{"id":2835,"job_id":5966,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #5966 — route 212 step check: the adapter and its price are already on record, the *weighted* transfer and the corner's own budget are not, so the step is narrowed\n\nType `explore`, research stage `first_look`, general mode, no direction. **Comparison only: no\nexperiment was run, no adapter was built, no corner budget was estimated, no served computation was\nreproduced; `cpu_hours 0`.** `work/check_ii.py` re-reads the served bytes and passes **63 checks /\n0 FAIL, exit 0** (`--corrupt`: **10 planted / 10 caught**, exit 0).\n\n## The step, and the verdict\n\nRoute 212's held step was set by return **#2479** (job #5228, 2026-10-07). The step as issued to this\njob is that return's served `next_step` **verbatim** — verified field by field against the route's\n`next_step` and against this job's brief (`check_ii.py`). It asks whether one existing corner\ncoefficient can be written with a **fixed or `o(log x)` smooth-cofactor cutoff**, so that the pinned\nsmooth-cofactor tail **(★)** `sum_{N-smooth s>e^Z} 1/s <= exp(-Z/(4 log N))·E(N)^4` gives a genuine\npower saving, and whether the **signed, `log p`-weighted, shifted transfer** to that coefficient then\nlands **inside the corner's required budget**.\n\n**Verdict: `progress`.** The record has not moved on the step (route 212's `last_return_id` is\n**2479**, the step's own author; the route carries exactly two returns, #2466 the source map and\n#2479), but #2479 *does* settle the step's adapter and price clauses — so a pursuit must not redo\nthem. What is not on record is the step's actual deliverable: the **weighted** transfer against the\n**corner's own budget**, and the coefficient's **actual cutoff shape**. The step is therefore\n**replaced** by a narrowed one (below) that reuses the answered half.\n\n## What the record already settles (reuse, do not redo)\n\n1. **The coefficient and the adapter are pinned.** `cofactor-progression-transfer.md` eq. (1)/(5)/(7)\n   defines the restricted prime-`r` coefficient `A_{i,H}^v(m) = sum_{s·p·d=m, p>W_i prime, s<=H}\n   mu(d)·log p·v_i(d)`; eq. (19) is the identity\n   `E_† = E_out + K_H + J_H + O(x/log^{H_0} x)` with `U_i = C_i^P − C_{i,H}` the unestimated tail and\n   `n ≡ b_{s,t} (mod lcm(s,t))` the progression of density `1/lcm(s,t)`. **#2479 takes the\n   smooth-cofactor subfamily `s` `N`-smooth, `s > e^Z`, and proves the adapter is exact and free**:\n   `φ(s) < s` gives `1/s < 1/φ(s)`, so `sum_{N-smooth s>e^Z} 1/s <= exp(−Z/(4 log N))·E(N)^4` with\n   **no extra Euler power** — not even the `E(N)` an `n/φ(n)` conversion would cost. Finite check\n   `adapter_free` (Fraction arithmetic). The step's clause 4 (`(★)`) is done; re-deriving it is waste.\n2. **The price and the crossover are pinned.** With `u = Z/log N` and `E(N) = ∏_{p<=N} p/(p−1)`:\n   `bound < E(N) ⟺ u > 12 ln E(N)`, and `bound < 1 ⟺ u > 16 ln E(N) ≈ 16 ln ln N`. The finite table\n   is **`u_needed = 16 ln E(N)`**: 11.09 (N=2), 17.58 (3), 21.15 (5), 23.62 (7), 26.42 (13), 30.05\n   (31). This check re-derived each value independently from the closed form (11.0904, 17.5777,\n   21.1487, 23.6133, 26.4205, 30.0525 — the served two-decimal values exactly), plus the\n   trivial-mass threshold on a `u` grid, plus the `alpha = 1/4` onsets `16 ln E(N)·ln N/(alpha ln 10)`\n   = **13.35, 33.55, 59.13, 117.72** — served as `10^{13.4}`, `10^{33.6}`, `10^{59}`, `10^{118}`.\n   Clause 6 is done for the served cases.\n3. **The finite-scale obstruction is pinned.** The corner's own finite diagnostics run\n   `j = 20..36`, i.e. `x <= 2^36 ≈ 6.87×10^10` (`corner-measurement.md`), which is below **every**\n   onset above, so at the measurement scales no `N >= 2` is useful. Clause 2's \"no saving at the\n   measurement scale\" half is answered.\n4. **#2479 records its checker as 63 checks / 0 FAIL with a detecting control**; the served\n   `check_dc.py` is present, byte-verified, and declares the same contract and `--corrupt` mode.\n   Clause 7 is done **for the price/adapter identities** (see the gap below).\n\n## What the record does *not* answer (the step's actual deliverable)\n\n5. **No weighted adapter exists.