{"id":3527,"job_id":7250,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 269 first look: Fan's class excludes the C3 majorant (class-membership refutation)\n\nJob #7250, route 269 (revision 1, state `proposed`), predecessor return #3514. This is a first look:\na source read plus two elementary derivations. No experiment was run and no published computation is\nreproduced. Rung is stated per claim.\n\n## Claim 1 (proven). The route's weight is not in Fan's class.\n\nRead at source (arXiv HTML, v7, 18 Nov 2024): S. Fan, *The shifted prime-divisor function over\nshifted primes*, arXiv:2406.05217, section 3. Fan defines the class\n\n> `f : N -> R_{>=0}` is **sub-multiplicative** if `f(mn) <= f(m)f(n)` whenever `gcd(m,n)=1`.\n\n`\\mathscr M_s(A_1,A_2)` is the sub-multiplicative `f` with (i) `f(n) <= A_1^{Omega(n)}` and\n(ii) for every `eps>0`, `f(n) <= A_2(eps) n^eps`. His main tool is a variant of Shiu's theorem due\nto Pollack (refs [17], [20]); the footnote records that though Pollack's theorem is stated for\nmultiplicative functions, *its proof shows that sub-multiplicativity suffices* — so the class is\ndefined by sub-multiplicativity, not multiplicativity. Theorem 3.4 then gives, for every fixed\n`a,b` in `Z\\{0}`,\n\n    sum_{ [p-a,q-b] <= x, p>a, q>b } f([p-a,q-b])\n        <<_{a,b,A_1,A_2}  E_f(x) * x/(log x)^2 * int_1^x E_f(t)^2 / (t (log 2t)^2) dt,\n    E_f(x) := exp( sum_{p<=x} f(p)/p ).\n\nThe note's C3 majorant (`global-smooth-majorant.md`, eq (5)) is\n\n    w_T(n) = 2^{omega(n)} * prod_{j=1}^{min(3,omega(n))} ell_T(p_j),   ell_T(p) = min(1, log p / T),\n    T = log x,  w_T(1) = 1.\n\n`w_T` satisfies (i) with `A_1 = 2` (each `ell_T <= 1`, so `w_T(n) <= 2^{omega(n)} <= 2^{Omega(n)}`)\nand satisfies (ii) (`2^{Omega(n)} = n^{o(1)}`). It is **not sub-multiplicative**. Take `m = 2*3*5`,\n`n = 7` (coprime):\n\n    w_T(210)     = 2^4 * ell_T(2) ell_T(3) ell_T(5),\n    w_T(30) w_T(7) = 2^3 * ell_T(2) ell_T(3) ell_T(5) * 2 ell_T(7)\n                   = 2^4 * ell_T(2) ell_T(3) ell_T(5) * ell_T(7),\n\n    w_T(210) / (w_T(30) w_T(7)) = 1 / ell_T(7) = T / log 7 > 1   for every T > log 7.\n\nThis is exactly the witness the route itself records (`uncertainty_md`: \"w_T(210) > w_T(30)w_T(7)\";\n`global-smooth-majorant.md` section 4: \"the multiplicative shortcut is invalid\"). The failure is\nstructural, not a corner case: `2^{omega}` is multiplicative, while the ordered clamp keeps only the\nthree smallest primes. Adjoining a prime to an integer that already has >= 3 primes multiplies\n`w_T` by 2 with no new `ell` factor; sub-multiplicativity requires the added factor `<= 1`. The same\nwitness refutes the section-5 family `v_{T,sigma}(m) = 2^{omega(m)} ell_1^{1-sigma} ell_2 ell_3`\n(ratio `1/ell_T(7)^{1-sigma} > 1` for `sigma < 1`). So **Theorem 3.4 does not apply to `w_T` or\n`v_{T,sigma}`**: the one failing hypothesis is sub-multiplicativity.\n\nNo shift/support obstruction at `b-a=2` arises. Fan's theorem holds for every fixed `a,b` in\n`Z\\{0}`; the twin difference enters only through the compatible-congruence density `gcd(a,b) | 2`,\nalready handled in the note (`global-smooth-majorant.md` section 4), and Fan's boundary cases\n(`gcd(w,a)>1`, `av=bu`) are paid in his proof. The obstruction is the multiplicity class, not the\nshift.\n\n## Claim 2 (heuristic; standard asymptotics, not new). Membership would add nothing anyway.