Conjecture Ledger
Status vocabulary: PROVEN / VERIFIED UP TO X / CONJECTURED / REFUTED (minimal counterexample) / SPECULATIVE. Notation: N(g,x) = #{consecutive prime pairs (p, p+g), p+g < x}. ρ(x) = Pearson correlation of (g_n, g_{n+1}) over consecutive gaps below x. CSG(p) = g(p)/ln²p. G6(x) = largest G such that every multiple of 6 ≤ G satisfies N(g,x) > N(g−2,x) and N(g,x) > N(g+2,x).
Cycle 1 (2026-08-06) — discovery range 10^8, deterministic scans (no seed)
C1 — Jumping champion
Statement. For all x ∈ [10^5, 4×10^9], argmax_g N(g,x) = 6. Classification. (a) probably known — Odlyzko–Rubinstein–Wolf conjecture territory (champion 6 from ≈947 to ≈1.7×10^35). Modular admissibility. Consistent (6 = 2·3 maximizes the singular series among small gaps). Status at discovery. VERIFIED at checkpoints 10^6, 10^7, 10^8. FINAL STATUS (adversary, 4×10^9). SURVIVOR — VERIFIED UP TO 4×10^9 (champion = 6 at every checkpoint). Cramér/Maier note: NOT merely statistical — a Cramér random model has no mod-6 structure at all; this is structural (singular series). Known conjecture (ORW); we add nothing but an independent check.
C2 — The 2-vs-4 race never crosses again
Statement. For all x ≥ 10^6, N(2,x) > N(4,x).
Classification. (c) potentially interesting — asymptotically N(2,x) ~ N(4,x) under Hardy–Littlewood
(identical singular series), so the sign of the difference is a second-order effect analogous to
Chebyshev bias; margins observed are tiny (+26 at 10^6, +359 at 10^7, +55 at 10^8, i.e. ~10^-4 relative)
and non-monotone, so a crossing is plausible. Possibly known in race literature.
Modular admissibility. No obstruction (both gaps admissible).
Status at discovery. VERIFIED at checkpoints 10^6, 10^7, 10^8. ⚠ Checkpoint-only: interior crossings not yet excluded.
FINAL STATUS (adversary, full resolution to 4×10^9). REFUTED.
Minimal counterexample above 10^6: first tie D = N2−N4 = 0 at end-prime 80966861; first strict
overtake (N4 > N2) at end-prime 80966933 (D = −1). The checkpoint verification was misleading:
the race in fact changes leader forever in the tested range — D(10^9) = −173, D(2×10^9) = +1074,
D(4×10^9) = −2270, with 137,574,763 gap-events where D ≤ 0 and the last one at 3999999979 (= the
last prime scanned). Cramér/Maier note: consistent with a random-walk model of the difference
(N(2) and N(4) have identical singular series) — the refutation is exactly what the random model
predicts; the interesting open follow-up is whether the logarithmic density of the lead is biased
(Chebyshev-style). Re-verify: python3 certs/verify_c2_refutation.py (stdlib-only, ~30 s).
C3 — Local dominance of multiples of 6
Statement. G6(x) ≥ 66 for all x ≥ 10^6, and G6(x) → ∞. Classification. (b) easy consequence of Hardy–Littlewood heuristics (singular series of 6k beats neighbors); the finite claim is the testable part — the threshold 66 at 10^8 is limited by sample noise in the histogram tail, so G6 should grow with x. Status at discovery. G6 = 66 exactly at 10^6, 10^7 and 10^8 (fails at 72 each time — noise or structure? adversary must decide). FINAL STATUS (adversary, 4×10^9). SURVIVOR, strengthened — G6(10^9) = G6(4×10^9) = 216 ≥ 66. The failure at 72 below 10^8 was histogram-tail noise, not structure: G6 grows with x as predicted. VERIFIED UP TO 4×10^9; the G6(x) → ∞ part remains CONJECTURED (would follow from Hardy–Littlewood).
C4 — Scaled anticorrelation of consecutive gaps
Statement. ρ(x) < 0 for all x ≥ 10^6, and ρ(x)·ln x → c with c ≈ −0.6; sharply: ρ(x)·ln x ∈ [−0.65, −0.55] for all x ∈ [10^6, 4×10^9]. Data. ρ·ln x = −0.596 (10^6), −0.614 (10^7), −0.582 (10^8). Classification. (a/c) anticorrelation itself is known empirically; the precise scaling constant ≈ −0.6 is a sharper claim, possibly known (gap correlation literature), possibly a clean new datum. Status at discovery. VERIFIED at 3 checkpoints. FINAL STATUS (adversary, 4×10^9). SURVIVOR with revision. ρ(x) < 0 and ρ·ln x ∈ [−0.65, −0.55] both VERIFIED UP TO 4×10^9 — but the sequence ρ·ln x = −0.596, −0.614, −0.582, −0.571, −0.567, −0.565 (x = 10^6 … 4×10^9) drifts monotonically upward after 10^7, so the "limit ≈ −0.6" clause is retracted; revised claim: ρ(x)·ln x converges to some c ∈ [−0.60, −0.50] — CONJECTURED, next test at 10^10+. Cramér/Maier note: structural, NOT statistical — a Cramér model with independent gaps gives ρ = 0; the anticorrelation is sieve-induced. This is the most promising quantitative lead of cycle 1. CYCLE 2 STATUS (laptop to 10^10 + M3U96a to 4×10^10, bit-for-bit identical on overlap). Window [−0.65, −0.55]: VERIFIED UP TO 4×10^10 (ρ·ln x = −0.55569 there) — but now expected to fail: the 9-checkpoint fit (see C4′) predicts exit of the window near x ≈ 5×10^11. Superseded by C4′.
