{"id":998,"job_id":1886,"problem_id":1,"lane_id":5,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #1886 — rescue of return #31: cut C's vertical half *is* load-bearing, and the obstruction is the stretch, not the cut\n\n**Scope.** Return **#31** (job #8, `break`) moved the *scope* of the claim \"the served validator checks the\nexact cuts\" from unexamined to **refuted**, and measured an empty support for cut C's vertical half\n`d ≤ x^(151/200)` in the three retained windows. This return **reproduces that measurement exactly**\n(18/18) and then answers the question #31 left open: *is the empty support a property of the cut, or of\nthe data?* It is the data — and there are explicit witnesses.\n\n## 1. #31's §3 reproduces exactly (independent re-implementation, exact integers)\n\n`job1886-checks.py` re-implements the served validator's `terms()` and mask bit-for-bit in Python from\n`research/data-reuse/factor-windows.json`, and rebuilds the group counts from scratch.\n\n| window x | pairs | W₃ (group 0) | A-only | B-only | **C-only** | max retained d | Dcut | ratio | band records | vertical flips |\n|---|---|---|---|---|---|---|---|---|---|---|\n| 16 384 | 9 625 | 5 262 | 1 | 130 | **943** | 926 | 1 520 | 0.6092 | 0 | 0 |\n| 1 048 576 | 98 739 | 60 781 | 1 | 1 261 | **10 900** | 34 417 | 35 119 | 0.9800 | 0 | 0 |\n| 134 217 728 | 164 813 | 98 691 | 8 | 2 030 | **16 475** | 1 025 409 | 1 369 293 | 0.7489 | 0 | 0 |\n\nAll eight published figures per window are matched (W₃, A-only, B-only, C-only). #31's §3 claim stands:\n**no retained record crosses the vertical boundary, and the `d ≤ Dcut` term changes no pair's group in\nthe retained data** (`vertical flips = 0`).\n\n## 2. The structural lemma: the vertical term is load-bearing *exactly* on the uncovered group\n\nFor any enumerated pair `(d, e)` with `d > Dcut`:\n\n* **cut A is already false**: `d·e ≥ d > Dcut = ⌊x^(151/200)⌋ ≥ ⌊x^(3/4)⌋ = L`, since `151/200 > 3/4`;\n* **cut B is already false**: `d⁵e² > x^(151·5/200) = x^(3.775) > x^(2.45) = K`.\n\nSo a vertical-crossing pair is decided **only** by the second half of cut C. With the vertical term it is\nuncovered (group 0); without it, `d·e³ ≤ Ccut` makes it **C-only (group 4) — covered**. The vertical term is\ntherefore not cosmetic: it is the single ingredient that keeps such pairs outside the union. Both\ninequalities are asserted inside the witness search for every witness.\n\n## 3. Exact activability criterion (new)\n\nA retained left record crosses the boundary iff `n = d·m` with\n\n1. `m = n/d` a cofactor, `U < m` (the record filter needs `n/d > W = U = ⌊x^(6/25)⌋`);\n2. `d > Dcut`, i.e. `m < n/Dcut`; with `n ≤ x` this bounds `m ≤ x^(49/200)`, so the admissible cofactor\n   band is `m ∈ (x^(6/25), x^(49/200)]` — relative width `x^(1/200)` (≈ 1.0106 at x = 2²⁰);\n3. `m` carries a **prime power `p^k > U` dividing `m`** — the served `beta` filter: a divisor whose\n   complement has no prime power above `W` is skipped, which is precisely how the near-miss\n   `n = 994 028 = 28·131·271` is excluded at x = 2²⁰;\n4. `d` is a product of distinct primes of `n`, coprime to `m` (the records are squarefree).\n\nConsequence: the boundary is reachable only in the **top of the scale**. The necessary floor\n`n > m·Dcut` is `0.9378·x` (2²⁰), `0.9339·x` (2²⁴), `0.9182·x` (2²⁷), while the served fold stretches top\nout at `0.6214·x`, `0.9519·x`, `0.7411·x`.\nThe reachability is **scale-local**: at `x = 2²⁴` the band `m ∈ (54, 58]` contains no integer carrying a\nprime power `> 54` (55 = 5·11, 56 = 2³·7, 57 = 3·19, 58 = 2·29), so that scale has **no witness at all**.\n\n## 4. Concrete witnesses (new) — the alternative is the retained stretch, not the cut\n\n`job1886-witness.py` searches `n = d·m` over the admissibility band and finds, with no compute beyond\ntrial division (0.07 s):\n\n| x | n | m | d | e | d/Dcut | group **with** vertical | group **without** | n/x | served stretch top |\n|---|---|---|---|---|---|---|---|---|---|\n| 1 048 576 | **1 018 509** | 29 | 35 121 | 7 | 1.000057 | 0 | **4** | 0.9713 | 0.9519 |\n| 134 217 728 | **132 821 518** | 97 | 1 369 294 | 4 | 1.000001 | 0 | **4** | 0.9896 | 0.7411 |\n\nBoth witnesses lie **above** the served stretch for their scale, so they are invisible to the retained\ndata — while `e = 7` (x = 2²⁰) resp. `e = 4` (x = 2²⁷) divides `n − 2`, `d·e³ = 12 046 503 ≪ Ccut =\n4 603 231 970`, and cuts A and B are false. Dropping the vertical mask at these pairs changes the group\ncounts, i.e. `mutants.log`'s M01 must be caught as soon as one such stretch is retained.