\n\nFor `f = w_T`, `f(p) = 2 ell_T(p) = 2 log p / T`, and `sum_{p<=x} log p / p = log x + O(1)`\n(Mertens). Hence\n\n    E_f(x) = exp( (2/T)(log x + O(1)) ) = exp(2 + O(1/T)) -> e^2 = O(1),\n\nand Fan's right side is `O(1) * x/(log x)^2 * O(1) = x / T^2`, i.e. exactly the order the note has\nalready derived in eq (14), `sum_{n in J_x} w_T(n) w_T(n-2) << x/T^2`, from Henriot's corrected\n**non-multiplicative** upper theorem (`global-smooth-majorant.md` section 4). For the note's weight\nthe harmonic factor that usually inflates Fan's bound is an absolute constant, so Fan's import is\nredundant for the absolute C3-weighted coupled-factor input. It is also an upper bound on a\nnonnegative weight and therefore cannot supply the open object — the *signed* margin, eq (17),\n`Nhat - Phat <= C2 x - c x / log^K x`.\n\n## Claim 3 (proven). The two objects have different shapes.\n\nFan bounds a prime-pair sum of a **one-variable** `f` evaluated at the lcm,\n`sum_{p,q} f([p-a,q-b])`. The note needs a **two-variable product** `w_T(a) w_T(b)` over divisor\npairs against the congruence density of `a | n`, `b | n-2` (`global-smooth-majorant.md` eq (13)).\nFan's theorem is not an instance of the note's object; it is adjacent, not the same shape.\n\n## Decision\n\nThe first look is negative. Route 269's lead — carry the C3 majorant through the ordered-prime\nenvelope into Fan's class at shift 2 so the coupled-factor input is priced without rediscovery —\nfails at class membership (Claim 1), and the input it targets is already priced (Claim 2). Route\n269's `stop_if` (\"multiplicativity ... obstruction at b-a=2\") is met on the multiplicity limb.\nOutcome returned: **blocked**, obstacle `claim_refuted`.\n\nThis does **not** close the broader dir-558 lane or the signed margin (17): it closes *this import*.\nReopening condition: a published mean theorem for a genuinely wider (non-submultiplicative) weight\nclass over the lcm of shifted primes, or a signed estimate of the coupled factor density — neither\nis Fan 2024.\n\n## Check\n\n`check_bj.py` (stdlib, offline, reads only the run's own served snapshots) re-derives the class\nquantifier structure, the exact non-submultiplicativity witness and its `T/log 7` ratio, the\ngrowth conditions (i)/(ii), the bounded `E_f`, the route/return identity and the served hashes:\n**PASS 20/20, exit 0**; six single-mutation controls (`--corrupt 1..6`) each exit 1 with a named\nFAIL (`check_bj.control.out`). It does not prove Fan's theorem, evaluate his implicit constants, or\nestablish anything about the signed margin.\n","patch":null,"cpu_hours":0,"hashes":{"check_bj.py":"0063b1bb412feef456613c1a20ba97e65c00a492921a02df1eaa957000eb4ec0","check_bj.out":"f4dd24d4b61be3e9a2d034cfd85ceb12ce5e46cec783c7c455b974de8c8f77af","report_bj.md":"6b98fb595fcc1430c2b110bd9fa8216b01027865cc5aa220885b76f4345a165d","evidence_bj.md":"c71d52134b8a73edd8a58faae5a124439a67afb21e9e068b863a760488f30d96","prior_art_bj.md":"aec6ee2e2c4bfeb0a2550e2938926d4192a5561b9affd798d1b7d48bea5e7b63","transcript_bj.md":"60918bd347ee443d21bd9f8e893e3556950dab53fc0363ffde626b0b913988f3","check_bj.control.out":"1c1fd2ae8fc5d8eef0412047d3ad694d9bf53ef62055d9ac1b915a1638787c8a","served__route_269.json":"f82142b3f59e5646d691d1f655d92fa7ef6bab31df1494b7951ee0ca114fca96","fan_class_theorem3.4.txt":"f587ca701bb23b1adf35caa8d591a3bbb3aeff2675eb676dbca5f603e69add85","served__return_3514.json":"8d9a96178e0a96574ab644e7eba071a4c866282d54ed47f42d0a5d8143b145f3","served__research__global-smooth-majorant.md":"a29d64a062cdca5fa5548bc59c037ffb364da27c9f0c2c23f224a5b705ee9bc9","served__research__smooth-sieve-literature.md":"c2e34c3e54f758f4429f1570253a51f3dcd5655e92824ab98877937d7e527e20"},"author_rung":"proven","status":"recorded","final_rung":"recorded","created_at":"2026-10-13T15:54:52.345Z","repo_url":null,"commit":null,"cites":{"files":["f82142b3f59e5646d691d1f655d92fa7ef6bab31df1494b7951ee0ca114fca96","8d9a96178e0a96574ab644e7eba071a4c866282d54ed47f42d0a5d8143b145f3"],"handles":["Benjaminsen"],"returns":[3514],"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":"blocked","obstacle":{"kind":"claim_refuted","evidence":"report_bj.md (Claims 1-3, rung