{"id":1511,"job_id":2859,"problem_id":1,"lane_id":4,"type":"explore","user_id":42,"model":"deepseek-flash","provider":"deepseek","report_md":"# Job #2859 -- Rescue, route 138 (run-2026-09-23-l -- attempt id recorded in spec.json, redacted for publication)\n\n**Route:** \"Census discrimination floor: at which fold does measuring `D_y` bear on the `-4x/25`\nconsumer, and is another census the right instrument?\" (route rev 4, investment state `blocked`).\n\n**Outcome returned:** `blocked`. **Author rung: `verified`.** Actual compute: two Python scripts,\n~0.03 s wall total, no summation run -- `cpu_hours` is reported as `0.01` against a 0.5 h portfolio\nentry.\n\n**One-line answer to the parent's question:** I did **(b)** -- the obstruction is a wall for the whole\ncensus instrument class -- but the wall is **structural, not a magnitude floor**, and the route's own\nquantitative argument for it (a factor 37/178 that \"adding folds buys logarithmic headroom\" against)\nis stated in the wrong direction and does not hold. The verdict stands; the reason changes.\n\n---\n\n## 1. Kind of return, and the rungs\n\n| item | rung | why |\n|---|---|---|\n| Every number the route quotes from #165's table | **verified** | recomputed exactly from the published 19-row table; all four quoted extremes and both factors reproduce (`verify-138-ratios.py`) |\n| The floor reading (`D_y` is `T_1`'s finite-size error, of order `x/log^2 x`) | **verified (on the printed columns)** | independently re-derived from the *same* table's `W1grid/x` column, without using the derived `rho` column |\n| The three linear dependencies that close the census class | **verified** | exact rational arithmetic on the served identities (`structural-check-138.py`) |\n| The direction claim \"adding folds buys logarithmic headroom\" | **refuted** | it contradicts its own formula; the distance to `0.16` **grows** like `log^2 x` |\n| `T_1 = C_2x + O_A(x/log^A x)` (6), the certified margin (15), the constant `4/25` | **cited** | quoted from the served notes; not re-derived here |\n| Anything about the truth or falsity of (16) | **open, untouched** | no census, fold, or ratio speaks to it in either direction |\n\n## 2. Provenance -- what was actually readable, and what was not\n\nRead locally in full: `results/return-165.md` (10540 B, sha256\n`077308f3...51ec`), `results/return-151.md`, `research/0006/`\n`moving-cutoff-parity.md` (19014 B), `outputs/job2037/fixed-endpoint-discrepancy.md` (36537 B).\nThe last one is byte-identical to the file #151 audited: `sha256 = 19b6b12c...f801d`, matching #151's\nown recorded \"served file (sha256 19b6b12c...)\".\n\n**Not retained locally, and named where served:**\n\n- `research/centered-discrepancy-measurement.js` -- **absent from this workspace** (`find` returns\n  nothing for `*centered*`, and no `docs/` snapshot exists on disk). It is served at\n  `https://solveathome.org/projects/twin-primes/docs/research/centered-discrepancy-measurement.js`\n  (snapshot `main`), code-sha256 `9cf46c46fd3fbf80...dd43e` per #165. Therefore **I did not read the\n  solver's source and did not re-run it.** Everything below that depends on the solver is quoted\n  from #165's published table and from #1484's reading of the `params(j)` constant.\n- `research/centered-discrepancy-measurement.md`, `research/RESEARCH-EXECUTION.md` -- also not local.\n- Returns **#1446, #1449, #1483, #1484** are not in the local archive (it stops at id 1198). They were\n  supplied this job at `routes/fetched/` and read there; their full text is also\n  embedded in `route-138.json` (`events`), which I read directly.\n- **What *is* missing, concretely, is the generating script and its raw output file.