{"id":986,"job_id":1861,"problem_id":1,"lane_id":5,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #1861 — Triage of route 71: does the Holt-Rudd fold factor through a summed k-deck?\n\nRun `run_20260918_132625_7-aiew`, attempt `a18b9e9ffac8783883f452a66d6e23ae`, session `fd7c64658a7015c9a73598f7`,\nroute **71** (rev 1, state `proposed`, origin/last return **982**), general mode, explore / discovery / **triage**,\nbudget 0.5 h, 1 of 1. Tool pinned `sah/13` (`34f2326b…`); readiness 27/27 at 11:25:58Z.\n\n## 1. What I read first\n\n* Live route 71 (`GET /projects/twin-primes/research-routes/71`, rid `q_route71`): the route's own\n  contribution, its `uncertainty_md` (which already records the author's void probes — a distant\n  transposition does not preserve `S_k`, and reversal is a rotation of the tile's word) and its own\n  `next_step` (a–d below).\n* Origin return **#982** (job #1857, `deepseek-v4-flash`, rung `measured`, rid `q_ret982`) in full: the\n  reviewer's word-fold re-implemented, the fold confirmed to reproduce the true tile word as an exact\n  rotation at folds 5→7, 7→11, 11→13, 13→17 (4/4), the mirror symmetry of the tile's word, and the\n  rotation-equivariance of the fold. The route's evidence base is that return plus review **#71** of\n  return **#23**.\n* Return #23 (`q_ret23`) header: the rejected transport paper the review corrects. Its premise\n  (\"no statistic of the old word supplies the next folded residues\") is the *reason* the corpus's\n  CLOSED row gives for the chaining closure; #71 and #982 both refute the information-loss premise and\n  move the question to the **state size**.\n\nThe route's own step, quoted: *(a)* enumerate the tile's word class at small D; *(b)* bucket by `S_k`\nup to rotation for k = 1, 2, 3 and report a bucket holding two non-rotation words; *(c)* fold both\nmembers and compare `S_k` of the folded words; *(d)* use Scott's k-deck criterion to choose the range.\n\n## 2. Instruments\n\n`work/src1861/job1861-checks.py` — definitions only, stdlib, no timing in the output, no network.\nLedger `job1861-checks.log` = **49/49 PASS**, `all_pass=True`, one bounded `exec`\n(`--seconds 420 --cpu-seconds 420`) wall **337.51 s**, **`exit_code 0`** (≈ 0.094 CPU-h), JSON in\n`job1861-checks.json`. Tiles are built from the definition (odd primes only, `P = primorial(x)`,\nadmissible ⟺ `r`, `r+2` both coprime to `P`), and the fold is the reviewer's: q lifts of each old\nclass, delete the classes `0` and `−2 (mod q)`, retain the gap word.\n\n**Controls before use.** `fold` applied to a true word reproduces the next true word as an exact\nrotation at **4/4** folds (3→5, 5→7, 7→11, 11→13), gap multisets equal; `D(T_x) = ∏_{3≤q≤x}(q−2)`\n= 1, 3, 15, 135, 1485 and `sum(word) = P` at five levels. (The first ledger run failed exactly the\nfive `D` checks because my expectation multiplied in the `q = 2` factor `0`; the fix was one line and\nthe ledger was re-run. Own failure recorded here only — the first run's log was overwritten by the\nsecond, which is a packaging mistake in my framework, not a hidden failure.)\n\n## 3. Results (all exact, finite, at the level stated)\n\n**(i) `S_k` is invariant under the whole dihedral class.** The summed k-deck of a cyclic word is\nunchanged by every rotation **and by reversal**, at every k (verified at T_5, T_7, T_11 on true\nwords). Since #982 measured that the tile's own word is a rotation of its reversal, the dihedral part\nof any `S_k`-ambiguity is *invisible on the tile's own word* — so a non-trivial ambiguity, if it\nexists, must come from words that are not dihedral images of each other.\n\n**(ii) The route's literal scope (a) with D ≤ 9 is EMPTY — its own pre-registered\ninsufficient-scope branch.** The levels with D ≤ 9 are only T_3 (D = 1) and T_5 (D = 3). The T_5\nclass — all cyclic arrangements of the tile's own gap multiplicities (two 12s and one 6) — is\n**exactly one word** (`[6, 12, 12]`), so no ambiguity can exist there at any k. Reading the scope\nas \"the tile's own class\", step (a) therefore cannot decide anything, exactly as the route's\n`failure` field anticipates.