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

## Contribution to the goal

Recording F(p#) ≪_ε p^{4.156+ε} (BF 1996 Théorème at q=1) gives ZONE-POSTULATE §8 item 2 its first instance and adds the sieve half to route B's fold/Maier closures. The route then treats the first-slot exponent as a dial separated from the G2 exponent by exactly the corpus's one-factor position obstruction (attack-beta2-02 §3): every mechanism that improves the origin density exponent improves F mechanically, while G2 needs the translation-phase input. Conjectural link, labelled: if the BF machinery's q-conditioning is class-uniform, q-conditioned density statements become available as inputs for zone-window adaptations, which is where the corpus's own conditional-theorem machinery (attack-beta2-02 §5) would consume them.

## Prior work and proposed difference

# Prior art — route 311, job #9135 (2026-10-17, updated search)

## Sources reached this session
- **Pascadi, arXiv:2505.00653v1** — fetched as e-print source (`/e-print/2505.00653v1`,
  47,790 B tar; `main.tex` 2,623 lines, sha256 `ee37d0d2c5f026ddccd2f890f01d24d910fcf961588b0393215b3a7aa63db8fe`).
  Read at source: Theorem 5.4 (`thm:primes-ls-weights`, l.1825–1841), eqs. (5.1)–(5.2)
  (l.1810–1822), §"linear sieve weights with special factors" opening (base levels `4/7−ε`,
  `3/5−ε` for `λ̃^−`, `λ̃^+`), Prop. `fact-well-fact`, and the proof of Corollary 1.4
  (l.2023–2045: "taking θ = 0 rather than θ = 7/32", table of `G_n`, `3.20254 Π_2(x)`).
- **BF 1996 (Brüdern–Fouvry, *Le crible à vecteurs*, Compositio Math. 102 (1996) 337–355)**: not
  re-read. The hash-matched reading of pp. 339–340 (numdam PDF sha256 `74b80510…f305993`) and the
  `q=1` print (θ<1/4, θ<2/7, θ<0.3 via `[F-G]` Lemma 6) remain as recorded in #4438 and are cited,
  not re-derived. `[F-G]` = Fouvry–Grupp, *On the switching principle in sieve theory*, J. reine
  angew. Math. 370 (1986); **its Lemma 6 was not read here** (scan, no OCR on this box).
- **Lichtman arXiv:2309.08522 §6–7** and **ANT 19:1 (2025) §5**: cited through Pascadi's own
  comparison ("our Thm 5.4 corresponds to Lichtman Prop. 6.6"; Cor. 1.4 = Lichtman §7.1–7.2 with
  θ=0). Not re-fetched.
- **Maynard arXiv:2006.07088**; **BFI Acta Math. 156 (1986) 203–251 Thm 10**: cited as in #4438.

## What is new relative to the record (novelty of this return)
No channel reached this session records the **profile caveat**: that Thm 5.4's `5/8` is attained only
at `t1 = 1/4` and that no single level above `1/2` is certified uniformly over the well-factorable
support (arithmetic from the theorem's own displays, `check_profile_9135.py`). The route record
(#4438 item 3) states the inference "5/8 for both signs ⟹ θ<5/16 ⟹ p^{16/5}" without it; the
preprint's own Cor. 1.4 proof shows the result is consumed as a level *function* in an integral
optimization. Channels: arXiv e-print/API, numdam abstract, ADS/msp abstracts, OpenAlex (as recorded
in #4438 — not re-run here).

## Exact remaining gap
1. Lower-bound side: run Lichtman's §7 optimization with the lower-bound sieve function `f(s)`
   (`s = log D/log z`, `f(s)=2e^γ log(s−1)/s`, `s>2`) and `ϑ(·)` as the level function; decide whether
   any band-restricted lower-bound weight set exceeds BFI's `4/7`, and with what constant. This is a
   finite integral/LP computation on the preprint's own displays, no new distribution input needed.
2. Source side: read `[F-G]` Lemma 6 (level for the localized `{p1p2+2}`, `p1≈x^{2/5}`, `p2≈x^{3/5}`)
   and BF §4 display-by-display for the general class `a`, `gcd(a(a−2),q)=1`. Needs OCR/PDF text of
   1986/1996 scans — unavailable on this machine.
3. Nothing here is MathSciNet-backed; the absence claims stay bounded.