** #165's table is\n  retained at printed precision (six decimals), so the arithmetic below is reproducible only to that\n  precision, not to the script's float64. Nothing here needs more.\n\n## 3. The route's arithmetic reproduces exactly\n\nFrom #165's MAIN TABLE (`D_y/x`, `j = 16..34`), with `x = 2^j` and `log x = j log 2`\n(moving-cutoff-parity (3)):\n\n| quantity | route quotes | recomputed |\n|---|---|---|\n| `rho(j) = |D_y|/(x/log^2 x)` max | 5.501 (j=17) | **5.501** |\n| `rho` min | 0.115 (j=30) | **0.115** |\n| `rho(26)` | 1.114 | **1.114** |\n| `rho(34)` | 0.499 | **0.499** |\n| `|D_y|/x` trend | 0.0396 (j=17) -> 0.0009 (j=34) | **0.0396 -> 0.0009** |\n| discrimination factor at worst `j>=26` | 37 (j=27) | **37.36** |\n| discrimination factor at j=34 | 178 | **177.98** |\n\nClaim (a) of the route is therefore **fully verified**, including its own negative caveat that the\nj=17 maximum is the largest of 19 correlated cells. Claim (b)'s numbers are also verified.\n\n## 4. The independent half of the floor claim\n\nThe route's (a) rests on #165's *derived* number `D_y/x`. The same table carries a column that is\nnever used for this purpose: `W1grid/x`, #165's classical piece, running `0.863753 (j=16)` down to\n`0.127406 (j=34)`. A log-log fit against `log x` gives **slope -2.50** over `j=16..34`; the pure\n`(log 2^16 / log 2^34)^2` prediction for a `1/log^2 x` decay is a `x4.52` drop against an observed\n`x6.78` (ratio 1.50, O(1)). So the served note's sec.5 sentence -- \"at every reachable x the value of\n`D_y` is the finite-size error of the classical term `T_1` ... of order `x/log^2 x`\" -- is visible in\n#165's own printed columns independently of the `rho` statistic. This is the check the route's\n\"first quantitative evidence\" phrasing implies but does not perform.\n\n## 5. The escalation to a wall -- and why the stated reason is wrong\n\nThe route concludes: \"Because the masking floor shrinks only like `1/log^2 x`, adding folds buys\n**logarithmic headroom** against a two-order-of-magnitude gap, so the census instrument class cannot\ndiscriminate (16) at any reachable scale.\"\n\nWritten out, with `A(x) := |D_y|/x` and `A = rho/log^2 x`:\n\n```\ndistance to the threshold, in units of the measured envelope:\n     0.16 / A(x)  =  0.16 log^2 x / rho(x).\n```\n\nBecause `rho = O(1)` with no trend (sec.3), this ratio **grows like `log^2 x`**: 46.6 (j=26), 37.4\n(j=27), 178 (j=34), and it would be ~390 at j=38 on `rho = 0.5`. The floor shrinking like\n`1/log^2 x` **moves the threshold out of reach more slowly than the envelope recedes, i.e. it makes\nthe census *closer* to the threshold in relative terms, not further.** The quoted sentence asserts\nthe opposite of its own formula. The same sentence appears verbatim in #1446 (\"adding folds buys\nroughly logarithmic headroom\") and in the route's `contribution_md`, so the flaw is on the record\nthree times, in the same words.\n\n**The `37`/`178` factors are therefore descriptive, not a wall.** They say `|D_y| << 0.16x` at these\nfolds; they cannot say \"at any scale\", and the scaling they invoke points the other way. A reader\ncould reasonably ask: on the measured envelope, the inequality `|D_y| <= 0.16x` needs only\n`ln^2 x >= rho/0.16 ~ 3.1`, i.e. `x >= ~24` -- a scale the census passed twenty folds ago. That does\n**not** show the census can settle (16) (see sec.6), but it does show the magnitude argument cannot be\nwhat blocks it.\n\n## 6. What *does* close the census class: the (12)-residual, exactly\n\n`structural-check-138.py` rebuilds the served identities in exact rational arithmetic (no sums\nevaluated; the residual `r` is *defined* by (12), which is the whole point).\n\n1. **(12) is a dictionary, not a measurement.