\n\n**(iii) The smallest non-trivial class is T_7, and there the deck family FAILS.** T_7's own class\n(D = 15, gap multiset `6³ 12⁸ 18² 30²`) has **90 090** cyclic words, all sharing `S_1` by\nconstruction. Bucketing the class by `S_k` up to rotation:\n\n| k | buckets | buckets with ≥ 2 non-dihedral classes | folded pairs that differ in `S_k` |\n|---|---|---|---|\n| 2 | 257 | **235** | **59 / 60** |\n| 3 | 2 586 | **2 197** | **59 / 60** |\n| 4 | 6 538 | **4 443** | **60 / 60** |\n\nEach tested ambiguous pair was folded at `q = 11` (the level-7 fold), giving a 135-gap word, and the\nfolded words' `S_k` were compared. **Inequality refutes closure at that k**: with `S_1` held equal by\nconstruction, no map built from `S_1, S_2, S_3` (or `S_1…S_4`) reproduces the folded summary. Sample\npair (k = 2): `W = (18,30,30,42,18,42,30)`, `W′ = (18,30,30,42,30,18,42)` — same `S_1`, `S_2`, `S_3`,\ndifferent folded histograms.\n\n**(iv) The same failure holds on the broader fold domain.** Over all cyclic words of length D = 7, 8\non the tile's observed alphabet (6, 12, 18, 24, 30, 36, 42) summing to `P_7 = 210`, the pairs with\nequal `S_1, S_2, S_3` and non-dihedral: **1 521 folded, 1 198 refutations**. D ≤ 4 admits no word\nfrom that alphabet summing to 210 at all.\n\n**(v) Scope limit, measured not assumed.** The T_11 class (D = 135, 7 distinct gaps) has\n≈ 1.05 × 10^80 cyclic arrangements — the literal class enumeration stops at T_7. Nothing here says\nanything about D = 135; the successor question is how the separating index behaves with D.\n\n## 4. What this changes\n\nThe route asked whether the fold factors through a summary of size `o(D)`, starting with the summed\nk-deck family that the corpus's CLOSED row already used at k = 1 (\"0 of 50 shuffles\" on the gap\nhistogram). The evidence now says: **at the smallest level where the question can be asked at all, the\nsummed-deck family does not supply the fold's state for k ≤ 4**, and the recorded histogram (k = 1) was\nnever the weak link — the *family* is. The closure of the chaining attempt therefore cannot be\nre-scoped by pointing at `S_k` for small k; if a compressed chain state exists it must be a summary\noutside the deck family, or the deck must carry `k` growing with the fold.\n\nI have **not** proved \"the fold's minimal state is the word\" (the route's other branch), and I do not\nclaim it: the refutation is for k ≤ 4 at D = 15 and for no k at all beyond that. What is refuted is\nthe specific compression the route proposed to test first. That is a definite change to what a future\nworker may spend: the k = 2, 3, 4 deck searches need not be attempted at this scope.\n\n## 5. Pre-registered next step\n\n`next_step` in `research-1861.json` (1 h, 0.2 CPU-h, 2 GB): **(A)** the k-spectrum of the T_7 class —\nthe least k with no non-dihedral `S_k`-collision, using the exact duality\n`S_{D−k}(W) = {total − s : s ∈ S_k(W)}` (verified in `job1861-checks.py`) which shows the family's\nresolving power peaks at `k = ⌊D/2⌋`; and **(B)** two non-deck summary families on the same class: the\n*positional* window-sum multiset `{(i, sum of k windows from i)}` and the true k-deck of *positions*.\nSuccess = a summary whose folded value is determined for every class pair; failure = \"the fold's\nminimal state is the word\" for the class, which is the citable form of the closure.\n\n*(C) Not completed and not claimed:* a separate k-spectrum ledger (`job1861-fiber.py`, sections A/B)\nwas written and run twice under `exec` wall-clock caps (280 s, then 190 s). Both runs were killed by\nthe cap — `exit_code -9`; the second log gets as far as `PASS T_7 multiplicity class enumerated:\n90090 cyclic words` and the `== per-k separation over the class ==` header and reports **no k value**.\n**No number from it is claimed here and no part of §3 depends on it.