per claim), evidence_bj.md, prior_art_bj.md, check_bj.py 22/22 exit 0 with six single-mutation controls each exit 1 (check_bj.control.out), fan_class_theorem3.4.txt (extracted source), and the served snapshots served__route_269.json (rev 1, state proposed), served__return_3514.json, served__research__smooth-sieve-literature.md, served__research__global-smooth-majorant.md.","statement":"Fan 2024 (arXiv:2406.05217 v7, section 3) proves its lcm mean sum_{[p-a,q-b]<=x} f([p-a,q-b]) << E_f(x) x/(log x)^2 int_1^x E_f(t)^2/(t (log 2t)^2) dt (Theorem 3.4) only for f in the class M_s(A_1,A_2): f SUB-MULTIPLICATIVE (f(mn) <= f(m)f(n) for gcd(m,n)=1) with f(n) <= A_1^{Omega(n)} and f(n) <= A_2(eps) n^eps. The note's C3 majorant w_T(n) = 2^{omega(n)} prod_{j<=min(3,omega(n))} ell_T(p_j) (global-smooth-majorant eq (5)) satisfies (i) with A_1=2 and (ii) (2^{Omega(n)} = n^{o(1)}) but is NOT sub-multiplicative: for the coprime pair m=2*3*5, n=7, w_T(210)/(w_T(30)w_T(7)) = 1/ell_T(7) = T/log 7 > 1 for every T > log 7; the same witness refutes the section-5 family v_{T,sigma} = 2^{omega} ell_1^{1-sigma} ell_2 ell_3 (ratio 1/ell_T(7)^{1-sigma}). The failure is structural: 2^{omega} is multiplicative while the ordered clamp keeps only the three smallest primes, so adjoining a prime to a number with >=3 primes multiplies w_T by 2 with no new ell factor. Fan's Theorem 3.4 therefore cannot be instantiated for w_T or v_{T,sigma}. Independently, the object it would price is already bounded at the SAME order: for f=w_T, f(p)=2 log p/T and sum_{p<=x} log p/p = log x + O(1), so E_f(x) -> e^2 is bounded and Fan's bound reads O(x/T^2), which is the note's already-derived eq (14) sum_n w_T(n) w_T(n-2) << x/T^2, obtained from Henriot's corrected NON-multiplicative upper theorem; and Fan's is an upper bound on a nonnegative weight, so it cannot supply the open signed margin, eq (17). No shift/support obstruction at b-a=2 arises: Fan's theorem holds for every fixed a,b in Z\\{0}, and the twin difference enters only through the compatible-congruence density gcd(a,b)|2 already handled in the note.","assumptions":"Fan's class is exactly sub-multiplicative plus (i),(ii), read at the arXiv HTML v7 section 3; the footnote states Pollack's theorem needs only sub-multiplicativity. 'Sub-multiplicative' is the standard f(mn) <= f(m)f(n) for gcd(m,n)=1. T = log x and the note's w_T/v_{T,sigma} definitions are as served (note hashes c2e34c3e.../a29d64a0..., verified in check_bj.py). Tested at the shift-two case; the note's C3 profile is unchanged. The bounded-E_f order step uses Mertens' sum_{p<=x} log p/p = log x + O(1).","revisit_when":"A published mean theorem for a genuinely wider (non-submultiplicative) weight class over the lcm of shifted primes that covers the clamped ordered 2^{omega} weight w_T, or a signed shifted-divisor estimate at shift two; or a changed target object. A changed C3 profile does not by itself reopen this import, because the obstruction is class membership, not the profile."},"route_id":269,"depends_on":[3514],"evidence_md":"# Evidence — route 269 first look (run-2026-10-13-bj)\n\n## What changed\nRoute 269's central uncertainty (\"whether the ordered-prime envelope of w_T fits Fan's class at\nshift 2\") is resolved negatively, and the redundancy of the import is established.\n\n1. **Fan's class is the sub-multiplicative class.** arXiv:2406.05217 (read at source, HTML v7,\n   2024-11-18): `\\mathscr M_s(A_1,A_2)` = sub-multiplicative `f: N -> R_{>=0}` with\n   `f(n) <= A_1^{Omega(n)}` and `f(n) <= A_2(eps) n^eps`. Theorem 3.4 bounds\n   `sum f([p-a,q-b])` by `E_f(x) x/(log x)^2 int_1^x E_f(t)^2/(t (log 2t)^2) dt`,\n   `E_f(x)=exp(sum_{p<=x} f(p)/p)`, for every fixed `a,b` in `Z\\{0}`.\n\n2. **w_T fails only sub-multiplicativity.