders = [sorted(qs), sorted(qs, reverse=True)] | |
| 59 | + for order in orders: | |
| 60 | + L = 1 | |
| 61 | + last_good = None | |
| 62 | + # increase L until greedy fails twice in a row (L must keep parity odd) | |
| 63 | + fails = 0 | |
| 64 | + while fails < 6: | |
| 65 | + cand = greedy_cover(qs, L, order) | |
| 66 | + if cand: | |
| 67 | + last_good = (L, cand) | |
| 68 | + fails = 0 | |
| 69 | + else: | |
| 70 | + fails += 1 | |
| 71 | + L += 2 # keep L odd: p=2 then covers odds, sparing 0 and L+1 | |
| 72 | + if last_good and last_good[0] > best[0]: | |
| 73 | + best = last_good | |
| 74 | + return best | |
| 75 | + | |
| 76 | + | |
| 77 | +def crt(classes): | |
| 78 | + """x with x = -a_p (mod p) for all p; 0 <= x < prod(p).""" | |
| 79 | + x, mod = 0, 1 | |
| 80 | + for p, a in sorted(classes.items()): | |
| 81 | + r = (-a) % p | |
| 82 | + # solve x + mod*k = r (mod p) | |
| 83 | + k = ((r - x) * pow(mod, -1, p)) % p | |
| 84 | + x += mod * k | |
| 85 | + mod *= p | |
| 86 | + return x, mod | |
| 87 | + | |
| 88 | + | |
| 89 | +def main(): | |
| 90 | + t0 = time.time() | |
| 91 | + m = 59 | |
| 92 | + qs = [int(p) for p in primes_upto(m)] | |
| 93 | + L, classes = max_cover(qs) | |
| 94 | + assert classes is not None, "no covering found" | |
| 95 | + print(f"prime set: primes <= {m} ({len(qs)} primes); covered interval length L = {L}") | |
| 96 | + baseline = int(primes_upto(200)[len(qs)]) - 1 # next prime after m, minus 1 | |
| 97 | + print(f"baseline (classic primorial construction): {baseline} composites -> improvement x{L/baseline:.2f}") | |
| 98 | + | |
| 99 | + # verify covering exhaustively before CRT (belt and braces) | |
| 100 | + for i in range(1, L + 1): | |
| 101 | + assert any(i % p == a for p, a in classes.items()), f"position {i} uncovered" | |
| 102 | + for p, a in classes.items(): | |
| 103 | + assert 0 % p != a and (L + 1) % p != a, f"class of {p} hits an endpoint" | |
| 104 | + | |
| 105 | + x, P = crt(classes) | |
| 106 | + print(f"P = {m}# = {P}") | |
| 107 | + # search smallest t with both endpoints prime; keep N + L + 1 under MR limit | |
| 108 | + t_max = (MR_LIMIT - x - L - 1) // P | |
| 109 | + found = None | |
| 110 | + for t in range(1, t_max + 1): | |
| 111 | + N = x + t * P | |
| 112 | + if is_prime(N) and is_prime(N + L + 1): | |
| 113 | + found = (t, N) | |
| 114 | + break | |
| 115 | + assert found, f"no prime endpoints for t <= {t_max}" | |
| 116 | + t, N = found | |
| 117 | + gap = L + 1 | |
| 118 | + merit = gap / log(N) | |
| 119 | + print(f"t = {t}; N = {N} ({len(str(N))} digits)") | |
| 120 | + print(f"CERTIFIED GAP: next_prime({N}) - {N} = {gap}, merit = {merit:.4f}, " | |
| 121 | + f"csg = {gap/log(N)**2:.5f}") | |
| 122 | + | |
| 123 | + # interior divisor table: smallest covering prime per position | |
| 124 | + divisors = {} | |
| 125 | + for i in range(1, L + 1): | |
| 126 | + q = min(p for p, a in classes.items() if i % p == a) | |
| 127 | + assert (N + i) % q == 0 and N + i > q | |
| 128 | + divisors[i] = q | |
| 129 | + | |
| 130 | + cert = { | |
| 131 | + "title": "Certified prime desert via covering system", | |
| 132 | + "author": "Simon-Pierre Boucher — contact@spboucher.ai", | |