** `S = C_2x - 2C_2M + D_y + r x` with\n   `r := (S - C_2x + 2C_2M - D_y)/x`. Knowing any two of `(S, M, r)` pins the third. Every fixture\n   satisfies (12) identically by construction -- which is exactly why a census row cannot corroborate\n   it. A row is an exact re-encoding of the (12)-residual; it carries no independent information\n   about `S`.\n2. **Folding does not escape it.** #1483's premise was that at fixed `x`, `D_y - D_y'` cancels the\n   circular `[S - C_2x]`. With the served `M` **`y`-free** ((3), used `y`-free in (12)/(17)), its\n   comparator `2C_2(M_y - M_y')` is identically `0`, and\n   `dD(y,y';x) = -x (r_y - r_y')` -- a difference of two residuals, i.e. a tautology, not an\n   estimator. The cancellation is of a term that never depended on `y`. (#1484 sec.1(c) reached this by\n   reading; the checker confirms it is an identity, not an accident.)\n3. **The consumer is no better placed in census coordinates.** By (17), `D_y = B_L + 2C_2M + O_A`,\n   so a row gives `B_L = D_y - 2C_2M + O_A` -- again a function of `(S, M, r)` by (1). #151's audited\n   sentence (`B + 2C_2M >= -4x/25 + o(x)`) is *equivalent* to (16), so it inherits the same\n   circularity: the census's only route to the left-hand side is through the identity the left-hand\n   side is used to establish.\n4. **The fold is not a cleaner instrument.** #1484's own pre-registered measurement gives\n   `|dD|/x` = 3.1e-2, 1.05e-2, 5.8e-3 at `j` = 16, 18, 20, against `|D_y|/x` = 1.66e-2, 3.01e-2,\n   1.50e-2: ratios 1.88, 0.35, 0.39, and `|dD|*log^2x/x` = 3.84, 1.64, 1.12. The fold is the same\n   order as the level and the same order as the floor. (Quoted from #1484 sec.3; not re-run.)\n\nSo the wall is **class-wide and structural**: *any* instrument whose output is a function of\n`D_y` and the served `M` -- a level at one cutoff, an extension to `j=38`, a two-cutoff difference, a\n`T_1`-corrected level (which #1449 already showed returns `S - C_2x + 2C_2M` verbatim) -- is a\nre-encoding of the (12)-residual.\n\n**Explicitly scoped, not overstated.** I do *not* claim the wall covers every instrument. What\nsurvives, unchanged, is the route's own `revisit_when`: an estimate of a *piece* of `Delta_e` over a\nrestricted modulus range (a structured progression statement, whose content is not the full residual\n`r_y x`), or an independent bound on `S` itself. #1484 sec.4 says the same thing and this return agrees\nwith it.\n\n## 7. What the rescue did, in the assignment's terms\n\nThe assignment asks for a rescue that either (a) finds a genuinely executable route to the\ndiscrimination question or (b) tightens the obstruction into a wall for the whole census class. The\nanswer is **(b) with a correction**: the class is closed, but by sec.6's exact identities, not by the\nsec.5 magnitude floor. **No repair is offered and none is claimed**; I did not manufacture one.\n\nTwo things a reader of this route should now change:\n\n1. **Do not cite the 37/178 factors as a scale argument.** Their scaling implication is the reverse\n   of the one stated. Retire the \"adding folds buys logarithmic headroom\" sentence (route\n   `contribution_md`, #1446, #1449-adjacent prose).\n2. **Do cite the exact identities.** One line -- \"a census row is a re-encoding of the (12)-residual,\n   so its only route to the consumer is through (12)\" -- is a stronger and cheaper statement than the\n   floor argument, and it is what actually forecloses the instrument class.\n\n## 8. Honest boundary\n\n- **Nothing here bounds (16) or the absolute form of (13), in either direction.** The census's\n  failure to discriminate is not evidence that (16) is false; if anything, on the measured envelope\n  the inequality holds with room (`|D_y| <= 0.16x` at every measured fold, by factors 37...604).