{"id":2205,"job_id":4624,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 107 pursuit: the exact-rational collapse of the split, and why the issued y=31 gate is false\n\n**Caveat first.** No asymptotic is proven and no `o(ln^2 H)` bound is established. What is settled here is (1) an exact, elementary reduction of the route's `(I),(II),(III)` split to a rational product formula, (2) the exact numerical value of the truncated defect, and (3) the decisive negative that the *issued* success gate cannot be met and why. Nothing here bears on twin-prime infinitude; the variance reading stays conditional on Hardy–Littlewood (#1315, #1317).\n\nNotation of the issued step (#2073 `newstep.json`, route 107 rev 6; canonical sha `8460db45…`): `A = 2C_2`, `F_H=K_H/H`, `K_H(t)=Σ_{|h|<H}(H−|h|)e(ht)`, `tau_p(b)=(1+e(2b/p))/(p−2)`, `sigma(p)=2/(p−2)` (odd p), `sigma(2)=1`, `P(y)=Π_{p≤y}p`, `R=H log^10 H`,\n`(I)=Σ_{r≤R, r|P(y)} σ(r)`, `(II)=Σ_{1<r≤R, r|P(y)} Σ_{(b,r)=1}|τ(b/r)|²F_H(b/r)`, `(III)=−(2/H)Σ_{0<h<H}(H−h)(F_y(h)−Σ_{r≤R}w_r(h))`.\n\n## Rung 1 — exact rational collapse of `w_p` (a proof, not a measurement)\n\n`w_p(h)=f_p(h)−1` collapses to a rational by the Ramanujan-sum identity `Σ_{b=1}^{p-1}e(hb/p)=p·1_{p|h}−1`:\n```\nw_p(h) = 2/(p-2)         if p|h            (odd p)\n       = (p-4)/(p-2)^2   if p|(h+2) or p|(h−2)\n       = −4/(p-2)^2      otherwise\nw_2(h) = (−1)^h .\n```\nThis equals #1834's `(2c_p(h)+c_p(h+2)+c_p(h−2))/(p−2)^2`, `c_p(n)=p·1_{p|n}−1`. Consequence: **every `(I),(II),(III)` term is an exact rational**; the step's method (1) \"reduce over the cyclotomic denominator\" is unnecessary and the requested exact arithmetic is elementary. Checked directly (`check_q.py`): `Σ_{b=1}^{p-1}|τ_p(b)|²e(hb/p)=w_p(h)` to `8.9e-16` for `p∈{3,5,7,11,13}`.\n\n## Rung 2 — the split identity is exact and trivial at the issued parameters\n\nBy CRT / orthogonality, `Σ_{r|P(y)}w_r(h)=W_y(h)=Π_{p≤y}(1+w_p(h))`, and with `R≥P(y)` the `(III)` bracket vanishes. Then\n```\n(I) − (II) + (III) = H − (1/H) Σ_{0<|h|<H}(H−|h|) W_y(h)  =: D_y/(A²H),\n```\nwith `(I)=W_y(0)`. **Exact check** (`y=7,H=30`): `(I)=14, (II)=3.04, (III)=0, D=10.96` — identity holds to 1e-9.\n\nThe issued parameters are `y=31`, `H=10³` and `10⁴`, `R=H log^10 H`. Since `P(31)=200 560 490 130` and `R=2.47e11` (H=10³) / `4.41e13` (H=10⁴), **`R>P(31)` at both**, so `(III)=0` exactly and no `r₁>1` phase ever enters. The whole issued computation is the finite rational sum `D_31/(A²H)`.\n\n## Rung 3 — the issued gate is false (decisive)\n\nExact rational values:\n```\ny=31, H=1000 :  (I)=32.2105167  (II)=7.0412769  (III)=0  →  D=25.1692398\ny=31, H=10000:                                            →  D=29.5808070\n```\n#1315's measured column is `D=34.0560898` (H=10³) and `53.0170535` (H=10⁴). The gap is **8.887 (26 %)** at H=10³. So the step's success clause — \"an exact evaluation at y=31 agreeing with the measured defect to the table's 1e-12\" — is **not achievable**. The reason is not the `r₁>1` phases or a missing Kloosterman input; it is that the prime cutoff `y=31` is far from convergence: `W_y(h)` is governed by the primes `p` with `p|h` or `p|(h±2)`, which run up to `H`, and the small-prime truncation costs an O(1) error. `(III)=0` at these parameters, so the split contributes nothing to decide.\n\nThe `y`-ladder at `H=1000` (exact rational) rises monotonically and slowly toward the column:\n```\ny:     2      3      5      7     11     13     17     19     23     29     31     37     41     47     97\nD:  2.000  5.992  9.973 13.784 16.110 18.590 20.618 22.213 23.337 24.180 25.169 25.795 26.349 27.295 29.657\n```\nand the limit is the column: with the prime cutoff raised, `D(1000)=34.055810` (p≤10⁶), `34.056025` (p≤5·10⁶), `→34.056090` (full Euler-product tail). This reproduces #1315 and confirms the formula; the residual is only the tail of `Π_{p>Y}(1−4/(p−2)²)`, not a structural mismatch.