# Cycle 2 Report — C4 escalation to 4×10^10 (dual-machine)

# Results

C4 refined into C4′ (main result). The consecutive-gap anticorrelation obeys, empirically, ρ(x)·ln x = c + d/ln x with c = −0.486 ± 0.010, d = −1.72 (9 checkpoints 10^8…4×10^10, max residual 0.0025, leave-one-out spread on c: [−0.494, −0.482]). Falsifiable prediction for cycle 3: ρ·ln x = −0.549 ± 0.005 at x = 10^12; the cycle-1 window [−0.65, −0.55] should be exited near x ≈ 5×10^11. Epistemic status: CONJECTURED (consistent up to 4×10^10).
First-order HL model rejected (PROVER). The triple-correlation model P(g₁,g₂) ∝ S({0,g₁,g₁+g₂})·e^{−(g₁+g₂)/λ} reproduces the sign and the c + d/λ drift form but gives ρ·λ ≈ −0.154, a factor ~3.2 too small. The anticorrelation is therefore dominated by interior-compositeness (inclusion–exclusion) effects, not the bare singular series. Open problem.
Dual-machine reproducibility. Laptop (10^10) and M3U96a (4×10^10) produced bit-for-bit identical values at every shared checkpoint — deterministic pipeline validated on independent hardware.
New conjecture C6. Lag-2 gaps also anticorrelate: ρ₂·ln x → c₂ ≈ −0.29 ± 0.03, drifting in the opposite direction to lag-1. Low-confidence fit; to be firmed up.
Free continuations. Champion = 6 (C1) and G6 ≥ 66 (C3) hold to 4×10^10 (G6 is non-monotone: 312 → 282 → 276 — tail noise). C2 race keeps changing leader (D = −804 at 4×10^10). All 37 maximal gaps below 4×10^10 match OEIS A002386 (web-checked); max CSG = 0.79535 (456 after 25056082087, known).

# Instructive failure

A hand-written "known table" check briefly claimed our maximal gaps disagreed with the literature; fetching OEIS A002386 showed the scan was right and the from-memory list wrong. Rule reinforced: never transcribe "known" values from memory — always fetch the source.

# Conjecture table (cumulative)

#	Statement (short)	Status
C1	Jumping champion = 6	VERIFIED UP TO 4×10^10
C2	N(2,x) > N(4,x) for x ≥ 10^6	REFUTED (cycle 1); leader still swapping at 4×10^10
C3	G6(x) ≥ 66	VERIFIED UP TO 4×10^10
C4	ρ·ln x ∈ [−0.65, −0.55]	VERIFIED UP TO 4×10^10, predicted to FAIL ≈ 5×10^11 — superseded
C4′	ρ·ln x = c + d/ln x, c = −0.486(10), d ≈ −1.72	CONJECTURED; falsifiable at 10^12
C5	Maximal-gap table reproduction	VERIFIED UP TO 4×10^10 (37/37 = A002386)
C6	ρ₂·ln x → c₂ ≈ −0.29	CONJECTURED (new, low confidence)

# Files and re-verification

src/cycle2_c4.py — deterministic scan; sanity: python3 src/cycle2_c4.py 1e8 sanity must reproduce cycle-1 checkpoints exactly.
data/cycle2_c4_1e10_laptop.json, data/cycle2_c4_4e10_M3U96a.json — raw results (identical on shared checkpoints; diff them to re-verify cross-machine agreement).
src/fit_c4.py → data/fit_c4.json — the C4′ fit; src/model_c4.py → data/model_c4.json — the rejected first-order model.

# Next precise action (one)

Run the C4′ test at 10^12 distributed core-proportionally across the MacLustr cluster (mergeable streaming sums make the scan embarrassingly parallel; ~284 cores ⇒ minutes), to accept/refute ρ·ln x(10^12) = −0.549 ± 0.005.

Sources: OEIS A002386, t5k.org GapsTable