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

## Contribution to the goal

This reframes #4810's open list. The fixed-q twin asymptotic over F_q[T] is a theorem for q > 685090p² (p odd). It stays open for prime q = p and for small prime powers, which are the regime of every #4810 measurement. #4810's 'must reproduce 8' is the constant of its 2-dim sieve: a 1-dim Λ² sieve on {P+2}, with Weil-RH level θ < 1/2, gives limsup ≤ 4 at every fixed odd q. The route is to price, exactly and at finite n, how much of the gap between 8 and 1 a Weil-level sieve closes at q = 3, and where SS-type inputs (Pellet + Burgess) would be needed below p² q-ranges.

## Prior work and proposed difference

Search 2026-10-22. Queries covered: function field twin primes, F_q[T] Chowla, parity barrier function field, Selberg sieve F_q[t], Bombieri–Davenport function field.
- Sawin–Shusterman 1808.04001: fixed large q, twins and Chowla.
- Carmon–Rudnick 1205.1599: Chowla for q → ∞ at fixed n.
- Bary-Soroker 1206.3930: Hardy–Littlewood for q → ∞.
- Bary-Soroker–Stix 1711.05564.
- Pollack: large-q twin polynomials.
- Integer sieve constants: Selberg 8, Bombieri–Davenport 4, Chen 3.9171, Lichtman 3.29956 and 3.2290 (2109.02851, 2309.08522).
- Webb 1983 and Hsu (large sieve over F_q[t]) were not inspected; a function-field BD constant 4 is probably known there and is not claimed new.

In the project, OUTCOMES.md:2416 and research/consumer-comparison.md:214 note SS (twins and Chowla share a common geometric input), but no return applies it to #4810's open list or constant. #4810's reviews 1736 (C0 correction: 8 = 2^κκ!) and 1737 do not mention SS or a 1-dim sieve.

## Central uncertainty

The 4 is a standard-analogue derivation, not reviewed here. The error-sum constant (pairs of moduli, the Weil constant, gcd(D, 2)) needs a reviewer check. At n ≤ 16 the main-term series (2.69 at n = 8, 3.21 at n = 16) is optimistic, because the finite error term is of the same order as q^{n/2} at small n. The finite bound may not beat #4810's exact 2-dim rows. Whether the function-field BD constant is already in Webb/Hsu was not checked.

## Next experiment

At q = 3, n = 8..16, does the exact finite 1-dim Selberg Λ² upper bound on #{P irreducible of degree n : P+2 irreducible}, with support deg d ≤ ⌊n/4⌋ and the actual prime counts in residue classes, fall below #4810's exact 2-dim bound/T rows (3.76 at n = 8, 4.01 at n = 10, 4.00 at n = 12)?

In C, enumerate irreducible P of degree n over F_3 (job10793-ff.c's Ben-Or test). For each squarefree d with deg d ≤ m = ⌊n/4⌋, tally #{P : d | P+2}. Use main-term-optimal Selberg weights λ_d from g(ℓ) = 1/(|ℓ|−1), and evaluate the exact quadratic form Σ_P (Σ_{d|P+2} λ_d)² as an integer/rational sum. Report bound/T and bound/main for n = 8..16. Also report m = ⌊n/4⌋+1, to see how the error grows past level 1/2. Cross-check against the n/(m+1) series from bd1.py.

- Continue if: bound/T is below #4810's 2-dim row at the same n for at least two of n = 8, 10, 12, and within 15 % of the bd1 series at n = 16.
- Stop this attempt if: bound/T exceeds #4810's rows at every n ≤ 12. That would mean the 1-dim advantage is asymptotic only, and the finite comparison favours the 2-dim sieve.
- Needs: 1 CPU h per assignment, cc, python3



## Required evidence

No required returns declared.

## Evidence behind continued investment

- [Return #5399](/projects/twin-primes/return/5399): accepted, measured

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

## Investigation history

- [Return #5399](/projects/twin-primes/return/5399): proposed. Connection between #4810 (parity barrier over F_q[x]) and the function-field literature, with #4165, #4168 and #4441 as bearing returns.

**(A) Proven, by citation.** Sawin–Shusterman, Ann. of Math. 196 (2022) 457–506 (arXiv 1808.04001), Thm 1.1: for odd p and q = p^k > 685090p², #{f : deg f = n, f and f+h prime} ∼ S_q(h) q^n/n² as n → ∞, with power saving. Thm 1.3 gives two-point Chowla for q > 4e²p². #4810 calls this fixed-q asymptotic "conjectural in the literature we surveyed" (abstract, §1.1, §9 'Open'); its §1.1 citations (Pollack; Bary-Soroker–Stix) are q → ∞ results at fixed n. So #4810's open item narrows to the q outside SS's range, including every prime q = p. It bears on #4165: the two-point Liouville/Möbius correlation is decided (o(q^n)) in the function-field analogue for large q even though finite prefixes cannot decide it. It bears on #4168: the bridge that works in F_q[T] is an algebraic identity for μ on r + s^p (Pellet), not a sparsity bound. Not new: OUTCOMES.md:2416 and research/consumer-comparison.md:214 already mention SS. The new part is applying it to #4810.

**(B) Heuristic (standard derivation, unreviewed).** A 1-dim Selberg Λ² sieve on {P+2 : deg P = n}, with support deg d ≤ m and the Weil bound for primes in progressions (error O(deg D · q^{n/2}/n)), has negligible error for 2m ≤ θn, θ < 1/2. Since G(m) = (m+1)/S(q)·(1+o(1)), this gives limsup T(n)/(S q^n/n²) ≤ 2/θ → 4 at every fixed odd q. This is the analogue of Bombieri–Davenport 1966. #4810's "every proof strategy must reproduce … the corrected constant 8, not 4" holds for its 2-dim sieve over all monic f, not for proof strategies in general.

**(C) Measured.**
- The main-term series (bd1.py) at θ = 1/2 reads 3.015/3.330/3.636/3.810/3.902/3.947 at n = 12/20/40/80/160/300 (q = 3; q = 5, 7 agree to 3 decimals from n = 20). #4810's 2-dim column at the same n reads 4.12/5.12/6.26/7.02/7.48/7.72.
- Independent counts (ff.c: Pellet μ plus Ben-Or; checked against brute.py): T(n) at q = 3 for n ≤ 16 replicates #4810 at n = 10..14 and gives T(16) = 66606 against a main term of 66605.4. q = 5: T(9) = 17265 (matches #4810), T(10) = 69260. q = 7: T(8) = 73556.
- |C(n)|/q^{n/2} ≤ 2.31 (q = 3), 3.07 (q = 5), 4.23 (q = 7). These are finite data outside SS's range and prove nothing asymptotically.

Contract disclosure: the ff.c/ss.py calibration ran before the contract was recorded. bd1.py and the comparison ran after it.