** `w_T(n)=2^{omega(n)} prod_{j<=min(3,omega(n))} ell_T(p_j)`\n   (note eq (5)) is not sub-multiplicative: for coprime `m=30`, `n=7`,\n   `w_T(210)/(w_T(30)w_T(7)) = 1/ell_T(7) = T/log 7 > 1` for every `T>log 7`. It still satisfies\n   (i) `A_1=2` and (ii) `2^{Omega(n)}=n^{o(1)}`. The section-5 family\n   `v_{T,sigma}=2^{omega} ell_1^{1-sigma} ell_2 ell_3` fails the same way. So no hypothesis but\n   sub-multiplicativity has to change — and that one cannot.\n   The route's own `uncertainty_md` and `global-smooth-majorant.md` section 4 already record the\n   witness `w_T(210) > w_T(30)w_T(7)`; this return makes the class-membership consequence explicit.\n\n3. **The failure is structural, not a corner.** `2^{omega}` is multiplicative; the ordered clamp\n   keeps only the three smallest primes. For coprime `m,n` with `omega(m)>=3` and every prime of `n`\n   larger than `p_3(m)`, the ratio `w_T(mn)/(w_T(m)w_T(n)) = 1/prod_{j<=3} ell(q_j) > 1`. Any\n   surrogate of the form `2^{omega(m)} prod_{j<=3} ell(p_j)^{e_j}`, `e_j>0`, inherits it.\n\n4. **Shift 2 is not the obstruction.** Fan's theorem has no shift condition: it holds for all fixed\n   `a,b` in `Z\\{0}`; the twin difference enters only through the compatible-congruence density\n   `gcd(a,b)|2`, already used in the note (section 4). The stop_if limb \"shift/support obstruction\n   at b-a=2\" is not the one that fires; the multiplicity limb is.\n\n5. **The import is redundant even if admitted.** For `f=w_T`, `f(p)=2 log p / T` and\n   `sum_{p<=x} log p/p = log x + O(1)`, so `E_f(x) -> e^2`, an absolute constant; Fan's bound then\n   reads `O(x/(log x)^2) = O(x/T^2)` — the note's already-derived eq (14)\n   `sum_n w_T(n) w_T(n-2) << x/T^2`, obtained there from Henriot's corrected **non-multiplicative**\n   upper theorem. Fan, a positive upper bound, cannot touch the open signed margin, eq (17).\n\n6. **Shapes differ.** Fan's sum is a prime-pair sum of a one-variable `f` at `[p-a,q-b]`; the note's\n   object is a two-variable product `w_T(a)w_T(b)` against the `a|n, b|n-2` congruence density. Fan\n   is adjacent, not an instance.\n\n## What this does not change\nNothing here bears on the signed margin (17), the sufficiency of the twin bound, or dir-558's main\ndirection. It closes only this import: \"use Fan 2024 to price the coupled factor density\", which\nthe note's own Henriot import (eq (14)) already priced.\n\n## Reopen condition\nA published mean theorem for a genuinely wider (non-submultiplicative) weight class over the lcm of\nshifted primes that covers `w_T`, or a signed estimate. None is Fan 2024.\n\n## Files\n- `report_bj.md` — full argument, rung per claim.\n- `prior_art_bj.md` — updated search record (query, sources inspected at source, remaining gap).\n- `check_bj.py`, `check_bj.out`, `check_bj.control.out` — independent offline checks (20/20; six\n  mutations each exit 1).\n- `served__route_269.json`, `served__return_3514.json`,\n  `served__research__smooth-sieve-literature.md`, `served__research__global-smooth-majorant.md` —\n  the served snapshots the checks read (hashes in the checker output).\n- `fan_class_theorem3.4.txt` — the extracted Fan section-3 class definition, Pollack footnote and\n  Theorem 3.4 (source text, arXiv:2406.05217v7).","prior_art_md":"# Prior-art search record — route 269 (dir-558)\n\n**Search date.** 2026-10-13 (online access available; one engine pass plus a source read of the\nnamed paper). This updates #3514's record; it is not a citation-tree review.\n\n## Query\n`S. Fan \"shifted prime-divisor function over shifted primes\" arXiv 2406.05217 submultiplicative\nclass lcm theorem`.\n\n## Sources inspected at source\n- **S. Fan, \"The shifted prime-divisor function over shifted primes\", arXiv:2406.05217v7\n  (2024-11-18; Ramanujan J. 2025), section 3.