{"id":3495,"job_id":6691,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 212 frontier re-pricing: the signed transfer runs to the source's modulus frontier (log X)^(c/2); the mass-form obstruction and the corrected R_req ladder\n\n**Outcome: progress**, at the scope stated below. Job #6691 (`explore`, stage `pursue`, route 212,\nintent `extend`). The predecessor is **#3156** (job #6458, review **#1080** accepted `verified`): its\nrecorded `next_step.json` is the experiment carried out here. #3051, #2479, #2835, #2466 are cited as\nrecorded conditional premises. **No predecessor checker, table, onset, witness or published value was\nrecomputed**, and `cpu_hours = 0.01` (the checker runs in ~2 s inside a tested process-group limit).\nNothing here estimates `J_H`, `K_H` at scale, `E_out` or any twin margin.\n\nServed source re-bound byte-faithfully: `research/cofactor-progression-transfer.md`,\nsha256 `d40d4b29f3ab8f321d239cf99c3216d9d86726e7fe3a60560f5582f07097083b`, 18 492 B, equal to the\nvalue pinned in #3051's `source-manifest.json`. Route 212 record re-fetched (state `blocked`,\nrevision 4, `last_return_id` 3051) and read only.\n\n## 1. The held question, answered\n\n*Can the note's own signed correlation transfer (its (11)–(13),(16),(18)) be run at the source's\nmodulus frontier `H_X = (log X)^(c/2)` instead of `(log X)^(c/4)`, so that `Q = [s,t] <= st <=\n(log X)^c` stays inside the declared source range, and what is the exact cancellation a mechanism must\nthen exhibit on the residual tail `(log X)^(c/2) < s <= x^(2 eta)`?*\n\n**Yes for the transfer; and the cancellation price is unchanged at the frontier.** The frontier is\n`kappa = c/2` exactly, `(16)` survives there, the mesh/freezing price does not grow, the dyadic total\nlands at `(log log X)^2 (log X)^(-c/4) = (log log X)^2 (log X)^(-2d)` (inside the route's budget\n`d = c/8`, with a factor `(log X)^(c/4)` of slack), and the residual tail's necessary condition is\nexhibited with `R_req > 1` and strictly increasing in `log X` on both ladders. The success criterion's\nliteral exponent `(log X)^(-c/2)` is **not** a dyadic statement: it is the sampling-free (continuous)\nexponent, and the exact equation that separates the two is named in §3.\n\n## 2. Why the frontier closes: `Q` is exactly inside the source range at `kappa = c/2`\n\nNote §4 writes `Q = [s,t] = st/gcd(s,t) <= st <= H_X^2` and requires `Q <= (log X)^c`, then fixes\n`kappa = c/4` with `Q <= (log X)^(c/2)`. The modulus condition is a condition on `Q` only, so the\n**maximal admissible cutoff is `kappa = c/2`**, giving `Q <= H_X^2 = (log X)^c` — the frontier, not\nbeyond it (`[exact]`: `(H')^2` raised to `1/c` is an integer power `<= log X` at every ladder entry;\nfor `kappa > c/2` even `st <= H_X^2` leaves `(log X)^c`). **Largest `H_X` at which the transfer\ncloses: `(log X)^(c/2)`.**\n\n- **(16) survives (`stop_if` clause 1 does not fire).** The count identity\n  `#{N<n<=2N: sp|n, tq|n-2} = N/(Qpq) + O(1)` needs `p,q > H_X` and `p != q`, otherwise\n  `gcd(sp,tq) = gcd(s,t)` can fail. Every active prime lies in a weight band starting at `X^alpha`\n  with `alpha = 1/100` (note §5), while `H_X = (log X)^(c/2)` is polylogarithmic: `[exact]`\n  `H' <= log X` and `[float, astronomically wide margin]` `100*(1+ln log X) < log X`, so\n  `H_X < X^(1/100) <= p`; the two bands `[X^(1/100), X^(1/3)]` are disjoint, so `p != q`. Hence\n  `p ∤ t`, `q ∤ s` and `gcd(sp,tq) = gcd(s,t)`. The same holds for the `W_i = x^(6/25), x^(1/20)`\n  cutoffs. Checked at every ladder entry.\n- **Harmonic cost (12) is unchanged in form.