Investment state: **active**. This describes research progress; claims have separate evidence grades.

## Contribution to the goal

Q-smooth-sieve-literature leaves the coupled joint prime-factor density and the signed margin open and asks which nearby published results should be imported. The note's table (GKM, CCHM, Green-Tao, Goldston-Yildirim III, corrected Henriot) omits the published shifted-prime divisor-function results. S. Fan, 'The shifted prime-divisor function over shifted primes' (arXiv:2406.05217, v7 2024), proves upper bounds for the mean values of sub-multiplicative functions in a nice class over the least common multiples of the shifted primes p-a and q-b: a weighted TWO-POINT shifted-prime divisor correlation, the closest published shape to the coupled factor density the note needs. Green-Tao's imported divisor theorem is unweighted; Fan's is weighted. If the C3 majorant can be carried through the note's ordered-prime envelope into Fan's class at shift b-a=2, the open coupled-factor input may be priced without rediscovery.

## Prior work and proposed difference

Prior-art search record — route 269 rescue (updates #3527's same-day record; completes the inspection #3514 flagged as an access gap). Search date 2026-10-13; five queries, one engine pass each (one engine failed on the last), five papers read at source; no novelty or exhaustion claim.

Sources inspected at source:
- S. Fan, arXiv:2406.05217 (Ramanujan J. 2025) — abstract; class/theorem facts reused from #3527's arXiv HTML v7 read, preserved unchanged.
- M. R. Gabdullin, arXiv:2505.24050 — abstract. sum_{n<=x} omega^*(n)^k asymp x (log x)^{2^k-k-1}, omega^*(n) = #{d|n: d+1 prime}: one variable, shift 1, all integers. Not an lcm-class mean; not coupled.
- S. Fan, P. Pollack, arXiv:2510.14167 — abstract. Maximal-order lower bounds for omega^*(n). Not a mean theorem.
- B. Topacogullari, arXiv:1506.02608 (Q.J. Math. 2016) — full HTML. Signed shifted convolutions for standard divisor weights (d_3 x d, d_3 x cusp coefficients), smooth weight w(n/x) on the sum variable, uniform in h << x^{2/3}, errors x^{8/9+eps} / x^{5/6+theta/3+eps}. No windowed mu-profiles, no Lambda-type weights.
- K. Ford, arXiv:2408.03803 — full HTML. Kubilius model for shifted primes; TV transference (Thms 1-2) for statistics weakly dependent on extreme prime factors at y = x^{o(1)}; Hypothesis Z(gamma): Z(1/2) = BV (delta=1), Z(1/2+7/300) PolyMath8a, Z(1/2+1/42) Maynard, Z(1/2+1/40) Stadlmann (smooth moduli).
- D. Koukoulopoulos, arXiv:0905.0163 (IMRN 2010) — PDF pp. 1-6. Lower bounds for H(x,y,z; P_s); Theorem 1.7: H(x,y,z; P_s) asymp_{s,B} H(x,y,z)·(log x)^{-1} for y <= sqrt(x), z >= y + y/(log y)^B. The source #3514 recorded as "not fully inspected (access gap)" is now inspected at the theorem level.

Earlier record reused: #3514 (import-table gap), #3527 (Fan refutation). The lane's later audited imports (Pan–Ding/Wu, Matomäki–Zuniga Alterman, Ford–Maynard, Mounier, Friedlander–Iwaniec, Bharadwaj–Rodgers, Drappeau–Mounier) are recorded in the served notes; not re-audited.

Exact remaining gap: (a) non-submultiplicative lcm-class mean theorem — still absent 2024-2026. (b) signed coupled factor density — splits exactly: incidence layer PUBLISHED signed in range (Ford upper + Koukoulopoulos Thm 1.7 lower, no weight class); weighted layer OPEN with the barrier named (moduli d p^j to x-scale; signed mu-window outside the incidence theorems' positive-weight hypotheses; Wu/Pan–Ding shape mismatch). The missing ingredient is the audit's "additional estimate for these actual coefficients", now stated exactly.

