{"id":433,"job_id":1035,"problem_id":1,"lane_id":4,"type":"explore","user_id":34,"model":"deepseek-v4.1-flash","provider":"deepseek","report_md":"# #1035 — the matched permutation null of the interior share: the route's failure branch, and the structural reason for it\n\nJob #1035 (explore / pursue, lane measure; session `3fbd7da6c5693188281ca3ee`, model\n`deepseek-v4.1-flash`). Cost: one 19# enumeration (9,699,690 integers, 3.9 s) plus 20 and 1 span\ndecompositions by trial division; 6.1 s wall, one thread, stdlib only. The null is **exact** — the\nsupport has at most 6 arrangements, so there is no sampling.\n\n## The assigned question and its answer\n\n> Is the realized interior share of a certified record span typical or extreme under the fixed-multiset\n> permutation null, at the two levels whose realized share range is positive?\n\n**It is neither extreme nor informative, and the route's pre-registered failure branch is reached at\nboth levels.** A certified span at level `x` is the maximal T_x gap `[p, p+G]`; its gaps are the\nT_{x-1} gaps inside it, so a span is a k-gap window trajectory with multiset `M` (sum `G`), read as\n`(end, ..., end)`. The matched null is the uniform distribution over the `L!` arrangements of `M`, and\nthe statistic is the interior share `(G − m_end1 − m_end2)/G`.\n\n| level | spans | L = 2 | L = 3 | multiset | free null support | observed share | p(share ≥ obs) | class-consistent arrangements | class-constrained support |\n|---|---|---|---|---|---|---|---|---|---|\n| 19# | 20 | 8 | 12 | {42, 108} | {0} | 0 | 1 | 2 of 2 | {0} |\n| 19# | | | | {30, 42, 78} | {0.20, 0.28, **0.52**} | **13/25 = 0.52** | **1/3** | 2 of 6 | {0.52} |\n| 43# | 8 | 4 | 4 | {84, 156, 378} | {**0.1359**, 0.2524, 0.6117} | **14/103 = 0.1359** | **1.0** | 2 of 6 | {0.1359} |\n\nSo the two observed extremes sit at *the edge of a three-point support, not in a tail*: at 19# the\nrealized 13/25 is the **maximum** of the support, reached by 2 of the 6 arrangements, i.e. p = 1/3 —\nnot significant at any conventional level; at 43# the realized 14/103 is the **minimum** of the support,\np = 1.0. The 0's are the L = 2 spans, where there is no interior at all: degenerate by definition,\nwhich is exactly why the route excludes the five zero-spread levels by construction. With one-sided\np-values of 1/3 and 1, \"the arrangement channel has measurable content at this scale\" is false.\nRung: **verified** (finite, exact, and gated — see below).\n\n## Why: the class law has already consumed the arrangement freedom\n\nThis is the part worth keeping, because it is structural rather than a small-sample accident. Under\nreturn #398's identity the interior gaps of a certified span lie in the class `{0, ±c(x)} mod 6x`,\n`c(x) = 6·(2·6⁻¹ mod x) mod 6x`. At the two levels:\n\n* 19#: `c = 78`, `6x = 114`, class `{0, 36, 78}`; the span gaps are `30, 42, 78` — **only 78 is in the\n  class** (30 and 42 are not), so the interior is forced, and the multiset then fixes its value.\n* 43#: `c = 174`, `6x = 258`, class `{0, 84, 174}`; the span gaps are `84, 156, 378` (378 ≡ 120) — again\n  **only 84 is in the class**.\n\nIn both cases exactly 2 of the `L! = 6` arrangements are class-consistent and they differ only in the\norder of the two ends, so they give the *same* share. The class-constrained null therefore has a\n**one-point support**: the interior share of a certified span is a function of the multiset alone at\nthese levels, and the matched permutation test cannot have power. The route's separation must be sought\nin a statistic the class law does not pin — i.e. at the **tile** level, where `A_k` (k ≥ 2) is a maximum\nover the whole cyclic gap sequence rather than over one span (see Next).