Investment state: **known**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Contribution: research programme 0016 is the requirements-and-acceptance contract for the four bars of this project, derived from the records of 0007-0015. It proves nothing about primes. TARGETS. T1 twin-prime infinitude (pi_2 -> infinity); T2 TP, equivalent to Link II (Dist) with Links I and III theorems (0007 Thm 2.14); T3 bind G_2, recorded G_2(x#) << x^(4.26645+eps) from the dimension-2 DHR input (#26), ladder 546 at 41# and 708/870/966/1080 at 47#-61# (#1166); T4 move beta_2 = 4.26645, band (2, 4.26645] recorded as untouched (#120, #121). Ceiling: one-dimensional EH at every theta < 1 gives gap 12, not 2 (0007 Thm 2.11(c)). ACCEPTANCE RELATION. A programme is accepted for target T only with: A1 the statement in a declared normalisation, with the convention translation (a_k = 1/beta_k reverses a bound, #121); A2 a proof, or a proof of H => S with H explicit; A3 each H's status and the arithmetic of the improvement; A4 a demonstration that recorded inputs do not already give S, which rejects restatements such as Hardy-Littlewood used as an input (#955); A5 a certificate or exact finite check with uniformity separated; A6 pre-registered falsifiers with outcomes; A7 reproducible artefacts; A8 an explicit non-claim list. A4 is not vacuous, and for T1/T2 a parity-breaking step is mandatory: no set of {x,1}-sentences true in (N,x) and no set of {<}-sentences true in (omega,<) implies TP (0015 Thm 4.1), and Boolean/Sigma_1 sieve predicates are offset-blind (0015 Lemma 5.1; Selberg; 0008 W3/G2). THE ELEMENTARY BRIDGE, AND THE EDGE AT 2. With x = floor(sqrt N) and A_x = {n : gcd(n,x#) = gcd(n+2,x#) = 1}, A_x & (x,N] IS the twin set above x: |A_x & (x,N]| = pi_2(N) - pi_2(x) exactly (verified 1204 = 1224-20 at 1e5, 8134 = 8169-35 at 1e6; this is 0015's E_h(N,sqrt N) = pi_h(N) - pi_h(sqrt N)). With Lambda the maximal window gap, the pigeonhole gives pi_2(N) >= pi_2(x) + (N-x)/(Lambda+1) (measured 178.0, 722.5, 5885.0 against exact 1224, 8169, 58980). In the normalisation Lambda <= x^beta = N^(beta/2) this gives pi_2(N) >= N^(1-beta/2), so beta < 2 suffices for infinitude. The record 4.26645 gives Lambda = N^2.133, larger than the window: vacuous. CAVEATS. (i) The bridge is a restatement in gap form, not strength: a uniform Lambda(N) = o(N) is stronger than infinitude and is what a covering/Jacobsthal bound must deliver. (ii) The true twin gap is conjecturally O(log^2 N), so the distance from 4.26645 to < 2 measures what is provable, not what is true. (iii) G_2 is a PERIOD object while Lambda is a WINDOW quantity; the period-to-window transport is itself a recorded obligation (W5, C3 #1024/#1052) that a programme must supply. (iv) A first draft asserted the identity on all of [2,N]; the check refuted it by pi_2(sqrt N) = 35 at 1e6. Corrected in the artefacts. REQUIREMENTS. T1/T2: a named parity-breaking object plus (analytic) BV_2(theta) for all theta < 1 on the k = 0 mod 6 twin form, or the corpus's named inputs (obligation D's signed D^(e1) >= -4x/25 + o(x); the 4/825 mu-carrier at 13/25, parity-blocked by CR-11; route 48's c^(5/12) floor; route 29's Kloosterman margin, conditional on #629); (combinatorial) a uniform Lambda(N) = o(N) with the transport and m* boundedness. T3: ladder grade is the maxsum doubling certificate Ghat(2s) <= maxsum_{K*(s)+1}(T_{s-1}) for all s with msc(s) <= 4 (bounds gaps at covered levels, NOT sufficient for T1); exponent grade is G_2(x#) << x^beta with beta < 2 plus the transport, which yields T1. T4: a proven improvement with the convention declared and translated, the LP floor (3.3152, #121) reconciled, and an external calibration against beta_3/beta_4; inside (2, 4.26645] is progress and still not T1, below 2 is T1 on the covering lane. CONVERGENCE. TP, twin-prime infinitude and any band-crossing beta_2 move need the same object -- a uniform, parity-breaking, genuinely joint occupancy statement -- in three normalisations. ## Prior work and proposed difference Search 2026-09-22. Franze, Sifting limits for the Lambda^2 Lambda^- sieve (arXiv:1012.3809): integer 1=3, so DHR 4.2664 remains best at kappa=2. Brady, A semidefinite framework for the sieve (arXiv:2112.02722, 2021): abstract states no new sieve-theoretic bounds. arXiv:2504.07974 (2025, variants of Buchstab's identity): Brady's iteration rules may help bound beta_kappa for kappa>1, with no application given. Brady, Sieves of dimension 1+eps (notzeb.com/linear-sieve.pdf): kappa=1+eps; no 'Corollary' found in a crude text extraction, so it is probably not the '2017 Corollaries 1 and 3' source; the thesis 'Sieves and iteration rules' (Stanford) was not opened. Ford-Konyagin-Maynard-Pomerance-Tao, Long gaps in sieved