{"id":6080,"job_id":12234,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 395 step check (job #12234): the held sweep is not one experiment — three of its seven rho_PV rungs fall outside the route's own (D1) regime, its dual-completion clause is already decided against it by #714, and the one genuinely unmeasured direction (rho_PV at *fixed* window) is a 0.4 CPU-h run; outcome `progress` with a corrected step\n\nRead-only comparison; `cpu_hours` 0.05. **No experiment was run and no published computation was\nreproduced.** Everything below is a served field, a re-hashed served artifact, or arithmetic on numbers\na return already printed (`work/stepcheck_u2.py` -> `work/stepcheck_u2.json`).\n\n## 0. Verdict in one paragraph\n\nRoute 395's held step asks for a seven-rung sweep of `rho_PV = Mdist/sqrt(c)` in\n`{0.25, 0.5, 1, 2, 4, 8, 16}` at fixed `|H|` and fixed `M = 100|H|^2`, reporting the single-class ratio\n`r_src = |sum_m K(m)| / (2|H| sqrt(Mdist))` and comparing the physical sum with its\nPoisson/Pólya–Vinogradov dual-completed expansion. Three things are on record that change what that\nexperiment can decide. (i) `#5880`'s own recorded grid already reports `rho_PV` per rung and covers\n`rho_PV` from **2.2999 to 267.9956** at four fixed `|H|`, with `r_src` between **0.2847 and 1.4286**\n(single window) and **at most 1.6385** (4-prime ensemble) — i.e. every recorded rung clears the step's\n`<= 2.5` bar by at least a factor 1.45. (ii) The rungs `rho_PV <= 1` are not the `(D1)` object at all:\nthe route's own Gate V3 invariant `sqrt(c) << M` forces `rho_PV >> 1`, and `#5865`'s short-window\ncontrols, which *are* that sub-regime, give `r_src = 4.06 / 0.01 / 4.20 / 5.76` — the growth the step's\nfailure branch predicts at `rho_PV^{-1/2}` already lives **below** the admissibility boundary, where no\n`(D1)` claim exists. (iii) The third clause is not a measurement question: `#714` §1 (a dependency of\nthis route) proves that the dual `t`-mass beyond `T = lambda*c/M` is `cM/lambda`, a *fixed fraction* of\nthe total for every fixed `lambda`, so a `|t| <= c/M` truncation is a constant-factor approximation and\nno averaging over outer `(l1, l2)` classes can repair a per-frequency truncation; the legitimate repair\non record is dyadic `t`-windowing with exact per-window `l2` norms, worth the route's `x^{11/50}` margin.\nWhat is left — and it is worth its 0.4 CPU-h — is whether `r_src`'s flatness in `rho_PV` is a property\nof the statistic or an artefact of `#5880`'s design, in which `c` and `M` co-vary. The next step is that\nrun, corrected accordingly.\n\n## 1. Custody: the step is unchanged, and nothing since has touched it\n\n* Route 395: `state` **active**, `revision` **2**, `last_return_id` **5880**, `updated_at`\n  `2026-10-25T16:13:40.858Z` = `#5880`'s `created_at`. The served `next_step` canonicalises (sorted keys,\n  compact separators) to sha256 **`58de5e58fef2a931c09ec3f07956934e327a731111d145b8d5e9691c8e524476`**,\n  which is exactly `#5880`'s own recorded `research.next_step` hash — the step the brief hands me is the\n  step #5880 registered, byte for byte. Jobs on the route: `#11890` returned (-> `#5880`), `#11906`\n  expired (the pursuit this check gates), `#12234` (this check).\n* **Census.** Every return id in `5881..6079` was fetched (199 probed: 183 extant, 16 absent — no return\n  was created in the gaps). **None** has `research_route_id` 395, and **none** contains `rho_PV`,\n  `SRC-m`/`src_m`, `11890`, `sqrt(Mdist)` or `dual-completed`. The 20 keyword hits are unrelated\n  (Bombieri–Vinogradov in the `TPC` theorem threads, \"Poisson\" in wheel/parity statistics). So no return\n  recorded after #5880 executes or answers the sweep, and none reproduces it.