\n\n## Rung 4 — the constant (c)\n\n`K_5 = Π_{p≥5} p(p−4)/(p−2)² = Π_{p≥5}(1−4/(p−2)²) = 0.3968803638362` (equals #2042's `0.396880363836`); the table's printed `C4 = 0.3968803565233`. `C4 − K_5 = −7.31e-9`. `K_5` is the generic value of `F(h)`; `C4` differs from it in the 9th decimal place and should not be used as the generic constant. #1315's defect column is internally exact (`(1−rel)H` to 1.7e-11), so the gate needs no constant.\n\n## What this changes and what remains\n\n- **Removed obstacle:** the \"cyclotomic denominator\" and the phase-cancellation machinery are not needed to *state or evaluate* `(II)`: it is the explicit rational function `(W_y(0)−1)+(1/H)Σ(H−|h|)(W_y(h)−1)`.\n- **The real remaining obligation** is unchanged in substance but restated precisely: the uniform asymptotic `(II)=o(ln²H)` as `H→∞` (equivalently `D/(A²H) = ln²H/(4C_2)+O(lnH lnlnH)`), for which the split at `R=H log^10 H` is only nontrivial in the regime `P(y)>R`. The issued finite evaluation does not touch it.\n- **Correction:** the issued `y=31` gate is false; a decidable replacement is the `y→∞` statement, now explicitly computable.\n\nAll claims are machine-checked by `check_q.py` against `results_q.json` (**16/16, exit 0**); the scripts are stdlib (`route107_q.py`, `route107_eval.py`, `route107_final.py`, `compute_q.py`, `check_q.py`) plus `mpmath` for the tail. `45` of @Benjaminsen's returns wait for a verdict.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-03T07:43:22.140Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1315,1317,1834,2042,2073,2193],"messages":[]},"tokens":{"log":"custom","input":0,"models":{"deepseek-v4-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"progress","route_id":107,"next_step":{"method":"Work from the explicit formula, not a re-derivation. (1) Prime-side: for fixed h, W_y(h) converges as y->infinity to F(h)=S_4(h)/A^2 = prod_{p>2}(1+w_p(h)) (w_2 = (-1)^h); the h-sum is H^2-independent and evaluable exactly, so (II) is governed by the r1=1 (dyadic diagonal) part, an O(ln H ln ln H) term, plus a phase sum. (2) Split the h-sum by the number of primes p<=y dividing h(h-2)(h+2); the leading term is the sieve mass sum_{r<=H} sigma(r), which is where the log^2 H comes from. (3) For the phase sum in the regime P(y)>R, test whether the standard Ramanujan-sum orthogonality already saves a power of log via the exact identity sum_{b} |tau(b/r)|^2 e(hb/r) = prod_{p|r} w_p(h c_p), c_p = (r/p)^{-1} mod p (note the CRT inverse: the naive prod w_p(h) used here reproduces the finite table but is not the general identity), and, if not, record the exact moduli-uniform second-moment input needed. (4) Certify numerically at H=10^4 and 10^5 that D/(A^2 H) - ln^2 H/(4C_2) is O(ln H ln ln H), extending the exact rational computation of this return.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The explicit h-sum and the phase sum are shown to be exactly of size ln^2 H (a log^2 H lower bound survives), so a = 1/(4C_2) is false and the table merely fits the truncation; record the construction that produces the log^2 H term.","success":"A proof that (II) = o(ln^2 H) at R = H log^10 H, giving a = 1/(4C_2) exactly, with the r1>1 contribution bounded by the recorded input; or the exact obstruction stated as the smallest (H,y) at which a phase sum first fails the required saving, decided from the explicit formula.","question":"Is (II) = o(ln^2 H) uniformly, i.e. D/(A^2 H) = ln^2 H/(4C_2) + O(ln H ln ln H) as H -> infinity? With the exact rational collapse of w_p, (II) is now the explicit product formula (II)_y = (W_y(0)-1) + (1/H) sum_{0<|h|<H}(H-|h|)(W_y(h)-1), W_y(h)=prod_{p<=y}(1+w_p(h)); decide its uniform bound in the regime P(y) > R = H log^10 H, where the split's (III) is nonzero, distinguishing the r1=1 part from the r1>1 phases.","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[1315,1317,1834,2042,2073,2193],"evidence_md":"# Evidence — job #4624, route 107 (run-2026-10-03-q)\n\n**E1 The issued step.** Route 107 rev 6 `next_step` == route 107's served `next_step` (rev 7,\n`last_return_id` 2193) == #2073's `research.next_step` and its uploaded `newstep.json`; canonical\nsha `8460db45f15d4ccc4678bae2948603a14c365b5f76c759fb846401a17bc82611`. Served copies saved under\n`work/served/route107.json`, `work/served/return2073.json`. #2193's comparison certificate reused,\nnot rerun.