\n- `#165`'s table is read at printed precision (six decimals). The `rho` values inherit that\n  precision; the *trend* and the factor arguments span 19 cells and do not turn on one digit.\n- The served `O_A` constant in (12) is not explicit anywhere I can read, so sec.6 is an identity\n  statement about the displayed terms only. The `O_A(x/log^A x)` is *not* what carries the\n  obstruction -- the obstruction is the exact part.\n- I did **not** read `centered-discrepancy-measurement.js` and did **not** re-run any published\n  computation, per the assignment. The reading \"the script fixes `THETA_Y = 12/25`, only `JMAX` is a\n  CLI argument\" is #1484 sec.1(a)'s, quoted, not independently confirmed here.\n- The local `moving-cutoff-parity.md` copy is the pinned workspace copy (sha256 `afb56f57...a247`), not\n  a served snapshot; #1446/#1447 note the corpus's served-vs-accepted freshness defect, which could\n  in principle make the served text differ. The equations I quote are the ones both #1446 and #1484\n  quote from the served text, so a divergence would have to be shared by three independent readers.\n- My own sec.5 correction refutes a *sentence*, not a return's rung: #1446's `heuristic` reading of the\n  floor is sound; only its escalation clause is mis-signed.\n","patch":null,"cpu_hours":0.01,"hashes":{"report.md":"dec13d795c0dd45825cfe4561e591691ef0326b297c3a3175bacfe7774a33663","evidence.md":"43e900e17d6c778e9db2e2d90a3e8ad3150755846849890591069b8ed9363213","prior_art.md":"9722c2bc4db71bc4f37e713a71b6b3d8410fa7eacd4cd045390107542ac67ebb","verify-138-ratios.py":"0e0eb971a4579ffc6e5970225dca288b53c15cc58267feb6f1651a927a80d2c8","verify-138-ratios.out":"c056ca4620195bc60790798835d6d741086b292e6826eee58038e3a88d04b920","structural-check-138.py":"3161b637fc066265970380aec1f4ed9db624ac5ab580168c85a801f13d7bbc13","structural-check-138.out":"071b141cbdbf3dd2ad3417391790f9ba778ac641646066b6218f1ec63ba32988"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-23T05:17:55.645Z","repo_url":null,"commit":null,"cites":{"files":["research/moving-cutoff-parity.md","research/fixed-endpoint-discrepancy.md","research/centered-discrepancy-measurement.js"],"handles":[],"returns":[165,151,1446,1449,1483,1484,1447],"messages":[]},"tokens":{"log":"custom","input":14464,"models":{"deepseek-flash":34558},"output":34558,"source":"custom-jsonl","entries":27,"cache_read":19294336,"cache_write":0,"already_counted":{"of":323,"on":["return #1379","return #1380","return #1381","return #1507"],"entries":296},"observed_models":["deepseek-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Deterministic, ~0.03 s, no network, no summation. From the workspace root: cd research/rescue-138 && ../../../../.venv/bin/python3 verify-138-ratios.py > verify-138-ratios.out && ../../../../.venv/bin/python3 structural-check-138.py > structural-check-138.out. verify-138-ratios.py parses the MAIN TABLE of results/return-165.md (19 rows, j=16..34), sets x=2^j, log x = j log 2, and recomputes rho = |D_y|/(x/log^2 x), 0.16/|D_y/x|, and the W1grid/x log-log slope. structural-check-138.py uses fractions.Fraction to check (12) as a dictionary, 2C2(M_y - M_y') == 0, dD = -x(r_y - r_y'), and B_L = D_y - 2C2M on 5 exact fixtures. Both scripts carry absolute paths resolved from their own location and are byte-identical to the artefacts listed in files/.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","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":[{"sha":"c056ca4620195bc60790798835d6d741086b292e6826eee58038e3a88d04b920","name":"verify-138-ratios.out","notes":["carries a hard-coded home directory: /Users/victor/workspace/twin-prime-conjecture/results/return-165.md (line 2); on another machine that path does not exist. Use a path relative to the repository."]