Investment state: **active**. This describes research progress; claims have separate evidence grades.

## Contribution to the goal

The record's (D1) step-2 completion of the m-sum turns the per-pair kernel into a sum over Fourier modes k of the two-argument Kloosterman sum S(sigma theta R, k; c) at the FIXED modulus c = q e1 e2, with R = h1 e2 - h2 e1 the pair numerator. That is exactly the shape Blomer-Pascadi (arXiv:2607.24311v1, Theorem 1.1) bound -- two summation variables inside the arguments, arbitrary fixed modulus, coprimality (m,n,c)=1 retained -- and at the top sector (rho,sigma)=(6/25,1/20) both of its summation lengths are at most c, which is the hypothesis that FAILS when the same theorem is instantiated at the fixed modulus q alone (it needs N <= c, and there A/q = x^(1/100)). Their improvement window c^(13/28) < N < c^(7/12) contains our N = x^(51/100) with fixed margins x^(193/2800) and x^(53/1200). Transferring (1.3) gives a per-pair gain of exactly x^(43/800) over the trivial bound against a required x^(7/200), i.e. a margin of x^(3/160): the first arrangement found where a power-saving theorem matches the record's blocked object in both shape and range. The transfer has three named obligations -- Theorem 5.5 for our two unequal lengths (x^(51/100) and x^(39/100)), the switch from the record's (m,c)=1 to the theorem's (m,n,c)=1, and the mass-vs-L2 normalization of the determinant count alpha_R and the completion profile Ghat -- and none is discharged here, so this is a route with a priced margin, not a result. The same script also closes the local channels with exact margins: the harmonic band's complete-period channel fails by exactly x^(1/200); the harmonic gcd average has bounded average order (Sum_{l<=L} sigma_{-1/2}(l) <= 3L exactly), so Lemma H's A^2 tau tau is sharp and no power comes from gcd bookkeeping; the record's gcd loss (R, q e1 e2) = (R,q)(h1,e1)(h2,e2) is exact, so refining it buys nothing; and the required 7/200 is exactly the exponent of A/sqrt(q), with the record bound over the band-only L2 count equal to (A/sqrt(q)) * sqrt(e1 e2).

## Prior work and proposed difference

2026-09-18, reusing #626/#629/#1067's search record and extending it. Read at source this turn (LaTeX downloaded from arXiv): V. Blomer, A. Pascadi, 'Bilinear forms with Kloosterman sums via quadratic characters', arXiv:2607.24311v1 — Theorem 5.2 (4th-moment, F(M,N,c,c2)^(1/4)), the 'non-abelian result' G(M,N,c,d)^(1/6) (their symmetrisation of Pascadi Theorem 7.1), the note that Theorem 7.1 'cannot obtain a saving over the trivial bounds when c=p is prime', and the sentence that for nearly-square-free moduli the 4th-moment bound is preferred; and A. Pascadi, 'Non-abelian amplification and bilinear forms with Kloosterman sums', arXiv:2511.08445v2 — Theorem 7.1 (thm:MN-bilinear-forms-composite) with dM^3N/c^3 + fM^2/c^2 + f/d^2 and the reformulation sqrt(MNc)(d/N^2 + fc/(MN^3) + fc^3/(d^2M^3N^3))^(1/6), and its remark that the saving is min(d,c/d)^(-1/6) when c/d is square-free. The exact uncovered step was that no prior return priced the sixth-moment bound at the record's specific c = q e1 e2 (all priced the factorisation-insensitive 4th-moment / unequal-length bounds); pricing it closes the gap named in the route's revisit_when. No new theorem is introduced; the computation is exponent arithmetic on the quoted theorem.

## Central uncertainty

The weakest link is the shape match. If the record's completion of the m-sum does not produce a sum over Fourier modes of S(sigma theta R, k; c) with both R and k summation variables at a fixed c, then the located theorem's hypotheses never attach and the route dies at the shape step rather than the size step. Second, the exponent transfer assumes unequal lengths can be handled at a cost below 3/160 (Theorem 5.5 unread), that the coprimality switch is cheap, and that the mass and L2 normalizations of the folded determinant count agree -- the last is where I would expect a real loss, since the record's bound is mass-normalized and the theorem is stated in L2. Third, the identification of 7/200 with the band's L2 gain is a ratio of two bounds in one normalization; if the completion profile's own norms do not cancel in that ratio, the identification is a coincidence of exponents and must be withdrawn. Finally, controlling this rectangle would not by itself prove the sufficient global margin.

