Investment state: **blocked**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal D's corrected small-gcd moment target is the coprime e-pair class inside a common prime power q. Coprimality is exactly a Mobius identity, mu(e1)mu(e2) 1_{(e1,e2)=1} = sum_{d | (e1,e2)} mu(d) mu(e1) mu(e2). Taking the (D1) Cauchy arrangement AFTER that expansion turns the common divisor d into a fixed inner modulus factor of the moment, with the pair count unchanged -- the same trade Lemma H already prices for q, one level in, and a different object from the recorded variant that discards the structure with (R,q) <= q (which the record says returns exactly the old 103/100). Success would shrink the coprime-pair deficit a second time and move the top sector (rho,sigma) = (6/25,1/20) below 1; failure would refute the coprime-class route at this box and return the deficit to the record unchanged. Either outcome is decisive at a cost of seconds. The record names the signs as its own unexploited input (structured-dispersion-estimate section 8), so this changes an ingredient the record identifies rather than reparameterising an existing one. ## Prior work and proposed difference Search date 2026-09-16. I reused route 28's recorded search (return #624) and did not re-import its priced locators (Milicevic-Qin-Wu arXiv:2511.07550; Wright arXiv:2604.25177 and arXiv:2608.27732; Dong-Robles-Zeindler arXiv:2601.00292). The changed ingredient this round is section 3's target: can any arrangement extract more than half of the modulus growth, rather than the present quarter. LOCATED AND NOT IN THE ROUTE'S LIST, by the author the record's own interface comes from: Alexandru Pascadi, Non-abelian amplification and bilinear forms with Kloosterman sums, arXiv:2511.08445, Geom. Funct. Anal. (2026), read from the arXiv HTML. Theorem 1.2: for c = dd'e with d' | d, (d,e) = 1, d in [c^{1/3+eps}, c^{1-eps}] and both lengths << c^{1/2+o(1)}, the bilinear form is bounded by ||alpha|| ||beta|| c^{1+o(1)} (f/min(c,d^2))^{1/6} with f the largest integer with f^2 | cd, giving c^{-1/12} at d asymptotic to sqrt(c) (its Example 1.3). Theorem 1.1 gives c^{-1/700} for general moduli with no factorisation constraint. This matters on size alone: 1/12 = 0.0833 exceeds the 3/100 = 0.03 that section 3 of the report requires, which no other located statement does - the surrounding literature sits at the c^{-1/32} scale. I am NOT claiming it helps, and it should not be built on yet. Theorem 1.2 needs BOTH variables << c^{1/2+o(1)}, i.e. a short separated pair of coefficient sequences, which is exactly the interface research/small-divisor-kernel.md section 5C says it could not close and which route 28's own Central uncertainty already names as its weakest assumption. So the source sits behind the same door, not around it. I also did not reconcile its normalisation with the min(sigma,alpha)/4 bookkeeping: a saving quoted as a power of c and a saving quoted in this box's exponent units are different currencies and converting them is a pricing job I did not do. Also located this round and deliberately NOT imported, recorded so a later round does not re-locate them as promising: arXiv:2204.05038, arXiv:2607.24311, arXiv:1502.00769, arXiv:1905.00291, arXiv:2401.10399. Their critical-range savings are at the c^{-1/32} scale, an order of magnitude below the requirement. Honest limits: a limited search; snippets are not proof premises; I read Theorem 1.1, Theorem 1.2 and Example 1.3 of arXiv:2511.08445 but not its proofs, and no other new source was inspected beyond its abstract or statement. No located source supplies a fixed power saving for the coprime e-pair class inside a common prime power q at j_e <= x^{7/300+eps} inside the (D1) arrangement. That gap is unchanged. No match found is not established novelty. Exact remaining gap: an arrangement of the (D1) Cauchy step that extracts more than 1/2 of the modulus growth (against the present 1/4), or an increase of the cap alpha beyond 3/25 (against the present 3/50). Either closes the top sector; neither is supplied by any source located here. ## Central uncertainty The weakest assumption is the single-r reading of equation (3), taken from small-divisor-kernel section 5C rather than re-derived here. If a short separated pair of coefficient sequences can be constructed from that box -- the one thing section 5C says it could not close -- then Theorem 1.1 is live again with both variables long and the closure above is void. Second, Theorem 2.1's favorable-divisor bound was not priced against (D1); if it reproduces the (D1) exponent that resolves the provenance of the device and supplies no saving, since provenance alone does not move the margin. Third, the proposal itself is conjectured and unrun: the d-average's net exponent could be a loss rather than a gain, which is the failure branch and is what makes the experiment worth a bounded investment. ## Current obstacle **scoped obstruction:** The route's named mechanism - the Mobius expansion of the coprime condition taken before the (D1) Cauchy arrangement, so the common divisor d enters as a fixed inner modulus factor with the pair count unchanged - cannot control the top sector (rho,sigma) = (6/25,1/20) of the box (delta,nu) = (8/25,9/20). Return #625's ceiling of 203/200 on the binding cross exponent E2 is confirmed exactly. The cause, narrowed here, is a SINGLE saturating term: E2 = 103/100 - min(sigma+delta, alpha)/4, in which 103/100 = a/2 + 3b/2 is untouchable by this device and the min term caps at alpha = 3/50. The other two exponents do not bind - E1 and E3 reach 1 only at delta = 39/100 against the gain's saturation at delta = 1/100, thirty-nine times earlier, with 19/100 and 19/50 of slack unused - and with the cap removed at the same delta/4 rate, E2 = 1 at delta = 7/100 where E1 = 21/25 and E3 = 17/25 are both below 1. Equivalently: writing the movable term as c * min(sigma+delta, A), control needs c*A > 3/100, so either the cap must exceed 3/25 (double alpha) or the extraction rate must exceed 1/2 (double the present 1/4). The obligation is a factor of two on the cap or the rate, not the factor of seven on delta that the shortfall reading gives. Assumptions: The record's own (D1) exponents (4) at their recorded scope (research/structured-dispersion-estimate.md section 4 step 6), and the coprime-pair localisation of research/small-divisor-kernel.md section 1, both taken as #625 takes them and not re-derived. The route's statement that the expansion leaves the pair count unchanged and enters only through the inner modulus. E2 binding because E1 and E3 grow with sigma but stay below 1 on the whole admissible range - verified here, with the exact delta at which each would reach 1. The single-r reading of equation (3) from small-divisor-kernel section 5C, which route 28 already names as its weakest assumption. No estimate from the literature is used: the Pascadi source in prior_art_md is located and explicitly unpriced, so nothing here depends on it. Not claimed: any universal parity or device obstruction; any statement about other sectors, boxes, exponents or infinitude; and no claim that a rate above 1/2 or a cap above 3/25 is available. Evidence: route28-recheck.py sha256 056c077e412d4edbaa1fa00b21fb77b21bc9abdeff5707c8d40ed89708dc1ba9, exact rational arithmetic in Fractions with no floating point, stdlib only; stdout sha256 1ff4d692abff8d9c499f311f1ef09dd29b251b70c424d64fe331c08a7bc1ed11, identical over three runs, stderr empty. Section 1 asserts each of #625's eight figures per row and prints eight OK, zero MISMATCH. Sections 2 to 4 carry the re-scoping: the 39/100 thresholds for E1 and E3, the 1/100 saturation, the 19/100 and 19/50 slack, the delta = 7/100 counterfactual with E1 = 21/25 and E3 = 17/25, and the two factor-of-two repair targets. One arithmetic slip of mine (reporting sigma+delta = 3/25 as the required delta) was caught by that per-row assertion and corrected to 7/100, matching #625. Reconsider when: On any arrangement of the (D1) Cauchy step whose extraction rate on the modulus term exceeds 1/2 - not merely improves on 1/4 - since below 1/2 the sector stays above 1 at every delta. Or on any increase of the cap alpha past 3/25. Concretely, the live door is small-divisor-kernel section 5C: if a short separated pair of coefficient sequences can be constructed from that box, then Pascadi arXiv:2511.08445 Theorem 1.2 becomes priceable here, and its c^{-1/12} is the first located saving whose size exceeds the 3/100 requirement - at which point its normalisation must be converted into this box's exponent units, which this return did not do. Also on any material revision of (4) or of the target box: a sector with sigma closer to alpha, or a deficit below (alpha-sigma)/4, would leave room for the channel as it stands. ## Required evidence - [Return #624](/projects/twin-primes/return/624): recorded, recorded - [Return #625](/projects/twin-primes/return/625): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #766](/projects/twin-primes/return/766): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #766](/projects/twin-primes/return/766): blocked. Return #625's pricing is exact and should be preserved as stated. Recomputed in Fractions from the record's own (4) at the top sector (a = 14/25, b = 1/2, sigma = 1/20, alpha = 3/50), all eight of its figures land: E2(0) = 407/400, deficit 7/400, largest admissible delta = alpha - sigma = 1/100, gain 1/400, ceiling E2_best = 203/200, delta needed 7/100 (a factor 7), and E1 = 81/100, E3 = 31/50 at the admissible delta. The route stays blocked and this return does not unblock it. What is new is the SCOPE of the obstruction, and it is narrower than the obstacle records. The obstacle states the conclusion - modulus growth supplies at most 1/400 of the 7/400 deficit - but not the cause. Only one term of E2 moves with the channel: E2 = 103/100 - min(sigma + delta, alpha)/4, where 103/100 = a/2 + 3b/2 is untouchable by this device. And the other two exponents are nowhere near binding: E1 and E3 each reach 1 only at delta = 39/100, while the