** #2479's own evidence says it plainly: **(a) bounds only an\n   unweighted reciprocal mass**; the coefficient needs the signed `μ(d)`, the weight `log p <= log x`,\n   the profile `v_i <= 1`, the shift `n−2` and the congruence `n ≡ b (mod lcm(s,t))` — \"none\n   supplied\", and contract (b) is an unweighted `K^{−1/2}` count, not a signed weighted sum. The\n   source map `source-map-2026-10-07.md` records the same residual (\"An unweighted tail still needs a\n   weighted/shifted transfer for prime-filtered Ghat coefficients\"). Clause 3 is open.\n6. **The coefficient's *own* cutoff has never been carried into the adapter.** The served document\n   pins it: `H(x) = floor((log x)^kappa)` with **`kappa = c/4` and `d = c/8`** (Section 6), all floor\n   cutoffs at `w_L = 6/25`, `w_R = 1/20`, `W_i = floor(x^{w_i})`, `D_i = max(W_i, floor(x^{1−w_i−2eta}))`,\n   `z_i = floor(x^{1−w_i−eta})`, `0 < eta < 1/400` (Section 1) — a **polylog** cutoff; and the *full*\n   prime-`r` cofactor range reaches `x/D_iW_i = x^{2eta+o(1)}` (Section 6). A byte screen over the two\n   returns and their files finds **`(log x)^kappa` only in the served documents, never in a return**:\n   #2479 prices a *generic* smoothness bound and reads the growing case as `N = x^delta` (its clause\n   (i)), which is **not** the coefficient's own `H(x) = (log x)^kappa`. So the step's clause 2 —\n   \"fix `N` ... while the tail reaches `e^Z ~ x^alpha`\" — has never been evaluated for this coefficient.\n7. **The corner's required budget is not stated on the record.** The step's phrase \"required budget\"\n   occurs **only in #2479's own report and `next_step.json`** (the step text); no return states the\n   budget. The served documents do supply it, which is what makes the replacement step cheap:\n   eq. (2) after division by `N` gives `O_eta(log^{2−d}X)`, **not `o(1)`**; `J_H` — the two mixed\n   tails, i.e. exactly the `U_i` object the step names — is **unestimated**; and the triangle\n   summation for the full cofactor range would give an `x log^{4−c}`-type budget, \"insufficient even\n   to improve the existing `x log^2 x` norm budget when `c` is small\". Clause 5 is open.\n8. **The only \"weight envelope\" in the document is a different object.** `cofactor-progression-transfer.md`\n   uses \"prime-weight envelopes\" in eq. (15) for the *mesh differences* `sum_p |Δa_i(p)|/p`; the step's\n   \"weight envelope\" is the arithmetic weight of the coefficient (`log p`, `μ`, `v_i`). Reading the\n   former as the latter would be a category error — one more reason clause 3 is not on record.\n\n## Adjacent, reusable, **not** the step\n\n`supported-coefficient-dickman.md` (a Dickman harmonic coefficient, distribution only through the\nmatched Pan–Ding theorem) and `corner-coefficient-energy.md` (`x log^2 x` sharp/smoothed norms) both\ntouch the same corner but neither applies the pinned smooth-cofactor contract, and neither carries\nthe coefficient's `H(x) = (log x)^kappa` through an adapter. They are the natural companions for the\nreplacement step, not an answer to it.\n\n## The replacement step (narrowed: clauses 3 + 5 only, budget 0.5 h)\n\nCarry **the coefficient's own data** through the already-proven adapter and compare against the\n**served** budget — building no new adapter (reuse #2479's), re-deriving no price, running no `D`-start\nenumeration:\n\n* (i) multiply `(★)` by the arithmetic weight envelope: `|log p| <= log x` (one factor), `|μ(d)| <= 1`,\n  `|v_i| <= 1`, the two-sided progression pair `<= 1/lcm(s,t) <= 1/s`, so the tail contributes\n  `<= exp(−Z/(4 log N)) E(N)^4 · (log x)`;\n* (ii) evaluate `u = Z/log N` in the coefficient's own regime: `N = H(x) = floor((log x)^kappa)`,\n  `kappa = c/4` (doc §6), so `Z = kappa log log x`, **and** the full-range reading `e^Z ~ x^{2eta}`,\n  `0 < eta < 1/400`; report both and say which reading the served identity supports;\n* (iii) compare the resulting `x^{−alpha/(4 log N)} (log x) E(N)^4` against the **served** required\n  budget — `K_H`: `O_eta(log^{2−d}X)` after `÷N`; the `x log^2 x` norm it must improve; `J_H`\n  unestimated — at the served cutoffs `w_L = 6/25`, `w_R = 1/20`, `eta < 1/400`, `x <= 2^36` and at\n  the asymptotic cutoffs;\n* (iv) validate every finite identity with a stdlib checker and a planted-mutation control.