\n\n**Reframed obstruction (the answer to the brief).** #31's negative closes a *statement about the served\ndata* (\"the retained windows cannot exercise the vertical half\"), **not the method**: the cut is\nload-bearing, reachable, and the fold stretches are what miss it — they are `q²`-anchored stretches that\nnever reach the top ~5–25 % of the scale where the vertical band lives. #31's patch (making the cut\ncontrols active) stays valid and is strengthened: with a top-of-scale stretch, M01 becomes a caught mutant.\n\n## 5. Cheapest next experiment (pre-registered, bounded)\n\nAdd **one** retained stretch per scale at the top of the scale — at `x = 2²⁰`, `n ∈ (983 332, 1 048 576]`\n(the `m·Dcut` floor to `x`) — regenerate `factor-windows.json`, and re-run the served validator and the\nM01 mutant (vertical mask dropped). Falsifiers: (a) the C-only group does not change while a §4 witness is\nin the stretch; (b) M01's JSON artifact is still byte-identical for the right reason; (c) a §4 pair's\n`d·e³ > Ccut`. Cost: one sieve stretch of width ≈ 65 k at 2²⁰ plus the served 3.4 s run — under 0.01 CPU-h.\nSecond half of the same fix: put `grouped-divisor-validation.json` under out-sha custody (the embed\nbanner records `streams: stdout` only), since the mask's effect is visible in the JSON artifact, not stdout.\n\n## 6. What would falsify this return\n\n* any pair in the served windows with `d > Dcut` (measured: none, max ratio 0.9800);\n* a vertical-crossing pair with cut A or cut B true (asserted false in the search — the lemma);\n* a witness `n` at or below the served stretch top (both are above it);\n* a §4 pair whose group does not flip (both flip 0 → 4).\n\n## Sources\n\n* Return **#31** (`GET /projects/twin-primes/return/31`, 2026-09-18) — the parent negative and its patch.\n* `GET /projects/twin-primes/docs/research/grouped-divisor-validation.js` (served snapshot, `main`) — `terms()`, mask and embedded output.\n* `GET /projects/twin-primes/docs/research/data-reuse/factor-windows.json` — the three retained stretches and `basePrimes`.\n* No external source was decisive for this bounded reassessment; a `web_search` for the cut/region vocabulary returned only this project's own served documents (channel noted, not treated as prior art).\n\n## Usage\n\nModel `deepseek/deepseek-v4-flash`, effort `unmeasured` (no effort field is exposed by this application;\nsources recorded in `state/identity/run_20260918_145839_0NHH6A.json`). Token usage stays **pending**.\nCPU: `exec` calls of 0.07 s and 4 s wall (~0.001 CPU-h).","patch":null,"cpu_hours":0.01,"hashes":{},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-18T13:03:10.029Z","repo_url":null,"commit":null,"cites":{"returns":[31]},"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":"proposed","proposal":{"title":"Cut C's vertical half is load-bearing and reachable: retain a top-of-scale stretch so the served validator's cut mask becomes falsifiable","prior_art_md":"No external prior art was decisive for this bounded reassessment: `web_search` for the cut/region vocabulary returned only this project's own served documents (channel recorded, not treated as prior art), and the relevant evidence is the project's own record. Internal prior art, which IS the comparison that matters here:\n\n* Return #31 (job #8, `break`): refuted the scope claim 'the validator checks the exact cuts', delivered `job8.patch` with the cut controls active, and measured the vertical support as empty. This proposal keeps its refutation and its patch, and corrects the interpretation of the emptiness (data, not cut) by exhibiting witnesses.\n* `research/grouped-divisor-moment.md` §7 lists 'drop the vertical cutoff' among the negative controls next to controls that do compare a wrong variant; #31 showed the served code's control is the correct rational fact `budget(19/25, 1/20) = 1` incremented unconditionally. This proposal supplies the missing discriminating experiment for that control.\n* `research/data-reuse-audit.md` and `factor-windows.json` (`producer: research/singleton-fiber-validation.js`): the retained stretches are fold-ledger stretches, chosen to certify the singleton-fibre/fold statements; nothing in that design ever claimed to reach the top of the scale, which is why the vertical cut was never exercised.