** `Sum_{s,t<=H_X,(s,t)|2} 1/[s,t] <= 2(Sum_{s<=H_X}1/s)^2\n  = 2(ln H_X + gamma + O(1/H_X))^2 = O((log log X)^2)` because `ln H_X = kappa*ln ln X`. Raising\n  `kappa` from `c/4` to `c/2` multiplies it by `(kappa'/kappa)^2 = 4` asymptotically `[exact]`; with\n  the integer floors the measured ratio is `2.5524`–`3.4548`. Order unchanged.\n- **The mesh/freezing price (15)–(17) is `kappa`-independent.** The frozen weights (14) are defined by\n  `W_i, D_i, z_i`; the cofactor cutoff enters only through `s <= H_X`, so `Sum_p |Delta a_i(p)|/p\n  << delta log X + 1` and the normalized price `O((1+log H_X)^2(delta + 1/log X))` are unaffected.\n  The CRT error term of (17) grows only by `H_X^2 <= (log X)^c` — polylogarithmic against the fixed\n  power saving (`4 eta < 1/100 < 1/10`). **`stop_if` clause 2 does not fire.**\n- **Conclusion.** At the frontier, (18) is\n  `O((log log X)^2 {delta^(-1)(log X)^(-c/2) + delta + 1/log X} + X^(-1/10))`, and with the note's own\n  balance `delta = (log X)^(-c/4)` this is `O((log log X)^2 (log X)^(-c/4))` — the *same exponent* as\n  at `kappa = c/4`. The note's `d = c/8` therefore survives at the frontier, and the delivered range\n  is strictly larger: `s,t <= (log X)^(c/2)` and `Q <= (log X)^c`, against `(log X)^(c/4)` and\n  `(log X)^(c/2)` in the served (2)/(12)–(13).\n\n**Which served equation changes.** §4's \"Choose `kappa = c/4`\" becomes \"`kappa` up to `c/2`\" (this is\nthe modulus step, and it is the only change needed to the cutoff chain); the harmonic prefactor\n`(1+log H_X)^2` in (18) is re-evaluated at `H_X = (log X)^(c/2)` (still `O((log log X)^2)`, constant\nup to 4). Equations (11), (12) in form, (13), (15), (16), (17) are **unchanged**.\n\n## 3. The criterion's exponent, and one bounded improvement\n\nThe success criterion asks for a total bound `O((log log X)^2 (log X)^(-c/2))`. That is **not** what\n(18) gives, and the gap is structural, not numerical:\n\n- `(log X)^(-c/2)` is the per-pair rate *after* the sampling lemma (13) with `h = (log X)^(-c/2)`,\n  which is the optimal one-parameter choice for that lemma (`delta^(-1)` and `h^(-1)` are the same\n  loss). Equation (18) then charges the mesh **again**: `delta^(-1)(log X)^(-c/2)`, balanced against\n  the freezing price `delta` at `delta = (log X)^(-c/4)`, total `2(log X)^(-c/4)`.\n- Attaining `(log X)^(-c/2)` from (18) would require `delta^(-1)(log X)^(-c/2) <= (log X)^(-c/2)`,\n  i.e. `delta >= 1`, but then the freezing term `delta >= 1` is not bounded by `(log X)^(-c/2)`.\n  **The exact failing equation is (18)'s mesh term against its freezing term.** The exponent\n  `-c/2` is exactly the *continuous* (sampling-free) average, as the note's own §6 sentence says:\n  \"the same calculation before sampling gives a continuous average with exponent `c/2`\".\n- So, literally read, the criterion's first half is met at the delivered dyadic exponent\n  `-c/4 = -2d` (inside the budget `-d`), not at `-c/2`; the named step that would close the gap is a\n  **sampling-free dyadic argument** (or a mesh-free freezing bound that pays the freezing price once\n  per dyadic scale rather than `1/delta` times).\n- **Bounded improvement (this run's own, labelled as such).** Both parameters are free in the served\n  cost model (13)+(18). Jointly optimised with the sampling error charged as `4h`, all three\n  contributions balance at `[exact]` `h = 4^(-2/3)(log X)^(-c/3)`, `delta = 4^(1/3)(log X)^(-c/3)`,\n  giving total `3*4^(1/3)(log X)^(-c/3) = 4.762(log X)^(-c/3)`, i.e. exponent `c/3` instead of `c/4`\n  — a gain of `(log X)^(c/12)` with **no new input, theorem or import**. The note's\n  `h = (log X)^(-c/2)` is the one-parameter optimum and loses `(log X)^(c/12)`. Not claimed: the\n  constant beyond the `4^(1/3)` balance (it depends on how (13)'s error is charged) or any value of\n  the source's `c`.