# 🔬 Prime Mystery Engine **A verification-first computational laboratory for prime gap statistics.**          An autonomous research pipeline that studies gaps between consecutive primes under a strict epistemic protocol: **no claim without a certificate**. Every statement is labelled `PROVEN` / `CERTIFIED` / `VERIFIED UP TO X` / `CONJECTURED` / `REFUTED`, every scan is deterministic and bit-reproducible across machines, and every headline object ships with a standalone re-verification script. --- ## 🏆 Headline results | Result | Status | |---|---| | **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) | | **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` | | **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 | | **Lag-2 law** — ρ₂(x)·ln x → c₂ ≈ −0.278, prediction confirmed at 10¹³ (−0.2545 ± 0.004 → −0.25247) | `CONJECTURED`, 1/1 hit | | **First-order Hardy–Littlewood triple model rejected** — right sign and drift form, but only ~32 % of the observed magnitude | `MODEL REJECTED` | | **All published tables reproduced** — 53/53 maximal gaps below 10¹³ (OEIS A005250/A002386), twin counts, π(10ᵏ), CSG maxima | `VERIFIED` | **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 --- ## ⚡ Verify the headline claims yourself (stock Python 3) ```bash # the certified 260-gap desert (independent Miller–Rabin implementation) python3 certs/verify_desert.py # the race-conjecture refutation (stdlib-only sieve, ~30 s) python3 certs/verify_c2_refutation.py # the core-primitives validation gate (37 checks, needs numpy) python3 src/test_core.py ``` ## 🗂 Repository layout ``` src/ core primitives, scans, distributed worker/merger, Metal GPU sieve, models certs/ certificates + standalone verifiers (the "trust nothing" layer) data/ raw checkpoint JSONs for every campaign (deterministic, re-mergeable) paper/ LaTeX manuscript (main.tex → main.pdf, 13 pp., full math detail) journal.md complete research journal: every cycle, decision, failure conjectures.md the conjecture ledger with epistemic statuses records.md state of the art (what would count as new; thresholds to beat) report_cycle{1..4}.md per-cycle syntheses ``` ## 🖥 Compute | Campaign | Range | Fleet | Wall time | |---|---|---|---| | Cycle 1 | 4×10⁹ | 1 laptop core | 15 s | | Cycle 2 | 4×10¹⁰ | laptop + M3 Ultra (1 core each) | ~2 min | | Cycle 3 | 10¹² | 12 Apple-silicon nodes, 162 CPU cores | ~5 min | | Cycle 4 | 10¹³ | 3 nodes: 58 CPU cores + **3 GPUs (custom Metal kernel)** | ~50 min | The distributed design uses *mergeable* streaming statistics (exact integers < 2⁵³ in float64 → **zero rounding**, bit-identical across machines) with checkpoint-aligned chunking and a junction buffer, so a 10 000-chunk merge equals the single-machine scan exactly. The Metal sieve kernel was accepted only after producing byte-identical prime sets to the CPU sieve; it processed 11.7 % of cycle 4. ## 📄 Paper Full write-up with all equations, proofs of the infrastructure lemmas, robustness checks and threats-to-validity: [`paper/main.tex`](paper/main.tex) → [`paper/main.pdf`](paper/main.pdf). ## 👤 Author **Simon-Pierre Boucher** · [contact@spboucher.ai](mailto:contact@spboucher.ai)