# Cycle 4 Report — C4′/C6 at 10^13, CPU + Metal GPU on 3 nodes

# Headline

Both out-of-sample predictions confirmed at 10^13. ρ·ln x(10^13) = −0.54264 (predicted −0.5432 ± 0.004 — C4′'s second consecutive hit); ρ₂·ln x(10^13) = −0.25247 (predicted −0.2545 ± 0.004 — C6's first hit). The scaling law ρ(x)·ln x = c + d/ln x now stands twice-tested; 16-point refit: c = −0.48354 (LOO ± 0.002), d = −1.7772 (max residual 0.0026).

# The compute

Fleet (user directive): M3U96a + M4M64a + M4M64b only — 58 CPU cores saturated + 3 GPUs via a custom Metal kernel (MLX fast.metal_kernel), which passed byte-identity gates against the CPU sieve before use and processed 1170 of 10,000 chunks (11.7%).
10,000 chunks of 10^9, measured-rate quotas, ~50 min wall, 147,738 core-seconds (~41 core-hours, incl. ~4% duplicated tail work from a late rebalance; duplicates byte-identical, tiling asserted).
Validation: 10,000 unique chunks tile [2, 10^13) exactly; all 10 overlapping checkpoints bit-identical to cycles 2/3; all 53 maximal gaps below 10^13 equal the published table (new in range: 582, 588, 602, 652, 674 — web-checked against t5k/Nicely).

# Conjecture table (cumulative)

#	Statement	Status
C1	Champion = 6	VERIFIED UP TO 10^13
C2	N(2,x) > N(4,x)	REFUTED (cycle 1); race still swinging at 10^13 (D = +8870)
C3	G6 ≥ 66	VERIFIED UP TO 10^13 (G6 = 576)
C4′	ρ·ln x = c + d/ln x	CONFIRMED at 10^12 AND 10^13; c = −0.48354 ± 0.002
C5	Maximal-gap table	VERIFIED UP TO 10^13 (53/53); max CSG 0.79754 (652 after 2614941710599)
C6	ρ₂·ln x → c₂	CONFIRMED at 10^13; c₂ = −0.27762, d₂ = +0.719

# Predictions on record (falsifiable)

ρ·ln x(10^14) = −0.5387 ± 0.004; ρ·ln x(10^15) = −0.5350 ± 0.004.
ρ₂·ln x(10^14) = −0.2553 ± 0.004.

# Files and re-verification

src/gpu_sieve.py — Metal kernel + its correctness/benchmark gate (python3 src/gpu_sieve.py).
src/gpu_runner.py, refactored src/worker_c4.py (CPU/GPU modes, identical outputs).
data/cycle4_c4_1e13.json — merged result with built-in cross-validation records.
Paper updated: paper/main.tex (12 pp., compiles clean).

# Next precise action (one)

Cycle 5: 10^14 (~10× the work; ≈ 8–9 h wall on the 3-node fleet — overnight run) to test ρ·ln x(10^14) = −0.5387 ± 0.004. If compute is deferred: attack the main open problem instead — the inclusion–exclusion correction to the HL triple model that should explain c ≈ −0.484.

Sources: t5k.org GapsTable, Nicely maximal gaps