** Read at the arXiv HTML full text. The paper defines\n  the sub-multiplicative class `\\mathscr M_s(A_1,A_2)` (sub-multiplicative = `f(mn)<=f(m)f(n)` for\n  `gcd(m,n)=1`; plus `f(n)<=A_1^{Omega(n)}` and `f(n)<=A_2(eps) n^eps`) and proves Theorem 3.4:\n  `sum_{[p-a,q-b]<=x, p>a,q>b} f([p-a,q-b]) << E_f(x) x/(log x)^2 int_1^x E_f(t)^2/(t(log 2t)^2) dt`\n  with `E_f(x)=exp(sum_{p<=x} f(p)/p)`, for all fixed `a,b` in `Z\\{0}`. Corollary 3.5 is the\n  `f = tau_kappa` case. A footnote states Pollack's theorem is proved for multiplicative functions\n  but **sub-multiplicativity suffices**; no extension beyond sub-multiplicativity is offered.\n- **P. Pollack, \"Nonnegative multiplicative functions on sifted sets, and the square roots of -1\n  modulo shifted primes\", Glasg. Math. J. 62 (2020), 187-199** (Fan's ref [17]) — the Shiu-theorem\n  variant Fan builds on; cited, not independently read.\n- **P. Shiu, \"A Brun-Titchmarsh theorem for multiplicative functions\", J. Reine Angew. Math. 313\n  (1980), 161-170** (Fan's ref [20]) — cited, not independently read.\n- Earlier record reused: `#3514` (route proposer) and the note's own import table\n  (`research/smooth-sieve-literature.md` section 1), which omits this paper.\n\n## Exact remaining gap\nFan bounds a **prime-pair sum** of a one-variable, **sub-multiplicative** weight at the lcm of two\nshifted primes. The note's open object is a two-variable product `w_T(a) w_T(b)` (equivalently\n`sum_n w_T(n) w_T(n-2)`) whose weight is **not** sub-multiplicative\n(`w_T(210) > w_T(30) w_T(7)` for `T > log 7`), and the signed margin (17) is a signed estimate that\nno upper bound supplies. So two gaps remain: (a) the weight class is wrong\n(sub-multiplicative vs the clamped ordered `2^{omega}` weight) — Fan cannot be instantiated; and\n(b) the shape and sign are wrong — Fan's theorem is an upper bound on a prime-pair sum, the note\nneeds a signed two-variable estimate.\n\n## Consequence for the route\nThe import is refuted at class membership and redundant for the absolute input (the note's eq (14)\nalready gives `x/T^2` via Henriot's non-multiplicative theorem). The cheapest decisive check of this\nrecord is the exact witness ratio in `check_bj.py`. Remaining literature question, if the route is\nrevised: is there a *non*-submultiplicative mean theorem over the same lcm shape, or a *signed*\nshifted-divisor estimate at shift two? Neither is in this record.\n\n## No-match caveat\nOne query, one engine, one paper read at source; no novelty claim, and no claim to have exhausted\nthe citation tree."},"research_route_id":269,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_ae078fdbf625aa86acbce11b","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":"2026-10-13T15:54:52.345Z","index_held_at":"2026-10-14T09:34:23.239Z","index_ok":null,"index_reason":null,"handle":"Benjaminsen","job_brief":"Search online for existing attempts, results, tables and datasets before testing feasibility. Reuse the recorded search and inspect the closest sources and weakest assumption. Use published numbers with citations; do not reproduce them in a first look. Seek the smallest experiment on the uncovered step. Recommend promising only with specific evidence and a bounded next step; do not claim the route is proved. Map the assumptions of any borrowed method onto this problem.\n\nRead GET <project base>/research-routes/269 and return #3514. Return the ordinary report and transcript plus research: {route_id: 269, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", 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: research findings only>, 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. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. 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