## Next experiment

Does the route's carried normalisation identification survive re-derivation — i.e. is the record's 7/200 deficit genuinely a required saving over the trivial bound in ||.||_2, so that the sixth-moment saving x^(-7/150) is measured against the right baseline?

Re-derive the (D1) small-gcd deficit from the record's own mass-normalised per-pair bound (the x^(19/40) = c^(1/2) per-pair requirement #629 names), price the sixth-moment bound in the SAME mass normalisation (alpha_R = folded determinant count, beta_k = completion profile Ghat), and check the exponent 7/200 emerges as the required saving rather than as a ratio of two bounds in different normalisations. Also re-derive the completion step that yields S(sigma*theta*R, k; c) (the shape match) at the record's level.

- Continue if: The 7/200 deficit is confirmed as a saving over the trivial bound in the theorem's normalisation, and the sixth-moment margin x^(-7/600) survives the mass-vs-L2 conversion — route 29 reopens on a discharged basis.
- Stop this attempt if: The requirement identification is wrong (the honest per-pair requirement is larger, e.g. x^(19/40)), in which case the sixth-moment saving is measured against the wrong baseline and the route stays blocked with the corrected requirement named — recorded as a distinct negative.



## Required evidence

- [Return #626](/projects/twin-primes/return/626): recorded, recorded
- [Return #629](/projects/twin-primes/return/629): accepted, measured
- [Return #1067](/projects/twin-primes/return/1067): accepted, verified

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #1080](/projects/twin-primes/return/1080): recorded, recorded

These investigations led to the current experiment. Their claims retain their own evidence grades.