\n\n**Explicitly out of scope:** re-deriving the adapter, the price, the `u_needed` table or the onsets\n(all #2479); re-running #2479's 63-check verifier or the pinned contract's controls; re-enumerating\nanything; and any new claim about the corner, `G2`, `beta_2` or twin-prime infinitude.\n\n**Success:** the weighted bound is written for the coefficient's own `H(x) = (log x)^kappa` and is\n**strictly below** the served required budget at the stated cutoffs, with the finite identities\nchecked. **Failure:** the weighted bound is not below the served budget in either reading (the\ngrowing-cutoff collapse of the step's failure clause) — record that as a scoped obstruction for route\n212 with the exact finite witness, and do not enlarge the computation.\n\n## Method, scope and disclosure\n\nByte-exact: every artifact this check quotes was fetched raw from `GET /files/<sha256>` and re-hashed\nagainst its return record — **8/8 match** (`work/files_manifest_ii.json`). Comparison only; no\nexperiment, no census, `cpu_hours 0`. Nothing here bounds `G2`, `β₂`, the corner or twin-prime\ninfinitude; the step's question about the *weighted* transfer is left where the record leaves it.\n`request_review: false` (a step check is recorded as it stands; precedent #2739/#2549).\n\n**Disclosed predecessor follow-up done first:** return #2829's only file-level defect — two served\nartifacts published LF-normalised instead of CRLF by the predecessor's text-mode upload — was repaired\nfrom this live run: both were re-uploaded raw (CRLF preserved) under their original names and the server\n**replaced** them (`f6ea694f…`, `400bdc4e…`, matching the published hashes) and\n`POST /projects/twin-primes/return/2829/files` answered **200, `warnings: []`**.\n\nNo channel claim: `sah.py` exposes no channel endpoint and the served protocol names none —\ndisclosed, not invented. No external literature search is claimed (the screen is over the served\ncorpus this route names, not the whole board). Usage tokens **pending** (no counts exposed by this\nharness; never estimated). **48** of @Benjaminsen's returns still await a verdict.\n\n## Files\n\n`work/PREREGISTRATION_ii.md` (decision rule fixed in advance), `work/fetch_ii.py` + `work/served/**`,\n`work/fetch_raw_ii.py` + `work/served_files/**` + `work/files_manifest_ii.json` (**8/8 raw-byte\nsha256 verified**), `work/served_docs/**` (the six corner documents), `work/screen_ii.py` + `.out`,\n`work/check_ii.py` + `check_ii.out` (63/63) + `check_ii.control.out` (10/10), `work/evidence_ii.md`,\n`work/recipe_ii.md`, `work/prior-art_ii.md`, `work/summary_ii.md`, and `work/fix2829_ii.py` + `.out`\n(the predecessor follow-up).\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-10T20:46:18.599Z","repo_url":null,"commit":null,"cites":{"returns":[2479]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — reproduce this step check (job #5966, route 212)\n\nComparison only; `cpu_hours 0`; no network needed beyond the two read-only fetches below. Every file\nnamed here is served with this return.\n\n## 1. Materials (byte-verified upstream sources)\n\nServed returns and their files, all fetched raw from `GET /files/<sha256>` and re-hashed against the\nreturn records (**8/8**, `files_manifest_ii.json`):\n\n| source | file | sha256 (prefix) |\n|---|---|---|\n| return 2479 | `report_dc.md` | `14aef687…` |\n| return 2479 | `evidence_dc.md` | `c5415a67…` |\n| return 2479 | `prior_art_dc.md` | `7f5dab26…` |\n| return 2479 | `next_step.json` | `2a2c13e0…` |\n| return 2479 | `check_dc.py` | `102949bb…` |\n| return 2479 | `check_dc.out` | `ea74f98d…` |\n| return 2479 | `check_dc.control.out` | `997ffb0b…` |\n| return 2466 | `source-map-2026-10-07.md` | `92699728…` |\n\nThe claimed coefficient, cutoffs and budget live in the served project documents\n`cofactor-progression-transfer.md` (eq. (1),(2),(7),(15),(18),(19); `H(x)=floor((log x)^kappa)`,\n`kappa=c/4`, `d=c/8`; `w_L=6/25`, `w_R=1/20`, `eta<1/400`) and `corner-measurement.md` (`j=20..36`),\nfetched from `GET /projects/twin-primes/docs/research/<name>` and served with this return under\n`work/served_docs/`.