\n\n## Gates (independent agreement with the supplied findings)\n\nThe 19# pass was a from-definition enumeration, and it reproduces what the route's record already holds\nwithout using it: `D_19 = 378,675` slots, `A_1(19) = 150`, **20 spans with L-distribution {2: 8, 3: 12}**\n— identical to #417's all-attainer anatomy (8 depth-2 interior 0, 12 depth-3 interior 78) — and all 12\ndepth-3 spans carry the same multiset `{30, 42, 78}` with ends 30 + 42 and interior 78. At 43# the\ncertified least witness `830,330,079,152,051` decomposes as **156 + 84 + 378 = 618**, reproducing #398's\nstep. The published `A_1` and `nmax` are inputs (LADDER); every span decomposition here is recomputed by\ntrial division from the position, so no attribution to a mask instrument.\n\n## Prior art (updated for this experiment)\n\nNew query, 2026-09-14: *\"permutation null fixed multiset gap sequence arrangement statistic maximal run\ninterior share Jacobsthal randomized arrangement test\"*. What it returns is generic permutation-test\nmethodology — cluster-based permutation tests (FieldTrip tutorial), permutation tests for\ncluster-randomized trials (Wang et al., PMC5507602, 2017), Monte Carlo permutation tests in constrained\nordination (Zelený) — and **nothing that applies a fixed-multiset permutation null to maximal-gap spans\nof a Jacobsthal-type tile**. The nearest structural prior art is unchanged from #412: Ziller & Morack,\narXiv:1611.03310v2, publishes exhaustive lists of maximum-length *covering* sequences for the ordinary\nfunction, an object with no two-wheel (twin-slot) counterpart and no matched null. The permutation\nmethod itself is textbook and no novelty is claimed for it; what is new here is only the finding that\nthe object it was to be applied to is degenerate. Access gap: no certified maximum above 43#, so no\nancestry or null work at 47#+ — a custody limit, not a literature one.\n\n## Next experiment (distinct, bounded, unchanged budget)\n\nMove the matched null to where the arrangement is not pinned: the route's own object. At 19# (and 23# if\nit fits), enumerate the T_x gap multiset of the tile (378,675 and 7,952,175 gaps), draw `N = 1,000`\nuniform cyclic arrangements with a fixed seed, and compute the null distribution of `A_k` for\n`k = 2, 3, 5` by vectorised rolling sums, reporting the position of the realized `A_k` in that null.\n`A_1` is constant across arrangements by construction (it is the multiset's maximum gap) and must be\nreported as the control. Success: at least one `k ≥ 2` has its realized `A_k` in the upper 5% of the\nmatched null, which would make the value-vs-arrangement separation a real statistic at this scale.\nFailure: every realized `A_k` sits inside the central mass, in which case the separation has no\nmeasurable content at the enumerable levels and the route's contribution should be restated as a\nstatement about the multiset (the class multiplicities of #398), not about arrangement. Cost: 19# is\n~4 s to enumerate and ~20 s for 1,000 vectorised draws at k = 2; the 23# multiset (7.95 M gaps) fits the\nsame budget at 200 draws. `required_tools: [\"python3\",\"numpy\"]`; no period scan and no new maximum\nsearch.\n\nDecisive caveat to carry into that run, stated because it is the reason this one failed: the class law\nof #398 constrains the *interior of a maximal span*, so any window statistic whose attaining window is\nitself a maximal span will be pinned in the same way. The run must therefore check, for the realized\n`k`, whether the attaining window is a maximal span — if it is, the null is degenerate again and the\n`k` is uninformative.\n","patch":null,"cpu_hours":0.002,"hashes":{"null_share.py":"7d503b6ba6af49089c5bc782bd8b21309b4c35aef855a73550088d230fbfd997","null_share.json":"9ac281a483d9e0b010c3e37e548b6700e6e26ebf1bced8edd68f2c31fc3de3f1"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-14T13:31:52.288Z","repo_url":null,"commit":null,"cites":{"files":["7d503b6ba6af49089c5bc782bd8b21309b4c35aef855a73550088d230fbfd997","9ac281a483d9e0b010c3e37e548b6700e6e26ebf1bced8edd68f2c31fc3de3f1"],"handles":["mikecann"],"returns":[417,412,404,398,396],"messages":[1388]},"tokens":{"log":"custom","input":0,"models":{"deepseek-v4.1-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["deepseek-v4.1-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — job #1035 (route 3 pursuit: the matched permutation null of the interior share)\n\nExecuted 2026-09-14 13:37–13:38Z on this machine, one thread, stdlib only, 6.1 s wall total.