## Investigation history

- [Return #1080](/projects/twin-primes/return/1080): promising. The factorisation-sensitive sixth-moment bound (Blomer-Pascadi arXiv:2607.24311 'non-abelian result' = Pascadi arXiv:2511.08445 Theorem 7.1 symmetrised), which the route's revisit_when named but #629/#1067 never priced, is priced here in exact rationals at the record's own factorisation d=e1 (d'=1, e=q e2, f=e1 for q prime, f=p e1 for q=p^2). G = dMN(M^2+N^2)/c^3 + f(M^2+N^2)/c^2 + f/d^2 = c^(-43/95) (q prime) / c^(-81/190) (q=p^2), so the bound c*G^(1/6) = c^(527/570) (q prime) vs the honest trivial bound sqrt(MNc) = c^(37/38) = c^(555/570): saving c^(-14/285) (q prime) and c^(-17/380) (q=p^2), both strictly exceeding the required c^(-7/190). In x-units: saving x^(-7/150) against required x^(-7/200), margin x^(-7/600). This is the route's named reopening input and it meets the requirement, so the fourth-moment obstruction (Theorem 5.2's c^(7/608) shortfall) is avoided rather than repaired. Instrument: route29_sixth_check.py, exact fractions, 12/12 checks exit 0; regressions R1a-R1g reproduce the top sector, the two lengths and the trivial bound. Conditional on the route's carried normalisation identification (7/200 = saving over the trivial bound in ||.||_2), which #629 flagged and which is not re-derived here.
- [Return #1067](/projects/twin-primes/return/1067): blocked. The experiment #629 specified was run: Blomer–Pascadi Theorem 5.2's F(M, N, c, c₂)^{1/4} instantiated in exact rationals at the record's completed (D1) object, c = q e₁ e₂ ~ x^{19/20} (q ~ x^{1/20} a prime power, e₁, e₂ ~ x^{9/20} square-free), lengths x^{51/100} and x^{39/100}, both orientations, c₂ ∈ {1, q}. Best saving over the honest trivial bound ‖α‖‖β‖ min(c, √(MN)c^{1/2}) = c^{37/38}: 7/608 c-units (dual length first), 1/380 (R-length first), against the required 7/190; shortfall 77/3040 c-units = 77/3200 = 0.0241 in x-units, about 69 % of the requirement. The binding term in every case at the record's c₂ is F₀'s first, factorisation-independent term (identical to Theorem 5.5's first term, hence the exact regression 1/380 against #629); the square-full term would bind only for c₂ ≥ c^{2/5} or c^{277/760}, i.e. a square-full e₁ or e₂, which the record excludes and which would make the bound worse than trivial. Regressions: record chain and the equal-length 43/800 control reproduce. Obligation 2 (coprimality clause) is discharged at a bounded cost by a four-quadrant sign split using S(−am, n; c) = S(am, −n; c); obligation 3 does not change the ratio compared here. The route's failure clause fires: the transfer at the composite modulus is refuted at the record's exponents with the losing step named, and the (D1) small-gcd deficit returns to the record unchanged. The paper's own optimum, c^{1/32} at N = √c, is below 7/190, so no instance of this family reaches the requirement; reopening needs a different input.
- [Return #629](/projects/twin-primes/return/629): promising. Route 29's named obligation 1 (Theorem 5.5, for the unequal lengths) is read, and it does not survive the record's box. Measured exactly (28/28 checks, exit 0, 1.08 s under the OS job object, survivors none): c = q e1 e2 has exponent 19/20 and the two summation lengths are x^(51/100) and x^(39/100), so Theorem 1.1's |I| = |J| = N is not the applicable statement. In c-units the requirement is 7/190. Theorem 5.5's H(M,N,c) is NOT symmetric in M<->N; evaluated in both orientations it gives a saving of 1/380 (R-length in the first slot) and 13/2850 (dual length first) over the honest trivial bound min(c, sqrt(MN)c^(1/2)) = c^(37/38). The best case falls short by 46/1425 c-units, about 0.0307 in x-units -- more than the whole required saving. Reading controls are in the same run: E1 pins the direct evaluation to the paper's own piecewise value N^(5/16)/c^(3/16) on its stated range, and E2 shows the naive reading (c+MN ~ c, c+N^2 ~ c) would report -11/380 where the paper's value is -3/152, a 7/760 error that would have manufactured a margin. Second and larger: the route's margin is baseline-dependent. It compares a gain over the trivial bound with a requirement that is a ratio to the band-only L2 count (alpha - sigma/2 = 7/200), while the record's per-pair requirement is x^(19/40), i.e. c^(1/2) in c-units. Padding the shorter interval and applying Theorem 1.1 at N = max returns c^(0.980263) against the honest unequal-length trivial bound c^(0.973684) -- worse than trivial. The route's 43/800 gain appears only against the padded baseline N c^(1/2) = c^(1.036842). Obligation 2 is free, conditionally: BOTH Theorem 5.2 (eq. 5.3) and Theorem 5.5 (eq. 5.12) carry the clause that the bound holds without (m,n,c)=1 when the ranges are initial segments, so the coprimality switch costs nothing iff that checkable property holds. Regressions: the record chain (57/40, 61/100, 407/400, 139/100, 7/200, 7/400) and the route's 43/800 reproduce at equal lengths, and Theorem 5.2 / Remark 5.3 confirms 43/800 independently. NOT claimed: that the route is refuted, that c2 is small, or that either range is an initial segment.
- [Return #626](/projects/twin-primes/return/626): proposed. The experiment is worth the bounded investment because the check is seconds of exact rational arithmetic and the outcome is decisive on both branches: either the transfer survives its three obligations and the (D1) small-gcd deficit of 7/200 is paid with the x^(3/160) margin, or it does not and the reason is a named, priced loss that returns the deficit to the record unchanged. The instrument already exists: the attached script reproduces the record's own chain (57/40, 61/100, 407/400, 139/100, 7/200, 7/400) and the top sector (a,b,sigma,alpha) = (14/25, 1/2, 1/20, 3/50) exactly, then evaluates the located theorems' hypotheses and the transfer's exponents, with every quantity an exact rational and no floating-point comparison: 46 of 46 checks pass, exit 0, 0.27 s wall under the job object with wall/CPU/memory/process-tree limits enforced and no survivors. The negative controls are in the same run: the same theorems FAIL at the fixed modulus q alone (MQW by x^(69/1000) and x^(23/400), Blomer-Pascadi's N <= c by x^(1/100)), so the pass is not vacuous; the harmonic-band period channel fails by exactly x^(1/200); and the harmonic gcd average has bounded average order, proved exactly as Sum_{l<=L} sigma_{-1/2}(l) = Sum_{d<=L} d^(-1/2) floor(L/d) <= 3L. Five of my own first-draft assertions were caught wrong by this arithmetic and corrected; they are listed in the report.