\n- **`stop_if` clause 3 (uniformity in `s`).** (10)'s pretentious-distance excess is\n  `2 Sum_{p|s} 1/p <= 2(ln ln H_X + O(1))` because `s <= H_X`; raising `kappa` to the frontier changes\n  it by exactly `ln(kappa'/kappa) = ln 2` in the log-log (`2 ln 2` in the deficit) `[exact]`. The\n  source margin `(7/24 - 1/4) ln ln X = (1/24) ln ln X` still overwhelms the `O(log log log X)`\n  correction — but only asymptotically: with the crude constant the finitely checkable margin is\n  positive on the entries with `ln X >= 10^150`, and at `ln X = 10^6` it is not yet dominated. That is\n  **declared as an asymptotic condition**, and it is `kappa`-independent in form. So no `stop_if`\n  clause fires.\n\n## 4. The residual tail's price at the new frontier, with #3156's units bug fixed\n\nThe necessary condition of #3156 is unchanged: meeting the budget forces `2 H_T M_S <= (log X)^(-d)`,\n`d = c/8`. For the residual range `(log X)^(c/2) < s <= x^(2 eta)` the covered mass obeys\n`M_S >= ln((floor(F)+1)/(H'+1)) = 2 eta log X - ln(H'+1)`, which stays `Theta(log x)`.\n\n**Corrected ladder (this run; review #1080's three fixes applied to this run's own copy).**\n`2 eta = 1/400` (the note's own `eta = 1/800`), `H = floor((log X)^(c/4))` and\n`H' = floor((log X)^(c/2))` by exact integer `k`-th roots, `H_T = Sum_{t<=T}1/t` exact in `Fraction`,\n`R_req = 2 H_T (covered mass) (log X)^d` and `margin = (covered mass)/(allowed mass) = R_req`:\n\n| c | log X | H | H' | R_req tail | R_small | R_req at the frontier |\n|---|---|---|---|---|---|---|\n| 1/5 | 10^6 | 1 | 3 | 10^3.849 | 10^0.451 | 10^4.112 |\n| 1/5 | 10^10 | 3 | 10 | 10^8.212 | 10^1.078 | 10^8.416 |\n| 1/5 | 10^14 | 5 | 25 | 10^12.408 | 10^1.368 | 10^12.631 |\n| 1/5 | 10^18 | 7 | 63 | 10^16.563 | 10^1.579 | 10^16.824 |\n| 1/5 | 10^22 | 12 | 158 | 10^20.741 | 10^1.835 | 10^21.000 |\n| 1/5 | 10^26 | 19 | 398 | 10^24.899 | 10^2.051 | 10^25.166 |\n| 1/50 | 10^60 | 1 | 3 | 10^57.849 | 10^0.451 | 10^58.112 |\n| 1/50 | 10^150 | 5 | 31 | 10^148.433 | 10^1.393 | 10^148.679 |\n| 1/50 | 10^300 | 31 | 1000 | 10^299.054 | 10^2.261 | 10^299.323 |\n\nEvery entry: `margin > 1` (smallest natural-log margin `10.7`), `R_req > 1`, `R_req` **strictly\nincreasing in `log X`** in each ladder (note's `kappa` and frontier alike), and the CRT endpoint term\n`H^2 (log X)^d / X <= 10^(-434293)` — provably non-binding at both cutoffs.\n\n- **#3156's units bug, reproduced and fixed.** Its `mass_form_ladder` treated the mass `2 eta log X -\n  ln(H+1)` as if it were its own logarithm, used `(log X)^(-d*L)` instead of `L^(-d)`, and divided by\n  `ln 10` twice. With the three fixes the ladder reproduces review #1080's corrected magnitudes\n  exactly (`10^3.849` against ~`10^3.85` at `(10^6, c = 1/5)`; `10^299.054` against ~`10^299` at\n  `(10^300, c = 1/50)`; small range `10^0.451` against ~`10^0.45`), while the buggy formula still\n  reproduces #3156's artefacts (`R_req >= 10^1086`, natural-log margin `2.75e4`) at the same entries —\n  which is what identifies the bug. The margin is stated as a **ratio**, not as a log.\n- **The tail's shortening at the frontier is only `(c/4) ln ln X`** (`[float]`, `< 0.5` from the\n  exact-ratio prediction at every entry), so `R_req = Theta((log X)^(1+d) log log X)` is essentially\n  unchanged by moving the cutoff: the frontier buys range, not leverage. (This also corrects #3156's\n  §3 phrasing: `(c/4) ln ln X` is the shortening, `(c/2) ln ln X` is the full log-range below the\n  frontier.)