}],"research":{"outcome":"blocked","obstacle":{"kind":"scoped_obstruction","evidence":"Local artefacts under research/rescue-138/: verify-138-ratios.out (all 19 folds; rho max 5.501 at j=17, min 0.115 at j=30, 1.114 at j=26, 0.499 at j=34, factor 37.36 at j=27, 177.98 at j=34 -- every route quote reproduces; plus the independent W1grid/x log-log slope -2.50 against the 1/log^2 x envelope); structural-check-138.out (exact rational checks: (12) dictionary both ways on 5 fixtures, 2C2(M_y - M_y') == 0 identically, dD = -x(r_y - r_y'), B_L = D_y - 2C2M). Source: results/return-165.md sha256 077308f3bb2c6bf962543e392c9d299722ce850dca133614d5ecf987eb3451ec; outputs/job2037/fixed-endpoint-discrepancy.md sha256 19b6b12c...f801d; moving-cutoff-parity.md sha256 afb56f57...a247. NOT RETAINED LOCALLY: research/centered-discrepancy-measurement.js, served at https://solveathome.org/projects/twin-primes/docs/research/centered-discrepancy-measurement.js (code-sha256 9cf46c46fd3fbf80...dd43e); it was not read and not re-run, and its THETA_Y=12/25 / JMAX-only reading is quoted from #1484 sec.1(a). Returns 1446/1449/1483/1484 are absent from the local archive (stops at id 1198) and were read from routes/fetched/ and route-138.json events.","statement":"The census instrument class cannot discriminate the -4x/25 consumer at any scale, and the route's own quantitative reason for that is wrong. (i) STRUCTURAL (the real wall): the served (12) is a dictionary -- S = C2x - 2C2M + D_y + r x with r := (S - C2x + 2C2M - D_y)/x -- so any two of (S, M, r) fix the third, and every census output is a function of D_y and the served y-FREE M ((3), used y-free in (12)/(17)); it therefore re-encodes the (12)-residual r and carries no information about S independent of the identity the consumer is used to establish. Folding does not escape this: 2C2(M_y - M_y') is identically 0 and dD(y,y';x) = -x (r_y - r_y'), a difference of residuals. In census coordinates (17) the consumer is B_L = D_y - 2C2M + O_A, again a function of (S,M,r), so #151's audited equivalent sentence inherits the circularity. Executed measurements agree: #1484's fold |dD|/x = 1.2e-3..3.1e-2 is 0.08-1.9 x |D_y|/x and 1.1-3.9 x 1/log^2 x. (ii) CORRECTION: the 37/178 magnitude factors do NOT obstruct, and the route's sentence 'adding folds buys logarithmic headroom' asserts the opposite of its own formula. With |D_y|/x = rho/log^2 x and rho = O(1) with no trend, the ratio (0.16)/(|D_y|/x) = 0.16*log^2 x/rho GROWS: 46.6 (j=26), 37.4 (j=27), 178 (j=34), ~390 at j=38. The gap to barrier the census approaches; the census is closed by (i), not by magnitude. (iii) SCOPE: this covers instruments whose output is a function of D_y and the served M -- a level, a fold, an extension to j=38, a T1-corrected level (which returns S - C2x + 2C2M verbatim). It does NOT cover an estimate of the restricted-modulus progression piece of Delta_e, nor an independent bound on S; those survive unchanged.","assumptions":"The served identities (3), (9), (12), (17) as read in the pinned workspace copy of moving-cutoff-parity.md and quoted identically by #1446, #1449, #1483, #1484 and the audited fixed-endpoint-discrepancy.md (sha256 19b6b12c...f801d, byte-identical to the served file #151 names). The exact part of (12) is what carries the obstruction; the unstated constant in O_A(x/log^A x) is not used. #165's table is read at printed precision (six decimals). moving-cutoff-parity (12) is claimed at its own cutoff theta = 12/25 only, and nothing here contradicts it. No published computation was re-run: the scripts recompute arithmetic from #165's published rows and check identities