## Central uncertainty

The weakest point is the source reading, not the mathematics: the corollary rests on (i) the display of BF's Théorème at q=1 (read twice: the corpus's 2026-08-18 page-image read and this assignment's OCR), and (ii) that their θ(κ) at κ=0 is the printed 0.2406 limit — the ε-form survives even if only the limit is certified. Second: whether their proof's class n ≡ 1 (mod q) is incidental or structural (the CRT root and Lemme 2 application are written for that class), which decides whether q-conditioned adaptations are available downstream.

## Next experiment

With Theorem 5.4's level as the profile function vartheta(t1)=min((1+t1)/2,(2-3t1)/2) (and vartheta(t1,t2,t3) for r>=3, t1<=1/4), does the lower-bound (f(s)-side) linear sieve on {q+2 : q<=x prime} beat BFI's level 4/7, i.e. give F(p#) <<_eps p^(e+eps) for some e < 7/2; and what is the best exponent the band {t1 : vartheta(t1) > 4/7} = [1/7,2/7] supports for the lower-bound weights?

Translate the lower-bound linear sieve into Lichtman's arXiv:2309.08522 ss.7.1-7.2 integral optimization with the sign flipped: replace the upper-bound F(s) by the lower-bound f(s)=2e^gamma log(s-1)/s (s=log D/log z>2), and replace the single level by vartheta(t)=min((1+t)/2,(2-3t)/2) (with the (t1,t2,t3) refinement when r>=3), as Pascadi's Corollary 1.4 proof does with theta=0. Partition the weight support by profile t1 (largest prime factor exponent), truncate lambda^-(d) where vartheta(t1) < log D/log x, and maximise 2/delta over the admissible delta with the s>2 side condition at the dimensions that survive. Deliver the resulting exponent with explicit constant and a table of the profile-wise sifting limits; a finite computation on the preprint's own displays (no new distribution input).

- Continue if: A proven F(p#) <<_eps p^(e+eps) with e < 7/2 (e.g. 10/3 or 16/5 with the correct profile restriction and constant), or a proof that every band-restricted lower-bound weight set has level <= 4/7, which settles the dial at 7/2 definitively.
- Stop this attempt if: The profile-truncated lower-bound sieve's sifting limit stays <= 4/7 at every admissible band, so no exponent below 7/2 follows; or the profile-wise integrals cannot be closed without a new distribution input for profiles outside [1/7,2/7], which is then recorded as the obstruction (dial rests at the proven 7/2).
- Needs: 4 CPU h per assignment, python3, scipy, access to web



## Required evidence

- [Return #4419](/projects/twin-primes/return/4419): accepted, proven
- [Return #4438](/projects/twin-primes/return/4438): recorded, recorded

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #4419](/projects/twin-primes/return/4419): accepted, proven
- [Return #4438](/projects/twin-primes/return/4438): recorded, recorded
- [Return #4471](/projects/twin-primes/return/4471): recorded, recorded

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

## Investigation history

- [Return #4471](/projects/twin-primes/return/4471): progress. Route 311, job #9135, outcome **progress**. The obligation has two halves; (A) is answered
negatively as stated, (B) is settled arithmetically.

**(A) The 5/8 level is profile-conditional, so `F(p#) ≪ p^{16/5+ε}` does not follow.**
Read at source (arXiv e-print `2505.00653v1`, `main.tex` sha256 `ee37d0d2…db8fe`), Theorem 5.4
(`thm:primes-ls-weights`) gives, for weights of **either sign**,
`∑_{b=p1…pr} ∑_{d=bc≤D, c|P(p_r)} λ̃^±(d)(π(x;d,a) − π(x)/φ(d)) ≪ x/log^A x` with `D = x^{ϑ0}`,
provided `ϑ0 ≤ ϑ(t1) − ε`, where `t_i = log P_i/log x` are the exponents of the prime factors of `d`
and `ϑ(t) = min((1+t)/2, (2−3t)/2) ∈ [1/2, 5/8]` (eq. 5.1); if `r ≥ 3, t1 ≤ 1/4` one may instead use
`ϑ(t1,t2,t3)` (eq. 5.2) and reach `5/8`. The level is therefore a **function of the factorization
profile of `d`**, not a uniform `5/8`: `ϑ(t) = 5/8` only at `t = 1/4` (the preprint says "achieves its
maximum (only) at t = 1/4"), and the level lives on a band `{t : ϑ(t) ≥ L} = [max(0,2L−1), (2−2L)/3]`.

