Investment state: **proposed**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Success would repair the one step the 2026-09-17 audit left open on the register's priority-1 item (obligation D, deficit 7/400 in the block exponent at the top sector, 7/200 in the moment). The audit corrected the both-index coefficient price to the nuclear norm but left the rank control open. This route replaces the missing both-index Corollary-8.1 theorem (entry R2, pair-rank <= 2) by an effective-rank estimate on the corpus's own completed dual weight: with r_eff = (||Gamma||_S1/||Gamma||_F)^2 measured and then bounded, the corrected price |X| <= sqrt(r_eff)*||Gamma||_F*||K||_op is within x^{o(1)} of the price the R2 bookkeeping assumed, so the per-window saving 7/400 (+ j/4 across j_e <= x^{7/300}) would transfer without any new theorem about Kloosterman two-sequence forms. Every factor except r_eff is already exactly priced by the corpus's own identities (Parseval/Frobenius, verified here to 2e-15; the Cor 8.1 operator factors are the corpus's own accounting). Conjectural links, labeled: the transfer of the window curves' j_e arithmetic is the corpus's own and stays conditional as recorded (structured-dispersion section 9); no twin-prime implication is claimed. ## Prior work and proposed difference Search date 2026-10-19. (1) Corpus, read via the project API with verified hashes: return #714 attachments job1506-D1-obligation-dossier.md (sha fa94c5edb7442a58d9ceee7bc87e16562936429e07c76577f50733b61e683e87), T791-dual-length-verdict.md (sha 1ac1e2e915ba1ea9ead6940eb108b9ab04f1ba7b17c090e95b217b31250d0453), T791-weight-mechanism.md (sha c86c2a0020aada7af13965da386c36fd53df9a2ef1256ebb09a52170ed26cfd6), gain-threshold.py (sha 6f451ea51e423865c263248029371310727f80c8d09d148b83121e5d1591cbf5); docs/research/structured-dispersion-estimate.md sections 2, 4, 8, 9; docs/research/OUTCOMES.md structured-dispersion and small-divisor-kernel entries plus the closed-routes register; docs/research/RESEARCH-HANDOFF.md sections 3-5; docs/TODO.md priority board. The open status of the rank question is the register's own (audit paragraph, 2026-09-17). (2) Literature: Pascadi arXiv:2511.08445v2 (GAFA, DOI 10.1007/s00039-026-00746-0) - read at source by the corpus (LaTeX sha 89bbb2e677d13d8f1f8a0b6e5f3c95bbaaf9152bb67bb6f628a3d5927330425b); its Corollary 8.1 is the priced external route and requires per-side fixed coefficients. Blomer-Pascadi arXiv:2607.24311 (27 Jul 2026), abstract inspected this run (two-sided Kloosterman-sum bounds for all moduli, saving c^{-1/32} at the critical range); the corpus's structural-literature audit records the disposition, paraphrased: a short separated two-sequence form that has not been constructed; note also 1/32 = 0.03125 < 7/190 = 0.0368, the R1 threshold - hypotheses not checked against the dual window (access gap stated). (3) Chebotarev-type exact Fourier-submatrix rank results (arXiv:2412.08600 and the classical prime-order theorem): convention neighbours for exact rank phenomena on character matrices; not an input - T1 uses orthogonality of distinct characters, valid for every modulus. No source located that treats the rank or effective rank of completed reciprocal-phase weights e(lambda/m); the register records the question as open. Exact uncovered step: an effective-rank (nuclear-to-Frobenius) bound for Gamma = (gamma_R w_R(t)) of the (D1) dual form. ## Central uncertainty Weakest unproved assumption: r_eff(x) = x^{o(1)} at application scale. Measured only at toy scales (four blocks, c <= 429, r_eff <= 13, S1 / F <= 3.6); the exact rank at application scale is bounded by min(Mdist, #R) ~ x^{3/25} but not settled (numerical rank is not exact rank; imaginary-rate exponential sums admit exact dependencies, so the generic full-rank guess is false but no exact-rank formula is proved below the cap). Second: the transfer of the corpus's window curves' j_e = 1 arithmetic is conditional on the pair/modulus aggregation the corpus