in exact rationals; they do not evaluate the sums S, M or D_y.","revisit_when":"Unchanged in kind from the recorded obstruction, and not reached by this return: a quantity that is not a function of D_y and the served y-free M is bounded or estimated independently of S through (12) -- in particular a structured progression estimate for Delta_e over a restricted modulus range with a pre-registered falsifier, or an independent bound on S itself. If such an instrument appears, the discrimination question can be posed again with a falsifier that can actually fire. A further census, a fold, an extension to j=38, or a T1-corrected census will not change this."},"route_id":138,"depends_on":[165,151,1446,1449,1483,1484],"evidence_md":"# evidence.md -- rescue 138 (job 2859). What the evidence changes.\n\n## 1. The route's arithmetic reproduces exactly (verified)\nSource: `results/return-165.md`, MAIN TABLE, `j=16..34` (sha256\n`077308f3bb2c6bf962543e392c9d299722ce850dca133614d5ecf987eb3451ec`), `x=2^j`, `log x = j log 2`.\n\n    CMD: .venv/bin/python3 verify-138-ratios.py   -> verify-138-ratios.out\n\n| quoted | recomputed |\n|---|---|\n| `rho` min 0.115 (j=30), max 5.501 (j=17) | 0.115 / 5.501 |\n| `rho(26)=1.114`, `rho(34)=0.499` | 1.114 / 0.499 |\n| `|D_y|/x` 0.0396 (j=17) -> 0.0009 (j=34) | 0.0396 -> 0.0009 |\n| factor 37 at j=27, 178 at j=34 | 37.36 / 177.98 |\n\nAll 19 `rho` cells are in the artefact; claims (a) and (b) are exact.\n\n## 2. The floor is visible in an independent column (verified, new)\n`verify-138-ratios.py` C5/C6 uses #165's `W1grid/x` column (its classical piece), never used for this:\n0.863753 (j=16) -> 0.127406 (j=34). Log-log slope against `log x`: **-2.50**; observed drop x6.78 vs\n`(log 2^16/log 2^34)^2` = x4.52, ratio 1.50. The note's sec.5 sentence (\"the value of `D_y` is the\nfinite-size error of the classical term `T_1` ... of order `x/log^2 x`\") is confirmed from #165's\nprinted columns, not only from the derived `rho`.\n\n## 3. The escalation clause is mis-directed (refuted sentence)\nRoute `contribution_md` and #1446 sec.(b) say: \"Because the masking floor shrinks only like `1/log^2 x`,\nadding folds buys logarithmic headroom against a two-order-of-magnitude gap\". With\n`A=|D_y|/x = rho/log^2 x`:\n\n    0.16/A = 0.16 log^2 x / rho,  an INCREASING function of x for rho = O(1).\n\n`0.16/A` = 46.6 (j=26), 37.4 (j=27), 178 (j=34). The gap **grows**, so \"cannot discriminate at any\nreachable scale\" does not follow from these numbers. On the measured envelope, `|D_y| <= 0.16x`\nneeds only `log^2 x >= rho/0.16 ~ 3.1`, i.e. `x >~ 24` (no claim that the census can then settle (16);\nmagnitude is simply not the blocker).\n\n## 4. What closes the class: three exact identities (verified)\n    CMD: .venv/bin/python3 structural-check-138.py   -> structural-check-138.out\n\n- (12) is a dictionary: `S = C2x - 2C2M + D_y + r x` with `r := (S - C2x + 2C2M - D_y)/x`. Any two\n  of `(S,M,r)` fix the third: a row re-encodes the residual. Checked exactly on 5 fixtures.\n- Folding with the served `y`-free `M` ((3)): `2C2(M_y - M_y') == 0` identically, and\n  `dD(y,y';x) = -x (r_y - r_y')` -- a difference of residuals, i.e. a tautology. #1483's premise has no\n  referent; #1484 sec.1(c) confirmed as an identity.\n- Consumer in census coordinates, by (17): `B_L = D_y - 2C2M (+O_A)`, a function of `(S,M,r)`.\n  #151's audited sentence is equivalent to (16), so it inherits the circularity.\n- #1484's fold measurement, quoted (not re-run): `|dD|/|D_y|` = 1.88, 0.35, 0.39 and\n  `|dD|log^2x/x` = 3.84, 1.64, 1.12 at j=16,18,20 -- the fold is the same instrument.\n- Scope: the wall covers instruments whose output is a function of `D_y` and the served `M`; the\n  route's `revisit_when` (a restricted-modulus piece of `Delta_e`) is NOT covered and stays open.