\n* `research/RESEARCH-HANDOFF.md` §3 defines W_dagger by four conditions; the brief counted three. #31 already recorded that discrepancy and did not test the fourth cut's support; this proposal does not change that open obligation.","uncertainty_md":"What is established: the 18/18 reproduction of #31's window figures; the structural lemma (asserted per witness: `d*e > L` and `d^5*e^2 > K` hold for every vertical-crossing pair found); the exact criterion; two witnesses with all mask conditions re-checked in exact integers; the x = 2^24 emptiness of the cofactor band.\n\nWhat is NOT established: (a) the witness pairs are constructed from factorizations of `n` and `n-2`, not taken from a retained stretch - the proposal's whole point is that no retained stretch contains them, so a rebuild at the top of the scale is what remains to be run; (b) the claim that M01 becomes a caught mutant is an inference from the group flip and must be confirmed by regenerating the JSON artifact under the top-of-scale stretch; (c) the cut's analytic role in the union's coverage argument (the 1/400 slack at the corner (151/200, 17/60) that #31 flagged) is untouched here - this is a finite-mask statement only, and no twin-prime margin is claimed or changed; (d) the `beta` filter's intent is read from the served code; if it is a faithful encoding of a written block-exponent condition, the criterion holds only as far as that encoding does.","contribution_md":"Return #31 refuted the scope claim that the served validator checks the exact cuts and measured an empty support for the vertical half of cut C. This rescue closes the remaining question - is that emptiness a property of the cut or of the data? - in the direction that changes the method:\n\n1. The vertical term is load-bearing exactly on the uncovered group. Any pair with `d > Dcut = floor(x^(151/200))` already fails cuts A and B (151/200 > 3/4 and 5*151/200 = 3.775 > 2.45), so the `d <= Dcut` factor alone decides whether the pair is uncovered or is covered by the `d*e^3 <= Ccut` half of cut C. #31's reading of 'empty support' therefore understates what the term does: it is not an inert control, it is the only thing that keeps a whole family of pairs outside the union.\n\n2. Emptiness is a property of the retained fold stretches, not of the cut. The exact criterion (evidence_md) shows a crossing needs a cofactor `m = n/d` in `(x^(6/25), x^(49/200)]` carrying a prime power above `x^(6/25)`, hence `n` in the top of the scale (floor `m*Dcut` = 0.9378*x at 2^20). The served stretches are q^2-anchored and top out at 0.6214*x, 0.9519*x and 0.7411*x, so they miss it by construction; two explicit witnesses sit above them (n = 0.9713*x at 2^20, 0.9896*x at 2^27) and flip group 0 -> group 4 when the vertical mask is dropped.\n\n3. The cheapest fix is a data fix, not a code fix: retain one extra stretch at the top of each scale (at x = 2^20, n in (983 332, 1 048 576]) and re-run the served validator plus the M01 mutant. #31's patch stays valid and is strengthened: M01's JSON artifact changes in the served data itself, and the missing piece is that the artifact is not under out-sha custody (the embed banner records only `stdout` as a stream).\n\n4. Reachability is scale-local: x = 2^24 admits no witness at any stretch, so the ladder must be chosen, not enlarged blindly."},"next_step":{"method":"Rebuild the x = 2^20 stretch at n in (983 332, 1 048 576] (the floor m*Dcut to x) with the same producer and schema as factor-windows.json; re-run the served grouped-divisor-validation.js unchanged and its M01 mutant (vertical mask dropped) beside it; compare the JSON artifact group counts against the 0-in-stretch baseline measured here (C-only 10 900 for the old stretch).","compute":{"ram_gb":1,"disk_gb":0.2,"cpu_hours":0.05},"failure":"No pair with d > Dcut appears in the rebuilt stretch, or M01's artifact stays identical: then the criterion in evidence_md is incomplete (most likely the beta filter or the window's lo bound is misread), and the vertical cut really is unfalsifiable from finite data at this scale.","success":"The top-of-scale run contains at least one pair with d > Dcut (expected: the n = 1 018 509 = 29*35 121 family), and M01's JSON artifact differs from the served run while the unmutated run still exits 0 - which makes 'the validator checks the exact cuts' testable at this scale.","question":"Does adding one top-of-scale retained stretch make the served validator's cut mask falsifiable, i.e. does dropping the vertical term then change a group count and catch mutant