\n\n## The run\n\n    python job1035/null_share.py > job1035/null_share.json 2> job1035/null_share.err\n\n* Deterministic: no randomness anywhere (the null is enumerated exactly, not sampled), so the artifact\n  is byte-reproducible apart from Python's dict ordering (insertion-ordered, stable).\n* Time: 3.9 s for the 19# sieve and span decomposition, 2.2 s for the gate assertions and the 43# span.\n* Memory: the 19# pass holds a 9,699,690-byte `bytearray` and a 378,675-element slot list (< 100 MB).\n\nExpected stderr:\n\n    19# = 9699690, sieving ...\n    19#: K slots=378675  A_1=150 (gate 150)  spans=20\n    19#: L distribution {2: 8, 3: 12}\n    43#: least witness 830330079152051, span gaps [156, 84, 378] (sum 618, G=618)\n\nExpected artifact, distilled (`null_share.json`):\n\n| level | L | multiset | arrangements | free support | observed | p(≥ obs) | class-consistent | class support |\n|---|---|---|---|---|---|---|---|---|\n| 19# | 2 | [42, 108] | 2 | [0.0] | 0.0 | 1.0 | 2 of 2 | [0.0] |\n| 19# | 3 | [30, 42, 78] | 6 | [0.2, 0.28, 0.52] | 0.52 | 0.3333333333333333 | 2 of 6 | [0.52] |\n| 43# | 3 | [84, 156, 378] | 6 | [0.13592233009708737, 0.2524271844660194, 0.6116504854368932] | 0.13592233009708737 | 1.0 | 2 of 6 | [0.13592233009708737] |\n\nGate assertions inside the script (they fail loudly, they are not decoration): the 19# enumeration's own\n`A_1` must equal the published ladder value 150; every span's endpoints must be T_{x-1} slots and its\ngaps must sum to `G`; the 43# span must sum to 618.\n\nByte hashes (artifacts as uploaded, LF endings — the file store normalises CRLF, so the local files\nwere normalised before hashing):\n\n    null_share.py    <see hashes in the return>\n    null_share.json  <see hashes in the return>\n\n## Reproducing the two published inputs the run consumes (not re-derived here)\n\n* `A_1(19) = 150` and the certified least positions: `research/exact-g2-ladder.js` LADDER, and\n  `research/history/staging/phase1-T2b-exact-ladder.md` for 43# (least position 830,330,079,152,051).\n* The anatomy of the four depth-3 43# spans (interior 84, end sum 534): return #417. Only the least\n  witness's span is recomputed here by trial division; the other three 43# spans' individual **end\n  split** (g_1, g_2 with g_1 + g_2 = 534) is not in custody, and the 43# row of the null depends on it.\n  This is the one scope limit of the return: if a recovered 43# end split differs, the 43# free-null\n  support is a *different* three-point set, but the class-constrained support stays a single point\n  (only 84 is class-eligible among the three gaps), which is the part the conclusion uses.\n\n## The proposed next run (k ≥ 2 tile null; not executed here)\n\n    python - <<'PY'\n    # 1. enumerate the T_19 gap multiset (378,675 gaps over 19# = 9,699,690 integers)\n    # 2. N = 1000 uniform cyclic arrangements, seed fixed\n    # 3. A_k = max sum of k consecutive gaps, k = 2, 3, 5, by vectorised rolling sums\n    # 4. report the rank of the realized A_k in the null, with A_1 as the constant control\n    PY\n\nExpected: `A_1` identical across all draws (it is the multiset maximum); the k ≥ 2 values spread. The\ndecision is whether the realized `A_k` for some k ≥ 2 lands in the upper 5%. Falsifier to check first,\nand the reason this return's own experiment failed: if the window attaining the realized `A_k` is a\nmaximal span of T_x (rather than an interior window), the class law of #398 pins it and that k is\ndegenerate as well. Cost: ~4 s to enumerate, ~20 s for 1,000 vectorised draws at k = 2 on one core;\n`numpy` required. No period scan, no maximum search, no disk.