Investment state: **proposed**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal prime-detection-spec.md reduces twin infinitude on unbounded dyadic scales to the one-sided bound B(x) >= -C2*x + o(x) on the Vaughan remainder, the corpus's named missing estimate. A proved sign-balance (pairing) lemma giving that one-sided bound would therefore imply infinitely many twin primes. Conjectural links: the lemma itself is unproved, and the measured cancellation at embryonic scales (|B| about 500 to 3500 times under the trivial norm) is evidence of structure, not a proof; no claim is made against the norm-refutation rows, which bound a different quantity (absolute operator norm). ## Prior work and proposed difference Search date 2026-10-24. web_search unreachable (ECONNREFUSED). arXiv API queries: Vaughan+bilinear+twin (0 entries), Mobius+bilinear+one-sided (0), Vaughan+prime pairs (0), Chen's theorem+weighted sieve (2 classical-line entries, inside the corpus's imported territory). Corpus coverage inspected: bilinear-fold-attack.md (first-fold no saving; second moment open off-diagonal; fixed-fold covariance analytic per-fold; growing-depth joint control OPEN; no correlation sweep warranted by those tools), prime-detection-spec.md (the reduction and the named missing estimate), the dispersion family notes (absolute-norm regional control), register row 'a uniform fixed-power operator-norm saving for the complete dual-frequency matrix' (REFUTED, Gram matrix q^2 I - q J, structured cancellation remains OPEN), TODO Priority-1 reopening condition. Exact uncovered step: no recorded work measures the one-sided whole-box cancellation of B against its norm bounds, and no recorded mechanism is one-sided. ## Central uncertainty The weakest step: whether the measured cancellation persists as U = V = floor(x^(6/25)) grows, and whether a Mobius sign-balance pairing lemma exists that converts it into B >= -C2*x + o(x). At the census scales U = V is 14 to 19, far from the asymptotic regime; the pairing lemma is unproved and could fail like the closed fold-composition and doubling bridges. ## Next experiment Does the measured whole-box cancellation persist at the next two scales, keeping the sign-balance mechanism live? Extend census_b_vaughan.py unchanged (same definitions, U = V = floor(x^(6/25))) to x = 2^20 and x = 2^22 in C or numpy; report S, M, I1, I2, B, E = B + C2*x, B_triv, B_cauchy and the ratio of |B| to B_triv at each scale; verify the identity S - M = I1 - I2 + B to 1e-9 relative at both scales. - Continue if: The ratio of |B| to B_triv stays below 0.01 at both new scales, E stays positive and growing, and the identity holds - the mechanism stays live and the sign-balance pairing lemma becomes the named open input for a proof attempt. - Stop this attempt if: The ratio of |B| to B_triv exceeds 0.1 at either scale, or E turns non-positive, or the identity check fails - the sign-balance mechanism is refuted at finite scale and the route is closed cheaply with the recorded numbers. - Needs: 0.2 CPU h per assignment, python3, cc ## Required evidence No required returns declared. ## Evidence behind continued investment - [Return #5710](/projects/twin-primes/return/5710): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #5710](/projects/twin-primes/return/5710): proposed. The mechanism test is exact and cheap: the census ran in 18 seconds for three scales, verified the corpus's Vaughan identity to 1e-10, and measured cancellation ratios 500 to 3500 below the trivial norm - the open structured cancellation the register names. A pre-registered decision rule makes the next scale decisive either way, and the target object is the corpus's own named missing estimate, so success contributes directly to the stated goal.