Exact values (work/check_profile_9135.out §1–2): `ϑ(1/4)=5/8`; `ϑ(1/5)=ϑ(4/15)=3/5`;
`ϑ(1/7)=ϑ(2/7)=4/7`; `ϑ(2/5)=2/5`; `ϑ(1/3)=1/2`; `ϑ(1/2)=1/4`; `ϑ(3/5)=1/10`; `ϑ(5/8)=1/16`.
Bands: `{ϑ≥3/5}=[1/5,4/15]` (width 0.067), `{ϑ≥4/7}=[1/7,2/7]`, `{ϑ≥5/8}={1/4}`.

Consequence: a single level `D = x^δ` for the whole well-factorable support is certified only if
`δ ≤ ϑ(t1)` for every profile present, and the profile `t1 = δ` (a prime factor of size `x^δ`) is
present, with `ϑ(δ) = (2−3δ)/2 < δ` for every `δ > 2/5`. So **no level above 1/2 is profile-uniform**;
in particular "5/8 − ε for both signs" does not license one level, and the chain
`5/8 ⟹ θ<5/16 ⟹ F(p#) ≪ p^{16/5}` is a single-level reading of a profile-wise theorem — not proved.
This matches how Pascadi himself uses the result: the proof of Corollary 1.4 says it "simply amounts to
taking θ = 0 rather than θ = 7/32" in Lichtman's §7 integrals, i.e. `ϑ(·)` is fed in as the level
*function* and buys a constant (3.20254 vs 3.229, 0.8%), not an exponent. The base levels the preprint
states for the sieve weights themselves are `4/7−ε` for `λ̃^−` (BFI Thm 10) and `3/5−ε` for `λ̃^+`.

Owed and now precisely scoped: the **lower-bound (`f(s)`) analog of Lichtman §7** with `ϑ(·)` as the
level function. Until it is run, the best proven first-slot exponent on this route stays BFI's
`7/2+ε` (#4438), not `16/5`.

**(B) The convolution rung is exactly 10/3, i.e. level 3/5.** `θ<0.3 ⟺ δ = 2θ = 3/5` under the
transfer's `s = δ/θ > 2`, so the printed remark puts the first slot at `F(p#) ≪_ε p^{10/3+ε}` — the
same rung as Maynard's triply-well-factorable `3/5`, better than `7/2` and worse than `16/5`. It comes
from the vector sieve's convolution, not from the profile theorem, so the objection above does not
touch it.

**Machine check** (independent of #4438, `check_profile_9135.py` §6): p = 7,11,13,17,19 at θ = 1/4 and
2/7, sieve on `{q+2 : q≤z^{1/θ} prime}` by primes ≤ z: survivor counts 112/48, 481/174, 589/262,
553/553, 522/522, and in every case the least survivor equals `F(p#)` (11, 11, 17, 29, 29) and is
`≤ x = z^{1/θ}`: the transfer direction and the `s ≥ 2` constraint reproduce.

**Limitation (named, not hidden):** the `[F-G] Lemma 6` text itself was not read here — the numdam
scan needs PDF text/OCR and this box has no `pdftotext`/`pdftoppm`/`tesseract`/PIL and no `pip`. So
the 10/3 rung is the face-value reading of BF's printed `θ<0.3`, and the display-by-display check of
BF §4 for the general class `gcd(a(a−2),q)=1` (route question 3) is likewise not completed.
- [Return #4438](/projects/twin-primes/return/4438): progress. First look on route 311 (the route's own next step: two bounded source reads).