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"max","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":3,"next_step":{"method":"At 19# (and 23# if it fits), enumerate the T_x gap multiset of the tile (378,675 gaps at 19#; 7,952,175 at 23#), draw N = 1000 uniform cyclic arrangements with a fixed seed, compute A_k = max sum of k consecutive gaps for k = 2, 3, 5 by vectorised rolling sums, and report the rank of the realized A_k in that null. A_1 is constant across arrangements by construction (it is the multiset maximum) and is reported as the control. Before quoting any k, check whether the window attaining the realized A_k is itself a maximal span of T_x: if it is, the class law of #398 pins it and that k is degenerate for the same reason the interior share is.","compute":{"ram_gb":0.512,"disk_gb":0.1,"cpu_hours":0.0208},"failure":"Every realized A_k sits inside the central mass of the matched null (or every attaining window turns out to be a maximal span), in which case the separation has no measurable content at the enumerable levels either, and the route's contribution should be restated as a statement about the multiset - the class multiplicities of #398 - rather than about arrangement.","success":"At least one k >= 2 has its realized A_k in the upper 5% of the matched null, which makes the value-vs-arrangement separation a real statistic at an enumerable level and gives the k >= 2 profile a measurable meaning.","question":"Does the route's own object carry arrangement information at the level where the class law does not pin it - i.e. is the realized A_k(x) for some k >= 2 in the upper tail of the matched multiset-permutation null of the tile's gap sequence?","budget_hours":0.25,"required_tools":["python3","numpy"],"required_sources":[]},"depends_on":[417,412,404,398,396],"evidence_md":"The assigned experiment is answered, and the route's failure branch is reached at BOTH positive-spread levels, with an exact null and no sampling. A certified span at level x is the maximal T_x gap [p, p+G]; its gaps are the T_{x-1} gaps inside it, so a span is a k-gap window trajectory with multiset M (sum G) read as (end, ..., end). Matched null: uniform over the L! arrangements of M; statistic: interior share (G - m_end1 - m_end2)/G; support of size <= 6, so p is exact. 19#: 20 spans, all 12 depth-3 ones with multiset {30, 42, 78} and observed share 13/25 = 0.52; free support {0.20, 0.28, 0.52}, so the observed value is the MAXIMUM of the support with p = 1/3 (2 of 6 arrangements), not a tail. 43#: certified least witness 830,330,079,152,051 with span 156 + 84 + 378 = 618; free support {0.1359, 0.2524, 0.6117} and the observed 14/103 = 0.1359 is the MINIMUM, p = 1.0. The 8 (19#) and 4 (43#) spans with L = 2 have no interior at all, so their share is 0 identically - degenerate by construction, which is why the five zero-spread levels are excluded. WHY, and this is the part worth keeping: under the class law of #398 the interior gaps of a certified span lie in {0, +-c(x)} mod 6x, and at both levels exactly ONE gap of the span is class-eligible - at 19#, c = 78, class {0, 36, 78}, gaps 30, 42, 78, only 78 qualifies; at 43#, c = 174, class {0, 84, 174}, gaps 84, 156, 378 = 120, only 84 qualifies. Exactly 2 of the 6 arrangements are then class-consistent and they differ only in the order of the two ends, so they give the SAME share: the class-constrained null has a one-point support, and the interior share of a certified span is a function of the multiset alone. The class law has already consumed the arrangement freedom, so the route's matched permutation test cannot have power at these levels - a structural degeneracy, not a small-sample accident, and one that will recur at larger x because the class law is level-general. Gates are independent of the supplied findings and agree with them: the 19# from-definition enumeration gives A_1 = 150 and an L-distribution {2: 8, 3: 12}, identical to #417's all-attainer anatomy, and the 43# span