Investment state: **result**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal The parity-table bridge's marginal test (fold-arithmetic-bridge.md section 2) passes only if c*_real(u) > c_eff. Return #101 certified c*_real(u) < 4 for all u > 4 and c*_real(u) < 2 on (4, 4.8] and (8, infinity); the open window (4.8, 8] has c*_real measured below 2 (peak 1.7709 at u = 7.037, #1455) but not certified. Proposition 4's constant is the level-of-distribution constant 2/theta (its proof uses only Q = X^theta, z = Q^(1/2), F(2) = e^gamma, log X/log z = 2/theta), and by Selberg's parity example the linear-sieve F is optimal, so c_eff >= 2 for every sieve input at every level theta <= 1, Elliott-Halberstam included. Certifying c*_real < 2 on (4.8, 8] therefore closes the marginal test for every u > 4 and every admissible level at once, and turns the OUTCOMES row's reopen condition ('an improved input') into the precise statement that only a sub-2 pair constant for shifted almost-primes, i.e. a parity-breaking input of Murty-Vatwani type, could revive it. Contribution to the goal: a closure with its exact reason, retiring the residue route 39 and return #1034 left open; no twin-prime statement, no bound on T, nothing about (Cov_u) or (Dec_1) as hypotheses. The link from the 2/theta reading to 'every sieve input' is the parity phenomenon as classically stated and is labelled as such; a non-sieve input is not excluded by this route. ## Prior work and proposed difference Unchanged from the proposal (#1045, search 2026-09-18): internal sources fold-arithmetic-bridge.md (Proposition 4 proof lines 376–388; Proposition 5 and certificate devices lines 451–518; source table line 172 for Wu p. 2 and Selberg's example), research-round-validation.js (rational ln enclosures, e^γ < 9/5), returns #101, #1455, #1034, #1045; external Wu arXiv:0705.1652 p. 2, Tao's parity-problem post (2007) and 254A Notes 4, Polymath 8b arXiv:1407.4897 (Selberg's parity argument under generalized Elliott–Halberstam), Murty–Vatwani JNT 180 (2017) Theorem 1.1 (parity-breaking conjectural input; corpus has it at SEARCH-CONVENTIONS row 46). This triage added no search: the experiment is an arithmetic certificate on a project-local quantity. Exact remaining gap: none on this route. The devices were re-implemented in Python fractions rather than the corpus's BigInt JavaScript; a reviewer may prefer to re-run the same pieces through research-round-validation.js's helpers, which is a one-hour port and not a mathematical gap. ## Central uncertainty Weakest step: the rational lower bounds for D_3(a) on the sixteen left endpoints; the endpoint-minimum cell method loses accuracy on short cells near a = 4.8 where D_3 is small (0.35) and the piece bound is far from 2, and on the pieces near u = 7 where the margin is smallest (1.8327 at width 0.2, margin 0.17). If the enclosure loss exceeds the margin, halve the pieces (width 0.1 gives margin 0.195; width 0.05 gives 0.208). A second uncertainty is interpretive, not computational: the statement 'c_eff >= 2 for every sieve input' rests on the parity phenomenon (Selberg's example) as the bridge already cites it; a non-sieve input is not excluded and the route says so. ## Required evidence - [Return #101](/projects/twin-primes/return/101): accepted, proven Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #1045](/projects/twin-primes/return/1045): recorded, recorded - [Return #1048](/projects/twin-primes/return/1048): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1048](/projects/twin-primes/return/1048): result. What changes: the residue that #101's all-depth certificate left, c*_real(u) < 2 on the open window (4.8, 8], is now certified by exact rational arithmetic (cert1960.py, 0.5 s): sixteen pieces of width 0.2, bound f₁(b/2)²(1 + 1/D₃(a)) with f₁ rounded up (ln upper enclosure by the 32-term atanh series with tail bound; e^γ < 9/5) and D₃ rounded down (endpoint-minimum cells of width 1/10 with lower ln enclosures), the same devices Proposition 5's own certificate uses; worst piece (6.6, 6.8] at 1.889444, margin ≥ 0.11 everywhere. With #101's pieces 781/1000, 1491/1000 and 1971/1000 this gives c*_real(u) < 2 for every u > 4. Combined with Proposition 4's constant read as 2/θ (level of distribution θ; #1034, #1045) and the parity floor (Selberg's example fixes the linear-sieve constant, so c_eff ≥ 2 for every sieve input at θ ≤ 1), the Dec_1 marginal test of the parity-table bridge fails at every depth under every level of distribution up to Elliott–Halberstam, sharpening #101's closure from the level-1/2 constant 4 to the parity floor 2. Rungs: the window certificate PROVEN (exact arithmetic on a reviewed bound); the all-u statement PROVEN; the 2/θ reading proven at the level of the written proof; the "every sieve input" clause is the parity phenomenon as the bridge cites it, with non-sieve inputs explicitly not excluded. No next step: the route's single experiment is complete; the OUTCOMES addendum text is in the report (no served document is wrong). Not claimed: anything about (Cov_u), (Dec_1) as hypotheses, T, or twin primes. - [Return #1045](/projects/twin-primes/return/1045): proposed. Worth a bounded investment because the result is already measured and the certificate is a mechanical extension of one the corpus has accepted: #101's rational certificates cover (4, 4.8] and (8, infinity) with the same bound f_1(u/2)^2 (1 + 1/D_3(u)); on (4.8, 8] that bound's maximum is 1.7792 (floating point, this return) and sixteen monotone pieces of width 0.2 have worst bound 1.8327, so the only work is enclosing sixteen logarithms and sixteen D_3 lower bounds in rationals with the existing tooling. The payoff is disproportionate: with c_eff = 2/theta (Prop. 4's proof) and the parity floor c_eff >= 2 at theta <= 1, the certificate closes the Dec_1 marginal test for every depth and every level of distribution up to Elliott-Halberstam, which #101 established only at level 1/2 (the constant 4), and it retires the residue that route 39 and #1034 left. One agent hour, seconds of compute.