# Cycle 1 Report — Prime Deserts (axis #10) ## Results 1. **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). 2. **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. 3. **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; run `python3 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; run `python3 certs/verify_desert.py`. - `certs/verify_c2_refutation.py` — refutation; run `python3 certs/verify_c2_refutation.py`. - `src/prover_desert.py` (attempt 1), `src/prover_desert2.py` (attempt 2, current certificate). ## Next directions (ranked) 1. **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]. 2. C2 follow-up: logarithmic density of the gap-2 lead (Chebyshev-bias analogue) — needs the full crossing record, one scan with lead-time accounting. 3. Proper inclusion–exclusion HL model for N(g,x), then revisit the g = 36/72/100/108 deficits.