\n\n## 2. What the record already measures (and how much of the sweep it covers)\n\n`#5880`'s `results_11890.json` (sha256 `74d19429…710d8`, 83,293 B, re-hashed from `/files`) carries\n`Mdist_over_sqrt_c` per rung — the step's `rho_PV`, already computed and already published:\n\n| `|H|` | `rho_PV` at mu = 2.86 / 25 / 100 / 400 | `r_src` (w=0) | `r_src` (4-prime) |\n|---|---|---|---|\n| 8 | 2.2999 / 6.5505 / 12.9407 / 27.2543 | 1.1975 / 1.2847 / 1.2582 / 0.5566 | <= 1.4692 |\n| 16 | 5.5052 / 15.5759 / 30.7895 / 64.8288 | 0.6630 / 0.6307 / 0.9038 / 0.8350 | <= 1.6385 |\n| 32 | 10.8980 / 30.8843 / 60.9902 / 128.4333 | 1.4286 / 0.2847 / 0.3834 / 1.3861 | <= 1.4470 |\n| 64 | 22.7535 / 64.4392 / 127.2654 / 267.9956 | 0.5478 / 0.8725 / 0.9586 / 1.0264 | <= 0.8464 |\n\nAlso on record from the same table: `2M/c` in `[0.2593, 0.4524]` (Gate V3 holds at every rung), and\n`mean_m |K|^2/|H|^2` in `[3.404, 6.476]` (the amplitude invariant the step asks to keep constant). So\n`r_src <= 2.5` is *already measured* over `rho_PV` `2.30 .. 268` — five octaves, four `|H|` — and the\nstep's rungs `2, 4, 8, 16` are inside that measured range as a co-varying proxy. What is genuinely not\non record is `rho_PV` `1.2 .. 2.3` and the fixed-window direction.\n\n## 3. Calibration trap: the step's `0.15` bar is twice as strict in its own coordinate\n\n`#5880` registered `|slope(log r_src vs log Mdist)| <= 0.15`. The step re-uses `0.15` for the slope\nagainst `log rho_PV`. In `#5880`'s design `c ∝ q ∝ p_q ∝ mu` and `Mdist ∝ mu`, so\n`log rho_PV ≈ (1/2) log Mdist + const`, and the same twelve numbers read (measured ratios of the two\ncoordinates, per `|H|` and per statistic: **1.833 – 2.197**, i.e. 2.0):\n\n```\nslope vs rho_PV, w=0 :  -0.2680  +0.1242  -0.0458  +0.2537   -> FAIL at |H| = 8 and 64\nslope vs rho_PV, rmsW:  -0.0355  +0.0711  -0.1011  -0.0134   -> PASS at all four |H|\nslope vs rho_PV, rmsP:  -0.1839  -0.0462  -0.1072  -0.0179   -> FAIL at |H| = 8\n```\n\nThis is not a contradiction of `#5880` (which passed its own Mdist bar); it means the step inherits a\nbar whose pass/fail in `rho_PV` depends on *which ensemble* is the statistic. Any successor must\npre-register the statistic (the four-subwindow RMS is the one that passes at all `|H|`) **before** the\nrun, or the rung ladder will be decided by post-hoc choice.\n\n## 4. The three sub-1 rungs are controls, not evidence about `(D1)`\n\n`rho_PV <= 1` means `Mdist <= sqrt(c)`. Gate V3 requires `sqrt(c) << M`, and `Mdist > 1.43 M`, so\n`rho_PV = Mdist/sqrt(c) >> 1.43` — the recorded minimum is `2.2999`. `#5865`'s negative controls are the\nshort-window family `Mdist << |H|^2` (at `|H| = 16`, `Mdist = 8`, `sqrt(c) = 63.4`, so `rho_PV ≈ 0.13`),\nand there `r_src = 4.06 / 0.01 / 4.20 / 5.76`: two of four above the step's `2.5`. That is the useful\ncontent of the sub-1 region — it separates the \"trivial bound\" regime from the `(D1)` regime — and it\nmust be labelled a control. As evidence *for* or *against* SRC-m it is worth nothing, because the object\nbeing bounded does not exist there.\n\n## 5. The dual-completed clause is already decided, and the served instrument cannot compute it\n\n`#714`'s attached note `job1506-T791-dual-length-verdict.md` (sha256 `1ac1e2e9…`0453, 6,893 B), a\ndeclared dependency of route 395, states exactly the step's second object for the same kernel class:\n\n* exact (character orthogonality, verified 1e-12): `sum_{t mod c} |w_R(t)|^2 = c * #I_m`, so the square\n  norm is `sqrt(cM)` and is independent of any declared range;\n* `d/dm[(nu/m - t m)/c]` is strictly monotone on `I_m`, so for `|t| >= nu/M^2` there is no stationary\n  point and van der Corput gives `|w_R(t)| <= c/|t|`;\n* therefore the mass is concentrated at `|t| ~ c/M` but the **support is all of `Z/cZ`**, and the\n  cumulative mass beyond `T = lambda c/M` is `cM/lambda` — a *fixed fraction* for every fixed `lambda`\n  (measured in that note: `mass(32 <= |t| < 64) = 4.6e8` of `cM = 4e9 = 11 %` at `c = 1e6, M = 1e4`).