Cycle 1 Report — Prime Deserts (axis #10)
Results
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REFUTATION (main result). Conjecture C2 — "N(2,x) > N(4,x) for all x ≥ 10^6" — is false. Minimal counterexample: the count of gap-4 pairs ties the count of twin pairs at end-prime 80966861 and strictly overtakes it at 80966933. Beyond that, the race changes leader endlessly up to 4×10^9 (last lead-change at 3999999979, i.e. still swinging at the scan boundary; difference −2270 at 4×10^9). Epistemic status: REFUTED, re-verifiable in ~30 s with a stdlib-only script. Consistent with the random-walk heuristic (equal singular series) — the right follow-up question is the logarithmic density of each leader (Chebyshev-bias analogue).
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CERTIFIED CONSTRUCTION. A prime desert of length 260 at 25 digits: N = 1116336781708038449369693 and N+260 are consecutive primes (merit 4.6955). Method: hybrid covering system mod primes ≤ 59 (243 positions composite for every CRT shift — proven; 16 holes certified by explicit factors/MR witnesses; endpoints by deterministic Miller–Rabin, below the 3.317×10^24 validity bound). ×4.3 over the classic primorial baseline. Not a record (records.md) — the certified, independently re-verifiable pipeline is the point.
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INDEPENDENT VERIFICATIONS of known tables (validation of the whole engine): all 32 maximal gaps below 4×10^9, twin counts at 10^6/10^7/10^8, π(10^k) k ≤ 8, max CSG ratio 0.73947 below 4×10^9 — all equal to published values.
Conjecture status table
| # | Statement (short) | Status |
|---|---|---|
| C1 | Jumping champion = 6 | SURVIVOR — VERIFIED UP TO 4×10^9 (known, ORW) |
| C2 | N(2,x) > N(4,x) for x ≥ 10^6 | REFUTED — min. counterexample 80966861/80966933 |
| C3 | Multiples of 6 local maxima; G6(x) ≥ 66, → ∞ | SURVIVOR, strengthened (G6 = 216 at 10^9..4×10^9) |
| C4 | ρ(x) < 0, ρ·ln x ∈ [−0.65, −0.55] | SURVIVOR with revision (drift → −0.565; limit clause retracted) |
| C5 | max CSG below 4×10^9 = 0.7395 | VERIFIED (matches published table; not new) |
Instructive failures
- Checkpoint-only verification is a trap: C2 "held" at every decade checkpoint while failing ~137 million times in between. Full-resolution (every-event) adversary scans are now mandatory for race-type conjectures.
- A first analysis run had segment boundaries misaligned with checkpoints (caught because twin counts disagreed with literature values — the validate-against-known-values rule paid off).
- The naive HL gap model exp(−g/ln t) is inadequate beyond g ≈ 40 (structured residuals at g = 36, 72, 100, 108): do not conjecture on it before building the inclusion–exclusion model.
Files and re-verification
src/core.py+src/test_core.py— primitives; runpython3 src/test_core.py(37/37 PASS, 0.3 s).src/explore_gaps.py,src/analyze_gaps.py,src/adversary_race.py— deterministic scans; outputs in/data(adversary_4e9.json,gapstats_1e8.*,maximal_gaps_1e8.csv, …).certs/desert_certificate.json+certs/verify_desert.py— desert; runpython3 certs/verify_desert.py.certs/verify_c2_refutation.py— refutation; runpython3 certs/verify_c2_refutation.py.src/prover_desert.py(attempt 1),src/prover_desert2.py(attempt 2, current certificate).
Next directions (ranked)
- C4 at 10^10 (cluster job): does ρ·ln x stabilize? Best quantitative lead — structural (absent from Cramér model), cheap to test, precise falsifiable target c ∈ [−0.60, −0.50].
- C2 follow-up: logarithmic density of the gap-2 lead (Chebyshev-bias analogue) — needs the full crossing record, one scan with lead-time accounting.
- Proper inclusion–exclusion HL model for N(g,x), then revisit the g = 36/72/100/108 deficits.