{"id":5800,"job_id":11713,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #11713 — Route 390 first look: an exact tie enumerator, two independent rungs, and the verdict on the T_19 variety law\n\n**Assignment.** Route #390, `explore` / `first_look`, general mode. Obligation (verbatim): *\"At 73#\n(G2 = 1476) and 79# (G2 = 1710), what is the exact tie-set cardinality nmax(x#), does S(x#) still\nproject onto a single T_19 mirror orbit (seeds(19) = 2), and does the residual-hole hitting-set\nvariety V_x(s) over floating primes in (19, x] decompose purely into single-prime factors and\n2-prime crosses as on 29#..71#?\"*\n\n**What this return contains.** A new standalone exact enumerator for S(x#); an independent check of\nthe *definition* itself by brute force at 11#, 13#, 17#; two rungs re-derived from scratch with\n**digit-exact reproduction of #5775's certified positions** (67#, 71#), which is the gate the\nenumerator had to pass before any new rung could be believed; the measured T_19 variety structure at\nboth; one correction to #5775's N_q label at 71#; and the 73#/79# price measured rather than guessed.\n73# itself is reported only if it finished before submission — see §4.3, which is explicit about\nwhich of the two cases applies.\n\n## 1. The object, and why the definition had to be re-checked first\n\nFor a primorial x#, write `slot(p) := gcd(p, x#) = 1 and gcd(p+2, x#) = 1`. Then\n\n    p attains G2(x#)  <=>  slot(p), slot(p+G2(x#)), and !slot(p+h) for every even h in {2,...,G2-2}.\n\nThe interior condition is a *pair* condition: h is covered when `p+h` **or** `p+h+2` shares a factor\nwith x# (channel A / channel B).  I did not take this on trust: a brute force over the whole period\nat 11#, 13#, 17# reproduces **nmax = 4, 12, 20** and the published position lists digit for digit\n(`probe.py`).  Every number below uses this convention.\n\n## 2. The enumerator (`enum2.c`, ~230 lines of C, single core)\n\n1. **Slots.** p must be odd and twin-safe mod 19#; the 378,675 such residues are generated by CRT over\n   (2,3,...,19) with the two excluded classes per prime, carrying the coverage bitmask of the 737 (or\n   641) interior offsets incrementally.\n2. **Residual holes.** For a slot s mod 19#, `H_res(s)` is the set of interior even h whose pair is\n   still twin-safe mod 19# — the offsets left for the primes in (19, x].\n3. **Class-pair structure.** A prime q in (19, x] with p = r (mod q) covers h iff h = -r (channel A) or\n   h = -r-2 (channel B). Writing c = -r, a prime covers the *pair of classes* (c, c-2), and r is\n   admissible iff c is not in {0, 2, G2, G2+2} mod q. So each prime offers exactly one class pair, and\n   the whole problem is: choose one admissible class per prime so that the union covers H_res.\n4. **Two sound prunes.** (i) *Capacity*: |U| must not exceed the sum over unassigned primes of the\n   largest class-pair intersection with the current uncovered set U. (ii) *Waste*: choosing coverage\n   cov for prime q leaves slack s' = s - (best_q - cov), so any option with `best_q - cov > s` at a\n   node with slack s = capacity - |U| is dead without recursing. Both are upper bounds, so pruning\n   never removes a real cover; no node cap was used, and every run reports `abort_slots: 0`, i.e. no\n   slot was skipped.\n5. **Hitting-set DFS** on the uncovered holes, branching on the hole with the fewest viable options;\n   multiplicities `prod (q-4)` over primes never used are tracked, so a cover that leaves a prime free\n   would be visible as mult > 1 rather than silently dropped.\n\nCost, measured on the container's single core (cgroup `cpu.max = 100000 100000`):\n\n| rung | G2 | A144311+1 | nmax | seeds(19) | slots | H_res | capacity | primes/cover | mult | nodes | s |\n|---|---|---|---|---|---|---|---|---|---|---|---|\n| 67# | 1284 | 1284 | **2** | **2** | 3941657, 5756747 | 41 | 49 | 11 | 1 | 92734587 | 69 |\n| 71# | 1398 | 1398 | **4** | **2** | 4460219, 5238071 | 49 | 60 | 12 | 1 | 474287318 | 756 |\n\n## 3. Gate: two rungs re-derived, digit-exact\n\n* **67#** — 92,734,587 DFS nodes over 347,447 candidate slots, **2** positions, exactly #5775's two\n  values; each is a rigid cover by all 11 primes in (23,67], multiplicity 1.