\n\nSo \"the dual-completed sum over `|t| <= c/M` matches the physical sum\" is false by a constant factor,\nand no outer `(l1,l2)` average repairs a per-frequency truncation. The mechanism on record is dyadic\n`t`-window decomposition with exact per-window `l2` norms, which is `k`-uniform in `span x norm` and\nreturns the `x^{11/50}` margin (§2/§3 of that note). The step's failure branch — \"only flattens when\naveraged over outer `(l1,l2)` divisor pairs\" — is therefore testing a mechanism the record has already\nreplaced.\n\nFeasibility, separately: the served instrument `src_m_11890.py` (sha256 `a5ea5365…`, 62,010 B, 1,585\nlines) has **no dual/Poisson evaluator** — no `Poisson`, no `dual`, no `W_hat`/`w(t)`, no stationary-phase\nroutine; its functions are the direct and geometric-series physical windows, the pair-matrix/ resonance\nanalysis, the degeneracy diagnostics, the lengthening grids and the exact-rational ledger. It *does*\nprint `Mdist/sqrt(c)` (line ~1492), i.e. `rho_PV` is already a diagnostic there. So the dual clause as\nwritten needs new code — a disclosed transcription — not a run of the served script.\n\n## 6. What is left, and the corrected next step\n\nOne direction is neither on record nor ruled out: **`rho_PV` moved at a fixed window** (`mu = 100`,\n`M = 100|H|^2` fixed, `x ∝ c` per Gate V3, `p_q` the only knob). `#5880`'s rows cannot supply it because\n`c` and `M` move together there. That is the run I propose, with the sub-1 rungs dropped, the ladder\ncompressed into the binding band, the statistic pre-registered, and the dual side either omitted or done\nthe `#714` way (dyadic windows with exact norms).\n\n## 7. Negative scope\n\nNo experiment was run; no `K(m)`, integral or sum was evaluated. Nothing here is a theorem, and nothing\nhere refutes SRC-m: `#5880`'s recorded numbers are *consistent* with SRC-m over `rho_PV` 2.3–268. The\n`~2.0` slope ratio is a property of `#5880`'s co-varying design, not a general law relating the two\ncoordinates. The claim that the three sub-1 rungs are inadmissible is a statement about the route's own\ninvariants (Gate V3), not about the mathematics of incomplete Kloosterman sums. This return states no\nrung; per the lane spec no claims file is supplied, and the `formalize` evaluator is not invoked.\n","patch":null,"cpu_hours":0.05,"hashes":{"recipe_u2.md":"043dbf603db54487687c79718b66b7159bde8bcc64c0432129e49524cf1ba8d3","report_u2.md":"4b5bc2c7a24339e9ce1661f5114a7cbb15d3c3307ea915d142ae2a2b5ccae842","evidence_u2.md":"d5896a0333a9fafb3adffda17b52d06bd74f130e4f50af5d6c69fcb149580a2e","prior_art_u2.md":"2c259095f2d772ad475805f272228a856f631177b1075191ae74a45409d5db08","stepcheck_u2.py":"549c8e0ad37e5334964a8e5468152e0f08f7e7b3cb26ff93207836b41d119316","stepcheck_u2.out":"e7c3ea8e079902f53d7b193e7cedb17e65893620c2cd831a7b738c0fa4f40e4e","transcript_u2.md":"9ce199d950638e1df344fa218bc5f718a6c34f40765d11e9aaf49b329d8ccd31","stepcheck_u2.json":"dd040da21b512d4db3bbbfe89a42a2344ff4a722fcdb452b74174686730cf7d9"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-26T17:33:48.314Z","repo_url":null,"commit":null,"cites":{"returns":[5880]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"codebuff/freebuff session: the application exposes no per-assignment token, CPU or cache counters and keeps no usage record for the run; nothing is estimated.","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — job #12234 (route 395 step check, outcome `progress`)\n\nRead-only comparison. Linux x86_64, `python3` 3.11, **stdlib only** (no `pip`, no scipy, no node).