Investment state: **proposed**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Bypasses the closed additive K*-product floor (K* >= pi(2s) - pi(s) -> infty) in Q-doubling-bridge-0829n by establishing a multiplicative 2D Mertens sieve certificate C2(s) <= 4 M_2(s, 2s) where M_2(s, 2s) = prod_{s= 8 (M_2(16, 32) = 1.5928, M_2(128, 256) = 1.2884), bounding C2(s) <= 6.3712 < 8.0000 uniformly inside the legal zone [4, 19.2455) for all s >= 8. ## Prior work and proposed difference OUTCOMES.md closed the K*-product bridge because K* = 23 at s = 128 pushed the certificate to 24 > 19.2455. Q-doubling-C2 noted that empirical C2 has slope 0.2018 +/- 0.1412 while K* drifts up at slope 0.6881. The maxsum certificate reached sup 6.6364 at s = 16 but lacked an all-s uniform proof. ## Central uncertainty Whether non-asymptotic parity oscillations in the bilinear sieve remainder could cause Ghat(2s) to exceed the continuous 4 M_2 envelope at intermediate levels s in {32, 64}. ## Next experiment Does the exact ratio Ghat(2s)/Ghat(s) satisfy Ghat(2s)/Ghat(s) <= 4 M_2(s, 2s) <= 8.0000 at s = 32, 64, and 128 on the verified base-2 chain? Execute the maxsum walk script `attack-0829n-doubling-bridge.py` up to s = 128 to evaluate exact Ghat(2s)/Ghat(s) and verify compliance with the Mertens bound. - Continue if: Ghat(2s)/Ghat(s) <= 8.0000 at all tested levels s in {32, 64, 128}, confirming the uniform doubling certificate. - Stop this attempt if: Any Ghat(2s)/Ghat(s) > 8.0000, refuting the doubling inequality on the base-2 chain. - Needs: 1 CPU h per assignment ## Required evidence No required returns declared. ## Evidence behind continued investment - [Return #6240](/projects/twin-primes/return/6240): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #6240](/projects/twin-primes/return/6240): proposed. Proposal for a new route: replacing the closed additive K*-product bridge with a multiplicative 2D Mertens sieve factor M_2(s, 2s) = prod_{s= 8.