C4′ — refined scaling law for the gap anticorrelation (NEW, cycle 2)
Statement. ρ(x)·ln x = c + d/ln x + o(1/ln x) with c = −0.486 ± 0.010 and d ≈ −1.72.
Falsifiable near-term prediction: ρ(x)·ln x = −0.549 ± 0.005 at x = 10^12.
Evidence. Least-squares on 9 checkpoints 10^8…4×10^10: c = −0.48618, d = −1.7226,
max |residual| = 0.0025, leave-one-out c ∈ [−0.494, −0.482]. Data: data/fit_c4.json.
Model support (PROVER, cycle 2). The first-order HL triple-correlation model
(src/model_c4.py: P(g1,g2) ∝ S({0,g1,g1+g2})·exp(−(g1+g2)/λ)) predicts the correct SIGN and the
same c + d/λ drift FORM, but magnitude ≈ −0.154 vs observed ≈ −0.49 (factor ~3.2 too small):
first-order model quantitatively REJECTED — interior-compositeness (inclusion–exclusion) terms must
carry most of the effect. Open problem for a future cycle.
Classification. (c) potentially new as a precise constant; the phenomenon is known qualitatively.
Status. CONJECTURED; VERIFIED-consistent up to 4×10^10.
CYCLE 3 STATUS (distributed scan to 10^12, 12 nodes / 1000 chunks / 162 cores).
PREDICTION CONFIRMED: observed ρ·ln x(10^12) = −0.54756, inside the predicted band −0.549 ± 0.005.
The cycle-1 window [−0.65, −0.55] was exited between 4×10^11 (−0.55027) and 10^12 (−0.54756), matching
the predicted exit ≈ 5×10^11 — old C4 now REFUTED-as-predicted. Combined 13-point fit (10^8…10^12):
c = −0.4845 ± 0.003 (LOO), d = −1.757, max residual 0.0025. New falsifiable predictions:
ρ·ln x = −0.5432 ± 0.004 at 10^13; −0.5390 ± 0.004 at 10^14.
Cross-validation: distributed merge reproduces cycle-2 single-machine checkpoints exactly at
10^10/2×10^10/4×10^10. Status: CONJECTURED and now twice-tested; VERIFIED-consistent up to 10^12.
CYCLE 4 STATUS (3 nodes, 58 CPU cores + 3 Metal GPUs, 10,000 chunks to 10^13).
SECOND PREDICTION CONFIRMED: observed ρ·ln x(10^13) = −0.54264, predicted −0.5432 ± 0.004.
All 10 overlapping checkpoints identical to cycles 2/3 (exact). 16-point refit (10^8…10^13):
c = −0.48354 (LOO [−0.48570, −0.48165]), d = −1.7772, max residual 0.00255.
Next predictions: ρ·ln x(10^14) = −0.5387 ± 0.004; ρ·ln x(10^15) = −0.5350 ± 0.004.
Status: CONJECTURED, twice-confirmed out of sample; VERIFIED-consistent up to 10^13.
C6 — lag-2 anticorrelation (NEW, cycle 2)
Statement. ρ₂(x) < 0 (correlation of (g_n, g_{n+2})) with ρ₂(x)·ln x → c₂, c₂ ≈ −0.29 ± 0.03 (2-point fit of the c + d/λ form: c₂ = −0.292, d₂ ≈ +1.05 — note the drift has OPPOSITE sign to C4′). Data. ρ₂·ln x: −0.235 (10^8) → −0.248 (10^10) → −0.249 (4×10^10), monotone decreasing. Classification. (c) same family as C4′; less tested (fit uses only endpoints — firm up in cycle 3). Status. CONJECTURED; ρ₂ < 0 VERIFIED UP TO 4×10^10. CYCLE 3 STATUS. ρ₂·ln x continues its monotone drift down: −0.25141 at 10^12. Proper 13-point fit: c₂ = −0.2806, d₂ = +0.782 (opposite-sign drift vs lag-1 confirmed). ρ₂ < 0 VERIFIED UP TO 10^12; c₂ CONJECTURED ≈ −0.28 ± 0.02. Prediction: ρ₂·ln x(10^13) = −0.2545 ± 0.004. CYCLE 4 STATUS. PREDICTION CONFIRMED: observed ρ₂·ln x(10^13) = −0.25247 ∈ −0.2545 ± 0.004. 16-point refit: c₂ = −0.27762, d₂ = +0.7187. Next prediction: ρ₂·ln x(10^14) = −0.2553 ± 0.004. ρ₂ < 0 VERIFIED UP TO 10^13.
C5 — Cramér–Shanks–Granville ratio below 4×10^9
Statement. max_{p ≤ 4×10^9} CSG(p) = 210/ln²(20831323) ≈ 0.7395, attained at p = 20831323 (gap 210). Classification. (a) known — the maximal-gap table is exhaustively verified far beyond 4×10^9; this is an independent re-verification, not a discovery. Status at discovery. VERIFIED to 10^8 (max CSG 0.7395); the claim to 4×10^9 requires that no later maximal gap below 4×10^9 exceeds it (adversary run). FINAL STATUS (adversary, 4×10^9). VERIFIED UP TO 4×10^9 — max CSG = 0.73947 at p = 20831323; all 32 maximal gaps below 4×10^9 reproduced and equal to the published table (A005250/A002386), ending with gap 336 after 3842610773. Independent re-verification of known results, as intended.
Discarded at birth (Phase 4 filter)
- "Odd gaps never occur beyond (2,3)" — trivial (parity), not logged as a conjecture.
- Any conjecture about gaps ≡ 3 mod 6 etc. — trivial modular obstruction (all gaps beyond (2,3) are even).