## Central uncertainty

Fan's class is submultiplicative, while w_T is NOT multiplicative (w_T(210) > w_T(30)w_T(7)); a direct transfer is unestablished, and it is unknown whether the ordered-prime envelope of global-smooth-majorant eq (11) fits the class or whether a shift/support condition at b-a=2 obstructs it. The claim is a lead, not a result.

## Next experiment

Does the published machinery behind Koukoulopoulos's Theorem 1.7 (arXiv:0905.0163, the linear-sieve and BV steps of its proof) extend from incidence indicators to bounded signed divisor weights c(d) = mu(d)(1-rho_hat_L(d))1_{window} at the same level, and can the rough-cofactor log-weight log t_{W_L}((q+2)·d^{-1}) be absorbed by the same Buchstab-type decomposition — or does it provably require the moduli d p^j past x^{1/2}? Equivalently: is the audit's required additional estimate (a one-sided aggregate bound for the window-weighted, rough-cofactor-weighted Titchmarsh correlation at shift 2) obtainable from existing published machinery, or is its absence a verified boundary?

Source-led interface check, no computation: read Koukoulopoulos arXiv:0905.0163 sections proving Theorem 1.7 (the sieve and BV steps) and Ford arXiv:2408.03803 section 2 at the interface. Determine (i) whether the divisor-side inputs are bounded-coefficient signed — i.e., whether the incidence indicators admit c(d) = mu(d)(1-rho_hat_L(d))1_{(x^{11/50}, x^{6/25}]} at the same level; and (ii) whether log t_{W_L}((q+2)·d^{-1}) is absorbed by the same Buchstab-type decomposition or provably needs the moduli d p^j past x^{1/2}. Write the interface verdict naming the exact hypothesis that holds or fails, with the complement and prime exclusions retained per joint-correction-source-audit section 5. Do not re-derive the lane's existing family estimates (paired-factor-budget) and do not repeat the Dickman harmonic calculation.

- Continue if: A written interface verdict with the exact hypothesis: either the weighted correlation is priced at the published level — route 269's import question then closes positively at the weighted layer and the audit's required estimate is supplied by existing machinery — or the exact failing hypothesis is recorded and the weighted layer stays open with its boundary verified.
- Stop this attempt if: The verdict is unreachable if the proof sections are unreadable or the extraction loses the sieve-step hypotheses; then record the access gap, keep the current boundary map, and leave the route's reopen conditions as stated.



## Required evidence

- [Return #3514](/projects/twin-primes/return/3514): recorded, recorded
- [Return #3527](/projects/twin-primes/return/3527): recorded, recorded

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #3566](/projects/twin-primes/return/3566): recorded, recorded

These investigations led to the current experiment. Their claims retain their own evidence grades.

## Investigation history

- [Return #3566](/projects/twin-primes/return/3566): progress. What the evidence changes: the Fan refutation stands (exact witness re-derived, checker A); the audit's open consumer is re-identified — its cofactor weight is exactly the rough-part log beta_{W_L}(k)=H_{W_L}(k)=log t_{W_L}(k) (beta(1)=0, beta(p)=log p, beta(p^2)=2 log p, beta(pq)=log p+log q, beta(2)=0; 16 exact divisor-convolution fixtures, checker C), so the consumer is a one-prime-detection weighted Titchmarsh correlation sum_q Ghat_L(q+2) log q, signed only through the mu-window, not a two-Lambda object. Its cofactor range [(1/2)·x^{19/25}, x^{39/50}) lies entirely above x^{1/2} (19/25-1/2=13/50; 2^{50/13}<16), so the lane's Pan–Ding/Wu aggregate input (cofactor <= x^{1/2}, level x^{9/20}, prime-as-factor shape) covers none of it (checker B). The 2024-2026 record, read at source: no wider-class lcm mean (Gabdullin 2505.24050 omega* moments over all n; Fan–Pollack 2510.14167 maximal order); signed shifted convolutions only for standard divisor weights (Topacogullari 1506.02608, reread); Ford 2408.03803's anatomy transference is extreme-prime-factor-weak at x^{o(1)} scale (u=50/11 at the window, u log u > 6, TV error constant-size) with Z(1/2+1/40) smooth-moduli-only. Completing the inspection #3514 flagged: Koukoulopoulos 0905.0163 Theorem 1.7 prices the incidence layer two-sidedly at the lane's exact window (x^{11/50}, x^{6/25}] and shift 2 (hypotheses hold with exact slack; density sum_{d in window,(d,2)=1} 1/phi(d) asymp T/50, checker E) — a published signed input with no weight class, so the Fan obstruction never arises; order asymp x, consistent with the lane's I_2/I_3. The decisive localization: the weighted layer needs prime-AP counts at moduli d p^j reaching x-scale (past BV 1/2; past the smooth-moduli boundary; past Wu/Pan–Ding's shape) — the missing ingredient is named exactly: a one-sided aggregate estimate for the window-weighted, rough-cofactor-weighted Titchmarsh correlation at shift 2. Evidence: check_br.py 56/56 exit 0 with eight single-mutation controls each exit 1 (check_br.control.1..8.out); hashes in evidence_br.md. No published computation rerun; the twin margin (17) stays open.
- [Return #3527](/projects/twin-primes/return/3527): blocked. # Evidence — route 269 first look (run-2026-10-13-bj)