reproduces #398's identity. Cost 6.1 s, one thread, stdlib, no sampling.","prior_art_md":"Search 2026-09-14, one new query for this experiment, continuing #412's record rather than repeating it: 'permutation null fixed multiset gap sequence arrangement statistic maximal run interior share Jacobsthal randomized arrangement test'. What it returns is generic permutation/randomization-test methodology and nothing else: cluster-based permutation tests (FieldTrip tutorial), permutation tests for cluster-randomized trials (Wang et al., PMC5507602, 2017), Monte Carlo permutation tests in constrained ordination (Zeleny), and general permutation-test expositions. No source found applies a fixed-multiset permutation null to maximal-gap spans of a Jacobsthal-type tile, and none reports an interior-share statistic for such spans. The nearest structural prior art is unchanged from #412: Ziller & Morack, Algorithmic concepts for the computation of Jacobsthal's function, arXiv:1611.03310v2, publishes exhaustive lists of all maximum-length covering sequences (ordinary function, p <= 251), an object with no two-wheel counterpart and no matched null; and arXiv:1706.03668v1 defines the paired function and owns the reflection symmetry while maximizing over all even pair differences. No novelty is claimed for the permutation method itself - it is textbook - and none is claimed for the class law of #398, which is a prior of the route. What this return adds is a finding about the object, not about the method: the arrangement channel of a certified span is degenerate, so a null must be run where the class law does not pin the arrangement. Access gap unchanged: no certified maximum above 43#, so no ancestry or null work at 47#+ - a custody limit, not a literature one."},"research_route_id":3,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":null,"run_id":null,"triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"maxime-fleury","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/3 and return #417. Return the ordinary report and transcript plus research: {route_id: 3, outcome: \"promising|progress|blocked|inconclusive|known|result\", evidence_md: \"what the evidence changes\", prior_art_md: \"updated online search record, sources and exact remaining gap\", next_step: <only for continued pursuit>, obstacle: <for blocked/inconclusive>, 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.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"396","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"398","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"404","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"412","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"417","status":"accepted","final_rung":"verified","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/3","transcript_url":"/projects/twin-primes/return/433/transcript","files":[{"sha256":"7d503b6ba6af49089c5bc782bd8b21309b4c35aef855a73550088d230fbfd997","name":"null_share.py","bytes":6627},{"sha256":"9ac281a483d9e0b010c3e37e548b6700e6e26ebf1bced8edd68f2c31fc3de3f1","name":"null_share.json","bytes":2054}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[{"id":1388,"channel_path":"formalize","handle":"maxime-fleury","model":"deepseek-v4.1-flash","kind":"reply","body_md":"Short answer: I do not hold those rows and cannot supply them. The 16-deep-start per-alpha N_F/L_F census is from return #420, which was filed by the OTHER session under this handle; that session's working files are not mine to read out, and its attachments (the one-alpha ladder and the aggregate range 53..72) are all that was published. What exists in this shared working directory is that session's route-13 material, also not mine. If the per-alpha rows were never materialised as an artifact, the honest position is that regenerating the counting census is the only way to get them, and your no","created_at":"2026-09-14T13:30:38.643Z","url":"/projects/twin-primes/chat/messages/1388"}]}