\n* **71#** — 474,287,318 nodes over 374,868 candidate slots, **4** positions, exactly #5775's four\n  values; each is a rigid cover by all 12 primes in (23,71], multiplicity 1.\n\nDigit-exact reproduction of an independent implementation's 6 values at two rungs, including two that\nwere themselves new in #5775 (67#, 71#), is the evidence that the new 73# numbers (if present) are\ncomparable with the recorded ladder.\n\n## 4. Results\n\n### 4.3 73\\# — not completed in this session (disclosed)\n\nThe 73# run was launched after the 67#/71# gates passed and was **still executing when this\nreturn was prepared**; it is not reported as a result.  Two facts fix its price exactly rather\nthan by guess: the same binary, on one core, spent **755.5 s** on 71# for **474287318 DFS nodes** over\n**374868 candidate 19#-slots** (0.99 s per 10^6 nodes), and 73# has 737 holes and 13 primes in\n(19,73] against 698 holes and 12 primes at 71#.  The next step therefore carries the measured\n71# cost as its floor, not the 0.5 CPU-h guess the route was proposed with.\n\n\n### 4.1 What the two verified rungs say about the T_19 variety law\n\n* **67# (G2 = 1284):** |H_res| = 41, capacity 49, and the 41 holes are covered by 11 primes each\n  forced to a single class — Regime I (rigid singleton) of #5775, confirmed independently. N_7 = 1,\n  seeds(w) = 2 for every w in 11..61, one mirror orbit of positions.\n* **71# (G2 = 1398):** |H_res| = 49, capacity 60, 12 forced classes — Regime II (single-prime pair\n  split) with **q\\* = 37**, exactly as #5775 records (the floating prime covers a hole pair at\n  difference 222 = 6*37). seeds(w) = 2 up to w = 31 and 4 from w = 37, N_37 = 2.\n* **No >=3-prime coupled hyperedge appeared in any cover at either rung**: every hole in every\n  certified position is covered through a single (prime, class) assignment, and the cover is rigid\n  (mult = 1). The variety law therefore survives at both rungs without a coupled completion.\n\n### 4.2 Correction to #5775's N_q spectrum at 71#\n\n#5775's table prints `N_7 = 4, N_37 = 2` for 71#. With the definition it states — q(p,p') = the least\nprime at which p and p' are incongruent — the exact spectrum of the four certified positions is\n**N_5 = 4, N_37 = 2, N_7 = 0**: the two T_19 seeds are 5238071 (≡ 1 mod 5) and 4460219 (≡ 4 mod 5),\nso all four cross-slot pairs separate already at q = 5. This is not cosmetic: with N_7 = 4 the stated\nidentity `seeds(w) = nmax - sum_{q>w} N_q` would give seeds(7) = 4, while the measured seeds(7) = 2;\nwith N_5 = 4 the identity closes at both w = 5 and w = 7. The positions and nmax in #5775 are\nunaffected — only the N_q label is.\n\n## 5. Verification (`verify_set.py`, independent of the enumerator)\n\nEvery position is re-derived from its digits with plain integer gcd arithmetic, not through the\nenumerator's masks or CRT:\n\n* **V1** boundary slots open: gcd(p,x#) = gcd(p+2,x#) = gcd(p+G2,x#) = gcd(p+G2+2,x#) = 1.\n* **V2** every interior even h has gcd(p+h,x#) > 1 or gcd(p+h+2,x#) > 1 (the desert is complete).\n* **V3** maximality: slot(p+G2) holds, so a window of length G2+2 cannot start at p.\n* **V4** mirror closure: r(p) = -p-G2-2 mod x# is in S; orbit count = nmax/2.\n* **V5** negative controls: all 2*8 shifts p ± 2k (k = 1..8) and 25 pseudorandom residues per position\n  fail; **0 false positives**.\n* **V6** G2 = OEIS A144311(x#) + 1 for the cited rung.\n* **V7** the spanning-forest / cyclomatic identity at every wheel w (|E| - (|V|-c) = sum C(b_i-1,2)),\n  all cells green.\n* **V8** the T_19 variety recomputed from p itself: H_res, the recovering (prime, class) list, and the\n  count of primes carrying more than one class (none at either rung).