| 133 | + "date": "2026-08-06", | |
| 134 | + "claim": f"The gap between consecutive primes N and N+{gap} is exactly {gap}.", | |
| 135 | + "N": str(N), | |
| 136 | + "gap": gap, | |
| 137 | + "merit": round(merit, 5), | |
| 138 | + "csg_ratio": round(gap / log(N) ** 2, 6), | |
| 139 | + "construction": { | |
| 140 | + "prime_set_max": m, | |
| 141 | + "primorial_P": str(P), | |
| 142 | + "crt_residue_x": str(x), | |
| 143 | + "shift_t": t, | |
| 144 | + "classes_a_p": {str(p): a for p, a in sorted(classes.items())}, | |
| 145 | + "note": "for EVERY t>=0, x+t*P+1 .. x+t*P+L are composite (covering); " | |
| 146 | + "this specific t makes both endpoints prime", | |
| 147 | + }, | |
| 148 | + "interior_divisors": {str(i): q for i, q in sorted(divisors.items())}, | |
| 149 | + "endpoint_primality": { | |
| 150 | + "method": "deterministic Miller-Rabin, bases 2..37 (12 bases)", | |
| 151 | + "validity_bound": str(MR_LIMIT), | |
| 152 | + "both_endpoints_below_bound": bool(N + gap < MR_LIMIT), | |
| 153 | + }, | |
| 154 | + "baseline_same_primes": baseline, | |
| 155 | + "epistemic_status": "PROVEN (interior compositeness, by covering) + " | |
| 156 | + "CERTIFIED (endpoint primality, deterministic MR). " | |
| 157 | + "NOT a record (records.md: merit record is 41.94); " | |
| 158 | + "value = fully certified pipeline.", | |
| 159 | + } | |
| 160 | + path = CERTS / "desert_certificate.json" | |
| 161 | + with open(path, "w") as f: | |
| 162 | + json.dump(cert, f, indent=2) | |
| 163 | + print(f"certificate written: {path} ({time.time()-t0:.1f}s)") | |
| 164 | + | |
| 165 | + | |
| 166 | +if __name__ == "__main__": | |
| 167 | + main() | |
added
src/prover_desert2.py
+179 −0
@@ -0,0 +1,179 @@ | ||
| 1 | +#!/usr/bin/env python3 | |
| 2 | +# ============================================================================= | |
| 3 | +# prover_desert2.py — Cycle 1 PROVER (attempt 2): hybrid covering desert | |
| 4 | +# Author: Simon-Pierre Boucher — contact@spboucher.ai | |
| 5 | +# ============================================================================= | |
| 6 | +# Upgrade over prover_desert.py: the covering system (primes <= 59) covers MOST | |
| 7 | +# of positions 1..L; the few uncovered positions are certified composite for | |
| 8 | +# the specific shift t by an explicit small factor (trial division) or an | |
| 9 | +# explicit Miller-Rabin witness (a valid compositeness certificate). | |
| 10 | +# Endpoints N, N+L+1 certified prime by deterministic MR (N+L+1 < 3.317e24). | |
| 11 | +# Deterministic search; no randomness. | |
| 12 | +# Output: certs/desert_certificate.json (overwrites attempt 1 - superseded) | |
| 13 | +# ============================================================================= | |
| 14 | + | |
| 15 | +import json | |
| 16 | +import time | |
| 17 | +from math import log | |
| 18 | +from pathlib import Path | |
| 19 | + | |
| 20 | +from core import is_prime, primes_upto | |
| 21 | + | |
| 22 | +CERTS = Path(__file__).resolve().parent.parent / "certs" | |
| 23 | +MR_LIMIT = 3_317_044_064_679_887_385_961_981 | |
| 24 | +TRIAL_PRIMES = [int(p) for p in primes_upto(1_000_000)] | |
| 25 | + | |
| 26 | + | |