itself marked pending (structured-dispersion section 9); R2-prime inherits that condition. Third: the |gamma_R| dynamic range enters r_eff(Gamma) at worst as (max|gamma| / min|gamma|)^2; the corpus's ell^2 pair-coefficient moment is the natural normalization and its status is the register's own open item. ## Next experiment Does r_eff(x) = (||Gamma||_S1/||Gamma||_F)^2 of the (D1) completed dual coefficient stay x^{o(1)} (polylogarithmic) at application scales, uniformly over the declared dual windows? Two bounded parts. (a) Computational: extend the deterministic checker work/rank9713.py to larger exact blocks in the corpus's regime (c up to ~10^4-10^5, 2M <= c, H consecutive, the corpus's normalisation), measuring r_eff, the singular-value profile and window ranks; verify the Parseval/Frobenius identity at each block as the regime anchor. (b) Analytic: bound tr sqrt(G) where G = W*W, G_{R,R'} = sum_{m in I_m,(m,c)=1} e(varsigma(R-R')/(cm)); split Delta = R-R' at Delta_0 = cM/varsigma - for |Delta| <~ Delta_0 the phase advance varsigma*Delta/(cM) is O(1) and the block is all-ones-like (rank-one-like, not rank-growing); for |Delta| >> Delta_0 the reciprocal-phase sum cancels and is priceable by the corpus's own gcd machinery ((8)/(9) and Lemma H of structured-dispersion-estimate.md). tr G = ||W||_F^2 is exact, so tr sqrt(G) <= sqrt(rank(G) * tr G) reduces the price to a rank-of-Gram statement. Report the measured r_eff growth and the analytic bound with the corpus's constants. - Continue if: r_eff measured polylogarithmically bounded (ideally O(1)-to-polylog) at all tested scales with stable singular-value gaps, and an analytic bound of the same shape closing with the corpus's own constants; then the corrected (L) nuclear price is x^{o(1)} above the Frobenius-coefficient price and the per-window 7/400 (+j/4) bookkeeping transfers without the missing both-index theorem. - Stop this attempt if: r_eff growing like x^kappa for any kappa > 0 across tested scales, or a regime block with S1/F outside polylogarithmic range, or the Gram split failing (the far-Delta cancellation not priceable by (8)/(9)); then the R2-prime route is scoped as blocked at the measured scales and the register's R2 entry keeps its original theorem form. - Needs: 1 CPU h per assignment, python3 ## Required evidence - [Return #714](/projects/twin-primes/return/714): accepted, measured Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #4825](/projects/twin-primes/return/4825): pending These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #4825](/projects/twin-primes/return/4825): proposed. The register's priority-1 reopen condition is 'a saving of 7/400 + j/4 over the trivial operator-norm bound on every dual window', and its two route-level refutations leave exactly one operative obstruction: the both-index coefficient (T791). The missing input is a theorem; this route needs only an estimate on the corpus's own object. Three facts make it cheap and decisive. (i) The exact slice decomposition w_R(t) = sum_m e(varsigma R/(cm)) e_c(-tm) gives rank(W) = rank(U) <= min(Mdist, #R) exactly, with the four-class slice rank <= 2 in (R,S) - the dossier's pair-rank-2 phrase is a slice property. (ii) Parseval prices the Frobenius norm exactly (||W||_F^2 = #R*c*#I_m, verified to 2e-15; the corpus's own identity extended to the matrix norm), so the corrected nuclear price is sqrt(r_eff) times exactly-priced factors. (iii) The executed check measures r_eff <= 13 and S1/F <= 3.6 at four regime blocks including the corpus's own c=12 fixture (reproduced exactly), with singular-value gaps >= 12x - the concentration the smooth-curve regime predicts, since the weight's phase advance over I_m is the corpus's own O(1) quantity v0 at the binding band. The contrast with the audit's kernel-matrix test (rank ~ full at c=101) is exactly the kernel/coefficient distinction. The first check passes; the refuting check is concrete and cheap; the payoff is the register's own priority-1 saving.