# Cycle 3 Report — C4′ tested at 10^12 (distributed, 12 nodes / 162 cores)

# Headline

C4′'s prediction was CONFIRMED. Cycle 2 predicted ρ(x)·ln x = −0.549 ± 0.005 at x = 10^12; the distributed scan measured −0.54756. The superseded cycle-1 window [−0.65, −0.55] was exited between 4×10^11 and 10^12, matching the predicted exit ≈ 5×10^11. This is the engine's first confirmed quantitative prediction.

# Results

Refined law. 13-point fit (10^8…10^12): ρ·ln x = c + d/ln x with c = −0.4845 ± 0.003 (LOO), d = −1.757, max residual 0.0025. New falsifiable predictions: ρ·ln x(10^13) = −0.5432 ± 0.004, ρ·ln x(10^14) = −0.5390 ± 0.004.
C6 firmed up. Lag-2: c₂ = −0.2806, d₂ = +0.782 (drift sign opposite to lag-1); prediction ρ₂·ln x(10^13) = −0.2545 ± 0.004.
Pipeline validation. The 1000-chunk distributed merge reproduces the cycle-2 single-machine checkpoints exactly (10^10, 2×10^10, 4×10^10) and all 48 maximal gaps below 10^12 equal the published table (tail spot-checked: 540 after 738832927927).
Free continuations. Champion = 6 up to 10^12 (C1). G6 grows to 480 (C3). The C2 twin-vs-quad race is still changing leader at 10^12: D = −238 out of ~1.5×10^10 gap events.
Cost. 17,719 core-seconds (≈ 4.9 core-hours), ≈ 5 minutes wall across the fleet.

# Instructive failures (logged for future cycles)

9/12 nodes crashed on launch: a Python-3.10-only annotation (np.ndarray | None) met Apple's Python 3.9. Fixed with from __future__ import annotations; rule: target the fleet's LOWEST interpreter and import-check per node class before launching.
5 nodes had no usable python3 at all (missing Xcode CLT), 1 was unreachable — the theoretical 228-core fleet was really 162 cores.
zsh word-splitting and modifier gotchas in orchestration (documented in journal).

# Conjecture table (cumulative)

#	Statement (short)	Status
C1	Jumping champion = 6	VERIFIED UP TO 10^12
C2	N(2,x) > N(4,x), x ≥ 10^6	REFUTED (cycle 1); race still swinging at 10^12 (D = −238)
C3	G6(x) ≥ 66	VERIFIED UP TO 10^12 (G6 = 480 and growing)
C4	ρ·ln x ∈ [−0.65, −0.55]	REFUTED-as-predicted (exited between 4×10^11 and 10^12)
C4′	ρ·ln x = c + d/ln x, c = −0.4845(30)	PREDICTION CONFIRMED at 10^12; next test 10^13
C5	Maximal-gap table reproduction	VERIFIED UP TO 10^12 (48/48 = published table)
C6	ρ₂·ln x → c₂ ≈ −0.28	CONJECTURED, firmed (13-point fit); prediction at 10^13 stated

# Files and re-verification

src/worker_c4.py, src/merge_c4.py — distributed chunk worker + merger. Re-verify the pipeline: run any subset of chunks locally and merge; the built-in cross-validation block compares against data/cycle2_c4_4e10_M3U96a.json.
data/cycle3_c4_1e12.json — merged result (checkpoints, maximal gaps, cross-val report).
Fits reproducible from the two checkpoint JSONs (script in journal / src/fit_c4.py pattern).

# Next precise action (one)

Cycle 4: test ρ·ln x(10^13) = −0.5432 ± 0.004 on the 3-node fleet only (M3U96a, M4M64a, M4M64b — user directive), saturated, with a Metal/GPU sieve path (benchmark MLX/Metal kernel vs numpy first; 10^13 is ~10× cycle 3's work, ~50 CPU core-hours, so GPU acceleration is the enabler).

Sources: OEIS A005250, OEIS A002386, t5k.org GapsTable