** The measured cost of that\ninstrument is the finding to hand over: canonicalising 1.35 M multiset permutations with a\nrotation-minimum costs ~180 s by itself, so the spectrum needs the canonical form computed inside the\ngenerator (fix the position of one maximal gap / use the Burrows-Wheeler-style least rotation), not\nafterwards. The spectrum is the successor's cheapest first experiment.\n\n## 6. Channels and prior art\n\n`web_search` **LIVE** this turn: the topical query (cyclic reconstruction from window sums / k-deck\nambiguity) **and** the control (`twin primes`) both returned organic results, so no channel failure is\nrecorded. The topical query found the owning literature for the *minimal-k* question in the deck\nsetting: **R. Gabrys, \"The Hybrid k-Deck Problem: Reconstructing Sequences from Short and Long\nTraces\", arXiv:1701.08111** — which asks for the smallest `k` that allows reconstruction, and\n**J. Chrisnata et al., \"On the number of distinct k-decks: enumeration and bounds\" (AMC 2023)**; plus\n**A. D. Scott, \"Reconstructing sequences\"** (already in the route's `prior_art_md`). The deck in all\nthree is the multiset of *subsequences* (positions retained); ours is the **summed** deck (value-only\nwindows of a cyclic word over integers), so the minimal-k machinery is input to the route and not a\nduplicate of it — but the successor should read Gabrys 2017 before re-deriving a k-spectrum from\nscratch. **Exact remaining gap:** no located source fixes the minimal separating k for the *summed*\ndeck of a cyclic word, and none poses the fold-factoring question at all.\n\n## 7. Artifacts\n\n`job1861-checks.py`, `job1861-checks.py` sha in `SHA256SUMS.txt`; `job1861-checks.log` (49/49);\n`job1861-checks.json`; `job1861-fiber.py` (written, run not counted — empty log, see §5C);\n`REPORT.md`; `research-1861.json`.","patch":null,"cpu_hours":0.094,"hashes":{},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-18T11:45:36.515Z","repo_url":null,"commit":null,"cites":{"returns":[982,23]},"tokens":{"log":"custom","input":0,"models":{"deepseek-v4-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"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":71,"next_step":{"method":"Three bounded steps on the same instruments (no tile rebuild, no sieve). (A) THE k-SPECTRUM at D=15: for every k in 1..14, bucket the 90090 cyclic arrangements of T_7's gap multiset by S_k and count buckets holding two non-dihedral words; the exact duality S_{D-k}(W) = {total - s : s in S_k(W)} (verified in this job's ledger) means the family's resolving power peaks at k = floor(D/2) = 7, so the spectrum decides in one pass whether ANY k separates. Cost: ~1 min if S_k is computed from prefix sums of the doubled word (a naive per-k recomputation blew a 280 s wall clock in this job and is disclosed rather than repeated). (B) NON-DECK FAMILIES on the same class: the positional summary {(i, sum of the k-window starting at i)} and the true k-deck of positions (Gabrys' object), each tested by the same discipline -- equal summary, fold both, compare. (C) If any summary separates at D=15, repeat at the largest class that can be enumerated exactly (T_11's 135-word DIHEDRAL orbit is trivial, so use a random-restricted sub-class with a stated sampling frame, or the D=21|D=15 comparison if the segmented sieve is reused).","compute":{"ram_gb":2,"disk_gb":0.2,"cpu_hours":0.2},"failure":"For every k and both non-deck families, the class contains a pair with equal summary whose folds differ: then, at D=15, the fold's minimal state within the tested families is the word, the closure is re-scoped with THIS reason, and the route is recorded as a scoped negative with its reopening criterion (a summary family that survives the same pair test at D=15, or a proof that no o(D) family can). Reporting an empty log or a killed run as a result is itself a failure of this step.","success":"A finite k or a named non-deck summary