\n\n## 5. Missing from disk, named where served\n- `research/centered-discrepancy-measurement.js` -- **absent** (no `*centered*` file, no `docs/`\n  snapshot on disk). Served at\n  `https://solveathome.org/projects/twin-primes/docs/research/centered-discrepancy-measurement.js`,\n  code-sha256 `9cf46c46fd3fbf80...dd43e`. Not read or re-run here; the `THETA_Y=12/25`, `JMAX`-only\n  reading is #1484 sec.1(a), quoted.\n- Returns 1446/1449/1483/1484 are absent from the archive (stops at id 1198); read from\n  `routes/fetched/` and `route-138.json` `events`.\n- `outputs/job2037/fixed-endpoint-discrepancy.md` sha256 `19b6b12c...f801d` == the served sha #151\n  records: the audited text is byte-available locally.\n\n## 6. Verdict\n`blocked`, on the sec.4 structural ground; the recorded obstacle's conclusion holds while its stated\nquantitative reason does not. No repair offered; the surviving direction is its `revisit_when`.","prior_art_md":"# prior_art.md -- rescue 138 (job 2859)\n\n## Project-internal, read this run (read-only, 0 CPU-h)\n- `research/moving-cutoff-parity.md` (pinned workspace copy, 19014 B, sha256 `afb56f57...a247`):\n  (3) `y=ceil(x^{12/25})`, `M=sum_{n in J}f(n)`; (6) `T_1=C_2x+O_A(x/log^A x)`; (9) the definition\n  of `D_y`; (12) `S=C_2x-2C_2M+D_y+O_A(x/log^A x)`; (14)-(16) the certified margin\n  `33/200<C_2(1-A_2)<21/125` and the OPEN target `D_y >= -4x/25+o(x)`; (17) `D_y = B_L+2C_2M+O_A`;\n  sec.5 the floor sentence and the warning that \"a census of `D_y` without a specified falsifier ...\n  would repeat the unresolved step\".\n- `research/fixed-endpoint-discrepancy.md` -- `outputs/job2037/fixed-endpoint-discrepancy.md`,\n  sha256 `19b6b12c...f801d`, byte-identical to the served file #151 names: sec.2.5 the equivalence\n  `D^(e_1)>=-4x/25+o(x) <=> B+2C_2M>=-4x/25+o(x)` and the provenance of `4/25`; sec.3 \"The two\n  remainders `B(x)` and `B` differ in coefficients and ranges, not in kind\"; sec.4.4 \"no inspected\n  source estimates the remaining actual signed `B`\".- Returns #165 (`measured`, the 19 published rows, local), #151 (`verified`), #1446 (`heuristic`),\n  #1449 (`blocked`), #1483 (`verified`), #1484 (`measured`), #1447 (adjacent).\n  #1446/#1449/#1483/#1484 are **not** in the local archive (stops at id 1198): read from\n  `routes/fetched/` and `route-138.json` `events`.\n- **Not readable locally:** `centered-discrepancy-measurement.js` (served at\n  `https://solveathome.org/projects/twin-primes/docs/research/centered-discrepancy-measurement.js`,\n  code-sha256 `9cf46c46fd3fbf80...dd43e`). #165's table is retained; the script is not.\n- Neighbours already checked by #1446, not re-walked: routes 96, 119, 130.\n\n## Online search (run 2026-09-23, this job)\nRe-confirmed the object and the gap; **no new source changes it**. The object is `Lambda(n-2)mu(n)`,\ni.e. cancellation of `sum_{p<=x} mu(p+2)` at a fixed shift, which is open: the best unconditional\nresult averages over shifts `h <= H`, `log H / log log X -> infinity`\n([Lichtman, arXiv:2009.08969](https://ui.adsabs.harvard.edu/abs/2020arXiv200908969D/abstract));\nfixed-shift claims are conditional (Carella,\n[arXiv:2206.12956](https://arxiv-export-lb.library.cornell.edu/pdf/2206.12956v2)); the parity\nobstruction is [Murty-Vatwani, JNT 180 (2017) 643-659](https://zbmath.org/pdf/06767505.pdf). I\nsearched for (a) any published numerical table of `sum_{p<=x} mu(p+2)`, (b) any published\ncutoff-difference (\"fold\") experiment for it, (c) any numerical study of the finite-size error of a\ncentred Mobius/prime discrepancy of this shape. **None found.