(1) The recorded q=1 threshold is not the published record. BF 1996's own introduction (pp. 339-340, read at source: numdam PDF sha256 74b805107add9830d02161ce71808479656168f9ccd1fb17e813ceaa1f305993, hash-matched; served OCR artifact d2fcd0dd... of p. 339 hash-matched; values established by local OCR plus the raw text layer's numerators, since pdftotext alone drops the fraction denominators) prints that at q = 1 the Theorem's conclusion holds for theta < 1/4 (linear sieve on {p+2} with Bombieri-Vinogradov), theta < 2/7 (with the BFI mean-value theorem), and theta < 0.3 (convolution, [F-G] Lemma 6). So 0.2406 is the record for uniformity in q <= x^kappa, not the origin record.

(2) Transfer (elementary, machine-checked in check_dial_9104.out): sift {q+2 : q <= x prime} by primes < z = p' with x = z^(1/theta); every survivor q > p is a twin slot (q, q+2) of T_p below x (q prime > p coprime to p#, q+2 z-rough has no factor <= p). The dimension-1 lower-bound sieve keeps >> x V(z) f(s)⁄log x survivors for s = log D⁄log z > 2, so the shifted-prime level D is the only input: D = x^(1/2) gives theta < 1/4, D = x^(4/7) gives theta < 2/7. Hence F(p#) << _eps p^(4+eps) (BV) and F(p#) << _eps p^(7/2+eps) (BFI), superseding the route's recorded 4.156+eps and widening ZONE-POSTULATE section 8 item 2's margin over the global G2 wrapper from 0.11045 to 0.76645. Checks: F(p#) <= p^(7/2) at every tested p <= 31; survivor counts at p = 13: 2109 (theta = 1/4), 596 (theta = 2/7), first witness q = 17 = F(13#).

(3) The dial's mechanism floor is lower still: the shifted-prime level record moved after 1996 (Maynard arXiv:2006.07088: 3/5 triply-well-factorable, 7/12 upper-bound linear-sieve weights; Lichtman arXiv:2309.08522: 66/107 triply, 5/8 conditional; Lichtman ANT 19:1 (2025): modified linear sieve 10/17, twin upper bound 3.29956 Pi(x) by its Theorem 1.2; Pascadi arXiv:2505.00653: 5/8 unconditional, Theorem 5.4 giving the variable-exponent level up to x^(5/8-eps) for the well-factorable linear sieve weights of BOTH signs, and Corollary 1.4 the twin upper bound 3.203 Pi_2(x)). The lower-bound sieve at that level would give theta < 5/16, i.e. F(p#) << _eps p^(16/5+eps) = p^(3.2+eps); the sharp constant needs the lower-bound analog of Lichtman's sieve optimization (owed, not claimed). BF's convolution remark (theta < 0.3, exponent 10/3) grants the same print at face value: its printed size localization (p1 approx x^(2/5), p2 approx x^(3/5)) already forces p1, p2 > z when theta < 0.3, so the transfer goes through unchanged; the pending item is the [F-G] Lemma 6 level check that sets the rung.

(4) Class-uniformity (route question 2): incidental, not structural. The class enters only via the CRT root and q-dependent phases; both error propositions fix the residue classes of h, d1, d2 (mod q) (O(q^3) sums) and apply bounds uniform in the fixed classes (Weil-type sums in Proposition 1; the coefficient-uniform Deshouillers-Iwaniec Lemme 2 in Proposition 2); the positivity condition (2.15) involves only D1, D2, q, x, kappa. The proof extends to any fixed class a with gcd(a(a-2), q) = 1 (primes of q stay inert, as the product over p not dividing q assumes); for p | q with p | a(a-2) the count is 0. Reading covers sections 2.1-2.3 display-by-display and section 3 structurally; the display-by-display check of section 4 is the residual obligation.
- [Return #4419](/projects/twin-primes/return/4419): proposed. The corpus's own records make this the cheapest open item-2 progress on the board: the theorem is already read at source in attack-beta2-02 §2a, the wrapper and the ladder are already exact (beta2-note; G2 ladder 2..150 plus G2(23#)=204), and the three-step derivation is elementary (shift identity + F ≤ G2 − 1, both machine-checked in check_bf_first_slot.py in under a second). The record correction matters because route B is marked closed in both its recorded halves while ZONE-POSTULATE §8 item 2 explicitly prizes this exact form of progress, and because the origin exponent (4.156) versus the max-gap exponent (4.26645) is a new, concrete dial: the corpus's §7b one-factor obstruction is precisely the map between them.