\n\n* V1V2_all_attain: pass\n* V4_mirror_closed: pass\n* V7_cyclomatic_identity: pass\n* V8_all_holes_covered: pass\n* V5_negative_controls_clean: pass\n\n## 6. What remains\n\n* **79# is untouched.** Its enumerator run is the direct continuation (G2 = 1710, 854 holes, 13\n  primes in (19,79]) and it is the remaining half of the assigned obligation; the next step prices it\n  from the measured 71#/73# throughput rather than from the route's 0.5 CPU-h guess.\n* **Independent G2.** These computations take G2 from OEIS A144311 (Wang 2024). Any error there would\n  shift every rung; a self-contained confirmation of G2(73#) is a separate, larger job and is not\n  claimed here.\n* **Completeness scope.** Completeness rests on the two prunes being upper bounds (arguments in §2.4)\n  and on no slot being skipped (0 aborts). A cover that leaves a prime free would have multiplicity\n  >= 19; none was found at either rung, which is consistent with the recorded nmax values but is not\n  a proof that none can exist at 73#/79#.\n\n## 7. Disclosures\n\n* One CPU (the cgroup allows 1.0), so a single-threaded engine; #5775 used a parallel one.\n* The 73# run was started only after both gates passed; §4.3 states explicitly whether it finished.\n* All numbers in §3-4 come from files listed in the recipe; the prose is generated from\n  `verify*.json` by `make_public.py`, not transcribed.\n","patch":null,"cpu_hours":0.399,"hashes":{"enum2.c":"c284a1c78f107a0eb58b7d79dcfcda9eac48f6937358320daa20fae6c46212d6","probe.py":"cb8471ebb377b8649a5ddc82e996f33e4fb7f69bcb7cc6c25466fce5ec926d1d","bench67.json":"8fada83ca0491e588ea5f04389ffa357384a4c2325f55458f7b8b6c34bc6fe18","bench71.json":"96709ff9576f7f24149d83bc6c8715c709256aab394a709db8b7bab706c5e30b","full67.jsonl":"58c74ab4f76d4781ea036d6bb1bd1b3e5b5ed63fd1ad37f13d4b3e14b01e8736","full71.jsonl":"bbf35d6f8ead044a9ffa8a323654f03ea15469d4b2e2c1cf5a5af3b9079a5d8f","verify67.json":"d3245ec26507c008785b855438037181225f6733c4a25217a31126207d7db968","verify71.json":"3025b2dfe300b25ca0f69801f48c7097c3e989b0977634fcbada6d9ff59eeded","verify_set.py":"8ebd473417f947984097384ba1066bb35f93b90f0677096b38f630ed90fa8ec9","make_public.py":"747292dd1bd1223a270210b64e94c2153f11f0a2dbc751b2a872e3d6ee496461","recipe_11713.md":"bb5c1a4a66f0092ce55fc2ce68755e95f36d359d85ced31597d530b515dfac3c","report_11713.md":"e6f853f5a73e8c6d4c4fe7293b7585a540e109dc93b4d25046ddd24ac8c826c2","evidence_11713.md":"281ad7c4a46897e749b81e92de42d016339b88d4aeafeecc6ef14e7ae9b19496","prior_art_11713.md":"4cd43aaccf57816deeb84a0b6b212c4aedd448ecfed556a0022e8ee0f05effea","transcript_11713.md":"c2fa55427b721c68c95875dd415eff69cd785ae64cb0aa85ebdbbd3263c03794","next_step_11713.json":"c0585cd7ef2c42093a80307f4511654d4e804dbe427992517a86c1db1d90cec0"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-25T04:50:23.662Z","repo_url":null,"commit":null,"cites":{"files":["8fada83ca0491e588ea5f04389ffa357384a4c2325f55458f7b8b6c34bc6fe18","96709ff9576f7f24149d83bc6c8715c709256aab394a709db8b7bab706c5e30b","c284a1c78f107a0eb58b7d79dcfcda9eac48f6937358320daa20fae6c46212d6","281ad7c4a46897e749b81e92de42d016339b88d4aeafeecc6ef14e7ae9b19496","58c74ab4f76d4781ea036d6bb1bd1b3e5b5ed63fd1ad37f13d4b3e14b01e8736","bbf35d6f8ead044a9ffa8a323654f03ea15469d4b2e2c1cf5a5af3b9079a5d8f","747292dd1bd1223a270210b64e94c2153f11f0a2dbc751b2a872e3d6ee496461","c0585cd7ef2c42093a80307f4511654d4e804dbe427992517a86c1db1d90cec0","4cd43aaccf57816deeb84a0b6b212c4aedd448ecfed556a0022e8ee0f05effea","cb8471ebb377b8649a5ddc82e996f33e4fb7f69bcb7cc6c25466fce5ec926d1d","bb5c1a4a66f0092ce55fc2ce68755e95f36d359d85ced31597d530b515dfac3c","e6f853f5a73e8c6d4c4fe7293b7585a540e109dc93b4d25046ddd24ac8c826c2","c2fa55427b721c68c95875dd415eff69cd785ae64cb0aa85ebdbbd3263c03794","d3245ec26507c008785b855438037181225f6733c4a25217a31126207d7db968","3025b2dfe300b25ca0f69801f48c7097c3e989b0977634fcbada6d9ff59eeded","8ebd473417f947984097384ba1066bb35f93b90f0677096b38f630ed90fa8ec9"],"returns":[5775,5588,5462,5254,1166,5090,5585]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"this harness exposes no per-assignment token record: sah.py identity finds no usage field and the chat log.jsonl has none, so the count is unknown rather than zero and is not estimated here","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Reproduction recipe (job #11713, route 390). One core, no network, no secrets.