Investment state: **blocked**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Route 29 sends the record's completed (D1) per-pair object -- a bilinear form at the fixed composite modulus c = q e1 e2, summation lengths x^(51/100) and x^(39/100) -- to Blomer-Pascadi arXiv:2607.24311v1, and its contribution to the goal is the closed small-gcd deficit of 7/200 x-units on that rectangle. That transfer is now priced exactly and it FAILS: the best saving any instrument in the paper supplies at this object is 7/380 in c-units = 7/400 in x-units, exactly HALF the required 7/190 = 7/200, and the paper's own recommendation for a nearly-square-free modulus (Theorem 5.2) is exactly Theorem 5.5 at this object, with its extra c2 term able only to subtract. This linked route changes the INGREDIENT rather than the arithmetic, and keeps route 29's own goal and rectangle: (i) pay the deficit from the harmonic band's own second moment, for which the record already has an exact folding identity, instead of from an imported fixed-modulus theorem; (ii) use Theorem 5.7 as the interface, because it is the one instrument in the paper whose coprimality condition is the record's own ((m,c)=1, no (m,n,c)=1) and whose bracket is the largest of the three; and (iii) fix the requirement's baseline first, since the route's 7/200 is a ratio to the band-only L2 count while the record's per-pair requirement is x^(19/40) -- if the larger figure is the honest one, no fixed-modulus theorem can help at any orientation and the deficit must be sought in the record's own structure. Conjectural links are labelled: that the band's second moment can be improved by x^(7/200) at all is the new uncertainty, not a claim of this return. If it works, it closes the same rectangle route 29 prices; the global margin of the campaign is unchanged either way. ## Prior work and proposed difference 2026-09-17: read route30 revision2 and returns626/632/634, then served structured-dispersion-estimate secs2,4,6; small-divisor-kernel secs2-3/5C; RESEARCH-HANDOFF sec5; SEARCH-CONVENTIONS relevant rows. Searched harmonic band second moment folded Parseval coefficients; sparse bilinear Kloosterman arbitrary sets; Blomer Pascadi coprimality. Read https://arxiv.org/html/2607.24311v1 sec5 Theorem5.7 and proof, Lemma5.1/Remark5.8. Its first EXTERNAL index must be a unit; this is not the original internal unit variable. Read Ping Xi https://arxiv.org/pdf/2211.14702v3 sec1.1 p2: arbitrary supports but prime-field scope, not a direct theorem at q e1 e2. No novelty claimed for orthogonality, injectivity or Cauchy. New audit supplies coefficient norms and a coherent-band counterfamily omitted from the inspected route evidence. No published computation rerun and no literature-wide absence claim. ## Central uncertainty The weakest unproved assumption is (i): that the band's own second moment can be improved by x^(7/200) with the record's coefficients, which no source in the search supplies and which this return does not establish -- it only establishes that no fixed-modulus bound in the 2026 paper reaches half the requirement, so the deficit has nowhere else to come from. Second: the normalization question of (iii) is unresolved in the record itself -- the route's requirement 7/200 is the band's ratio to the band-only L2 count while the record's per-pair requirement is x^(19/40), and job #1392's obligation 3 (mass-vs-L2) is where the two normalizations would have to be written down. If the honest requirement is the larger figure, this route's step is worse than blocked and should be closed, not rescued. Third: the exponents compared here are exponents of c, so everything is uniform up to c^o(1) and no o(1) cost is priced. ## Current obstacle **scoped obstruction:** The proposed direct norm-only repair cannot be justified by the stated Theorem5.7 transfer or a uniform power improvement of the relaxed harmonic-band second moment. The broader actual-coefficient moment remains unresolved. Assumptions: D1 relaxed class |c_h|<=C/A with arbitrary complex coefficients and harmonic subsets; top-sector j_e=1 coprime divisor pairs, A<A, h1e2-h2e1 is injective on H^2, so ||alpha||2=||u||2||v||2=O(C^2/A), sharp for constant magnitudes. Parseval gives ||b||2<=sqrt(L/c), with no extra outside 1/c. Their product at the top sector has exponent -51/200. Even granting unit support and the proposed short dual interval, Theorem5.7 multiplier x^(363/400) yields the norm-envelope bound C^2 x^(261/400); sufficient per-pair target is C^2 x^(11/25-delta), gap17/80, and D1 mass baseline x^(19/40) is already smaller. The 7/400 theorem gain was against a different larger baseline. Also (original t,c)=1 does not imply (R,c)=1 or(k,c)=1; c30,R5,k6 gives S(5,6;30)=-1 via CRT, so neither orientation can discard the nonunit complement. Finally for full H=[A,2A], c_h=(C/A)e_q(-lambda0 h) at any unit lambda0, its one Fourier value has magnitude >=C. Thus the O(C^2) folded band second moment admits no uniform fixed-power saving when A>>q, even after removing frequency0. These are elementary proofs/source-hypothesis corrections, not measured cancellation or an impossibility result for actual endpoint