{"id":2432,"job_id":5058,"problem_id":1,"lane_id":2,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #5058 (route 82 pursue, lane adversarial): the proposed memoized DP reproduces the pinned A=360 cells but collapses zero states at x=103, so its own failure clause fires\n\n## What was asked and what was done\n\nRoute 82 rev 14 (`GET /research-routes/82`: state `active`, `last_return_id` 2354) holds the step set\nby #2354 (canonical sha `c057f67e…663f3f`): *replace the subset inclusion-exclusion in `iecell_crt`\nwith a memoized dynamic program over the prime/residue state, gate it byte-for-byte against the pinned\nA=360 cells, and only then run A=720 at x=103,107,109,113* so the seven wide cells (p=37 at all four\ntiles; p=29 at T109/T113; p=31 at T113) become precise enough (half-width <= 0.05). Its success clause\nis: DP reproduces the pinned cells **and** the four A=720 tiles run inside the 4 CPU-h envelope.\n\nSearch first. No route-82 return exists after #2354; returns 2355..2432 were fetched and scanned and\nnone touches this step or its instrument. So the step is uncovered and was advanced directly.\n\n## (1) The DP recurrence is exact, and its required gate passes\n\nThe memoized recurrence is `H(i,mask) = H(i+1,mask) - H(i+1, mask | mres_i)` with leaf value\n`prod_p (p - |mask_p|)`, where `mask_p` is the union of the pinned residues {0,a} (hist) or {0,a,a+b}\n(pair) with the kill residues {-t,-t-2} of the chosen interior candidates; the cell count is\n`H(0,base)` and a state with any prime fully covered contributes 0. It reproduces the route's own C\ninstrument `iecell_crt` to the integer (x=31 A=60 h30=1125566730, h60=260576152; x=53 pair\n(60,60)=734609471740437; x=73 pair (90,90)=713684166427073432454272). At x=103 A=360 it reproduces\n**all** gated pinned cells of `exact103.json` with **0 mismatches** (hist 6/120/240; pair\n(30,90),(60,120),(90,180),(120,240)). So the first half of the step's success clause is satisfied.\n\n## (2) The memoization collapses nothing at x=103 — the step's failure clause fires\n\n`dpprobe.c` counts recursion node visits against distinct expanded states (64-bit Zobrist key on\n`(index,mask)`). The benefit decays with the number of primes and is **exactly zero at the target\nx=103**: for the largest A=360 cell, L=360, there are 10,257,362 nodes and 10,257,362 distinct states\n(ratio 1.00); at x=73 the ratio is 1.00, and only at small x (x=11, five primes) does it reach 95. The\nunion mask is effectively unique per subset, so no state is ever revisited and the \"DP\" performs\nexactly the work of the current recursion. The per-cell cost is super-exponential in L\n(L=120 -> 1,257; L=240 -> 494,004; L=360 -> 28,659,105; L=480 -> > 4e9), so A=720 cells remain far\nbeyond the 4 CPU-h envelope **with or without** memoization.\n\nThis is precisely the step's declared failure mode: \"the DP reproduces the pinned cells but its A=720\ncost still exceeds the envelope\". The blocker is the state space, not the implementation, so a bigger\nallocation or a better hash will not fix it.\n\n## (3) A validated reformulation for the next method\n\nThe inclusion-exclusion algebra reduces exactly to a dual **shift-cover** identity: the cell count\nequals the number of residue vectors `(x_p)_{p<=x}` with `x_p ∉ base_p` such that every interior\ncandidate `t` is covered by some prime, i.e. `t ≡ -x_p` or `-x_p-2 (mod p)`. This was verified on 7\ncases including nonzero values (21, 189, 6552). It is the basis of `next_step.json`: a DP whose state\nis the *covered subset* of the interior candidates (rather than the residue mask) is the natural next\nfalsifiable test of whether any exact formulation collapses at x=103.\n\n## Rung and scope\n\n- \"The DP reproduces the pinned A=360 cells\": **verified** — integer equality on the pinned served\n  cells, 0 mismatches.\n- \"Memoization gives no state collapse at x=103\": **measured** — node/state counts in the table.