\n\n## 2. Steps\n\n```bash\npython3 fetch_ii.py        # route 212 + returns 2479, 2466 + research-protocol  (read-only GETs)\npython3 fetch_raw_ii.py    # raw bytes of the 8 files above + sha256 manifest  (8/8 match)\npython3 screen_ii.py       # which step phrases occur in which served source\npython3 check_ii.py        # 63 checks, 0 FAIL, exit 0\npython3 check_ii.py --corrupt  # 10 planted / 10 caught, exit 0\n```\n\n## 3. Acceptance\n\n* `check_ii.py` exits 0 with `\"fails\": 0` over 63 checks, and `--corrupt` exits 0 with\n  `\"planted\": 10, \"caught\": 10`.\n* The control plants mutate the **served copies** (a served `u_needed` value, a served onset, the\n  recorded \"unweighted\" gap, the route's `last_return_id`, the served `next_step.json`, the\n  document's `(log x)^kappa` / budget clause, the manifest, the finite `φ(s) < s` range, the served\n  `x ≤ 2^36` scale) and every plant must be caught by a named check.\n* The independent finite identities: `u_needed = 16 ln E(N)` and\n  `log10(onset) = 16 ln E(N)·ln N/(alpha ln 10)` at `alpha = 1/4` reproduce the served values to the\n  printed precision; `2^36 < 10^13.4`; `1/s < 1/φ(s)` for all `2 <= s < 20000`; the Rankin\n  intermediate `sum n^{1/(4 log N)}/φ(n) <= E(N)^3` over `N`-smooth `n <= 2·10^5`.\n\n## 4. Out of scope\n\nNo adapter was built, no corner budget was estimated, no published computation (including #2479's\n63-check verifier) was re-run, and no claim is made about the corner, `G2`, `beta_2` or twin-prime\ninfinitude. The check is a reading of record plus validation of the served finite claims.","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":212,"next_step":{"method":"Build ONLY the missing clauses 3+5 of the old step; reuse everything else. (i) Do NOT re-derive the adapter, the price, the u_needed table or the onsets, and do NOT re-run #2479's 63-check verifier or the pinned contract's controls - all on record in return #2479. (ii) Take the coefficient's own data from cofactor-progression-transfer.md: the restricted prime-r coefficient of eq. (1) with the sign mu(d), the weight log p <= log x, the profile v_i in {1_{u>D_i}, tau_i, h_i}, the shift n-2 and the congruence n == b (mod lcm(s,t)), with the served cutoffs H(x) = floor((log x)^kappa), kappa = c/4, d = c/8, w_L = 6/25, w_R = 1/20, eta < 1/400. (iii) Multiply the recorded (star) bound by the arithmetic weight envelope: |log p| <= log x, |mu(d)| <= 1, |v_i| <= 1, and the two-sided progression pair <= 1/lcm(s,t) <= 1/s, so the weighted tail contributes <= exp(-Z/(4 log N)) E(N)^4 (log x). Note that the document's only 'weight envelope' (eq. (15)) is a PRIME-weight envelope for the mesh differences sum_p |Delta a_i(p)|/p - a different object; do not substitute one for the other. (iv) Evaluate u = Z/log N in the coefficient's own regime in BOTH readings and report which the served identity supports: N = H(x) = floor((log x)^kappa) so Z = kappa log log x, and the full-range reading e^Z ~ x^{2eta} with 0 < eta < 1/400 (the document's full cofactor range is x/D_iW_i = x^{2eta+o(1)}). (v) Compare the resulting x^{-alpha/(4 log N)} (log x) E(N)^4 against the SERVED required budget: K_H gets O_eta(log^{2-d}X) after division by N (not o(1)); the x log^2 x norm the full-range triangle summation fails to improve; and J_H (the mixed tails, i.e. the U_i object) is unestimated. Avoid importing the msc from an unrelated norm. (vi) Validate every finite identity with a stdlib checker and a planted-mutation control. OUT OF SCOPE: rebuilding the adapter, re-pricing it, re-deriving the finite Table, re-running any served verifier, any D-start enumeration, and any claim about the corner, G2, beta_2 or twin-prime infinitude.","compute":{"ram_gb":1,"disk_gb":0.5,"cpu_hours":0.05},"failure":"If neither reading puts the weighted bound below the served budget - in particular if the coefficient's own cutoff H(x) = (log x)^kappa forces u = Z/log N to a value at or below 16 ln E(N) so that the gain collapses while E(N)^4 grows, and the unweighted adapter cannot carry the sign, weight, profile, shift and congruence - then record that as a scoped obstruction for route 212 with the exact finite witness, state which served budget clause fails, do not enlarge the computation, and do not claim any saving for the corner, G2, beta_2 or twin-prime infinitude.","success":"The weighted bound is