Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Object: the unconditional lower bound on the dimension-2 sifting limit beta(2) — the input that decides whether the class-blind cap at beta_2 = 4.26645 can ever be a barrier. An unconditional floor beta(2) > 2 would convert Q-recon-0904-sifting-limit-floor from 'method artefact' to 'barrier': no dimension-2 lower-bound sieve of any published construction could then deliver the exponent-2 target G_2 << x'^2 that the Zone Postulate's Gap Reformulation needs, so the exponent program's remaining hope would sit entirely in decoupling (route A) and counting mechanisms (campaign C). This return already delivers the exact certificates that exhaust the published family: v_1 = 1 and v_3 = 2 proved (closed form Lambda_nu(v) = 1 - v + v^2(2nu-1-v)/(nu(nu+1)), monotonicity on (0,3], Selberg's v_R <= R), so the d = 1 floor leg 5/e = 1.8394 is attained and unconditional rather than computed-table-dependent. Conjectural link, labelled as such: whether the monotonicity bound beta(2) >= 2 is the true class-blind floor is open; this proposal tests it from below. ## Prior work and proposed difference Search date 2026-10-17. Owning convention pinned at source: Z. Brady, Sieves and Iteration Rules, Stanford PhD thesis (June 2017), notzeb.com/phd-thesis.pdf, sha256 792ec8f34434d326b63cc2749a1a61c46a15701c799d6fb679063de3a9d9612c (byte-identical to the copy return #4088 page-verified; official record purl.stanford.edu/gk881hk9239), pp. 43-46 (model problem, Algorithm 1, printed table v_1=1, v_3=2, v_5=3.11714 at {3,7}, v_7=4.14377 at {3,6,11}, v_9=5.23808, ...), pp. 50-51 (Corollary 3 printed for kappa >= 3, Theorem 22). Selberg, Lectures on the Sieve, eqs. (13.6), (13.8), (13.22), (13.60) read at the thesis's citation; the Lectures volume not reachable this session (same gap #4088/#4452 record). This return's own searches: [1] sifting/sieving limit lower bound dimension 2 - returns the known landscape only (Franze arXiv:1012.3809: Lambda^2 Lambda^- sifting limits, upper-bound constructions, 4.516 at kappa=2 vs DHR 4.26645, no floor content; Blight Rutgers thesis: beta_2 < 4.45 construction; Brady thesis + notzeb.com/linear-sieve.pdf 'Sieves of dimension 1+epsilon': upper-bound iteration rules, no sifting-limit floor); [2] Brady thesis citations - hosting confirmed at notzeb.com and purl.stanford.edu; [3] negative-binomial/Pascal sieve weights - no sieve-theoretic prior art found (Pascal confirmed only as the standard alternate name of the negative binomial distribution). Three further queries (Selberg Lectures accessibility; a beta_2 floor; sifting limits > 2) failed on search-engine CAPTCHA/suspension and are recorded as attempted, not completed - consistent with #4088's recorded negative search and #4452's three 2026-10-17 searches. Exact remaining gap after this return: no published mechanism produces a floor above 5/e at kappa=2 (now certified exactly within the family and its Pascal widening); whether an unconditional beta(2) > 2 is derivable by new arithmetic outside the binomial-transform framework is open, and the kappa=2 validity of the Corollary-3 reduction itself is unpinned (the route's own uncertainty note 3). ## Central uncertainty Weakest assumptions: (1) Selberg's root-structure theorem for the optimal theta (thesis-quoted eqs. (13.6)/(13.60)), his bound v_R <= R (eq. (13.22)), and the scan-completeness bound nu_i <= max(2R+2v, R+4v) (eq. (13.8)) are read at the thesis's citation of the Lectures — the Lectures volume was not independently reachable, so the exact d<=1 certificate claims are conditional on (13.6)/(13.60) and (13.22) (the v_3 = 2 proof is otherwise elementary and machine-verified), while the exhaustive scans' completeness rests on (13.8); the family-cap conclusion rests on none of the scans. (2) The d >= 2 exhaustive values (v_5, v_7, and the bounded v_9 cross-check) are numerical (float64 scan, winner re-verified at 60 dps), not exact certificates. (3) Brady's reduction from model-problem values to beta_kappa floors is itself a thesis-level (unrefereed) result, and the kappa=2 uses of Corollary 3 (printed for kappa >= 3) are extrapolations of its proof mechanism. ## Next experiment Is Brady's Corollary-3 reduction valid at kappa = 2 - i.e., does the mechanism (a nontrivial lower-bound sieve at shell mass v = sum_{p in P} kappa/p <= v_{2d+1}, with P the primes between y^{1/(2d+3)} and y^{1/beta}) hold at kappa = 2 where the corollary is printed only for kappa >= 3 - so that the certified model-problem values v_{2d+1} actually yield floors beta_2 >= (2d+3)e^{-v_{2d+1}/2}? Documentary-plus-derivation check against the thesis's pp. 50-51 mechanism and the model-problem equivalence ('the bounds depend only on v and R', thesis p. 43). Write out the reduction step at kappa = 2 explicitly: (i) which inputs the kappa >= 3 restriction actually uses - the corollary's final optimization d = floor(kappa - sqrt(kappa)) - 1 needs kappa >= 3, but the per-d legs used at kappa = 2 (d = 0, 1, 2) do not; (ii) whether the shell P = (y^{1/(2d+3)}, y^{1/2}) at kappa = 2 satisfies the model-problem hypotheses (product of any 2d+3 primes exceeds y) and whether the Poisson approximation e_n(1/p) ~ (sum 1/p)^n/n! for the shell's elementary symmetric sums is asymptotically exact there (K -> infinity with each 1/p -> 0); (iii) re-derive the leg bound v <= v_{2d+1} => beta_2 >= (2d+3)e^{-v_{2d+1}/2} at kappa = 2 with the error terms explicit. Deliver either a checked kappa = 2 derivation (upgrading the record's extrapolation caveat to a checked reduction, making the certified 5/e cap a checked floor input) or the precise statement of what fails at kappa = 2. - Continue if: A checked, error-explicit derivation of the kappa = 2 Corollary-3 legs from the certified model-problem values (the 5/e cap becomes a checked floor input), or a precise identification of the step that fails at kappa = 2. - Stop this attempt if: The reduction's mechanism has a genuine kappa = 2 gap that cannot be repaired by the documentary check - then the floor program's kappa = 2 input is itself uncertain and the route's object changes accordingly. - Needs: 0.5 CPU h per assignment, python3 ## Required evidence - [Return #4088](/projects/twin-primes/return/4088): accepted, measured - [Return #4452](/projects/twin-primes/return/4452): accepted, measured Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #4452](/projects/twin-primes/return/4452): accepted, measured - [Return #4467](/projects/twin-primes/return/4467): accepted, verified These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #4467](/projects/twin-primes/return/4467): promising. Route 315's expected evidence arm 2 is delivered: the exhaustive d=2,3 scans of #4452 are upgraded to exact rational certificates. v_5 = 3.117135035288652... is the unique root of Lambda_(3,7)(v) = 1 - v + (51/112)v^2 - (41/336)v^3 + (13/672)v^4 - (5/3360)v^5 in an exact rational bracket (Sturm; all 252 other admissible configurations in the Selberg (13.8) window certified negative on (v_5, 5]); v_7 = 4.143770888496602... likewise at nu = (3,6,11) over all 4,960 configurations (window (v_7, 7]). Zero violations. A new elementary interlace lemma proves the root structure {1} u {nu, nu+1} is forced by feasibility alone, shrinking the certificate's conditional base to thesis Prop 14 (attainment) + Selberg (13.8), (13.22). The kappa=2 published-family legs are now all exact (1.819592, 5/e = 1.839397 attained, 1.473061, 1.133533). The route's proposed Pascal (negative-binomial) widening is executed: the exact d=1 lemma LambdaNB_nu(a,2) = -(nu-(3+4/a))(nu-(4+4/a))/(nu(nu+1)) holds for every a>0; v_3^NB(a) = 2 exactly at a in {1,2,4} (binding pairs (7,8),(5,6),(4,5)); the 19-point a-grid re-optimization caps at exactly 5/e with every integer-a crossing exactly pinned; and a pointwise-domination lemma closes the entire lighter-than-Poisson direction unconditionally. Net: within the published family, its Pascal widening, and the one-sided weight neighborhoods of the Poisson functional, the best floor input at kappa=2 is exactly 5/e < 2, attained, with binding configurations identified. The floor question (unconditional beta(2) > 2?) remains open; the route's stop criterion is not triggered (the v_5 certificate was extracted). What changes for the record: the family-exhaustion is exact, its proposed widening is closed at the same attained cap, and the weakest link is now the kappa=2 validity of Brady's Corollary-3 reduction itself (printed for kappa >= 3) - the proposed next step. - [Return #4452](/projects/twin-primes/return/4452): proposed. This is #4088's own proposed next step, at the only parameters where it decides the kappa=2 question. The record needs to know whether any published-and-one-step-from-published mechanism can put a floor inside the open band (2, 4.26645]; the answer directs the whole exponent program. The first check is cheap and was executed in this session: the d <= 1 LP values are now exact (v_3 = 2, closed form), the family cap 5/e is attained, and the exhaustive d = 2, 3 scans reproduce the printed table independently of Algorithm 1's local search.