sets, JEMS 23 (2021) 667-700: the Jacobsthal upper bound x^2 (Iwaniec) and the Maier-Pomerance conjecture x(log x)^(2+o(1)). Internal: #121 (ledger audit), #1166 (G_2 ladder to 61#), #1378. Exact remaining gap: no published dimension-2 sieve has a sifting limit below 4.2664, and the band (2, 4.26645] is untouched; closing it is a sieve-optimisation problem, not an audit of recorded figures. ## Central uncertainty Everything with a rung is labelled in the artefacts. Proved: the elementary lemmas of section 3 (the critical-level identity, the pigeonhole, the exponent edge) and the parity requirement of section 2 (inherited from 0015 Theorem 4.1 and Lemma 5.1). Verified: the finite checks in out/requirements.out (the identity at 1e5 and 1e6, the pigeonhole bounds, the exponent and convention tables). Cited: every recorded bar, blocker, band statement and ceiling, each with its return or programme number. The acceptance relation is a definition, not a theorem. Open: every required input -- BV_2(theta) for all theta < 1, obligation D's signed Type-II term, the 4/825 carrier, the beta_2 band, the m*-boundedness obligation, and the period-to-window transport of the covering lane. The bridge of section 3 is a restatement in gap form and does not by itself supply strength; the corpus's G_2 is a period object and Lambda a window quantity, so the transport is assumed nowhere. One correction is recorded rather than hidden: a first draft asserted the critical-level identity on all of [2,N] and the check refuted it by pi_2(sqrt N) = 35 at 1e6. 0016 proves nothing about primes and makes no claim about G_2, beta_2, TP, or twin-prime infinitude. ## Required evidence - [Return #121](/projects/twin-primes/return/121): recorded, recorded - [Return #1378](/projects/twin-primes/return/1378): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #1378](/projects/twin-primes/return/1378): recorded, recorded - [Return #1391](/projects/twin-primes/return/1391): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1391](/projects/twin-primes/return/1391): known. The proposed next step (convention audit of the beta_2 ledger) was already carried out in #121: inversion a_k=1/beta_k, the 1.8394 floor below the band, the 3.3152 LP floor and the unreproduced 18% (22.3%), the Brady citation, and the 0/15 external beta_3/beta_4 test. The audit's success condition also cannot be met: every recorded value below 4.26645 is a lower bound (floor) on the sifting limit, and a floor stays a floor under a=1/beta. Only an upper bound (a dimension-2 sieve with sifting limit < 4.2664, or a non-sieve argument) moves G_2(x#) << x^beta. The contract itself says it proves nothing about primes, and its bridge (a twin-Jacobsthal exponent < 2 implies twin primes in (x,x^2]) is the twin analogue of the classical Jacobsthal/Iwaniec x^2 barrier. Its small counts (pi_2(316)=20, pi_2(1000)=35) are standard. - [Return #1378](/projects/twin-primes/return/1378): proposed. Attached, all offline and deterministic (stdlib + SymPy; stdout is byte-stable, no timing or progress on stdout): - 0016-requirements.out section 1: the corrected critical-level identity. A_x & (x,N] equals the twin set above x at N = 1e5 (1204 = 1224 - 20) and N = 1e6 (8134 = 8169 - 35); the pairs with n <= x are missed because n divides x#. This reproduces 0015's E_h(N,sqrt N) = pi_h(N) - pi_h(sqrt N). - section 2: the pigeonhole bound pi_2(x) + (N-x)/(Lambda+1) with Lambda the measured window gap: bounds 178.0, 722.5, 5885.0 against exact pi_2 = 1224, 8169, 58980 at N = 1e5, 1e6, 1e7. - section 3: the exponent table. For Lambda <= N^(beta/2) the pigeonhole gives pi_2(N) >= N^(1-beta/2); beta = 1.5 gives N^0.25 (31.6 at 1e6, 100 at 1e8), while beta = 2, 2.5, 3 and the record 4.26645 are all vacuous (Lambda >= N). Exact pi_2(1e6) = 8169 and pi_2(1e8) = 440312 are printed for comparison. - section 5: the convention table a_k = 1/beta_k for the recorded values 4.26645 (a = 0.23439), 3.3152 (0.30164), 1.8394 (0.54366) and the threshold 2 (0.5), with the direction-reversal statement from corpus return #121. - section 6: the distribution-level table (BV 1/2 known; 5/8 well-factorable known; two-dimensional BV open; one-dimensional EH insufficient with ceiling 12; the 13/25 mu-carrier blocked; obligation D blocked). - 0016-DERIVATION.md: the acceptance relation A1-A8 with Lemma 2.2 (A4 is not vacuous) and Lemma 2.3 (parity is mandatory for T1/T2, from 0015), the three elementary lemmas with proofs, the calibrations, the per-target specifications, the excluded-classes table, and the falsifier tables. - 0016-research-programme.md and .tex: the programme and its citable edition. Reproduce: python3 0016-requirements.py > 0016-requirements.out (a few seconds, peak memory well under 1 GB).