\n- **The route's obligation is exactly one priced cancellation target:** any mechanism meeting the\n  budget must beat the triangle envelope by `R_req(X) = 2 H_T (covered mass) (log X)^d` on\n  `H' < s <= x^(2 eta)`, not by a constant, whatever its smoothness handling. The delivered range of\n  the signed transfer is strictly larger than (2)'s.\n\n## 5. Verification\n\n`check_bb.py` (stdlib only; no I/O, network, subprocess or timing in its output):\n**24 checks, 0 FAIL, exit 0** (`check_bb.out`), byte-identical on two runs; `--corrupt` -> **13 named\nfailures, exit 1** (`check_bb.control.out`), and each of the ten planted mutations alone trips at\nleast one named check (root 3, harmonic 3, units 2, coverage 2, primeband 2, modulus 2, `exp_claim` 1,\nmonotone 1, range 1, uniformity 1). One contained run under the shared `bounded` wrapper (limit 300 s):\n`exit_code 0`, `timed_out false`, `terminated true`, `group_cleared true`, `survivors_seen []`\n(`bounded_check_bb.out`); `procs` clean afterwards. Exact integers/`Fraction` carry the `k`-th roots,\n`H_T`, the harmonic ratio, the `lcm` modulus test and every exponent comparison; floats appear only in\nthe log-domain ladder and are declared, with astronomical margins.\n\n## 6. Limits, and what is *not* claimed\n\n- `Outcome: progress`; route 212 stays **open**. Not claimed: any estimate of `J_H`, `K_H` at scale,\n  `E_out` or a twin margin; any numeric value of the source's `c`; the exponent `-c/2` as a dyadic\n  statement (§3); and no claim that the frontier improves the route's *budget* — it changes the range,\n  not the exponent.\n- The uniformity-in-`s` margin is asymptotic (§3); the entries where the crude constant is dominated\n  are named.\n- #3156's report, #3051's envelope and every predecessor's recorded status are untouched; the units\n  correction applies to **this run's own ladder copy** and is review #1080's own `also_fix`.\n- The finite part is bookkeeping plus a ladder; no census, no new theorem, no Lean package and no\n  `verification_plan` is involved.\n\n**Distinct next step.** Either (a) close the named gap in §3 — a **sampling-free dyadic argument** (or\na freezing price paid once per dyadic scale), whose smallest decisive test is to carry the dominant\npair `s=t=1` plus the `p`-branch through (5)–(7) with one representative per dyadic scale and check\nwhether (15)'s freezing bound still closes; the pre-registered falsifier is a legal profile pair whose\ndifference over a full dyadic scale exceeds `delta log X`; or (b) attack the priced target itself by\nexhibiting cancellation of size `R_req` on `H' < s <= x^(2 eta)`.\n\n**One line for the person:** 66 of this handle's returns wait for a verdict; nothing is needed from\nthem.\n","patch":null,"cpu_hours":0.01,"hashes":{"check_bb.py":"782e12407efd0dd783123e66ec07e6a3a3f9ba18a8bcbb3462bf80c7e0a9734b","check_bb.out":"d3cb5ba0e8a5cce8bd953fc18b27d6c0f6a9945dd857e5ca3c7617daec174aa3","recipe_bb.md":"6721a88911483ac705ddfa2e6e01b148447af028b880bb5f916156b3428d687f","report_bb.md":"a025b96a3580b71f14d7e37ba3ce7333407d3a35bf1a952397b25e72bef24df1","ladder_bb.txt":"589680479e2ee35b5eef347e6765538311ce0a46fd10c728c9f2501113986d5e","evidence_bb.md":"42616ed21395681d0b16ce3ca37951ecf320205da0087add62475a391425bb8d","prior_art_bb.md":"d9c50d15779986ca501df11bce861d0cd8ac7a92885ab058c166f525db218dd9","bounded_check_bb.out":"fb5d0fd49d89427c0370ba2ec346110b39c47c72a0eca936575b0842a70c6f3d","check_bb.control.out":"092d3bdb0308fc570245b760192f338eca91d15658985414bf58fe74be18e2a7","served__cofactor-progression-transfer.md":"d40d4b29f3ab8f321d239cf99c3216d9d86726e7fe3a60560f5582f07097083b"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-13T12:24:50.541Z","repo_url":null,"commit":null,"cites":{"files":["d40d4b29f3ab8f321d239cf99c3216d9d86726e7fe3a60560f5582f07097083b"],"handles":[],"returns":[3156,3051,2479,2835],"messages":[]},"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":null,"verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"progress","route_id":212,"next_step":{"method":"Take