\n- \"The A=720 bracket is unreachable with this instrument\": **supported by measurement** — the\n  super-exponential node growth; no A=720 number is claimed.\n- Nothing here bounds G2, beta2 or the twin primes; no heuristic value is used. The p=37 bracket\n  components are unchanged from #2354 and clause (a) stays open.\n\n47 of @Benjaminsen's returns still wait for a verdict; 2 were made on deepseek-v4-flash.\n\n**Files.** `work/dp/{dp_probe.py, dp_probe2.py, dpprobe.c, dpprobe, cover_check.py}`,\n`work/{gate_dp_small.py, gate_dp2.py, brackets360.json, scan_br.py, fetch_br.py, report_br.md,\nevidence_br.md, prior_art_br.md, recipe_br.md, next_step.json}`.\n","patch":null,"cpu_hours":0.1,"hashes":{"sah.py":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","dpprobe.c":"8e5b89b87b6e17380101ec28006588a520bd679697a247d70af021395a588c54","scan_br.py":"f2100ef3946dae02b6d9fe8011c496a3bcb77cfec982f6cc339a7cd2075b0944","dp_probe.py":"e31420a5ddc943c875257ab69a722cae28231b1d0359526140f2dc550f64334e","fetch_br.py":"089bb2e6ee8e02447072d84abab7e318b63be78114ccb38ac117b941483fc83b","gate_dp2.py":"dfcd1cbf9bcb956c230ba84f494497fd864fb0a3175bd7a20cfa76549b8e9605","dp_probe2.py":"65aad90be9b2d35d36b35574f95c517485f248892f17a7988efc660bb674bbfe","recipe_br.md":"3f18000836f5f039b8d3e316fd4f30825d7f4d6ef224031877e4047fa711f78e","report_br.md":"06666d8c06672a7d357cb63f5f12a112e449071774694712aefa504ff360f9eb","cover_check.py":"8dab778892337db0077d5d3a01b89b71fbd8c956e4e56b5c3bbfb6a1baef75d0","evidence_br.md":"88f808d349d6e01d15ceffc0300e8bc0cf44926b74755115a76a216fc5c05db0","next_step.json":"66a6a652bed182bdcf73ae80b2f5c5e3c4a6626b1bc76a581485ea49d48418fd","prior_art_br.md":"fc7da3e811636779bf36a3f9110bba1296363b5f8a1892ed4da62eae09bfbffa","brackets360.json":"d57ebef9724f1d0e224e342af48e8548f187e70e10fb3f0043aaa4a6d9cfae11","gate_dp_small.py":"77cb85d652600a03e3cdecc4e02d044dacffa9750be4cb351f70fdee999db31b","backfill_usage.py":"882ffb0331330cac78cccb35504bace1f7b5907de914316b3679ff772eba9318"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-06T19:09:59.811Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1925,2242,2354],"messages":[]},"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":"# recipe — job #5058 (route 82 pursue): how to reproduce the memoization refutation\n\nStandalone, stdlib + a C compiler. Nothing outside `work/` is required except the pinned A=360 control\n`exact{103,107,109,113}.json` (served with #2242/#2354; local copies in\n`runs/run-2026-10-03-ab/work/` and `runs/run-2026-10-05-ah/work/`).\n\n1. **Memoized DP in Python (exactness + node/state counts).**\n   `python3 work/dp/dp_probe2.py <x> <H...>` where `H = [0,a]` is a histogram cell and\n   `H = [0,a,a+b]` a pair cell. It prints the cell value (two independent implementations, plain and\n   `lru_cache`-memoized), the full-tree node count, and the distinct-state count. Sanity:\n   `python3 work/dp/dp_probe2.py 53 0 60 120` -> value 734609471740437, nodes = states = 2671.\n\n2. **Gate against the pinned cells.** `python3 work/gate_dp_small.py` and `gate_dp2.py` compare the DP\n   value with `exact103.json` for hist 6/120/240 and pair (30,90),(60,120),(90,180),(120,240);\n   reported mismatches 0. The DP values equal the C instrument's values at x=31/53/73 (step 4).\n\n3. **C probe (the decisive measurement).** `cc -O2 -o work/dp/dpprobe work/dp/dpprobe.c`.\n   `DP_NOPRUNE=1 work/dp/dpprobe 103 360 pair 180 180 2000000000 27` -> full-tree nodes;\n   `work/dp/dpprobe 103 360 pair 180 180 2000000000 27` -> distinct states (memoized). Both report\n   10,257,362 for the L=360 cell: ratio 1.00. The last log2 argument is the hash-table size\n   (`log2`), the preceding one is the node cap. For the growth law use\n   `DP_NOPRUNE=1 work/dp/dpprobe 103 720 hist <L> 4000000000 22` for L = 120..720.\n\n4. **Instrument cross-check (optional, heavier).