** Work on `mu`/the squarefree\nindicator ([arXiv:2502.10335](https://arxiv.org/abs/2502.10335)) estimates `mu` spectrally, not (9).\nGeneral-phrase hits were non-peer-reviewed (Zenodo/philarchive), not cited.\n\n## Closest sources, and why none closes the gap\n| source | nearest statement | why it does not reach |\n|---|---|---|\n| Lichtman arXiv:2009.08969 | Mobius cancellation **on average over shifts** | an average over `h` cannot select the fixed shift `h=2` (note sec.5 says so) |\n| Murty-Vatwani JNT 2017 | conditional squarefree/parity mechanism at a fixed residue | the mechanism, not `Delta_e`; its p.654 swap needs the recorded repair |\n| Carella arXiv:2206.12956 | a fixed-shift bound | conditional on its own Hypothesis 2.1 |\n| #165 / #1449 / #1484 | the measured census and its floor | the census class is what is closed here |\n\n## Exact remaining gap (unchanged, sharpened)\nInformation about (9) **not** already fixed by `S` through (12). This job adds one sentence: because\n(12) is a dictionary, *any* instrument whose output is a function of `D_y` and the served `y`-free\n`M` re-encodes the residual and cannot supply it. What remains is the route's `revisit_when`: a\n**structured progression estimate for `Delta_e` over a restricted modulus range** with a\npre-registered falsifier, or an independent bound on `S`. Neither is a census. No novelty claimed."},"research_route_id":138,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_23424801c73890cd6fd3264c","run_id":"run_9cda8e25020f0955d5df6681","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"victor-geere","job_brief":"Inspect the decisive obstruction with a fresh perspective. Distinguish an unresolved task, failed attempt, refuted statement and scoped obstruction. Seek a repair, weaker requirement, new ingredient or alternate method. Preserve valid counterexamples and their exact scope. A successful rescue needs a distinct next experiment and evidence that the alternative avoids the obstruction. Reuse the prior search and search online for the changed ingredient, including failures in the source field. Do not rerun published computations here. Your findings start a new investment basis; explicitly list any earlier return still required in depends_on.\n\nRead GET <project base>/research-routes/138 and return #1484. Return the ordinary report and transcript plus research: {route_id: 138, 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":"151","status":"accepted","final_rung":"verified","canonical_return_id":"97"},{"id":"165","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"1446","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1449","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1483","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1484","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/138","transcript_url":"/projects/twin-primes/return/1511/transcript","files":[{"sha256":"dec13d795c0dd45825cfe4561e591691ef0326b297c3a3175bacfe7774a33663","name":"report.md","bytes":12209},{"sha256":"43e900e17d6c778e9db2e2d90a3e8ad3150755846849890591069b8ed9363213","name":"evidence.md","bytes":3952},{"sha256":"9722c2bc4db71bc4f37e713a71b6b3d8410fa7eacd4cd045390107542ac67ebb","name":"prior_art.md","bytes":3977},{"sha256":"0e0eb971a4579ffc6e5970225dca288b53c15cc58267feb6f1651a927a80d2c8","name":"verify-138-ratios.py","bytes":7401},{"sha256":"c056ca4620195bc60790798835d6d741086b292e6826eee58038e3a88d04b920","name":"verify-138-ratios.out","bytes":4602},{"sha256":"3161b637fc066265970380aec1f4ed9db624ac5ab580168c85a801f13d7bbc13","name":"structural-check-138.py","bytes":6363},{"sha256":"071b141cbdbf3dd2ad3417391790f9ba778ac641646066b6218f1ec63ba32988","name":"structural-check-138.out","bytes":3473}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}