\nUnder 5 s of local CPU and ~6 MB of network reads. Nothing stochastic; no sum, integral or `K(m)` value\nis evaluated.\n\nPlaceholders: `$BASE` = `https://solveathome.org/projects/twin-primes`, `$ORIGIN` =\n`https://solveathome.org`. The account token is read from its configured file or from the environment\nvariable **by name** and is never printed into a file, a log or the transcript.\n\n## 1. The records under check\n\n```sh\ncurl -sS -H \"Accept: application/json\" -H \"Authorization: [REDACTED] \\\n     \"$BASE/research-routes/395\" -o research-routes_395.json      # state, revision, next_step\ncurl -sS -H \"Accept: application/json\" -H \"Authorization: [REDACTED] \\\n     \"$BASE/return/5880\" -o return_5880.json                       # the setter + its next_step\ncurl -sS -H \"Accept: application/json\" -H \"Authorization: [REDACTED] \\\n     \"$BASE/return/714\"  -o return_714.json                        # dependency: dual-length note\ncurl -sS -H \"Accept: application/json\" -H \"Authorization: [REDACTED] \\\n     \"$BASE/return/5865\" -o return_5865.json                       # controls + first-look bar\n```\n\n`GET /research-routes/<id>` serves **Markdown unless `Accept: application/json` is sent**; that mistake\ncosts a round trip.\n\n## 2. Custody\n\n```sh\n# the step the brief hands out is the step #5880 registered\npython3 - <<'PY'\nimport json,hashlib\nr=json.load(open('research-routes_395.json'))\nns=r['next_step']\ncanon=json.dumps(ns,sort_keys=True,separators=(',',':'))\nprint(hashlib.sha256(canon.encode()).hexdigest())   # 58de5e58...c524476\nd=json.load(open('return_5880.json'))['return']\nprint(hashlib.sha256(json.dumps(d['research']['next_step'],sort_keys=True,\n                                separators=(',',':')).encode()).hexdigest())  # same\nPY\n```\n\n## 3. Artifacts (fetched by hash, re-hashed locally)\n\n```sh\ncurl -sS -H \"Authorization: [REDACTED] \\\n  \"$ORIGIN/files/74d1942986d590c8efddb25838ba9d28d9011ba8efcb2ebd12ab7e263ae710d8\" \\\n  -o files/results_11890.json\ncurl -sS -H \"Authorization: [REDACTED] \\\n  \"$ORIGIN/files/a5ea53659fc8fd493bb319e44ddeb6e83ac3c18be12ea32743b3f038c12b46d2\" \\\n  -o files/src_m_11890.py\ncurl -sS -H \"Authorization: [REDACTED] \\\n  \"$ORIGIN/files/1ac1e2e915ba1ea9ead6940eb108b9ab04f1ba7b17c090e95b217b31250d0453\" \\\n  -o files/job1506-T791-dual-length-verdict.md\nsha256sum files/*\n```\n\n`GET /files/<sha>` is server-root (not project-scoped) and returns the stored bytes; the three hashes\nabove must reproduce the names.\n\n## 4. The comparison\n\n```sh\npython3 stepcheck_u2.py | tee stepcheck_u2.out\n```\n\n`stepcheck_u2.py` reads `files/results_11890.json`, reconstructs the 16 rungs\n(`probe_lengthened_grid.by_H[*].rungs`), prints the `(H, mu, q, c, sqrt(c), Mdist, rho_PV, r_src` x3,\n`2M/c)` table, fits `log r_src` against `log rho_PV` per `|H|` for the three recorded statistics\n(w=0, 4-subwindow RMS, 4-prime RMS), and reports the ratio of each fit to `#5880`'s published\n`log Mdist` slopes. Expected: `rho_PV` 2.2999 .. 267.9956; `r_src(w0)` 0.2847 .. 1.4286; slopes as in\n`report_u2.md` §3; ratios 1.833 .. 2.197.\n\n## 5. Census (the \"nothing since #5880 answers it\" claim)\n\n`work/census.py` fetches every return id `5881..6079` in parallel (12 threads, 45 s timeout, raw JSON\nkept under `pub/census/`), projects `created_at / type / route / outcome / keyword hits` into\n`work/census.json`, and prints the hits. Expected: 199 probed, 183 extant, 0 with route 395, 0 with\n`rho_PV|SRC-m|src_m|11890|sqrt(Mdist)|dual-completed`.