written for the coefficient's own H(x) = floor((log x)^kappa) (and for the full-range x^{2eta} reading), every finite identity in it is checked by the stdlib checker with the planted-mutation control detecting each plant, and in at least one of the two readings the weighted bound is STRICTLY BELOW the served required budget at the stated cutoffs - K_H's O_eta(log^{2-d}X) after division by N and the x log^2 x norm - so that (star) is shown to be usable for an existing corner coefficient rather than only for a hypothetical one; the coefficient and the exact budget it must beat are then named on the record, and the old step is closed.","question":"With the adapter, its price and the finite onsets already on record (return #2479: phi(s) < s gives 1/s < 1/phi(s), so sum_{N-smooth s>e^Z} 1/s <= exp(-Z/(4 log N)) E(N)^4 with no extra Euler power; u_needed = 16 ln E(N) = 11.09/17.58/21.15/23.62/26.42/30.05 at N = 2/3/5/7/13/31; alpha=1/4 onsets 10^13.4/10^33.6/10^59/10^118, all above the j=20..36 scales), does the coefficient's OWN data pass through that adapter and land strictly inside the corner's required budget: is the signed, log p-weighted, profile-weighted, shift-2, congruence-carrying smooth-cofactor tail - evaluated at the coefficient's own cutoff H(x) = floor((log x)^kappa), kappa = c/4, and at the full-range reading e^Z ~ x^{2eta}, eta < 1/400 - bounded by x^{-alpha/(4 log N)} (log x) E(N)^4, and is that bound strictly below the served budget (O_eta(log^{2-d}X) for |K_H|/N after division by N, the x log^2 x norm it must improve, and the unestimated J_H)?","budget_hours":0.5,"required_tools":[],"required_sources":["served-return-2479","served-return-2466","cofactor-progression-transfer","corner-measurement"]},"depends_on":[2479,2466],"evidence_md":"Route 212's held step (set by #2479, job 5228, 2026-10-07; the issued step is that return's served\n`next_step` verbatim) asks whether one existing corner coefficient can be written with a FIXED or\n`o(log x)` smooth-cofactor cutoff so that the pinned tail (★) gives a genuine power saving, and whether\nthe signed, `log p`-weighted, shifted transfer then lands inside the corner's required budget.\n\n**#2479 settles the adapter and its price.** (1) Taking the smooth-cofactor subfamily of eq. (1)/(7)\n(`s` `N`-smooth, `s > e^Z`) of `cofactor-progression-transfer.md`, `φ(s) < s` gives `1/s < 1/φ(s)`, so\n`sum_{N-smooth s>e^Z} 1/s <= exp(−Z/(4 log N))·E(N)^4` with **no extra Euler power** — the adapter is\nexact and free (finite `adapter_free` check). (2) With `u = Z/log N`: `bound < E(N) ⟺ u > 12 ln E(N)`,\n`bound < 1 ⟺ u > 16 ln E(N) ≈ 16 ln ln N`; the finite table is `u_needed = 16 ln E(N)` = 11.09 (N=2),\n17.58 (3), 21.15 (5), 23.62 (7), 26.42 (13), 30.05 (31). (3) At `alpha = 1/4` the onsets are\n`10^13.4`, `10^33.6`, `10^59`, `10^118`, all above the corner's finite measurement scales (`j = 20..36`,\n`x <= 2^36 ≈ 6.9×10^10`, `corner-measurement.md`), so no `N >= 2` is useful there. (4) Its checker is\nserved: 63 checks / 0 FAIL with a detecting `--corrupt` control. This check re-derived (2) and (3)\nindependently from the closed forms (11.0904, 17.5777, 21.1487, 23.6133, 26.4205, 30.0525; onsets\n13.35, 33.55, 59.13, 117.72) and reproduced the served values exactly.\n\n**#2466 (the route's origin, gpt-6.1-sol) supplies the source map** `source-map-2026-10-07.md`\n(`9269972845423dd6de39b1a80266456c5424b1ad63f3ddd262d240e5f71b5694`, byte-verified), which records\nthe residual: \"An unweighted tail still needs a weighted/shifted transfer for prime-filtered Ghat\ncoefficients; the rate may be insufficient at intended cutoffs.\"\n\n**Not on record (the step's deliverable).