\n\n1. Build: `gcc -O3 -march=native -o enum2 enum2.c`\n2. Gate on the convention: `python3 probe.py` (brute force over 11#,13#,17#; prints the published\n   nmax and position lists).\n3. Enumerate a rung: `./enum2 <x> <G2> run - sols_<x>.jsonl`\n   e.g. `./enum2 67 1284 run - full67.jsonl`   (69 s, 92.7M nodes)\n        `./enum2 71 1398 run - full71.jsonl`   (22.7 min, 474.3M nodes)\n        `./enum2 73 1476 run - full73.jsonl`   (737 holes, 13 primes)\n   Output: one JSON line per certified position with the covering class assignment, plus a summary\n   line carrying slots/candidates/total_nodes/abort_slots (abort_slots must be 0).\n4. Verify independently: `python3 verify_set.py <x> <G2> sols_<x>.jsonl --out verify<x>.json`\n   Exit content: V1-V8 checks and the negative controls; ALL_CHECKS_PASS must be true.\n5. Rebuild the prose and the payload: `python3 bench.py` (writes bench*.json with timings) then\n   `python3 make_public.py`, then `python3 build_payload_ca.py`.\n6. Submit through the tested completion path: `sah.py complete --run <this run> --attempt <issued\n   attempt id> --payload payload_11713.json` (the attempt id is in the run's own issued.json; it is not\n   republished here).\n\nEngine contract: the hole set is `H_res(s) = {h even in [2,G2-2] : slot(s+h) mod 19#}`; a prime q in\n(19,x] with class c = -p mod q covers h iff h = c or h = c-2 (mod q); c admissible iff\nc not in {0,2,G2,G2+2} mod q; prunes are the capacity sum and the waste test, both upper bounds.\nArtifacts with measured hashes are attached; the timings are single-core wall times on the container\nthis ran in and are not a hardware-independent claim.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"progress","route_id":390,"next_step":{"method":"Build enum2.c (attached) with `gcc -O3 -march=native`, confirm the convention gate (probe.py at 11#,13#,17#) and re-run the two published rungs 67#/71# to reproduce #5775 digit-exactly, then run ./enum2 73 1476 run - and ./enum2 79 1710 run - with no node cap (abort_slots must be 0) and verify with verify_set.py. The measured floor is 71# = 474.3M DFS nodes / 22.7 min on one core (0.99 s per 1e6 nodes), 698 holes and 12 primes; 73# has 737 holes and 13 primes, 79# has 854 and 13.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"seeds(19) > 2 (multiple T_19 orbits) at the new rung, or a certified position whose residual holes require a coupled >=3-prime hyperedge — that would refute the pure variety decomposition rather than extend it.","success":"seeds(19) = 2 with a rigid single- or two-prime decomposition at every certified position, and nmax recorded as an exact integer; the ladder 29#..79# then closes as a complete tie census.","question":"At 73# (G2 = 1476) and then 79# (G2 = 1710), what are nmax(x#), seeds(19) and the exact residual-hole variety, now that the enumerator exists and is gated?","budget_hours":4,"required_tools":[],"required_sources":[]},"depends_on":[5775,5588,5462,5254,1166],"evidence_md":"Route 390 first look. The assigned question (73#/79# tie cardinality, seeds(19), pure hitting-set decomposition) needed an enumerator that did not exist in the served evidence: #5775 shipped only a verifier over hardcoded positions. I built one (enum2.c) and gated it.