| 27 | +def greedy_cover_partial(qs, L): | |
| 28 | + """One class per prime, greedy max coverage of 1..L; classes may not touch | |
| 29 | + positions 0 or L+1. Returns (classes, uncovered_list).""" | |
| 30 | + uncovered = set(range(1, L + 1)) | |
| 31 | + classes = {} | |
| 32 | + for p in sorted(qs): | |
| 33 | + best_a, best_hits = None, -1 | |
| 34 | + for a in range(p): | |
| 35 | + if a == 0 or a == (L + 1) % p: | |
| 36 | + continue | |
| 37 | + hits = sum(1 for i in uncovered if i % p == a) | |
| 38 | + if hits > best_hits: | |
| 39 | + best_a, best_hits = a, hits | |
| 40 | + classes[p] = best_a | |
| 41 | + uncovered = {i for i in uncovered if i % p != best_a} | |
| 42 | + return classes, sorted(uncovered) | |
| 43 | + | |
| 44 | + | |
| 45 | +def crt(classes): | |
| 46 | + x, mod = 0, 1 | |
| 47 | + for p, a in sorted(classes.items()): | |
| 48 | + r = (-a) % p | |
| 49 | + k = ((r - x) * pow(mod, -1, p)) % p | |
| 50 | + x += mod * k | |
| 51 | + mod *= p | |
| 52 | + return x, mod | |
| 53 | + | |
| 54 | + | |
| 55 | +def mr_witness_for(n): | |
| 56 | + """Return a base a proving n composite (strong witness), or None.""" | |
| 57 | + d, r = n - 1, 0 | |
| 58 | + while d % 2 == 0: | |
| 59 | + d //= 2 | |
| 60 | + r += 1 | |
| 61 | + for a in (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37): | |
| 62 | + x = pow(a, d, n) | |
| 63 | + if x == 1 or x == n - 1: | |
| 64 | + continue | |
| 65 | + for _ in range(r - 1): | |
| 66 | + x = x * x % n | |
| 67 | + if x == n - 1: | |
| 68 | + break | |
| 69 | + else: | |
| 70 | + return a | |
| 71 | + return None | |
| 72 | + | |
| 73 | + | |
| 74 | +def small_factor(n): | |
| 75 | + for q in TRIAL_PRIMES: | |
| 76 | + if q * q > n: | |
| 77 | + return None | |
| 78 | + if n % q == 0: | |
| 79 | + return q | |
| 80 | + return None | |
| 81 | + | |
| 82 | + | |
| 83 | +def main(): | |
| 84 | + t0 = time.time() | |
| 85 | + m = 59 | |
| 86 | + qs = [int(p) for p in primes_upto(m)] | |
| 87 | + # choose the largest odd L whose greedy covering leaves <= UMAX holes | |
| 88 | + for UMAX in (16, 12, 8): | |
| 89 | + chosen = None | |
| 90 | + for L in range(91, 300, 2): | |
| 91 | + classes, unc = greedy_cover_partial(qs, L) | |
| 92 | + if len(unc) <= UMAX: | |
| 93 | + chosen = (L, classes, unc) | |
| 94 | + if chosen is None: | |
| 95 | + continue | |
| 96 | + L, classes, unc = chosen | |
| 97 | + print(f"UMAX={UMAX}: L={L}, holes={len(unc)} at {unc}") | |
| 98 | + x, P = crt(classes) | |
| 99 | + t_max = (MR_LIMIT - x - L - 1) // P | |
| 100 | + hit = None | |
| 101 | + for t in range(1, int(t_max) + 1): | |
| 102 | + N = x + t * P | |
| 103 | + if not is_prime(N): | |
| 104 | + continue | |
| 105 | + if not is_prime(N + L + 1): | |
| 106 | + continue | |
| 107 | + if all(not is_prime(N + i) for i in unc): | |
| 108 | + hit = (t, N) | |
| 109 | + break | |
| 110 | + if hit: | |
| 111 | + break | |
| 112 | + print(f" no t <= {t_max} works; relaxing") | |
| 113 | + assert hit, "hybrid construction failed at all UMAX levels" | |
| 114 | + t, N = hit | |
| 115 | + gap = L + 1 | |
| 116 | + print(f"t={t} N={N} ({len(str(N))} digits)") | |
| 117 | + print(f"CERTIFIED GAP {gap}, merit {gap/log(N):.4f}, csg {gap/log(N)**2:.5f} " | |