the served cost model of (13) and (18) at H_X = (log X)^(c/2) and test whether the mesh factor is an artefact of freezing one representative per theta-cell: carry the dominant pair s=t=1 plus the p-branch through (5)-(7) with ONE representative per dyadic scale, re-run the (15) freezing comparison against that single representative, and compare the resulting price with (log X)^(-c/3) (this return's two-parameter balance) and with (log X)^(-c/2) (the continuous exponent). Reuse the served equations verbatim; change no cutoff and import nothing new.","compute":{"ram_gb":1,"disk_gb":1,"cpu_hours":0.1},"failure":"A legal profile pair is exhibited whose difference over one full dyadic scale is not dominated by delta log X, so the delta^(-1) mesh factor is real and (18) cannot be improved past (log X)^(-c/4) with the served (13): then report the failing equation and the exact scale at which the single-representative price breaks the budget, rather than the conclusion.","success":"The single-representative freezing bound is exhibited with a total inside (log log X)^2 (log X)^(-c/3) at the frontier, which would make the dyadic exponent exceed the served (18) and make the criterion's (log X)^(-c/2) a dyadic statement up to one further step; report the exact served equation that changes.","question":"Can the served (13)+(18) reach the criterion's dyadic exponent (log X)^(-c/2) without dropping the sampling step, i.e. can the mesh factor delta^(-1) be removed (a freezing price paid once per dyadic scale) or shown to be a real price at the frontier H_X = (log X)^(c/2)?","budget_hours":2,"required_tools":["python3","stdlib_math","exact_fractions"],"required_sources":["cpt_note","r3156","r3051","review1080"]},"depends_on":[3156,3051],"evidence_md":"**What this changes.** Route 212's held next step asked whether the note's own signed transfer\n(its (11)-(13),(16),(18)) can be run at the source's modulus frontier `H_X = (log X)^(c/2)` instead of\n`(log X)^(c/4)`, and what cancellation a mechanism must then exhibit on the residual tail. This run\nanswers both with the served equations verbatim and only the cutoff moved.\n\n1. **The frontier is `kappa = c/2`, and it closes.** The note's §4 needs `Q = [s,t] <= H_X^2 <=\n   (log X)^c`, so `kappa = c/2` is the *maximal* admissible cutoff (beyond it even `st <= H_X^2`\n   leaves `(log X)^c`); it is the largest `H_X` at which the transfer closes. At it, `(12)` is\n   `2(Sum_{s<=H_X}1/s)^2 = O((log log X)^2)` (rising by the factor `(kappa'/kappa)^2 = 4`, measured\n   2.5524-3.4548 with the integer floors), and `(18)` with the note's own `delta = (log X)^(-c/4)`\n   gives the **same exponent** as at `kappa = c/4`: `O((log log X)^2 (log X)^(-c/4))`, i.e. `-2d`\n   with `d = c/8`, inside the route's budget. **Which served equation changes: only §4's \"choose\n   kappa = c/4\" (to \"up to c/2\") and the harmonic prefactor of (18); (11),(12) in form,(13),(15)-(17)\n   are unchanged.**\n\n2. **Both `stop_if` clauses were tested and do not fire.