spb/prime-mystery-engine

1<!--2report_cycle4.md — Prime Mystery Engine: Cycle 4 synthesis (Phase 7)3Author: Simon-Pierre Boucher — contact@spboucher.ai4Date: 2026-08-065-->67# Cycle 4 Report — C4′/C6 at 10^13, CPU + Metal GPU on 3 nodes89## Headline1011**Both out-of-sample predictions confirmed at 10^13.**12ρ·ln x(10^13) = **−0.54264** (predicted −0.5432 ± 0.004 — C4′'s second consecutive hit);13ρ₂·ln x(10^13) = **−0.25247** (predicted −0.2545 ± 0.004 — C6's first hit).14The 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).1617## The compute1819- Fleet (user directive): M3U96a + M4M64a + M4M64b only — 58 CPU cores saturated + **3 GPUs via a20  custom Metal kernel** (MLX `fast.metal_kernel`), which passed byte-identity gates against the CPU21  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 checkpoints25  bit-identical to cycles 2/3; all 53 maximal gaps below 10^13 equal the published table26  (new in range: 582, 588, 602, 652, 674 — web-checked against t5k/Nicely).2728## Conjecture table (cumulative)2930| # | 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 |3839## Predictions on record (falsifiable)4041- 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**.4344## Files and re-verification4546- `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).5051## Next precise action (one)5253Cycle 5: 10^14 (~10× the work; ≈ 8–9 h wall on the 3-node fleet — overnight run) to test54ρ·ln x(10^14) = −0.5387 ± 0.004. If compute is deferred: attack the main open problem instead —55the inclusion–exclusion correction to the HL triple model that should explain c ≈ −0.484.5657Sources: [t5k.org GapsTable](https://t5k.org/notes/GapsTable.html),58[Nicely maximal gaps](https://faculty.lynchburg.edu/~nicely/gaps/gaps.html)59