| 118 | + f"[{time.time()-t0:.1f}s]") | |
| 119 | + | |
| 120 | + # build interior certificates | |
| 121 | + interior = {} | |
| 122 | + for i in range(1, L + 1): | |
| 123 | + cov = [p for p, a in classes.items() if i % p == a] | |
| 124 | + if cov: | |
| 125 | + q = min(cov) | |
| 126 | + assert (N + i) % q == 0 and N + i > q | |
| 127 | + interior[str(i)] = {"type": "covering_factor", "q": q} | |
| 128 | + else: | |
| 129 | + f = small_factor(N + i) | |
| 130 | + if f: | |
| 131 | + interior[str(i)] = {"type": "trial_factor", "q": f} | |
| 132 | + else: | |
| 133 | + a = mr_witness_for(N + i) | |
| 134 | + assert a is not None, f"no witness for position {i}?!" | |
| 135 | + interior[str(i)] = {"type": "mr_witness", "a": a} | |
| 136 | + n_types = {} | |
| 137 | + for v in interior.values(): | |
| 138 | + n_types[v["type"]] = n_types.get(v["type"], 0) + 1 | |
| 139 | + print("interior certificate types:", n_types) | |
| 140 | + | |
| 141 | + cert = { | |
| 142 | + "title": "Certified prime desert via hybrid covering system", | |
| 143 | + "author": "Simon-Pierre Boucher — contact@spboucher.ai", | |
| 144 | + "date": "2026-08-06", | |
| 145 | + "claim": f"N and N+{gap} are consecutive primes (gap exactly {gap}).", | |
| 146 | + "N": str(N), | |
| 147 | + "gap": gap, | |
| 148 | + "merit": round(gap / log(N), 5), | |
| 149 | + "csg_ratio": round(gap / log(N) ** 2, 6), | |
| 150 | + "construction": { | |
| 151 | + "prime_set_max": m, | |
| 152 | + "primorial_P": str(crt(classes)[1]), | |
| 153 | + "crt_residue_x": str(x), | |
| 154 | + "shift_t": t, | |
| 155 | + "classes_a_p": {str(p): a for p, a in sorted(classes.items())}, | |
| 156 | + "holes": unc, | |
| 157 | + "note": "positions with a covering factor are composite for EVERY " | |
| 158 | + "shift t (proven); hole positions are certified composite " | |
| 159 | + "for THIS t by explicit factor or strong MR witness", | |
| 160 | + }, | |
| 161 | + "interior_certificates": interior, | |
| 162 | + "endpoint_primality": { | |
| 163 | + "method": "deterministic Miller-Rabin, bases {2..37} (12 bases), " | |
| 164 | + "valid for n < 3.317e24 (Sorenson-Webster 2015)", | |
| 165 | + "validity_bound": str(MR_LIMIT), | |
| 166 | + "both_endpoints_below_bound": bool(N + gap < MR_LIMIT), | |
| 167 | + }, | |
| 168 | + "epistemic_status": "CERTIFIED construction; NOT a record (see records.md: " | |
| 169 | + "merit record 41.94, largest maximal gap 1854). " | |
| 170 | + "Baseline with same primes: classic primorial gives gap 61; " | |
| 171 | + "pure covering (attempt 1) gave gap 90.", | |
| 172 | + } | |
| 173 | + with open(CERTS / "desert_certificate.json", "w") as f: | |
| 174 | + json.dump(cert, f, indent=2) | |
| 175 | + print("certificate written:", CERTS / "desert_certificate.json") | |
| 176 | + | |
| 177 | + | |
| 178 | +if __name__ == "__main__": | |
| 179 | + main() | |
added
src/test_core.py
+93 −0
@@ -0,0 +1,93 @@ | ||
| 1 | +#!/usr/bin/env python3 | |
| 2 | +# ============================================================================= | |
| 3 | +# test_core.py — Phase 0 validation: DO NOT proceed while any test fails | |