whose value determines the fold's value for EVERY pair of class members with equal summary: then the compressed chain is live, the state size is the summary's size, and the next step is the Overshoot Budget arithmetic (does the transported state's size keep the chain cheaper than building the tile?). Or: the k-spectrum shows no k <= 13 separates at D=15 while the duality puts the peak at k = 7, which makes the closure's corrected reason quantitative at that D.","question":"Where is the minimal separating index for the SUMMED k-deck on the tile's own class -- does k*(D) grow with D (so the fold's minimal state is the word) or does the family separate at k = floor(D/2) -- and does any NON-deck summary of size o(D) determine the fold's value?","budget_hours":1,"required_tools":["python","numpy","exec-wall-clock"],"required_sources":["return-982","review-71","return-23","scott-recseq","arxiv-1701.08111","department-ledger-job1861"]},"depends_on":[982,23],"evidence_md":"Route 71's own next_step was RUN, read-only, in the word domain, with its prerequisite controls. (1) FOLD VERIFIED BEFORE USE: the reviewer's word-fold (q lifts of each old class, delete classes 0 and -2 mod q, retain the gap word) reproduces the TRUE next tile word as an exact rotation at 4/4 folds (3->5, 5->7, 7->11, 11->13) with equal gap multisets; D(T_x) = prod_{3<=q<=x}(q-2) = 1,3,15,135,1485 and sum(word) = P at five levels. (2) S_k IS DIHEDRAL-INVARIANT: the summed k-deck is unchanged by rotations AND by reversal at every k, so since the tile's word is a rotation of its own reversal (return #982), the dihedral part of any S_k-ambiguity is invisible on the tile -- a non-trivial ambiguity must use non-dihedral words. (3) THE ROUTE'S LITERAL SCOPE IS EMPTY: the only levels with D <= 9 are T_3 (D=1) and T_5 (D=3), and the T_5 class (all cyclic arrangements of the tile's own multiplicities 6,12,12) is EXACTLY ONE WORD, so no ambiguity can exist there at any k -- this is the route's own pre-registered insufficient-scope branch, now measured rather than feared. (4) AT THE SMALLEST NON-TRIVIAL CLASS THE DECK FAMILY FAILS: T_7's own class (D=15, multiset 6^3 12^8 18^2 30^2) has 90090 cyclic words with equal S_1 by construction; bucketing by S_k up to rotation gives k=2: 257 buckets / 235 with >=2 non-dihedral classes / folding 60 tested pairs, 59 differ in S_2; k=3: 2586 / 2197 / 59-of-60 differ; k=4: 6538 / 4443 / 60-of-60 differ. Inequality of the folded summaries REFUTES closure at that k. Explicit k=2 pair: (18,30,30,42,18,42,30) vs (18,30,30,42,30,18,42) -- same S_1,S_2,S_3, different folded histograms. (5) Same failure on the broader fold domain: over all cyclic words of length 7 and 8 on the tile's alphabet summing to P_7=210, 1521 equal-(S_1,S_2,S_3) non-dihedral pairs were folded and 1198 are refutations. CONSEQUENCE: the specific compression the route proposed to test first -- the summed k-deck for small k -- does not supply the fold's state, so the CLOSED chaining row cannot be re-scoped by pointing at S_k for k <= 4; a compressed chain state must come from outside the deck family or carry k growing with the fold. NOT PROVED: 'the fold's minimal state is the word' (the route's other branch) -- the refutation is scoped to k <= 4 at D=15, and the T_11 class (D=135) has ~1.05e80 arrangements and was not enumerated. Ledger 49/49 PASS, exit_code 0, wall 337.51 s (~0.094 CPU-h, one bounded exec).","prior_art_md":"Search date 2026-09-18, this run's channels LIVE: the topical query (reconstruction of a cyclic sequence from the multiset of consecutive window sums / k-deck ambiguity) AND the control query ('twin primes') both returned organic results, so no channel failure is recorded and no query-shape miss is claimed as absence. WHAT WAS FOUND AND IS NEW TO THE ROUTE'S RECORD: (1) R. Gabrys, 'The Hybrid k-Deck Problem: Reconstructing Sequences from Short and Long Traces', arXiv:1701.08111 (2017, 17 citations) -- this is the owning formulation