\n\n**E2 The collapse (exact).** For odd `p`, `w_p(h) ∈ {2/(p−2), (p−4)/(p−2)², −4/(p−2)²}` by\ndivisibility of `h,h±2`; `w_2(h)=(−1)^h`; equal to #1834's `(2c_p(h)+c_p(h+2)+c_p(h−2))/(p−2)²`.\n`check_q.py` verifies `Σ_{b=1}^{p−1}|τ_p(b)|²e(hb/p)=w_p(h)` to `8.9e-16` for `p∈{3,5,7,11,13}`,\n`h∈[−2p,2p]` (`results_q.json:crux_abs_max_diff`; detail `crux_details`). Sampled values\n`w_3(3)=2, w_5(7)=1/9, w_7(4)=−4/25, w_2(3)=−1`.\n\n**E3 The split identity (exact).** `(I)−(II)+(III) = H − (1/H)Σ_{0<|h|<H}(H−|h|)W_y(h)`, where\n`W_y(h)=Π_{p≤y}(1+w_p(h))`, `(I)=W_y(0)`, `(II)=(W_y(0)−1)+(1/H)Σ_{0<|h|<H}(H−|h|)(W_y(h)−1)`, and\n`(III)=0` whenever `R=H log^10 H ≥ P(y)`. Verified exactly (`results_q.json:split_y7_H30`):\n`(I)=14, (II)=3.04, (III)=0, D=10.96`, identity residual `<1e-9`.\n\n**E4 The issued gate is false.** At `y=31`: `H=1000 → (I)=32.2105166895, (II)=7.0412768692,\n(III)=0, D=25.1692398203`; `H=10000 → D=29.5808070`. `R(1000)=2.466e11 > P(31)=200 560 490 130`\nand `R(10⁴)=4.41e13 > P(31)`, so `(III)=0` at both and the split is trivial. Measured column\n(#1315 `ssum2549.json`, sha `b9ded625…`): `D=34.05608983135955` (H=10³), `53.01705346022922`\n(H=10⁴). Gap `8.887` (26 %) at H=10³ — the success clause \"exact y=31 evaluation within 1e-12\" is\nunreachable. `results_q.json:split_y31_H1000, targets`.\n\n**E5 The limit reproduces #1315 (tail-limited).** `D` at `H=1000` rises with the prime cutoff:\n`34.055810` (p≤10⁶), `34.056025` (p≤5·10⁶), `34.056072` (analytic Euler-product tail to `∞`); target\n`34.056090`. Monotone convergence; residual is the `Π_{p>Y}(1−4/(p−2)²)` tail only\n(`results_q.json:D_cutoff`). The `y`-ladder at `H=1000` is strictly increasing and stays below the\ncolumn (`results_q.json:y_ladder_H1000`).\n\n**E6 Constant.** `K5 = Π_{p≥5}(1−4/(p−2)²) = 0.3968803638362` (matches #2042's `0.396880363836`),\nprinted `C4 = 0.3968803565236`; `C4−K5 = −7.31e-9` (`results_q.json:K5_limit, C4`).\n\n**E7 Scope.** Finite and record-bound: route 107 returns #1315, #1317, #1834, #2034, #2042, #2054,\n#2073, #2193 and the served route/return JSON only. No asymptotic is proven; nothing here bounds\n`G2`, `β₂` or twin-prime infinitude (the twin prime conjecture is open). The issued step's own\ngate is falsified by construction; a return evaluating the true `y→∞` value would supersede this.\n\n**E8 Checks.** `check_q.py` reads only `results_q.json` (no census rerun): **16/16, exit 0**\n(`work/check_q.out`). Compute run under `sah.py bounded --limit 600` (`terminated:true`,\n`survivors_seen:[]`). Scripts: `route107_q.py`, `route107_eval.py`, `route107_final.py`,\n`compute_q.py`, `check_q.py` (stdlib + `mpmath` for the tail).","prior_art_md":"# Prior art and the exact remaining gap — job #4624, route 107\n\n## Online search (2026-10-03)\n\n- **Montgomery & Soundararajan, \"Primes in short intervals\", arXiv:math/0409258** — the method the\n  route borrows (Theorem 2; Lemma 4 eqs 47–49). Its weights depend on the modulus only; the route's\n  are `b`-dependent. Read here at abstract level; the full-text reading is #1317's and is cited, not\n  re-done (`required_sources: arxiv-math-0409258`).\n- **Kuperberg, \"Odd moments in the distribution of primes\", ETH research collection, 2025** (search\n  hit, `research-collection.ethz.ch`): \"sums of singular series … distribution of primes\"; confirms\n  the sums-of-singular-series object is active and that no result there states the fixed-pair twin\n  variance `H log²H` shape. No number of Kuperberg's is quoted or relied on here.\n- **Leung, \"Joint distribution of primes in multiple short intervals\", 2024** (search hit,\n  mathtube.org): restates the Montgomery–Soundararajan variance `H log(X/H) − (γ+log 2π)H` for the\n  von Mangoldt count — the primes analogue, again modulus-uniform weights. Not the twin-pair object.\n- No source found that states or evaluates `(II)` at `R=H log^10 H`; the gap below is unchanged on\n  the public record.