| 4 | +# Author: Simon-Pierre Boucher — contact@spboucher.ai | |
| 5 | +# ============================================================================= | |
| 6 | +# Validates every core building block against independently known values: | |
| 7 | +# pi(10^k), maximal prime gaps below 10^6, strong-pseudoprime traps, | |
| 8 | +# Carmichael numbers, cross-agreement sieve vs MR vs BPSW. | |
| 9 | +# ============================================================================= | |
| 10 | + | |
| 11 | +import sys | |
| 12 | +import time | |
| 13 | + | |
| 14 | +import numpy as np | |
| 15 | + | |
| 16 | +from core import (bpsw, gaps_in_range, is_prime, prime_count, primes_upto, | |
| 17 | + sieve_segment) | |
| 18 | + | |
| 19 | +FAILURES = [] | |
| 20 | + | |
| 21 | + | |
| 22 | +def check(name, got, expected): | |
| 23 | + ok = got == expected | |
| 24 | + print(f" [{'PASS' if ok else 'FAIL'}] {name}: got {got}, expected {expected}") | |
| 25 | + if not ok: | |
| 26 | + FAILURES.append(name) | |
| 27 | + | |
| 28 | + | |
| 29 | +t0 = time.time() | |
| 30 | +print("== pi(x) against known values ==") | |
| 31 | +check("pi(10^2)", len(primes_upto(10**2)), 25) | |
| 32 | +check("pi(10^4)", len(primes_upto(10**4)), 1229) | |
| 33 | +check("pi(10^6)", len(primes_upto(10**6)), 78498) | |
| 34 | +check("pi(10^7)", len(primes_upto(10**7)), 664579) | |
| 35 | +check("pi(10^8) [segmented]", prime_count(10**8), 5761455) | |
| 36 | + | |
| 37 | +print("== segmented sieve == full sieve on random windows ==") | |
| 38 | +full = primes_upto(2_000_000) | |
| 39 | +for lo, hi in [(0, 1000), (999_000, 1_001_000), (1_500_000, 1_600_000)]: | |
| 40 | + seg = sieve_segment(lo, hi) | |
| 41 | + ref = full[(full >= lo) & (full < hi)] | |
| 42 | + check(f"segment [{lo},{hi})", int(np.array_equal(seg, ref)), 1) | |
| 43 | + | |
| 44 | +print("== maximal prime gaps (known table, gaps starting below 10^6) ==") | |
| 45 | +# (gap, prime after which it occurs) — classical table (OEIS A005250/A002386) | |
| 46 | +KNOWN_MAXIMAL = [(4, 7), (6, 23), (8, 89), (14, 113), (18, 523), (20, 887), | |
| 47 | + (22, 1129), (34, 1327), (36, 9551), (44, 15683), (52, 19609), | |
| 48 | + (72, 31397), (86, 155921), (96, 360653), (112, 370261), | |
| 49 | + (114, 492113)] | |
| 50 | +best = 0 | |
| 51 | +found = [] | |
| 52 | +for p, g in gaps_in_range(2, 1_400_000): | |
| 53 | + if g > best: | |
| 54 | + best = g | |
| 55 | + found.append((g, p)) | |
| 56 | +found = [(g, p) for g, p in found if g >= 4 and p < 10**6] | |
| 57 | +check("maximal gaps < 10^6", found, KNOWN_MAXIMAL) | |
| 58 | + | |
| 59 | +print("== Miller-Rabin deterministic: traps and cross-check ==") | |
| 60 | +# strong pseudoprimes base 2 must be caught | |
| 61 | +for n in [2047, 3277, 4033, 4681, 8321, 15841, 29341]: | |
| 62 | + check(f"spsp(2) {n} composite", is_prime(n), False) | |
| 63 | +# Carmichael numbers | |
| 64 | +for n in [561, 1105, 1729, 41041, 825265, 321197185]: | |
| 65 | + check(f"Carmichael {n} composite", is_prime(n), False) | |
| 66 | +# known primes, including large | |
| 67 | +check("2^31-1 prime", is_prime(2**31 - 1), True) | |
| 68 | +check("2^61-1 prime", is_prime(2**61 - 1), True) | |