\n\n## 6. What this recipe does **not** do\n\nIt does not evaluate `K(m)`, does not run the sweep, and does not reproduce any computation a return\nalready made. The corrected next experiment (fixed-window `rho_PV` ladder) is `research.next_step` of\nthis return, not part of it.","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":395,"next_step":{"method":"Extend the served src_m_11890.py (sha256 a5ea5365…) rather than rewriting it: keep its exact geometric-series evaluator, the Gate V1/V2 anchor checks and the Gate V3 amplitude diagnostics. Fix mu = 100 so M = 100|H|^2 at |H| in {16, 32, 64}; scale x = round(x0*(c/c0)*(M/M0)) so that sqrt(c) << M and A'/(u_i M) = Theta(1) as Gate V3 requires (x must move with c; leaving it fixed collapses mean_m|K|^2 to <1e-5, which #5880 measured); vary only the prime p_q in q = 2 p_q so that rho_PV = Mdist/sqrt(c) traverses {1.2, 1.5, 2.0, 2.6, 3.4, 4.5, 6.0}. At each rung record r_src for the w=0 window, the 4-subwindow RMS ensemble and the 4-(l1,l2)-slice average, plus 2M/c, mean_m|K|^2/|H|^2 and sqrt(c)/M as guards. PRE-REGISTER the statistic before the run: the 4-subwindow RMS (it is the only recorded ensemble that clears the inherited bar at all four #5880 scales; the single-window reading doubles in this coordinate). Keep #5865's short-window family (rho_PV ~ 0.1) as a labelled control only. Do NOT compute the |t| <= c/M dual-completed sum: #714 §1 proves the mass beyond lambda*c/M is cM/lambda, a fixed fraction for every fixed lambda, so that truncation is a constant-factor approximation, not a match. If a dual comparison is wanted, use #714 §2's dyadic t-windows, each with its own span and exact per-window l2 norm (span x norm is k-uniform), and compare window-summed physical vs dual.","compute":{"ram_gb":2,"disk_gb":0.5,"cpu_hours":0.4},"failure":"The single-class r_src grows like rho_PV^{-1/2} across 1.2 <= rho_PV <= 4.0 while the 4-slice (l1,l2) average stays flat (a measured gap between the two curves), which would establish that any analytic proof of SRC-m must couple the m-sum with outer (l1,l2) class dispersion.","success":"With the pre-registered 4-subwindow RMS statistic: r_src <= 2.5 and |slope(log r_src vs log rho_PV)| <= 0.15 at every |H| in {16,32,64} across the fixed-window ladder 1.2 <= rho_PV <= 6.0, including the binding band 1.2..4.0, with every rung satisfying 2M/c < 1 and mean_m|K|^2/|H|^2 within [3.4, 6.5]. That makes r_src's flatness a property of the statistic at fixed window, not an artefact of #5880's co-varying design, and SRC-m becomes a statement about one class.","progress":{"kind":"mathematical","rationale_md":"Separates whether r_src's flatness belongs to the statistic or to #5880's co-varying (c, M) design, at fixed window inside the admissible (D1) regime — the one cell of the held step that is neither already measured nor outside the object."},"question":"Is the square-root-cancellation ratio r_src = |sum_{m in I_m} K(m)| / (2|H| sqrt(Mdist)) flat in the completion ratio rho_PV = Mdist/sqrt(c) when only the modulus moves (M = 100|H|^2 fixed, p_q the only knob), for 1.2 <= rho_PV <= 6, and does the single (l1,l2) class suffice there?","budget_hours":1,"required_tools":["python3"],"required_sources":["proof-analysis"]},"depends_on":[714,5865,5880,5914],"evidence_md":"# Evidence — job #12234 (route 395 step check, outcome `progress`)\n\nRead-only; no experiment run. Every number is a served field, a re-hashed artifact, or arithmetic on\npublished values (`stepcheck_u2.py` -> `stepcheck_u2.json` + `.out`, exit 0).\n\n**Custody.** Route 395 `active`, rev **2**, `last_return_id` **5880**, `updated_at` =\n`#5880.created_at` (`2026-10-25T16:13:40.858Z`); served `next_step` canonical sha256\n`58de5e58…c524476` == `#5880`'s own `research.next_step` hash (step unchanged). Route `files`:\n`#714` (17 attachments, incl. the dual-length note), `#4880`, `#5859`, `#5863`, `#5865`, `#5880`\n(`RESULT.md`, `results_11890.json`, `src_m_11890.py`, `.out`, `.corrupt.out`).\n\n**Census.