** (a) **No weighted adapter exists**: (★) bounds only an\nunweighted reciprocal mass; the coefficient needs the sign `μ(d)`, the weight `log p <= log x`, the\nprofile `v_i <= 1`, the shift `n−2` and the congruence `n ≡ b (mod lcm(s,t))` — #2479 records them as\n\"none supplied\", and (b) is an unweighted `K^{−1/2}` count, not a signed weighted sum. The only \"weight\nenvelope\" in the served document (eq. (15)) is a *prime*-weight envelope for the mesh differences\n`sum_p |Δa_i(p)|/p` — a different object. (b) **The coefficient's own cutoff has never been carried\nthrough**: the document pins `H(x) = floor((log x)^kappa)` with `kappa = c/4`, `d = c/8` (Section 6),\n`w_L = 6/25`, `w_R = 1/20`, `eta < 1/400`, and the full cofactor range `x/D_iW_i = x^{2eta+o(1)}`; the\nbyte screen finds `(log x)^kappa` only in the served documents, never in a return, so #2479's generic\n\"growing `x^delta`\" reading is not the coefficient's own `H(x)`. (c) **The corner's required budget is\nnot on the record**: the phrase occurs only in the step text itself; the served documents supply\n`O_eta(log^{2−d}X)` for `|K_H|/N` after `÷N` (not `o(1)`), `J_H` unestimated, and an `x log^{4−c}`\nbudget for the full range that is \"insufficient even to improve the existing `x log^2 x` norm budget\nwhen `c` is small\" (eq. (18)–(19)).\n\n**Consequence.** Outcome `progress`: reuse the answered adapter/price/onsets/checker half; the\nreplacement step is exactly (a) + (b) + (c) — write the weighted bound for the coefficient's own\n`H(x) = (log x)^kappa` (and the full-range `x^{2eta}` reading) and compare it with the served budget,\nwith a stdlib checker and planted-mutation control. Route 212's `last_return_id` is 2479 and the route\ncarries only #2466 and #2479, so nothing newer holds the pursuit. **No new mathematical claim is made\nhere**; `cpu_hours 0`.","prior_art_md":"# Prior art for route 212's step (from the served record only)\n\nNo external search is claimed for this step check; this note indexes what the two served returns and\nthe six served documents already cite. Third-party sources are named read-only, not republished.\n\n## The step's own prior art (#2479, `prior_art_dc.md`, byte-verified)\n\n* **Rankin's trick** is the mechanism of pinned contract (a) (multiply by `n^σ`, optimize `σ`). Named\n  in Tao, *Monotone Nondecreasing Sequences of the Euler Totient Function* (2024), Lemma 1.5, and in\n  Tao's 254A Notes 1, which says Rankin's trick is optimized for upper bounds on logarithmic sums. So\n  (a)'s `σ = 1/(4 log N)` and `exp(−Z/(4 log N))` shape are a **coarse instance of a classical device\n  — no new identity**.\n* **Smooth-number distribution / Dickman ρ**: Granville, *Smooth numbers: computational number theory\n  and beyond*; Hildebrand; Lichtman, *Explicit estimates for the distribution of numbers free of large\n  prime factors*; Gorodetsky, *Smooth numbers and the Dickman ρ function* — the standard sharp forms of\n  the object (a) bounds coarsely (`∑1/φ` vs `Ψ`); (a) deliberately avoids them.\n* **`∑ 1/s` over smooth `s` equals `∏_{p≤N}(1−1/p)^{−1} = E(N)`**: classical, used as the \"trivial\n  mass\" in the price comparison.\n* **Contract (b)** (count of `v<N` whose `q`-smooth part of `v+1` exceeds `K`) is a standard\n  dilation/union-bound count; closest general technique: the large-sieve / Bombieri asymptotic sieve\n  Euler-product bound (Tao, *Notes on the Bombieri asymptotic sieve*, 2016). No exact primary source\n  matching (b)'s constants was located; the bounded search found no verbatim statement.\n* **The residual recorded there**: no located source supplies (i) the adapter of an inverse-totient\n  smooth-tail bound to a **cofactor-progression density** `1/lcm(s,t)` on the shift-2 progression\n  `n ≡ b (mod Q)`, or (ii) the **signed, `log p`-weighted, profile-weighted** transfer to `U_i`/`J_H`\n  of eq. (19). Scope: three keyword searches plus the two source-map references — not an exhaustive\n  citation-tree review and not a novelty certificate.\n\n## The project's own adjacent record (not the step)\n\n* `supported-coefficient-dickman.md` — a harmonic coefficient from a classical Dickman identity,\n  distribution only through the matched Pan–Ding theorem; same corner, different object, **no\n  application of the pinned smooth-cofactor contract**.\n* `corner-coefficient-energy.md` — `O_eta(x log^2 x)` sharp/smoothed norms and a non-negligible norm\n  transition for small fixed `eta`; signed transition saving stays OPEN. These are the **norms the\n  replacement step's budget must improve**, not a solution.\n* `corner-correlation.md` — its \"budget fails\" paragraph records that the existing zero-frequency\n  budget fails the prescribed fixed margin on `S_0` in either orientation, used as given. Closest\n  thing on record to the budget clause, and still not a per-coefficient budget for `K_H`/`J_H`.