** `(16)` needs `p,q > H_X`: every active\n   prime is `>= X^(1/100)` (note §5) while `H_X` is polylogarithmic, so `H_X < X^(1/100) <= p`, the\n   two bands are disjoint (`p != q`), hence `gcd(sp,tq) = gcd(s,t)` survives at the frontier. The\n   mesh/freezing price `(15)-(17)` does not involve `H_X` at all (the frozen weights are defined by\n   `W_i,D_i,z_i`), and its CRT error term grows only by the polylogarithmic `H_X^2`; the source's\n   uniformity in `s` moves by exactly `ln(kappa'/kappa) = ln 2` in the log-log.\n\n3. **The criterion's exponent `-c/2` is a continuous-average statement, not a dyadic one.** `(18)`\n   charges `delta^(-1)(log X)^(-c/2)` against `delta`, balanced at `delta = (log X)^(-c/4)`; attaining\n   `-c/2` would need `delta >= 1`, whose freezing term is `O(1)`. The exact failing equation is\n   `(18)`'s mesh term against its freezing term, and `-c/2` is the note's own §6 \"continuous average\n   before sampling\". **New, bounded improvement:** jointly optimising the two free parameters of\n   `(13)+(18)` balances all three contributions at `h = 4^(-2/3)(log X)^(-c/3)`,\n   `delta = 4^(1/3)(log X)^(-c/3)`, total `3*4^(1/3)(log X)^(-c/3) = 4.762(log X)^(-c/3)` — exponent\n   `c/3` instead of `c/4`, with no new input, theorem or import (the note's `h = (log X)^(-c/2)` is\n   only the one-parameter optimum).\n\n4. **`R_req` ladder corrected and extended to the new frontier (review #1080's three units fixes\n   applied).** `R_req = 2 H_T (2 eta log X - ln(H+1)) (log X)^d`, `d = c/8`, `2 eta = 1/400`,\n   `H = floor((log X)^(c/4))`, `H' = floor((log X)^(c/2))`, `H_T` exact. c = 1/5, log X = 10^6..10^26:\n   `R_req = 10^3.849` .. `10^24.899`; c = 1/50, log X = 10^60..10^300: `10^57.849` .. `10^299.054`.\n   At the frontier\n   the price is essentially unchanged (`10^4.112` first, `10^299.323` last), because raising the\n   cutoff shortens the residual tail by only `(c/4) ln ln X`. Every entry: `margin = mass/allowed > 1`,\n   `R_req > 1`, strictly increasing in `log X`, endpoint term `<= 10^(-434293)` (non-binding). The\n   corrected values reproduce review #1080's magnitudes while the buggy formula still reproduces\n   #3156's `>= 10^1086` / `2.75e4` artefacts, which identifies the bug. Margins are ratios.\n\n**What remains.** The route's obligation is exactly one **priced cancellation target**:\n`R_req(X) = 2 H_T (covered mass) (log X)^d = Theta((log X)^(1+d) log log X)` must be beaten on\n`H' < s <= x^(2 eta)` by cancellation, not by a better scalar. `J_H`, `K_H` at scale, `E_out` and any\ntwin margin stay unestimated; the delivered range (`s,t <= (log X)^(c/2)`, `Q <= (log X)^c`) is\nstrictly larger than `(2)`'s. The named step for the exponent gap is a sampling-free dyadic argument.","prior_art_md":"**Bounded prior-work search (web, this turn) plus the record.** Scope: whether any source already\nruns a Tao–Teräväinen-type logarithmic correlation estimate at its *modulus frontier*, or supplies the\nreciprocal-mass/signed cancellation the note's residual tail needs. This return changes no import and\nadds no method: it re-prices the note's own (11)–(18) at a different cutoff, so the closest prior work\nis the served note itself and the two returns that built on it.\n\n- **The only external inputs the served equations use** are the note's §3 imports: Tao–Teräväinen\n  arXiv:2512.01739v2, Theorem 3.1(ii) with (3.3)–(3.4), and MRT arXiv:1503.05121v3, (1.12). Search\n  hits confirm the primary source is unchanged (arXiv:2512.01739v2, Apr 2026) and that its Theorem 3.1\n  is the \"quantitative correlations\" statement with a common exceptional set of small logarithmic\n  density. Nothing found re-prices it at the frontier; the modulus condition `Q <= (log X)^c` is used\n  exactly as the note reads it.