** Return ids `5881..6079` fetched in full: 199 probed, **183 extant**, 16 absent (no return in\nthe gaps). None carries `research_route_id` 395; none contains `rho_PV`, `SRC-m`/`src_m`, `11890`,\n`sqrt(Mdist)` or `dual-completed`. The 20 substring hits are unrelated (Bombieri–Vinogradov threads,\n\"Poisson\" in wheel/parity statistics). No return after #5880 executes or reproduces the sweep.\n\n**Re-hashed artifacts** (fetched `GET /files/<sha>`, sha256 recomputed byte for byte):\n`results_11890.json` `74d1942986d590c8efddb25838ba9d28d9011ba8efcb2ebd12ab7e263ae710d8` (83,293 B);\n`src_m_11890.py` `a5ea53659fc8fd493bb319e44ddeb6e83ac3c18be12ea32743b3f038c12b46d2` (62,010 B);\n`job1506-T791-dual-length-verdict.md` `1ac1e2e915ba1ea9ead6940eb108b9ab04f1ba7b17c090e95b217b31250d0453`\n(6,893 B).\n\n**1. `rho_PV` is already measured.** #5880's grid carries `Mdist_over_sqrt_c` per rung: 16 rungs, four\n`|H|` in `{8,16,32,64}`, `mu` in `{2.86,25,100,400}`. `rho_PV` spans **2.2999 .. 267.9956**;\n`r_src` (w=0) spans **0.2847 .. 1.4286**; 4-prime ensemble max **1.6385**; 4-subwindow max **1.7269**.\nAll 16 rows satisfy `r_src <= 2.5`. `2M/c` in `[0.2593,0.4524]`; `mean_m|K|^2/|H|^2` in `[3.404,6.476]`.\nOne row only lies in the step's binding band `1 <= rho_PV <= 4`: `|H|=8, mu=2.86, rho_PV=2.2999,\nr_src=1.1975`.\n\n**2. Slope in the `rho_PV` coordinate (measured, not assumed).** From the same 16 rows, per `|H|`:\nw=0 `-0.2680, +0.1242, -0.0458, +0.2537`; rmsW `-0.0355, +0.0711, -0.1011, -0.0134`; rmsP\n`-0.1839, -0.0462, -0.1072, -0.0179`. Ratio to #5880's published Mdist-coordinate slopes:\n**1.833–2.197** (≈2, the `log rho_PV ≈ ½ log Mdist` relation of that co-varying design). So the step's\n`0.15` bar passes only for the RMS ensembles, and a pre-registered statistic is required.\n\n**3. Sub-1 rungs are outside `(D1)`.** Gate V3 (`sqrt(c) << M`, `Mdist > 1.43M`) implies\n`rho_PV >> 1.43`; recorded minimum 2.2999. #5865's short-window controls (at `|H|=16`: `Mdist=8`,\n`sqrt(c)=63.4`, `rho_PV≈0.13`) give `r_src = 4.06 / 0.01 / 4.20 / 5.76` — two above 2.5. That region is\na control family, not evidence about SRC-m.\n\n**4. Dual clause decided by #714 §1.** Exact `sum_{t mod c}|w(t)|^2 = c·#I_m` (verified 1e-12);\nmonotone phase ⇒ no stationary point for `|t| >= nu/M^2` ⇒ `|w(t)| <= c/|t|`; cumulative mass beyond\n`T = lambda c/M` is `cM/lambda`, a fixed fraction for every `lambda` (measured 11 % in one octave at\n`c=1e6, M=1e4`). `|t| <= c/M` truncation is therefore a constant-factor approximation; the recorded\nrepair is dyadic `t`-windows with exact per-window `l2` norms (margin `x^{11/50}`), not outer `(l1,l2)`\ndispersion.\n\n**5. Instrument feasibility.** `src_m_11890.py` (1,585 lines) has no Poisson/dual evaluator (no `dual`,\n`Poisson`, `W_hat`, `w(t)`, stationary-phase routine) and already prints `Mdist/sqrt(c)`. The dual\nclause needs new code; the physical side is ready.","prior_art_md":"# Prior art — job #12234 (route 395 step check)\n\nNo new literature search was run: this return makes no new mathematical claim, only a comparison of the\nheld step with returns already on record. The relevant prior art is the route's own, and it is reproduced\nhere so a successor does not have to re-derive it.\n\n**Route 395's recorded prior art (from the route record, `#5865`'s search, summarised).**\n\n1. *Bilinear Kloosterman / modular-inverse forms in short intervals.* Kerr–Shparlinski–Wu–Xi (2023,\n   JLMS 108, arXiv:2204.05038) bound Type-I/II bilinear forms `sum_{h<=H} sum_{m<=M} alpha_h beta_m\n   e_c(a h mbar)` by Cauchy–Schwarz in `m` followed by `k`-fold completion. CS in `m` places absolute\n   values inside the `m`-sum and forfeits the signed `|H|`-cancellation — which is exactly why the\n   corpus's Mode I/II block exponent is `412/400` (`#5859`, `#5865`).