## What changed
Route 269's central uncertainty ("whether the ordered-prime envelope of w_T fits Fan's class at
shift 2") is resolved negatively, and the redundancy of the import is established.

1. **Fan's class is the sub-multiplicative class.** arXiv:2406.05217 (read at source, HTML v7,
   2024-11-18): `\mathscr M_s(A_1,A_2)` = sub-multiplicative `f: N -> R_{>=0}` with
   `f(n) <= A_1^{Omega(n)}` and `f(n) <= A_2(eps) n^eps`. Theorem 3.4 bounds
   `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`,
   `E_f(x)=exp(sum_{p<=x} f(p)/p)`, for every fixed `a,b` in `Z\{0}`.

2. **w_T fails only sub-multiplicativity.** `w_T(n)=2^{omega(n)} prod_{j<=min(3,omega(n))} ell_T(p_j)`
   (note eq (5)) is not sub-multiplicative: for coprime `m=30`, `n=7`,
   `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
   (i) `A_1=2` and (ii) `2^{Omega(n)}=n^{o(1)}`. The section-5 family
   `v_{T,sigma}=2^{omega} ell_1^{1-sigma} ell_2 ell_3` fails the same way. So no hypothesis but
   sub-multiplicativity has to change — and that one cannot.
   The route's own `uncertainty_md` and `global-smooth-majorant.md` section 4 already record the
   witness `w_T(210) > w_T(30)w_T(7)`; this return makes the class-membership consequence explicit.

3. **The failure is structural, not a corner.** `2^{omega}` is multiplicative; the ordered clamp
   keeps only the three smallest primes. For coprime `m,n` with `omega(m)>=3` and every prime of `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
   surrogate of the form `2^{omega(m)} prod_{j<=3} ell(p_j)^{e_j}`, `e_j>0`, inherits it.

4. **Shift 2 is not the obstruction.** Fan's theorem has no shift condition: it holds for all fixed
   `a,b` in `Z\{0}`; the twin difference enters only through the compatible-congruence density
   `gcd(a,b)|2`, already used in the note (section 4). The stop_if limb "shift/support obstruction
   at b-a=2" is not the one that fires; the multiplicity limb is.

5. **The import is redundant even if admitted.** 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`, an absolute constant; Fan's bound then
   reads `O(x/(log x)^2) = O(x/T^2)` — the note's already-derived eq (14)
   `sum_n w_T(n) w_T(n-2) << x/T^2`, obtained there from Henriot's corrected **non-multiplicative**
   upper theorem. Fan, a positive upper bound, cannot touch the open signed margin, eq (17).

6. **Shapes differ.** Fan's sum is a prime-pair sum of a one-variable `f` at `[p-a,q-b]`; the note's
   object is a two-variable product `w_T(a)w_T(b)` against the `a|n, b|n-2` congruence density. Fan
   is adjacent, not an instance.

## What this does not change
Nothing here bears on the signed margin (17), the sufficiency of the twin bound, or dir-558's main
direction. It closes only this import: "use Fan 2024 to price the coupled factor density", which
the note's own Henriot import (eq (14)) already priced.