| 69 | +check("2^67-1 composite (Mersenne)", is_prime(2**67 - 1), False) | |
| 70 | +check("10^18+9 prime", is_prime(10**18 + 9), True) | |
| 71 | +# cross-check MR vs sieve over [0, 40000) | |
| 72 | +ref_set = set(primes_upto(40000).tolist()) | |
| 73 | +mismatch = sum(1 for n in range(40000) if is_prime(n) != (n in ref_set)) | |
| 74 | +check("MR == sieve on [0,40000)", mismatch, 0) | |
| 75 | + | |
| 76 | +print("== BPSW: same traps + cross-check ==") | |
| 77 | +for n in [2047, 3277, 561, 1105, 1729, 41041]: | |
| 78 | + check(f"BPSW {n} composite", bpsw(n), False) | |
| 79 | +check("BPSW 2^61-1 prime", bpsw(2**61 - 1), True) | |
| 80 | +check("BPSW 10^18+9 prime", bpsw(10**18 + 9), True) | |
| 81 | +mismatch = sum(1 for n in range(3, 40000) if bpsw(n) != (n in ref_set)) | |
| 82 | +check("BPSW == sieve on [3,40000)", mismatch, 0) | |
| 83 | +# random 64-bit cross-check BPSW vs deterministic MR (fixed seed) | |
| 84 | +rng = np.random.default_rng(42) | |
| 85 | +vals = [int(x) for x in rng.integers(2**40, 2**62, size=300)] | |
| 86 | +mismatch = sum(1 for n in vals if bpsw(n) != is_prime(n)) | |
| 87 | +check("BPSW == det-MR on 300 random 62-bit (seed 42)", mismatch, 0) | |
| 88 | + | |
| 89 | +print(f"\nTotal time: {time.time()-t0:.1f}s") | |
| 90 | +if FAILURES: | |
| 91 | + print(f"FAILED: {FAILURES}") | |
| 92 | + sys.exit(1) | |
| 93 | +print("ALL TESTS PASSED") | |
added
src/worker_c4.py
+112 −0
@@ -0,0 +1,112 @@ | ||
| 1 | +#!/usr/bin/env python3 | |
| 2 | +# ============================================================================= | |
| 3 | +# worker_c4.py — Cycle 3: one distributed chunk of the C4' scan | |
| 4 | +# Author: Simon-Pierre Boucher — contact@spboucher.ai | |
| 5 | +# ============================================================================= | |
| 6 | +# Scans gaps whose END prime lies in [lo, hi) and emits MERGEABLE partial sums: | |
| 7 | +# S1/S2 (lag-1/lag-2 Pearson streaming sums), gap histogram, D = N2-N4, | |
| 8 | +# chunk-local record gaps. A 5000-wide sieve buffer below lo recovers the | |
| 9 | +# two gaps preceding the first counted one (max gap < 1e12 is ~540, and any | |
| 10 | +# 540-window below 1e12 contains a prime, so the buffer always suffices). | |
| 11 | +# Deterministic. Usage: python3 worker_c4.py <lo> <hi> <out.json> | |
| 12 | +# ============================================================================= | |
| 13 | + | |
| 14 | +import json | |
| 15 | +import platform | |
| 16 | +import sys | |
| 17 | +import time | |
| 18 | + | |
| 19 | +import numpy as np | |
| 20 | + | |
| 21 | +from core import primes_upto, sieve_segment | |
| 22 | + | |
| 23 | +BUFFER = 5000 | |
| 24 | +SEG = 50_000_000 | |
| 25 | + | |
| 26 | + | |
| 27 | +def run(lo, hi, out_path, sieve_fn=None, tag=""): | |
| 28 | + sieve = sieve_fn or sieve_segment | |
| 29 | + t0 = time.time() | |
| 30 | + base = primes_upto(int(hi ** 0.5) + 1) | |
| 31 | + start = 2 if lo <= 2 else lo - BUFFER | |
| 32 | + | |
| 33 | + hist = np.zeros(6000, dtype=np.int64) | |
| 34 | + S1 = dict(n=0, x=0.0, y=0.0, xx=0.0, yy=0.0, xy=0.0) | |
| 35 | + S2 = dict(n=0, x=0.0, y=0.0, xx=0.0, yy=0.0, xy=0.0) | |
| 36 | + D = 0 | |
| 37 | + records = [] | |
| 38 | + best = 0 | |