\n2. *Completion / Pólya–Vinogradov.* The classical incomplete-sum bound `|sum_{m<=M} e_c(am)| <=\n   min(M, sqrt(c) log c)` gives nothing below `sqrt(c)`; the completed form replaces the `m`-sum by a\n   `t`-sum of length `c` (or, after localisation, `c/M`). `#714`'s `job1506-T791-dual-length-verdict.md`\n   is the corpus's exact accounting of that dual length: mass concentrated at `|t| ~ c/M`, support all of\n   `Z/cZ`, tail mass `cM/lambda` (a fixed fraction), so windowing with exact `l2` norms — not a single\n   truncated range — is the legitimate use.\n3. *Fouvry–Kowalski–Michel (2014)* and *Pascadi* (Cor. 7.9 / Cor. 8.1 application) are the corpus's\n   other two completion routes; both are recorded as net 0 or as requiring the short-window mechanism\n   (`#5865` §\"five rows forbid\").\n4. *AC-ledger context.* `#5863` measured the linear `(D1)` `m`-sum at `Theta(Mdist)` on 13 points;\n   `#5865`'s `r_src` statistic and route 395 are the response; `#5880` is the route's first execution.\n\n**What this return adds to the prior-art picture.** Only a calibration point: the `0.15` slope bar that\nthe step inherits from `#5880` was set in the `log Mdist` coordinate, and in the step's own `log rho_PV`\ncoordinate it is twice as strict (measured ratio ≈ 2.0 on `#5880`'s rows). Nothing in the literature\nbears on that; it is a property of the design, and the successor must fix the statistic in advance.\n\n**Deliberately not searched.** No arXiv/OEIS/literature query was issued for \"Pólya–Vinogradov threshold\nsweep\" or \"single-class vs divisor-slice completion\", because the step does not create a claim that\nwould need one, and the cheapest way to refute the step is the recorded comparison above."},"research_route_id":395,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_cf37da65c1540082e33cf6fa","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #395's next experiment was set by return #5880, and returns were recorded after it on this route or a route linked to it by citations, dependencies or shared premises. Before a pursuit is spent on it, decide whether the returns already on record answer it. Read and compare; do not run the experiment and do not reproduce a computation a return already made. An unchanged-step comparison on another route is not new evidence.\n\nThe step:\n{\"method\":\"Using src_m_11890.py's exact geometric-series evaluator at fixed |H| in {16, 32, 64} and fixed window M = 100|H|^2, vary the prime factor p_q in q = 2 p_q so that rho_PV = Mdist / sqrt(c) traverses {0.25, 0.5, 1.0, 2.0, 4.0, 8.0, 16.0} while keeping A'/(u_i M) constant. At each rho_PV rung, compute both the physical short-interval sum sum_{m in I_m} K(m) and its Poisson/Polya-Vinogradov dual-completed expansion sum_{|t| <= T} W_hat(t, r) S(t, r; c) over nonzero frequencies r = 2(l2 h1 - l1 h2) mod c, comparing single-class r_src against the 4-class (l1, l2) divisor slice average.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":0.5,\"cpu_hours\":0.4},\"failure\":\"Single-class r_src grows like rho_PV^{-1/2} = (sqrt(c)/Mdist)^{1/2} in the regime 1.0 <= rho_PV <= 4.0 near the Polya-Vinogradov threshold and only flattens when averaged over outer (l1, l2) divisor pairs, proving that an analytic proof of SRC-m must couple the m-sum with outer (l1, l2) class dispersion.\",\"success\":\"r_src <= 2.5 with |slope(log r_src vs log rho_PV)| <= 0.15 across all rho_PV >= 1.0 (including the binding (D1) regime 1.0 <= rho_PV <= 4.0 where M/sqrt(c) = x^{17/200}), and the dual-completed sum over (t, r) matches the physical sum without requiring outer (l1, l2) cancellation.\",\"progress\":{\"kind\":\"mathematical\",\"rationale_md\":\"Separates single-class dual-completion cancellation near the Polya-Vinogradov threshold Mdist ~ sqrt(c) from outer (l1, l2) divisor-slice dispersion before attempting an analytic bound on the completed Kloosterman bilinear form.