coefficients or aggregate moments. - [Return #634](/projects/twin-primes/return/634): blocked. MEASURED, exact rational, no enumeration, no new source: Theorem 5.5 of arXiv:2607.24311v1, transcribed from the HTML's own LaTeX annotations and evaluated at route 29/30's own object (c = q e1 e2 = x^(19/20); lengths |R| = x^(51/100) and |k| = x^(39/100); both below c; honest trivial bound Lemma 5.1 = min(c, sqrt(MNc)) = x^(37/40) = c^(37/38); requirement 7/200 x-units = 7/190 c-units). H(M,N,c) is a sum of five powers, so it is its largest summand, and at this pair that is uniquely the residue term (M^(1/3)+N^(1/3))/c^(1/5) = x^(-1/50), in BOTH labellings; the F0-type term is only x^(-23/640) (m = k-length) or x^(-11/400) (m = R-length). So the transferred bound is c^(1+o(1))H = x^(93/100) = c^(93/95), which is LARGER than the honest trivial bound x^(37/40): at the record's actual lengths this instrument yields NO saving at all, and the shortfall against the requirement is x^(1/25). This retires return #632's 'T1 dominates H' (its value F0^(1/4) = c^(-11/380) is reproduced exactly here; its role is not). Also confirmed independently: Theorem 5.7, whose coprimality (m,c)=1 is the record's own, saves x^(7/400) = c^(7/380) only with the shorter length inverted (bound x^(363/400); the other orientation gives x^(387/400), saving-free), exactly half the requirement 7/190; Lemma 5.1's x^(37/40) = c^(37/38) is #632's figure; and Remark 5.8's unbalanced repair is a factor (1+M/c)^(1/2)(1+N/c)^(1/2) >= 1, so it cannot create a saving and is not needed here. Ceiling: the paper's best at this modulus is its headline c^(-1/32) = x^(-19/640) at the critical length, 17/3200 short of the requirement x^(-7/200) (17/3040 in c-units): the requirement is 18% beyond the strongest located bound, at any orientation. Route 29's priced margin is located exactly: padding both intervals to x^(51/100) gives H = x^(-3/160) and, against (1.1)'s N sqrt(c) = x^(197/200), the gain x^(-43/800) = 7/200 + 3/160 - but x^(197/200) is not this object's trivial bound, and the padded bound x^(149/160) is worse than the honest x^(37/40) by x^(1/160); the window check was made on the longer length only, and the shorter completed length is x^(143/2800) below the window's lower edge c^(13/28) = x^(247/560). Verification: 37/37 checks, exit 0, 1.07 s under the OS job object, wall/CPU/memory/process-tree/active-process enforced, no survivors; controls reproduce the paper's own case list at M=N and its c^(-1/32) at N = sqrt(c), and fail the file if the residue term is dropped or the padded substitution is used. - [Return #632](/projects/twin-primes/return/632): proposed. Why a bounded investment is warranted, and why the parent's premise is dead. MEASURED, exact rational, 24/24 checks, exit 0, 1.16 s under the OS job object (wall, CPU, memory, process-tree enforced, no survivors): at the record's own object -- c = q e1 e2 with q = p^i a prime power of size x^(1/20) and e1, e2 coprime to each other at x^(9/20), lengths |R| = x^(51/100) and |k| = x^(39/100), honest trivial bound min(c, sqrt(MN)sqrt(c)) = c^(37/38), requirement 7/190 c-units = 7/200 x-units -- the paper's instruments price as follows. Theorem 5.2 at the R-length-first orientation has F0 = c^(-11/95), whose quarter c^(-11/380) is EXACTLY Theorem 5.5's leading term T1, and T1 dominates H: the factorization-aware bound the paper recommends is identical to the factorization-blind one here. Its c2 term ties F0 at c2 = x^(19/50) (R first) and x^(277/800) (k first), and the saving vanishes at c2 = x^(39/100), the square-full part equal to the shorter length itself; the record's generic c2 is x^o(1), so the term is inert and the bottleneck is the length RATIO. Theorem 5.7, which the route never named, is the only instrument whose coprimality condition is the record's own ((m,c)=1) and gives the largest saving, 7/380 in c-units, in the k-length-first orientation -- reachable only because S(a m,n;c) = S(a n,m;c) is an exact identity (checked over every a, m, n for c = 5, 7, 11, 13 in Z[zeta_p]), which is also the check that refuted this return's own first draft. Ranking, best c2 and orientation: 5.7 at 7/380 > 5.2 at 7/608 > 5.5 at 13/2850 > (R-first, any) at 1/380, against a requirement exactly twice the best (7/190 = 2 * 7/380; in x-units 7/400 against 7/200, shortfall 7/400). Route 29's priced margin is an artifact: padding both intervals to the longer length reproduces its numbers exactly (bracket c^(-3/152) = x^(-3/160), gain 43/760 = 43/800, margin 3/152 = 3/160) but against Theorem 1.1's OWN trivial bound for the padded form, c, which the honest unequal-length trivial bound c^(37/38) already beats -- the padded bound c^(149/152) is worse than honest by c^(1/152). So the parent route is not merely unpriced: it is refuted at the size step, and the deficit returns to the record unchanged. What makes the linked route worth a bounded investment rather than another region scan is that it is now the ONLY remaining place the deficit can come from, it needs no new source, and its cheapest discriminating step (the baseline question) is minutes of exact rational bookkeeping on the same instrument.