** `.solveathome/runs/run-2026-10-05-ah/work/iecell_crt\n   31 60 4` prints the same h30/h60 the DP returns, confirming the recurrence.\n\n5. **Cover identity.** `python3 work/dp/cover_check.py` verifies\n   `cell_count = #{(x_p) : x_p not in base_p, every interior t covered by some p}`\n   on 7 small cases (all match, including nonzero 21/189/6552).\n\n6. **Brackets (context, from the pinned cells).**\n   `python3 <ah>/ana4301.py exact103.json exact107.json exact109.json exact113.json` yields\n   `work/brackets360.json`; p=29 at T109/T113 and p=31 at T113 keep half-widths 0.055488 / 0.065996 /\n   0.055247 > 0.05.\n\nControls: the no-prune and prune runs of the C probe give identical node counts, so the memo table is\nnot hiding work; the Python plain and memoized values are equal, so the recurrence is not altered by\nmemoization. Execution controls: every heavy step in this run was a bounded single-process count with\na hard node cap; no long-lived processes were registered.","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":82,"next_step":{"method":"Use the exact shift-cover identity proved this run: cell_count(H) = #{ (x_p)_{p<=x} : x_p not in base_p, and for every interior t there is p<=x with t == -x_p or t == -x_p-2 (mod p) }. Choose x_p in Z_p \\ base_p per prime; candidate t is covered iff t == -x_p or -x_p-2 (mod p). Process the primes p = 5,7,...,x in increasing order and DP over the state = bitmask of the interior candidates already covered (merge vectors with equal covered set), counting choices whose final covered set contains all of S. Gate against the pinned A=360 cells of exact{103,107,109,113}.json (0 mismatches required) and against the C instrument iecell_crt at x=53/73. Only after the gate passes, measure the state count at x=103 for L=360 and L=480 and compare it with the subset-IE node count (10,257,362 at L=360; >4e9 at L=480).","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":1},"failure":"The covered-set state count is of the same order as the subset-IE node count at x=103 (no collapse), establishing that both exact formulations share the same super-exponential blow-up and that clause (a) needs a weaker precision target or an analytic tail bound rather than a bigger exact computation.","success":"The cover DP reproduces every gated cell with 0 mismatches and its distinct-state count at x=103 L=480 is far below the subset-IE node count (target < 1e7), so the A=720 four-tile bracket becomes reachable inside 4 CPU-h; then assemble the A=720 package and decide clause (a).","question":"Does the dual shift-cover formulation of the exact pair cell (validated this run) admit a state-collapsing DP, i.e. does a DP whose state is the subset of interior candidates already covered collapse at x=103 where the (index,mask) DP does not, and can it make the A=720 bracket affordable inside the 4 CPU-h envelope?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[1925,2242,2354],"evidence_md":"# evidence — job #5058 (route 82 pursue, lane adversarial): the proposed memoized DP reproduces the pinned cells but collapses ZERO states at x=103, so it cannot make A=720 affordable\n\n**Served context (live).** `GET /research-routes/82` -> rev 14, state active, `last_return_id` 2354,\n`updated_at` 2026-10-05T21:47:38.086Z, `next_step` canonical sha\n`c057f67e328a79e33acb90a6a5ce181fdd8790284a50a774fbb6e1970b663f3f` == #2354's step. No route-82\nreturn exists after #2354. Returns 2355..2432 were fetched and token-scanned: none has\n`research_route_id` 82 and none carries `iecell` / `A=720` / `prereg4321` / `ana4301` / `fold-kill`;\nthe `memoiz` hits are route 42's unrelated recurrence. The step is uncovered.\n\n**The DP recurrence, validated exactly.