| 39 | + n_gaps = 0 | |
| 40 | + prev = None | |
| 41 | + tail = [] # up to 2 gaps immediately preceding current segment's first gap | |
| 42 | + s = start | |
| 43 | + while s < hi: | |
| 44 | + e = min(s + SEG, hi) | |
| 45 | + primes = sieve(s, e, base) | |
| 46 | + s = e | |
| 47 | + if len(primes) == 0: | |
| 48 | + continue | |
| 49 | + if prev is not None: | |
| 50 | + primes = np.concatenate(([prev], primes)) | |
| 51 | + if len(primes) >= 2: | |
| 52 | + gaps = np.diff(primes) | |
| 53 | + ends = primes[1:] | |
| 54 | + starts = primes[:-1] | |
| 55 | + inr = ends >= lo # gaps counted by this worker (end prime >= lo) | |
| 56 | + gin = gaps[inr] | |
| 57 | + n_gaps += len(gin) | |
| 58 | + hist += np.bincount(gin, minlength=len(hist))[: len(hist)] | |
| 59 | + D += int((gin == 2).sum() - (gin == 4).sum()) | |
| 60 | + # pair sums: pair (g_{n-1}, g_n) / (g_{n-2}, g_n) counted at g_n | |
| 61 | + gl = np.concatenate((np.array(tail, dtype=gaps.dtype), gaps)) | |
| 62 | + k = len(tail) | |
| 63 | + idx = np.nonzero(inr)[0] + k # positions of counted gaps in gl | |
| 64 | + i1 = idx[idx >= 1] | |
| 65 | + a = gl[i1 - 1].astype(np.float64); b = gl[i1].astype(np.float64) | |
| 66 | + S1["n"] += len(a) | |
| 67 | + S1["x"] += float(a.sum()); S1["y"] += float(b.sum()) | |
| 68 | + S1["xx"] += float((a * a).sum()); S1["yy"] += float((b * b).sum()) | |
| 69 | + S1["xy"] += float((a * b).sum()) | |
| 70 | + i2 = idx[idx >= 2] | |
| 71 | + a2 = gl[i2 - 2].astype(np.float64); b2 = gl[i2].astype(np.float64) | |
| 72 | + S2["n"] += len(a2) | |
| 73 | + S2["x"] += float(a2.sum()); S2["y"] += float(b2.sum()) | |
| 74 | + S2["xx"] += float((a2 * a2).sum()); S2["yy"] += float((b2 * b2).sum()) | |
| 75 | + S2["xy"] += float((a2 * b2).sum()) | |
| 76 | + # chunk-local records among counted gaps | |
| 77 | + if len(gin) and int(gin.max()) > best: | |
| 78 | + sin = starts[inr] | |
| 79 | + for i in np.nonzero(gin > best)[0]: | |
| 80 | + g, p = int(gin[i]), int(sin[i]) | |
| 81 | + if g > best: | |
| 82 | + best = g | |
| 83 | + records.append([g, p]) | |
| 84 | + tail = [int(v) for v in gaps[-2:]] | |
| 85 | + prev = primes[-1] | |
| 86 | + | |
| 87 | + nz = np.nonzero(hist)[0] | |
| 88 | + out = { | |
| 89 | + "lo": lo, "hi": hi, "n_gaps": int(n_gaps), "D": D, | |
| 90 | + "S1": S1, "S2": S2, | |
| 91 | + "hist": {int(g): int(hist[g]) for g in nz}, | |
| 92 | + "records": records, | |
| 93 | + "host": platform.node() + tag, | |
| 94 | + "seconds": round(time.time() - t0, 1), | |
| 95 | + } | |
| 96 | + with open(out_path, "w") as f: | |
| 97 | + json.dump(out, f) | |
| 98 | + print(f"done [{lo},{hi}) on {out['host']} in {out['seconds']}s", flush=True) | |
| 99 | + | |
| 100 | + | |
| 101 | +def main(): | |
| 102 | + lo, hi = int(float(sys.argv[1])), int(float(sys.argv[2])) | |
| 103 | + out_path = sys.argv[3] | |
| 104 | + if len(sys.argv) > 4 and sys.argv[4] == "gpu": | |
| 105 | + from gpu_sieve import sieve_segment_gpu | |
| 106 | + run(lo, hi, out_path, sieve_fn=sieve_segment_gpu, tag="/gpu") | |
| 107 | + else: | |
| 108 | + run(lo, hi, out_path) | |
| 109 | + | |
| 110 | + | |
| 111 | +if __name__ == "__main__": | |
| 112 | + main() | |
| 113 | ||