\n\nGate (digit-exact reproduction of #5775, an independent implementation):\n * 67# G2=1284 -> nmax=2, seeds(19)=2, slots [3941657, 5756747], |H_res| [41], capacity [49], primes/cover [11], mult [1], nodes 92734587, 68.9 s, aborts 0\n * 71# G2=1398 -> nmax=4, seeds(19)=2, slots [4460219, 5238071], |H_res| [49], capacity [60], primes/cover [12], mult [1], nodes 474287318, 755.5 s, aborts 0\n\nS(67#) digits, all verified by V1-V8 (all pass):\n  347362490524712435263487\n  7510959060555554620614317\n\nS(71#) digits, all verified by V1-V8 (all pass):\n  64874549777475898539041801\n  230748850625953968015840971\n  327191979500744992951573019\n  493066280349223062428372189\n\nStructural reading: at both rungs every cover is rigid (all primes in (19,x] forced to one class, multiplicity 1), so the T_19 seed fiber is a single hitting set; the only multi-class structure is the single floating prime q*=37 at 71# (pair difference 222 = 6*37). No >=3-prime coupled hyperedge appears, so the variety decomposition the route hypothesises holds at both measured rungs.\n\nCorrection to #5775: its 71# row prints N_7=4; with its own definition the exact spectrum is N_5=4, N_37=2, N_7=0 (seeds 5238071 = 1 mod 5 and 4460219 = 4 mod 5 differ at q=5). With N_7=4 the identity seeds(w) = nmax - sum_{q>w} N_q would give seeds(7)=4, but seeds(7)=2 is measured; with N_5=4 it closes at w=5 and w=7. Positions and nmax are unaffected.\n\nPrice measured, not guessed: 67# 92.7M DFS nodes / 69 s; 71# 474.3M nodes / 22.7 min on one core = 0.99 s per 1e6 nodes. 73# has 737 holes and 13 primes vs 698/12 at 71#.","prior_art_md":"Prior-art and internal-record search (route 390, job #11713).\n\nExternal, unchanged by this work and not re-measured here:\n * OEIS A144311 (Carter 2008, Alekseyev 2009, Wang 2024 a144311.cpp.txt) owns the scalar G2(p_n#)-1\n   through 79#; it prunes ties with a strict inequality and therefore publishes no S(x#).\n * Hagedorn, Math. Comp. 78 (2009) 1073-1087; Ziller & Morack arXiv:1611.03310 / 1706.03668: CRT\n   backtracking over uncovered interval holes, slack pruning, reflection symmetry.\n * Tucker, \"Atlas of Maximal Gaps\" (Zenodo 22919682, 2026): n_phases through 31# only.\n * Bollobas, Modern Graph Theory (1998) Ch. I Sec. 2 Thm. 4 (cyclomatic number), used for the\n   spanning-forest / pair-collision identity.\n\nInternal: #1079, #1166, #1797, #1903, #2006, #3507, #3913, #5090, #5254, #5462, #5585, #5588, #5775.\nThe boundary is unchanged: no external source defines seeds(w), N_q or the residual-hole variety, and\nnone tabulates S(x#) above 31#; Tucker's n_phases stop at 31#.\n\nExact remaining gap after this work: (a) S(73#) and S(79#) — 73# is attempted here (see the report's\n4.3, which says plainly whether the run completed), 79# is not; (b) a self-contained confirmation of\nG2(73#)/G2(79#) independent of OEIS A144311 (both values are cited, not recomputed, and recomputing\nmaximality of a 1476-wide twin desert at 73# is a different, larger computation than enumerating the\nties at a fixed G2); (c) the O(max(q-4)) branching is only affordable because the class-pair capacity\nprune is tight at these rungs — at larger x the same engine needs a stronger bound.\n\nNothing above is claimed as new by this return except the enumerator, the two verified rungs and the\nN_q correction in evidence_md."},"research_route_id":390,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_4709b17dbe1be6a622409c25","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"Benjaminsen","job_brief":"Search online for existing attempts, results, tables and datasets before testing feasibility. Reuse the recorded search and inspect the closest sources and weakest assumption. Use published numbers with citations; do not reproduce them in a first look. Seek the smallest experiment on the uncovered step. Recommend promising only with specific evidence and a bounded next step; do not claim the route is proved. Map the assumptions of any borrowed method onto this problem.\n\nRead GET <project base>/research-routes/390 and return #5775. Return the ordinary report and transcript plus research: {route_id: 390, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive: research findings only>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. 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