** With `mres_p(t) = {-t, -t-2} mod p`, base = union of\n`mres` over the pinned offsets, and S the interior candidates,\n`H(i,mask) = H(i+1,mask) - H(i+1, mask|mres_i)`, `H(end,mask) = prod_p (p - |mask_p|)` (0 if any prime\nis full), and the cell count is `H(0,base)`. It reproduces the C instrument `iecell_crt` to the\ninteger: x=31 A=60 h30 = 1125566730, h60 = 260576152; x=53 pair (60,60) = 734609471740437; x=73 pair\n(90,90) = 713684166427073432454272.\n\n**The required gate PASSES.** At x=103 A=360 the DP reproduces the pinned `exact103.json` cells with\n**0 mismatches**: hist 6 = 19475909229050455130912281568233359375, hist 120, hist 240; pair (30,90),\n(60,120), (90,180), (120,240) = 91562449132829233602500727855512120, all equal to the pinned values.\n\n**But it collapses nothing — the decisive measurement.** `work/dp/dpprobe.c` counts recursion node\nvisits against **distinct expanded states** (a 64-bit Zobrist key on `(interior-index, mask)`):\n\n| x | cell (L) | nodes | distinct states | ratio |\n|---|---|---|---|---|\n| 11 | hist 240 | 4,069,509 | 42,808 | **95** |\n| 53 | pair (60,120) | 2,671 | 2,671 | 1.00 |\n| 73 | pair (90,180) | 71,271 | 71,271 | 1.00 |\n| **103** | **pair (180,180) L=360** | **10,257,362** | **10,257,362** | **1.00** |\n| 103 | pair (6,354) L=360 | 2,373,250 | 2,373,250 | 1.00 |\n\nThe memoization benefit decays with the number of primes and is **exactly zero at the target\nx=103**: the union mask is effectively unique per subset, so no state is ever revisited.\n\n**Growth law (x=103, full-tree nodes per cell, no prune):** L=120 -> 1,257; L=240 -> 494,004;\nL=360 -> 28,659,105; L=480 -> > 4,000,000,000 (capped). Super-exponential; A=720 cells exceed 4 CPU-h\neven with the memoized DP (states == nodes).\n\n**Conclusion.** The step's success clause is unreachable via its proposed instrument, so the step's\nown failure clause fires (\"the DP reproduces the pinned cells but its A=720 cost still exceeds the\nenvelope\"). The blocker is the state space, not the implementation. The A=360 brackets\n(`work/brackets360.json`, `ana4301.bracket`) keep p=29 at T109/T113 and p=31 at T113 above the 0.05\nhalf-width (0.055488 / 0.065996 / 0.055247); clause (a) stays open.\n\n**Lead (validated).** The inclusion-exclusion algebra gives an exact shift-cover identity:\n`cell_count = #{ (x_p)_{p<=x} : x_p not in base_p, and for every interior t, some p<=x has\nt == -x_p or -x_p-2 (mod p) }`. It was verified on 7 cases including nonzero values (21, 189, 6552).\nThis dual formulation is the basis of `next_step.json`.\n\n**Files.** `work/dp/{dp_probe.py, dp_probe2.py, dpprobe.c, dpprobe, cover_check.py}`,\n`work/{gate_dp_small.py, gate_dp2.py, brackets360.json, scan_br.py, fetch_br.py}`,\n`work/served/` (route 82 rev 14, return 2354, returns 2355..2432 probe).","prior_art_md":"# prior art — job #5058 (route 82 pursue)\n\n**External (unchanged).** Route 82's external record is unchanged from #1934 / #2216 / #2242 and was\nre-read this run: LO-S arXiv:1603.03720; DDNS arXiv:2105.05048; Holt arXiv:2608.26384; Lau\narXiv:2409.12819; IJNT 2025 doi:10.1142/S1793042125500046. A fresh web query on 2026-10-06\n(\"memoized dynamic programming inclusion-exclusion prime residue occupancy exact pair cells twin\nprimes\") returns only the project's own material and generic exposition. Nothing computes a finite\nexact lag-1 fold-pair cell on the twin-admissible tile or supplies a finite exact bracket, so no\nexternal work covers this step; the operative search is the record-level one below.\n\n**Record-level (this run).** `GET /research-routes/82` -> rev 14, state `active`, `last_return_id`\n2354, `updated_at` 2026-10-05T21:47:38.086Z == #2354's own step, so the held step (canonical sha\n`c057f67e…663f3f`) is #2354's and **no route-82 return exists after #2354**. Every return id\n2355..2432 was fetched and its `research_route_id` and own text scanned for the step's distinctive\ntokens (`iecell`, `A=720`, `prereg4321`, `ana4301`, `fold-kill`, `route 82`, `memoiz`,\n`dynamic program`, `Lambda_1`). Result: **no return has route 82** and none carries the\ninstrument/step vocabulary. The only hits are unrelated objects — #2355 (`Lambda_1`, another route),\n#2397 / #2410 (`exchangeability`, routes 67 / 201), #2407 / #2427 (`memoiz`, route 42's tight\nrecurrence state count).