Investment state: **proposed**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Route 59 asks whether the Mobius signs can buy the 7/200 moment deficit at the small-gcd box (8/25, 9/20). This return does not buy it. It contributes: (1) five corrections to claims circulated in this session, one of them measured - the band's centring is e/phi(e), not 1, so any earlier finite test of the band discrepancy measured a principal term of size x^(1/2) log log x; the band class is REDUCED (-2 mod e, gcd(-2,e)=1) and what blocks Bombieri-Vinogradov there is the exponent, not the class; the small-gcd sector excludes the diagonal, so its kernel has trace 0 and its positivity is undetermined; R=0 forces j > E/(2A) = x^(39/100)/2, so the rescue class is empty at this box and no positive-decomposition gain transfers. (2) Lemma A: levels of distribution do not compose - one summation, one error, and with rho_mu = 0 the composed object IS its error. With route 59's accepted Lemma 1 (mu(d)mu(e)=mu(de) on the coprime class, re-verified on 718,240 pairs) this closes the family 'combine BV for Lambda with BV for mu'. (3) Lemma B: the mean square over the modulus window is bounded by (X + |E|) with X the ambient length, so it is independent of E and cannot save; this closes 'Cauchy-Schwarz then large sieve over the window'. (4) Lemma C: the band's exact-valuation structure is CRT-multiplicative and its weight is the e-term of a product that vanishes once n has two prime factors in the window - a near-vanishing mask, not an oscillation. The surviving lever is the one route 59 already names: a signed estimate for the oscillating kernel K_q at q <= x^(1/20). ## Prior work and proposed difference No new literature search was run this session; the record's own searches were used (route 59: 2026-10-23, two queries on coprime-pair level of distribution and on power-saving mu cancellation, hitting Basak arXiv:2312.17435 and the classical Davenport bound; route 59's exclusion of arXiv:2604.25177v2 at Q >= sqrt(X) against Q = x^(1/20)). Inspected on the platform, at their own statements: return #5516 (Lemma 1 collapse, the finite falsifier's non-discriminating power, the x*(c) arithmetic showing a 7/200 power saving is asymptotically stronger than any sub-power mu-exponential bound), #5510 (decomposition + common bracket + triangle costs at least the Frobenius norm), #5506 (the 95/400 shortfall; BC sections 3-5 unread), #4333 (the exponent ledger). Local notes read: fixed-endpoint-discrepancy (B and H_B), structured-dispersion-estimate (D1), research-round-validation section 10. Access gaps: no PDF of Basak, of Davenport or of 2604.25177v2 was read; the sub-power statement is used in its standard published form. Exact uncovered step, unchanged by this return: a signed bilinear estimate for the oscillating-kernel block at q <= x^(1/20) that saves x^(7/200). No absence claim is made about that step. ## Central uncertainty The three lemmas are hand proofs of scope statements; the one with real content, Lemma A, is a negative composition result and says nothing about methods outside 'distribution in progressions + Cauchy or mean square'. The unproved assumption that remains load-bearing is the one Lemma 0 leaves open: that the negative part of mu is not merely small on average but provably smaller than the positive part on a structured set, which is what a one-sided consumer would need. Nothing here bounds the oscillating-kernel block, and the box remains STALLED WITH CURRENT INPUTS rather than closed. The finite checks are evidence about finite ranges only. ## Next experiment Does the oscillating kernel K_q at modulus q <= x^(1/20) carry a signed saving of x^(7/200) on the obligated e-pair class, once the signed block is compared with a sign-free block sharing the same kernel - and is that ratio decay a property of the kernel rather than of the support? Pin the kernel and coefficients exactly from the served notes (structured-dispersion-estimate (1), endpoint-fourier (4)-(9), grouped-divisor-moment section 1), recording every display used. At x = 10^4, 10^5, 10^6 evaluate the same block twice with identical kernel, weights, phases and support: T_signed with beta(e) = -mu(e)e^(-s)1_J(e), T_signfree with beta(e) = -|mu(e)|e^(-s)1_J(e). Report rho(x) = |T_signed|/|T_signfree| and delta = -dlog rho/dlog x with two one-interval slopes. Controls: (i) the sign-free arm must reproduce the sign-free block of the pinning fixture; (ii) a seeded permuted-sign arm on the same support; (iii) the note's exact small fixture as a convention check. No decomposition and no triangle: the block is evaluated directly, so #5510's Frobenius obstruction does not apply. Note the recorded caveat from #5516: a random-sign model over the same support already produces a ratio far below the 7/200 bar, so the test must report the permuted-sign arm and the support-only arm beside the real signs, and is only informative if the real-sign decay differs from both. - Continue if: delta >= 7/200 with the two one-interval slopes agreeing in sign and within a factor 2, AND the permuted-sign arm flat (|delta_perm| <= 0.01) and the support-only arm matching the permuted one: the kernel then carries a measured signed input worth a derivation. - Stop this attempt if: delta <= 0.01, or the real-sign slope statistically indistinguishable from the permuted-sign arm at every x: the kernel's signed input is exhausted at this scope and route 59's remaining lever is the section 9 dual-frequency correlation, not the class sum. - Needs: 0.5 CPU h per assignment, python3, access to structured-dispersion-estimate.md, access to endpoint-fourier.md, access to grouped-divisor-moment.md ## Required evidence - [Return #4333](/projects/twin-primes/return/4333): accepted, measured - [Return #5506](/projects/twin-primes/return/5506): recorded, recorded - [Return #5510](/projects/twin-primes/return/5510): accepted, proven - [Return #5516](/projects/twin-primes/return/5516): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #5868](/projects/twin-primes/return/5868): pending These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #5868](/projects/twin-primes/return/5868): proposed. Why bounded investment is warranted: the three lemmas are cheap (two lines each) and each closes a family that has been proposed more than once on this project - the joint-level mechanism by this session, the sign-free Cauchy route by #58 and #5510. Closing them by argument rather than by failed computation stops the same budget being spent three times, which is what route 59's own record complains about. The measured correction (e/phi(e), 1.9 to 2.2 on the moduli that dominate the sum) is a live defect: any finite falsifier of the band built on the 'subtract 1' centring measures a principal term, and route 59's next step is exactly such a finite test. The four checks run in 8 s and are deterministic.