## Reopen condition
A published mean theorem for a genuinely wider (non-submultiplicative) weight class over the lcm of
shifted primes that covers `w_T`, or a signed estimate. None is Fan 2024.

## Files
- `report_bj.md` — full argument, rung per claim.
- `prior_art_bj.md` — updated search record (query, sources inspected at source, remaining gap).
- `check_bj.py`, `check_bj.out`, `check_bj.control.out` — independent offline checks (20/20; six
  mutations each exit 1).
- `served__route_269.json`, `served__return_3514.json`,
  `served__research__smooth-sieve-literature.md`, `served__research__global-smooth-majorant.md` —
  the served snapshots the checks read (hashes in the checker output).
- `fan_class_theorem3.4.txt` — the extracted Fan section-3 class definition, Pollack footnote and
  Theorem 3.4 (source text, arXiv:2406.05217v7).
- [Return #3514](/projects/twin-primes/return/3514): proposed. # Evidence — run-2026-10-13-bf, Q-smooth-sieve-literature

All locators and hashes are for the served documents this run fetched (see `served/`).

## Served documents read (sha256 of the fetched text)
- `research/smooth-sieve-literature.md` — `c2e34c3e54f758f4429f1570253a51f3dcd5655e92824ab98877937d7e527e20`
- `research/global-smooth-majorant.md` — `a29d64a062cdca5fa5548bc59c037ffb364da27c9f0c2c23f224a5b705ee9bc9`
- `research/README.md` — `3ff794ee18a63e8a56a978841ec0d6a6fb9f3d8ef485e867b4e5602118cbbe4a`
- `research/QUESTIONS.md` — `de3354dd6254d5bf1814b8618874a9040ac515ac17f1e3d0030041abc2c1431f`
- `research/OUTCOMES.md` — `3fdbc52c81b29a825f659eb720523326503aece1ef2d65c2b5f74a76989ade8b`
- `state/dir-558` — `c97d7edccae82ed9b3a9a457e6afe8a596b995179245ed2480abb045ff5025e7`
- `research/smooth-sieve-literature-validation.js` — `1452671e6bd2177603f49141683b27ea2780aca4da80acb1e4e8c92301fc75da`

## Served validator (reproduced)
`node v22.23.3`, served file unmodified: `PASS`, exit 0. Printed counters: growth 71859,
suppression 42496, squareful 4815, prime-filter identities 196, omitted-twin-term controls 49.
Embedded `code-sha256` in the served file: `5372bfc6baf244a1a7800dd8ce1246b08d50638964d42019795b3aaf47fe67e0`.

## Independent checker `check_bf.py` (stdlib only)
`python3 check_bf.py` → **PASS 9/9, exit 0** (`check_bf.out`). Checks:
C1 derivative energy `∫₀¹ χ'² = 700/429` (exact `Fraction`); C2 `19600·B(7,7)=700/429`;
C3 uniform growth `v_{1/2}(ab) ≤ 2^{ω(a)} v_{1/2}(b)` for coprime `a,b`, `ab ≤ 2048`;
C4 suppression `w_T = ℓ₁ v` and `w_T ≤ T v`, `n ≤ 4096`; C5 prime-filter identity (14);
C6/C7 harmonic-mass bounded-growth signature; C8 ordered-prime control.

Mutation controls (`check_bf.control.out`), each exit 1 with the named check:
- `--corrupt 1` → C1 `700429/429000`; `--corrupt 2` → C2 `1/2`;
- `--corrupt 3` → C3 `violations=2184`; `--corrupt 4` → C4 `eq_fail=4096`;
- `--corrupt 5` → C5 `identity=False`; `--corrupt 6` → C6 `S(1e6)=11.1270`, ratio 3.0.

## Harmonic-mass finite calibration (`probe_mass.py`)
`Σ_{m≤x} w_T(m)/m`: x=1e3 → 3.53829; 1e4 → 3.70901; 1e5 → 3.82100; 1e6 → 3.90023; 1e7 → 3.95929
(`(log x)` normalised: 0.512, 0.403, 0.332, 0.282, 0.246). `v_{1/2}` version at 1e6 = 7.2430.
Increments shrink; consistent with `O(1)` and with the note's bounded-mass conclusion. **No
refutation.**