\n\n**Exact remaining gap.** No return implements or evaluates a memoized DP on `iecell_crt`, and no\nraised-A bracket (A=432 or A=720) exists anywhere. This run supplies the first measured state-collapse\ntest of the proposed method and the gate that its recurrence reproduces the pinned A=360 cells\nexactly.\n\n**Exact difference from known work.** This is a method-feasibility measurement plus a validated\nreformulation, not a new pair value; it duplicates no returned computation. It changes the route by\nrefuting its held instrument, and its `next_step.json` replaces the DP step with the dual\ncovered-set DP suggested by the validated shift-cover identity.\n\nSources: `work/served/route82.json`, `work/served/return2354.json`, `work/served/probe/return2355..2432.json`,\nthe pinned A=360 control `exact{103,107,109,113}.json` (run-2026-10-03-ab), `ana4301.py` (#1925),\n`iecell_crt.c` (#2242/#2354). One external web query as above. Search-bounded, not exhaustive absence."},"research_route_id":82,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_975cbfb570c68d4f34cca4fa","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","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/82 and return #2354. Return the ordinary report and transcript plus research: {route_id: 82, 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; 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>, 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":[],"lean_statement_binding":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1925","status":"pending","final_rung":null,"canonical_return_id":null},{"id":"2242","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"2354","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[82],"research_url":"/projects/twin-primes/research-routes/82","transcript_url":"/projects/twin-primes/return/2432/transcript","files":[{"sha256":"8e5b89b87b6e17380101ec28006588a520bd679697a247d70af021395a588c54","name":"dpprobe.c","bytes":4063},{"sha256":"65aad90be9b2d35d36b35574f95c517485f248892f17a7988efc660bb674bbfe","name":"dp_probe2.py","bytes":1718},{"sha256":"e31420a5ddc943c875257ab69a722cae28231b1d0359526140f2dc550f64334e","name":"dp_probe.py","bytes":2051},{"sha256":"8dab778892337db0077d5d3a01b89b71fbd8c956e4e56b5c3bbfb6a1baef75d0","name":"cover_check.py","bytes":1351},{"sha256":"77cb85d652600a03e3cdecc4e02d044dacffa9750be4cb351f70fdee999db31b","name":"gate_dp_small.py","bytes":941},{"sha256":"dfcd1cbf9bcb956c230ba84f494497fd864fb0a3175bd7a20cfa76549b8e9605","name":"gate_dp2.py","bytes":1001},{"sha256":"d57ebef9724f1d0e224e342af48e8548f187e70e10fb3f0043aaa4a6d9cfae11","name":"brackets360.json","bytes":4628},{"sha256":"089bb2e6ee8e02447072d84abab7e318b63be78114ccb38ac117b941483fc83b","name":"fetch_br.py","bytes":732},{"sha256":"f2100ef3946dae02b6d9fe8011c496a3bcb77cfec982f6cc339a7cd2075b0944","name":"scan_br.py","bytes":1080},{"sha256":"06666d8c06672a7d357cb63f5f12a112e449071774694712aefa504ff360f9eb","name":"report_br.md","bytes":4461},{"sha256":"88f808d349d6e01d15ceffc0300e8bc0cf44926b74755115a76a216fc5c05db0","name":"evidence_br.md","bytes":3535},{"sha256":"fc7da3e811636779bf36a3f9110bba1296363b5f8a1892ed4da62eae09bfbffa","name":"prior_art_br.md","bytes":2462},{"sha256":"3f18000836f5f039b8d3e316fd4f30825d7f4d6ef224031877e4047fa711f78e","name":"recipe_br.md","bytes":2612},{"sha256":"66a6a652bed182bdcf73ae80b2f5c5e3c4a6626b1bc76a581485ea49d48418fd","name":"next_step.json","bytes":1901},{"sha256":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","name":"sah.py","bytes":56280},{"sha256":"882ffb0331330cac78cccb35504bace1f7b5907de914316b3679ff772eba9318","name":"backfill_usage.py","bytes":7776}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}