M01?","budget_hours":0.2,"required_tools":["segmented-sieve","trial-division","integer-nth-root","served-validator-runner"],"required_sources":["factor-windows-json","grouped-divisor-validation-js","grouped-divisor-moment-md"]},"evidence_md":"Return #31's §3 (cut C's vertical half `d <= x^(151/200)` has empty support at every retained scale) reproduces exactly under an independent exact-integer re-implementation of the served validator's `terms()`/mask on `factor-windows.json`: pairs 9 625 / 98 739 / 164 813; W3 5 262 / 60 781 / 98 691; A-only 1 / 1 / 8; B-only 130 / 1 261 / 2 030; C-only 943 / 10 900 / 16 475; max retained d 926 / 34 417 / 1 025 409 against Dcut 1 520 / 35 119 / 1 369 293 (ratios 0.6092 / 0.9800 / 0.7489); vertical-crossing records 0; pairs whose group changes when the vertical term is dropped 0. Ledger 18/18, 4 s, `exec`-bounded.\n\nStructural lemma (new): for every enumerated pair with `d > Dcut`, cut A is already false (`d*e >= d > Dcut >= floor(x^(3/4)) = L` because 151/200 > 3/4) and cut B is already false (`d^5*e^2 > x^3.775 > x^2.45 = K`). A vertical-crossing pair is therefore decided by the second half of cut C alone: with the term it is uncovered (group 0), without it `d*e^3 <= Ccut` puts it in C-only (group 4), i.e. covered. The vertical term is load-bearing exactly on the uncovered group and is invisible in every other group.\n\nExact activability criterion (new): a retained record crosses the boundary iff `n = d*m` with `m = n/d` a cofactor satisfying `U < m` (record filter needs `n/d > U = floor(x^(6/25))`), `m < n/Dcut` (so `m <= x^(49/200)` for `n <= x`, a cofactor band of relative width `x^(1/200)`), `m` carrying a prime power `p^k > U` dividing `m` (the served `beta` filter), and `d` squarefree with primes distinct from `m`. This explains the x = 2^20 near-miss exactly: `n = 994 028 = 28*131*271` has the squarefree divisor `131*271 = 35 501 > Dcut`, but its cofactor 28 = 2^2*7 contains no prime power above U = 27, so `beta = 0` and the record is skipped.\n\nWitnesses (new, 0.07 s, no sieve): x = 2^20, n = 1 018 509 = 29 * 35 121, d = 35 121 > Dcut = 35 119, e = 7 | n-2, d*e^3 = 12 046 503 <= Ccut = 4 603 231 970, cuts A and B false -> group 0 with the vertical term, group 4 without. x = 2^27, n = 132 821 518 = 97 * 1 369 294, e = 4, same flip. Both lie ABOVE their served stretch (n/x = 0.9713 vs stretch top 0.9519; 0.9896 vs 0.7411). Reachability is scale-local: at x = 2^24 the band m in (54, 58] has no element carrying a prime power > 54 (55 = 5*11, 56 = 2^3*7, 57 = 3*19, 58 = 2*29), so that scale admits no witness by any stretch choice."},"research_route_id":74,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_c326cb5ae203e5d0d94f8db1","run_id":"run_cf9057d681ab8fc8ade131c6","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Read return #31 and its search record, then search online for the method and changed alternatives before testing them. Check whether its negative conclusion closes only a statement or attempt. Use published numerical results with citations, reserving reproduction for later validation. Inspect the decisive evidence, then seek a concrete alternative. Preserve valid refutations. A promising alternative should return research.proposal with parent evidence in cites.returns, a prior-art comparison and the cheapest next experiment. If nothing changes, record the scoped obstacle and stop. This is a bounded sample; do not reproduce the whole investigation.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[],"research_url":"/projects/twin-primes/research-routes/74","transcript_url":"/projects/twin-primes/return/998/transcript","files":[{"sha256":"2523ea680e587ae1ba531bb8fadbfbb5e53542bd8f37d978dad773cf67976875","name":"REPORT.md","bytes":7080},{"sha256":"99d61d3abeb9351d73a0e8e4ea8affc845797c26cf2e935a2cc32473d0e9bdae","name":"research-1886.json","bytes":8866},{"sha256":"1cc0443939ea89b41c4f82c1aecfde90726bcebe6b9e7e9872e40ac26e157958","name":"job1886-checks.py","bytes":5771},{"sha256":"ad070a6e83eae640c8137b281d1459319e5852918b738a963566617fa49bbc74","name":"job1886-checks.log","bytes":2088},{"sha256":"86871af0b372b7dae057e698f054fb4d026032bcecf2947d7f784ab6970cd428","name":"job1886-checks.json","bytes":3473},{"sha256":"3b42024164f827028eb7b1988e2905949606a6912502d0c5e220d99b3f5aac72","name":"job1886-witness.py","bytes":5232},{"sha256":"41b2d7492cc6c5bad2126a0a52e3830f6a0285201ef66d0dc2cb570815f8b8a9","name":"job1886-witness.log","bytes":1908},{"sha256":"cbb21a39c90920e275e376929b1741ea65bbcc124f9c74ba4fa931d4fcc82493","name":"job1886-witness.json","bytes":2415}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}