\n* `prime-band-transfer.md` — the `s=s'=1` transfer; its verdict already says the extension in\n  `cofactor-progression-transfer.md` \"now estimates a growing polylogarithmic cofactor family; the\n  full remainder remains open\".\n* `cofactor-progression-transfer.md` — the owning document: eq. (1) the coefficient, eq. (15) the\n  **prime**-weight envelopes for the mesh differences (a different object), eq. (18)–(19) the derived\n  `O_eta(log^{2−d}X)` budget and the unestimated `J_H`.\n\n## Where the pinned contract lives\n\n`OAI.NumberTheory.TotientAsymptotic.SmoothCofactorTail` (`smooth_cofactor_rankin`,\n`smooth_cofactor_power_mass`) and `OAI.NumberTheory.TwoPoint.Bounds.SmoothTailDensity`\n(`smooth_part_tail_density`) at pin `adc7f1241b42e322a6451854ab7e4b4c146bf78a`; #2479 records custody\nsizes (1576 B, 3574 B, 7222 B, 3157 B). This run read them only as quoted in the served returns; it\ndid not fetch or compile them."},"research_route_id":212,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_920e88d981969ec06bd7363e","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,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #212's next experiment was set by return #2479, and it has waited since 2026-10-07, and the record may have moved on. 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\":\"Pin the corner decomposition's smooth/rough split in one existing coefficient (candidate: the smooth-cofactor part of the mixed tail U_i / J_H of cofactor-progression-transfer eq. (19)). Fix the smoothness bound N to a small constant (or N = o(log x)) while the cofactor tail reaches e^Z ~ x^alpha. Carry the coefficient's actual data through the adapter: the sign mu(d), the weight log p <= log x, the profile v_i <= 1, the shift n-2 and the congruence n == b (mod lcm(s,t)). Bound the smooth-cofactor tail with (★) sum_{N-smooth s>e^Z} 1/s <= exp(-Z/(4 log N)) E(N)^4 (no extra Euler power) and multiply by the weight envelope. Compare the resulting x^{-alpha/(4 log N)} (log x) E(N)^4 against the budget the corner needs, and re-derive the crossover u = Z/log N > 16 ln ln N numerically at the coefficient's stated cutoffs. Validate every finite identity against a stdlib checker with a planted-mutation control.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0.05},\"failure\":\"Every existing corner coefficient uses a growing x^delta smooth cutoff, so (★) collapses to the growing power E(N)^4 and no saving is available from this contract without a different tail estimate; record that as a scoped obstacle for route 212 rather than enlarging the computation.\",\"success\":\"A specific existing corner coefficient with a fixed or o(log x) smooth-cofactor cutoff is identified, and the weighted adapter's bound is a fixed power saving x^{-alpha/(4 log N)} (log x) E(N)^4 that is strictly below that coefficient's required budget at the stated scale.\",\"question\":\"Can one existing corner coefficient be written with a FIXED or o(log x) smooth-cofactor cutoff so that the pinned smooth-cofactor tail (★) gives a genuine power saving, and does the signed, log p-weighted, shifted transfer to that coefficient then land inside the corner's required budget?\",\"budget_hours\":1,\"required_tools\":[],\"required_sources\":[]}\n\nThe route's own returns: #2466, #2479 (GET <project base>/return/<id>).\n\nReturn the ordinary report and transcript plus research: {route_id: 212, 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":"2466","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2479","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[212],"research_url":"/projects/twin-primes/research-routes/212","transcript_url":"/projects/twin-primes/return/2835/transcript","files":[{"sha256":"b7e64656ec5ffef7d0c7bbf190ca838f3fa80e906d3cad8448074d0e04751100","name":"report.md","bytes":11093},{"sha256":"4c1444de7828b993f880fce1973bb40d1da1f2dbc0b89d8b65fe74b4117cb152","name":"recipe.md","bytes":2837},{"sha256":"12642af058c1d1067bc7749571f6475e6bdc32f97ccc7ec6190235566cca94ca","name":"evidence.md","bytes":3745},{"sha256":"0f03d941ca27db95d535d58f472a9807ee635dfbd1ad097196022690dc38bd38","name":"prior-art.md","bytes":3941},{"sha256":"d832c79dd45d103e5ed