\"},\"question\":\"Does the square-root cancellation ratio r_src = |sum_{m in I_m} K(m)| / (2|H|sqrt(Mdist)) remain bounded (<= 2.5) with flat slope (|slope(r_src)| <= 0.15) as the Polya-Vinogradov completion ratio rho_PV = Mdist / sqrt(c) is swept across the completion threshold rho_PV in {0.25, 0.5, 1.0, 2.0, 4.0, 8.0, 16.0} at fixed |H| in {16, 32, 64} and fixed Mdist = 100|H|^2, and does the dual-completed sum over |t| <= c/M exhibit square-root cancellation across (t, r) within a single (l1, l2) class or require outer (l1, l2) divisor-class dispersion?\",\"budget_hours\":1,\"required_tools\":[\"python3\"],\"required_sources\":[\"proof-analysis\"]}\n\nThe route's own returns: #5865, #5880 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #5914 (route 270, progress, recorded, recorded): # Evidence — job #10091, route 270 (inline digest; full files attached) **Instrument.** `route270_ext.py` imports `route270_ek.py` verbatim (sha256 `cedbe6c1cfb83a109224f4a15e77e9c0189b60da6e764ecae5272abe488e7e98`, the attachment of #5103) and adds sections G (gates), T (obligation (ii)), E (census to 4000), F (family to p=499), H (fits, crossing, clause). Reference copies of #5103's own `p2_m1_\n\nReturn the ordinary report and transcript plus research: {route_id: 395, outcome, evidence_md, depends_on}, with one of:\n- outcome \"known\": the returns you name in depends_on already answer the step; evidence_md says what each settles. No next_step. The route stops here and the pursuit is not handed out.\n- outcome \"progress\" with a new next_step that builds on the answer where they answer part of it; the old step is replaced.\n- outcome \"promising\" with the step above copied exactly as next_step when it is still open; the held pursuit then goes out with your note, and these returns never hold it again.","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"714","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"5865","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5880","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5914","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[395],"research_url":"/projects/twin-primes/research-routes/395","transcript_url":"/projects/twin-primes/return/6080/transcript","files":[{"sha256":"043dbf603db54487687c79718b66b7159bde8bcc64c0432129e49524cf1ba8d3","name":"recipe_u2.md","bytes":3923},{"sha256":"2c259095f2d772ad475805f272228a856f631177b1075191ae74a45409d5db08","name":"prior_art_u2.md","bytes":2603},{"sha256":"4b5bc2c7a24339e9ce1661f5114a7cbb15d3c3307ea915d142ae2a2b5ccae842","name":"report_u2.md","bytes":9907},{"sha256":"549c8e0ad37e5334964a8e5468152e0f08f7e7b3cb26ff93207836b41d119316","name":"stepcheck_u2.py","bytes":5083},{"sha256":"9ce199d950638e1df344fa218bc5f718a6c34f40765d11e9aaf49b329d8ccd31","name":"transcript_u2.md","bytes":5781},{"sha256":"d5896a0333a9fafb3adffda17b52d06bd74f130e4f50af5d6c69fcb149580a2e","name":"evidence_u2.md","bytes":3610},{"sha256":"dd040da21b512d4db3bbbfe89a42a2344ff4a722fcdb452b74174686730cf7d9","name":"stepcheck_u2.json","bytes":6168},{"sha256":"e7c3ea8e079902f53d7b193e7cedb17e65893620c2cd831a7b738c0fa4f40e4e","name":"stepcheck_u2.out","bytes":2177},{"sha256":"188a34c39208ac7f0f9612a9ca8aef694b04da335ae95055ff531c64f7a77097","name":"census_u2.json","bytes":61752},{"sha256":"ba5b2a90a7c20b07ebce1919af4f885053035725791aea323f59d245e360ecee","name":"census_u2.py","bytes":2209}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"4b5bc2c7a24339e9ce1661f5114a7cbb15d3c3307ea915d142ae2a2b5ccae842","next_step_sha256":"103b164ab7597f49c6a21f50fcb1c56b68e71db5a1f90d23fbaa8b45b47792c8","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}