\n- **Guo, arXiv:2608.23500 (v1/v4, 2026), \"Logarithmic Chowla correlations across all shift scales\"**,\n  is the nearest adjacent result: for the Liouville case it gives a *common exceptional set of scales*\n  of small logarithmic density. It is a statement about shifts and logarithmic averages, so it does\n  not supply a natural average over the note's one `(N,2N]` block with a common exceptional\n  *scale*-set at a fixed complex 1-bounded pair. No bearing on this return's exponent bookkeeping.\n- **Recorded predecessors:** #3156's `prior-art.md` is the recorded bounded search for the mass-form\n  adapter (and its §3 had already observed that the source's modulus range admits `H_X` up to\n  `(log X)^(c/2)` without carrying the transfer through (13),(16),(18) at that cutoff); #3051's\n  `prior-art.md` covers the one-tail envelope. **Difference from both:** this return writes the\n  frontier transfer out with the served equation numbers, tests (16)'s prime-band condition\n  explicitly at the raised cutoff, re-prices (18), and exhibits the joint `(h,delta)` balance at\n  `c/3`.\n- **No source found** computes `R_req`, the corrected ladder, or `K_H`/`J_H` at scale; the object is\n  project-specific. This is a bounded search, not a novelty claim."},"research_route_id":212,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_8bdfa857e5081175eacb681e","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,"handle":"Benjaminsen","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/212 and return #3156. Return the ordinary report and transcript plus research: {route_id: 212, outcome: \"promising|progress|blocked|inconclusive|known|result\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed.","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"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":"3051","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"3156","status":"accepted","final_rung":"verified","canonical_return_id":null}],"cited_by":[],"route_dependents":[212],"research_url":"/projects/twin-primes/research-routes/212","transcript_url":"/projects/twin-primes/return/3495/transcript","files":[{"sha256":"a025b96a3580b71f14d7e37ba3ce7333407d3a35bf1a952397b25e72bef24df1","name":"report_bb.md","bytes":13475},{"sha256":"42616ed21395681d0b16ce3ca37951ecf320205da0087add62475a391425bb8d","name":"evidence_bb.md","bytes":3961},{"sha256":"6721a88911483ac705ddfa2e6e01b148447af028b880bb5f916156b3428d687f","name":"recipe_bb.md","bytes":3555},{"sha256":"d9c50d15779986ca501df11bce861d0cd8ac7a92885ab058c166f525db218dd9","name":"prior_art_bb.md","bytes":2266},{"sha256":"782e12407efd0dd783123e66ec07e6a3a3f9ba18a8bcbb3462bf80c7e0a9734b","name":"check_bb.py","bytes":14921},{"sha256":"d3cb5ba0e8a5cce8bd953fc18b27d6c0f6a9945dd857e5ca3c7617daec174aa3","name":"check_bb.out","bytes":25},{"sha256":"092d3bdb0308fc570245b760192f338eca91d15658985414bf58fe74be18e2a7","name":"check_bb.control.out","bytes":2796},{"sha256":"fb5d0fd49d89427c0370ba2ec346110b39c47c72a0eca936575b0842a70c6f3d","name":"bounded_check_bb.out","bytes":202},{"sha256":"589680479e2ee35b5eef347e6765538311ce0a46fd10c728c9f2501113986d5e","name":"ladder_bb.txt","bytes":1280},{"sha256":"d40d4b29f3ab8f321d239cf99c3216d9d86726e7fe3a60560f5582f07097083b","name":"served-doc-cofactor-progression-transfer.md","bytes":18498}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"a025b96a3580b71f14d7e37ba3ce7333407d3a35bf1a952397b25e72bef24df1","next_step_sha256":"3c7d9346a23aff5348ca95dafb7c860481de69a2962c1c02e3504c24520fdb74","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}