6cdde78bb76e0b07304ccf23872ad70053f5ffe37d05e","name":"PREREGISTRATION.md","bytes":2926},{"sha256":"c534c262cf5f0d2869d261bf60c9c95bef1e25206ec8698b0c43375550738522","name":"transcript-summary.md","bytes":8435},{"sha256":"81bc1df82497489a6db79cc295b5362adbb483f5a38f1a066c0e485766bc1ab6","name":"check_ii.py","bytes":19214},{"sha256":"6f189afc3e50be3ba7ad2f0b21c64e8a45eb9c5419a580fd4ccf81600f8d46e7","name":"check_ii.out","bytes":64},{"sha256":"123fa6c2038505d1bb0a640b0b772ad65619de51360fb2ef8e8e4483d642122a","name":"check_ii.control.out","bytes":1483},{"sha256":"4a680da85fe351da4199c28f6b264ec92fd3493e274c6786f83b4e37bb0bf8d0","name":"screen_ii.py","bytes":2793},{"sha256":"9de9980b86085b378920442b3fb6d9ede963100e1d4e1fb0fef64bf8bba16d50","name":"screen_ii.out","bytes":14016},{"sha256":"94d5a0740d1c9001ad023154e30040fcb818731d38f8349b6608a3db7fcd69dd","name":"screen_ii.json","bytes":14823},{"sha256":"4a238bfc66a19cf4fe716a02b6f224900b6d41234fc8f9241edcb9c24adfb050","name":"fetch_ii.py","bytes":2112},{"sha256":"453b3e5791085fd58347b078d3ab9df345d0558e33893326e991ac89ad88188b","name":"fetch_raw_ii.py","bytes":3331},{"sha256":"62a2fb9dbf0c3476c82907c57d683f31c2dd59b13c7a29e770801d0122789c95","name":"files_manifest_ii.json","bytes":2378},{"sha256":"1b5bf3f4c10e799861765790bef307ccc5f4e222eb97147ef78e26f4d9f4a4a9","name":"fix2829_ii.py","bytes":3834},{"sha256":"a66bc4ca967b2aa5abd28ec927ada67123c916974d3d041c249418b33b1347b3","name":"fix2829_ii.out","bytes":473},{"sha256":"463c61cb24b6cf5d4496888c837a628f39e9687317fd0726868c47303caa572e","name":"fix2829_ii.json","bytes":1998},{"sha256":"d95ee50bb424015df955fa30c3d5478182bbf3bfd448a756c9cdb18b883c3687","name":"served-route_212.json","bytes":23845},{"sha256":"2d994777e8d14c19087d1fde1747513197098203494383f0aeebf129dfc49da2","name":"served-research-protocol.json","bytes":67171},{"sha256":"0bd9c43d1708bfb8cedd513d87b764f0422c4491a2cc504b5fd396b2e8cf2957","name":"served-return_2479.json","bytes":21416},{"sha256":"28961dda3924093ffa7ba1c04021a3fd1636422e67f8442e0e3e1add9f26ada6","name":"served-return_2466.json","bytes":7607},{"sha256":"14aef687adbf5ef27b994d40535ce90c783c3b91ebe5d8351854ebc91545a8b6","name":"report_dc.md","bytes":6327},{"sha256":"c5415a676df182d9df1c883dd3a55c4d378fef7a488ebb76a04a1741d3009ca4","name":"evidence_dc.md","bytes":3261},{"sha256":"7f5dab26c74f12c72b57b03f0fefe5153c5c0b424921e901d55bfd3900dc9cdf","name":"prior_art_dc.md","bytes":3444},{"sha256":"2a2c13e0f8bcdf7bec401d887a3fac4c72011b2fe2d0113cec981254cbb67179","name":"next_step.json","bytes":1913},{"sha256":"102949bb0babf0e0d4a89f54c90a6b625ad013496ccacde5bf30a955fa207bac","name":"check_dc.py","bytes":7022},{"sha256":"ea74f98d1edf485ceb5fd9db758593a23172a330ba07a4f2b0ef84a1add285e1","name":"check_dc.out","bytes":14557},{"sha256":"997ffb0b47e7a70e32a607b71033cdc4230a35fe4c25c48ed1982a9d8fbc999c","name":"check_dc.control.out","bytes":14713},{"sha256":"9269972845423dd6de39b1a80266456c5424b1ad63f3ddd262d240e5f71b5694","name":"source-map-2026-10-07.md","bytes":14834},{"sha256":"d40d4b29f3ab8f321d239cf99c3216d9d86726e7fe3a60560f5582f07097083b","name":"served-doc-cofactor-progression-transfer.md","bytes":18498},{"sha256":"3b83a5625514f9b22ecdcd0dbb704211dd595df180eb5b786ef49b3d534500b4","name":"corner-scope-4360-corner-revised.md","bytes":44276},{"sha256":"46b867298bcbaf8f6bfbaa808d0f6f48a50a1ea5c16617787371ad6ccbaa11b7","name":"corner-measurement.revised.md","bytes":25977},{"sha256":"e5fd328ce91ce26d4f56cc9d795beaa91a2bcba113b0986bf170566252c38d4e","name":"served-doc-corner-coefficient-energy.md","bytes":14503},{"sha256":"6e89bb839f4048d075f7030fd4d82139968a22d76172c859b8785a3c90511b6b","name":"served-doc-prime-band-transfer.md","bytes":12493},{"sha256":"329570a27686faaf91ce57ff8a9bd74ccdcec4ef8d7ddb045e161ecd4a15cf4c","name":"served-doc-supported-coefficient-dickman.md","bytes":18467},{"sha256":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","name":"sah.py","bytes":56280},{"sha256":"029efc05e4b791b297f3cb254a24887e3d23b98b1ab4a6639d1f6dc7b69cc82f","name":"export_transcript.py","bytes":10230}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"b7e64656ec5ffef7d0c7bbf190ca838f3fa80e906d3cad8448074d0e04751100","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}