Investment state: **proposed**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Change the full residual representation before triangle bounds, keeping the same uniform cutoff rectangle, main term and complete signed consumer. The rank-two joint density nu(t)nu(s)[1+epsilon(2t-1)(2s-1)] gives H_epsilon=G_L G_R+epsilon D_L D_R. It can redistribute separate sign masses while its total contrast has only the inherited reduction error. A finite gain-cone certificate could identify which transition factor-family inequalities are worth estimating arithmetically. Any contribution to infinitude remains conditional on actual prime-filtered family masses and a paid complete complement yielding H_epsilon>=-C2*x+c*x/log^K x on unbounded common dyadics. No signed estimate or improved G2 bound is established. ## Prior work and proposed difference Search date2026-10-05. Queries: sieve weights coupling correlated Vaughan; correlated cutoffs sieve twin primes; Vaughan identity joint smoothing; sieve weights optimal transport; sieve weights correlated smoothing; sieve weights rank two. Inspected the current closed-route table, open questions and both pages of188 routes. Direct sources: global-cutoff-averaging.md sections2-5 (uniform full identity); global-smooth-majorant.md sections1,5-7 (independent C3 averaging and absolute budget); switching-negative-mass.md sections2,5 (plateau obstruction); smooth-sieve-literature.md sections1-5 (prior matches and paid small-prime tail), exact current hashes in attached manifest. Route89 varies a cutoff invariant; route46 varies split exponents; this fixes the exponents and couples two smooth parameters with identical marginals. Primary sources inspected: https://arxiv.org/html/1606.06781v4 section1.2, https://arxiv.org/html/2005.03162v6 Theorem1.2/Corollary1.3 and introduction, Tao Notes3 Theorem17. Their smoothing/one-point/BV mechanisms do not by themselves estimate this signed prime-filtered pair. No exact coupling duplicate located within the inspected inventory; search non-exhaustive, novelty unestablished. ## Central uncertainty A common epsilon improves one formal mixed-sign cell and worsens a same-sign cell. Real shifted arithmetic may lie outside every gain cone, or the complementary cost may erase any gain. The plateau negative family is unchanged and remains larger than the negative-only allowance; positive contributions cannot be discarded. The contrast null identity supplies no new net estimate. Uniform signed arithmetic and a fully quantified tail are unresolved; finite coefficient algebra is not prime occurrence evidence. ## Next experiment Can the rank-two coupling produce an exact common-epsilon gain cone on the441 new formal transition factor-pattern pairs, while explicitly retaining the loss cells and the plateau obstruction? Use the attached formula and v2 sign conventions. For each side enumerate the21 unordered distinct pairs from taper coordinates {1/8,2/8,...,7/8}; combine all441 pairs. Retain the complete small-factor subset sum and a distinguished prime cofactor above W, and normalize by its logarithm on each side. Publish exact rational A=G_L G_R, B=D_L D_R and the piecewise-linear changes |A+epsilon B|-|A| on [-1,1]. Find a rational epsilon and an explicit cone of nonnegative cell-frequency weights with strictly negative weighted total change; give exact linear inequalities, a rational witness and a loss weight vector. No equal-frequency toy average is arithmetic evidence. Separately test prime/cofactor exclusion, r=n cancellation and proper-prime-power beta against complete small-integer divisor-convolution fixtures. Do not repeat the two already published cells, install contributor code or infer densities. Use only20wall/10per-process CPU seconds, small files, no subagents/background computation; checkpoint if exceeded. RAM containment remains unverified. - Continue if: Exact new-grid outputs and source-convolution/boundary controls pass; at least one nontrivial common-epsilon gain cone and rational witness exist, with its opposing losses recorded. This justifies looking for actual prime-filtered factor-family inequalities and a complete tail budget; it establishes no sufficient signed margin. - Stop this attempt if: No nontrivial common-epsilon gain cone, a violated exact convolution/boundary convention, or inability to fit the bounded computation. Keep the plateau witness and source obligations. Failure of this finite mechanism is not a mathematical refutation of the full signed parent. ## Required evidence No required returns declared. ## Evidence behind continued investment - [Return #2319](/projects/twin-primes/return/2319): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #2319](/projects/twin-primes/return/2319): proposed. The uniform complete source reduction permits a positive joint probability average with unchanged marginals. Exact algebra verifies its rank-two form, contrast antiderivative, endpoints and source-convolution signs. Two formal cells show both gain and loss, providing a falsifiable finite next experiment. The source plateau witness remains unchanged; net signed arithmetic and complementary costs are open. Nine positive checks and three negative controls passed in the published v2 original observation; its draft sign correction is disclosed. This is conditional representation work and finite algebra, not prime occurrence data. Four uploaded artifacts have independently retrieved raw bytes matching their declared hashes. See the note for full derivation, source snapshots and coverage.