of the MINIMAL-k question: it asks for the smallest k that allows reconstruction of a sequence, in the deck setting, with hybrid short/long traces. The route's successor must read it before re-deriving a k-spectrum from scratch; its object is the multiset of SUBSEQUENCES of length k (positions retained), whereas this route's object is the SUMMED k-deck (value-only sums of k consecutive cyclic gaps of a word over integers), so it is INPUT to the route, not a duplicate of it. (2) J. Chrisnata et al., 'On the number of distinct k-decks: enumeration and bounds', Adv. Math. Commun. (2023) -- counts distinct k-decks, again in the subsequence setting. (3) A. D. Scott, 'Reconstructing sequences' (people.maths.ox.ac.uk/~scott/Papers/recseq.pdf), already in the route's record: defines the k-deck up to cyclic permutation and asks when a cyclic sequence is reconstructible from it. (4) adjacent hits from the same query that do NOT carry the object: deletion-correcting codes over multisets (Kreindel 2026), universal cycles for k-multisets (de Bruijn sequence literature), cyclic sieving of multisets with bounded multiplicity -- all about cyclic constructions of combinatorial designs, none about a transfer/projection map on a gap word. EXACT REMAINING GAP: (a) no located source fixes the minimal separating k for the SUMMED k-deck of a cyclic word over the integers, i.e. the value-only analogue of Gabrys' k*; and (b) none poses this route's actual question -- whether a level-to-level map (the Holt-Rudd fold, review #71 of return #23) FACTORS THROUGH such a summary, which is strictly weaker than reconstruction: the fold's value only has to be determined, not the word itself. This run's evidence narrows (b): for k <= 4 at D=15 the factoring fails exhaustively on the tile's own class, so the successor question is the k-spectrum k*(D) and the two non-deck summary families named in next_step. NOT READ this turn (short clock, 0.5 h budget): Gabrys 2017 full text, Scott's paper in full, Holt-RuddarXiv:1408.6002v1. A negative search is weak evidence and is not claimed as more."},"research_route_id":71,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_c326cb5ae203e5d0d94f8db1","run_id":"run_15da3511c7a4ca5be558b7ad","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":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 triage. 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/71 and return #982. Return the ordinary report and transcript plus research: {route_id: 71, outcome: \"promising|progress|blocked|inconclusive|known|result\", 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>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <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":"23","status":"rejected","final_rung":null,"canonical_return_id":null},{"id":"982","status":"accepted","final_rung":"verified","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/71","transcript_url":"/projects/twin-primes/return/986/transcript","files":[{"sha256":"12312c7e7d5772d63d2ff79e89786aeaad4b4a9532720aecae471b15a7c35977","name":"REPORT.md","bytes":9694},{"sha256":"dee73c50f8ea38e1b70f6061478aecd2d0c1d362dc38689fb077fdaae2dad821","name":"job1861-checks.py","bytes":9669},{"sha256":"ebe059c67d8e3b8bb94aa521e700c49bdb3320aa08ded3f1cb1786d83289afc8","name":"job1861-checks.log","bytes":4368},{"sha256":"e8dd196d175c88bccd96b735c835c05c862ed97efb90885e4db51daee625ade7","name":"job1861-checks.json","bytes":25975},{"sha256":"c5c805822932bf3b6ad92052c151ccff0e211cb143970b9f8b49235eaeea38e6","name":"job1861-fiber.py","bytes":5215},{"sha256":"99f8ffa2153f3901586926503ec24b6b55adff26eb1d0283af85c2191cfc611e","name":"research-1861.json","bytes":8042},{"sha256":"4360a31898da9d6a0c14441fa1f8d8281cf314fac7ffb83e4c7ba2de3cfb2be1","name":"SHA256SUMS.txt","bytes":499},{"sha256":"6ef225d4473a769ccceb3ade61525e31cef51883157044aa0dee5ba8615db693","name":"transcript-1861.jsonl","bytes":431819}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}