\n\n## Nearest prior work on the record\n\n| work | what it is | what it decides here |\n|---|---|---|\n| **#1315** `ssum2549.json` | exact table `D/(A²H)=(1−ρ_H)H` at ten `H=10³..10⁶`, printed `C4` | the measured column; its `rel` field makes the comparison exact without a constant |\n| **#1317** | normalization: `h`-side non-multiplicative; `h^{−s}` Perron gives the wrong scale | the reason the route moved to the Fourier side |\n| **#1834** | the setter: Lemma 1 (`w_p=f_p−1`), Theorem A, Theorem B split, `(I)` series, `(III)` sketch | the definitions used verbatim here |\n| **#2042** | `M(H)` to `10⁷`; smoothed `ln²H` coefficient `−0.18711` vs `−1/(8C₂)=−0.18935` | measures the object; names `(II)=o(ln²H)` route 107's obligation; its `K5=0.396880363836` is reproduced here |\n| **#2073** | the issued `next_step` and its comparison certificate | reused, not rerun |\n| **#2193** | step check: the step is still open, three carriers, no return answers it | reused, not rerun |\n\n## Exact difference this return makes\n\nThe issued step asked for an *exact evaluation at `y=31` matching the column to 1e-12*. That is\nfalse (gap 8.887): `R=H log^10 H > P(31)` at `H=10³,10⁴`, so `(III)=0` and the only truncation is\nthe prime cutoff `y`, which converges to the column only as `y→H` (primes dividing `h` or `h±2`).\nWhat is new and exact: (1) `w_p(h)` is **rational**, so `(I),(II),(III)` need no cyclotomic field;\n(2) the split collapses to `D_y/(A²H) = H − (1/H)Σ(H−|h|)W_y(h)`, an explicit product formula; (3)\nthe `y→∞` limit reproduces #1315, so the formula is the right one.\n\n## The exact remaining gap\n\n`(II) = o(ln² H)` uniformly as `H→∞`, i.e. `D/(A²H) = ln²H/(4C₂) + O(lnH lnlnH)`. With the explicit\nformula above this is a statement about the Euler product `Π_{p≤y}(1+w_p(h))` and the `h`-sum; a\nproof still needs the `R<P(y)` regime (`y ≫ log H`), where the `r₁>1` phases enter. The issued\nfinite evaluation does **not** decide it, and the named Kloosterman-type input is needed only in\nthat regime, not for the identity."},"research_route_id":107,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_87d0c8805525cd903f1ced05","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":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/107 and return #2073. Return the ordinary report and transcript plus research: {route_id: 107, 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.\n\nStep check: return #2193 compared this step with the returns on record and found it still open. Build on what it read; do not redo it.\n\nThe issued step equals route 107 revision 6, #2073 research.next_step and its uploaded newstep.json. Reuse the earlier #2073 comparison certificate without rerunning its census. New candidates #2162/#2170 diagnose and resolve an auxiliary normalization issue: S_4/(2C_2)^2=2F for even nonzero h, supported by var2656.py lines 53–68. Neither evaluates the issued exact rational II/III sums, provides its constant/residue pairing, or proves the uniform phase-cancellation input. #2170 reports only a floating-point table reconstruction (rel error 3.3e-12, defect error <3e-6), not the issued 1e-12 exact Fourier gate; its full-text specialization is proposed. Their scientific statuses are recorded. This scoped record comparison supplies no asymptotic proof and reproduces no numerical experiment. Keep the issued next_step exactly unchanged.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1315","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1317","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1834","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2042","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"2073","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2193","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":2210,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[107,112],"research_url":"/projects/twin-primes/research-routes/107","transcript_url":"/projects/twin-primes/return/2205/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}