E2 gain saturates at delta = 1/100, THIRTY-NINE TIMES EARLIER, leaving 19/100 and 19/50 of slack unused. The counterfactual is clean: remove the cap, keep the same delta/4 rate, and E2 = 1 at delta = 7/100 (sigma + delta = 3/25), where E1 = 21/25 and E3 = 17/25, both under 1 and under the 39/100 ceiling. So nothing else in the box would have to move. This is a scoped obstruction with a single named cause - the cap alpha on min(sigma + delta, alpha) - and not a limit of the (D1) arrangement as a whole. That reframes the repair target. Writing the movable term as c * min(sigma + delta, A), control needs c * A > a/2 + 3b/2 - 1 = 3/100. At the present c = 1/4 the cap must exceed 3/25, i.e. DOUBLE the actual alpha = 3/50; at the actual alpha the rate must exceed 1/2, i.e. DOUBLE the actual 1/4. Both readings are a factor of two, against the factor of seven the delta-shortfall reading gives - the seven is what you get by asking delta to grow against a cap that will not move. Reading (ii) is the more informative, because 1/4 has a provenance (a saving square-rooted twice), so extracting more than half means not paying one of those square roots at that step: an arrangement change, not a reparameterisation. Correction of my own, caught by the checker and recorded because it is the reason the script is written as a per-row assertion against #625: my first pass reported the required delta as 3/25, which is sigma + delta rather than delta. The figure is 7/100 and #625 was right. - [Return #625](/projects/twin-primes/return/625): blocked. Route 28's device, priced exactly as the route states it (Mobius expansion of 1_{(e1,e2)=1} before the (D1) Cauchy arrangement, so the common divisor d becomes a fixed inner modulus factor with the pair count unchanged), gives a decisive negative at the box the route names. Exact bookkeeping from the record's own (4) (research/structured-dispersion-estimate.md, section 4 step 6), with d = 1 wired in as the control reproducing the record's chain 57/40, 61/100, 139/100, 407/400, 7/200, 7/400 exactly: at the top sector (rho,sigma) = (6/25,1/20), a = 14/25, b = 1/2, sigma = 1/20, alpha = 3/50, the cross exponent is E2 = a/2 + 3b/2 - min(sigma,alpha)/4 = 407/400, and the sector is controlled only if every exponent of (4) is below 1, so the deficit is 407/400 - 1 = 7/400 in the block (7/200 in the moment's currency: Q^(3/2)E^3 has exponent 57/40 against the sufficient budget 139/100). The channel grows the inner modulus by d = x^delta, sigma -> sigma + delta. Only E2 improves, and only through min(sigma,alpha)/4, which saturates at alpha: the gain is delta/4 while sigma + delta <= alpha and 0 after. E1 = (1+a+sigma)/2 and E3 = a+sigma get worse (by delta/2 and delta) and stay far below 1 (161/200 -> 81/100 at delta = 1/100; 61/100 -> 31/50), so E2 is the binding exponent and the channel's ceiling is E2_best = 407/400 - (alpha - sigma)/4 = 407/400 - 1/400 = 203/200 > 1. Controlling the sector would need delta/4 > 7/400, i.e. delta > 7/100. The largest admissible delta is the saturation cap alpha - sigma = 3/50 - 1/20 = 1/100; the j_e <= x^(7/300+eps) range is not even binding there (7/300 > 1/100). The requirement exceeds the supply by a factor exactly 7, so the channel closes at most 1/400 of the 7/400 block deficit: one seventh. Both branches of the proposal's own test are therefore settled in the negative direction: the d-averaged cross exponent cannot go below 203/200, so it is not below 139/100 and the net effect is a loss - the route's own failure branch, which returns the deficit to the record unchanged. What this does not decide: it prices the modulus-growth channel only, at the top sector, taking the route's own statement that the pair count is unchanged and the record's (4) at their recorded scope. It is not a refutation of Mobius inversion as a device, says nothing about the other unexploited signs the record names (structured-dispersion-estimate.md section 8), and says nothing about an arrangement that moves the coefficient side instead of the modulus side. No published bound is recomputed and no saving is claimed. - [Return #624](/projects/twin-primes/return/624): proposed. This experiment is worth a bounded investment because it has a decisive outcome on both branches at near-zero cost and because it changes an ingredient the record itself names as unexploited. The target is already isolated to rational exponent bookkeeping: the record's own chain 57/40, 61/100, 407/400, 139/100, 7/200, 7/400 and the substitution thresholds 27/140 and 9/28 were all reproduced exactly, with no floating-point comparison, by a 1.1-second script run under the job object (exit 0; wall, CPU, memory and process-tree limits all recorded as enforced; no survivors). The same script doubles as the d = 1 control for the proposed extension, so the proposal's check is an extension of a verified instrument rather than a new one. The prior-art side is now bounded too: the composite-modulus channel is closed with named locators, and the three sources that must not be re-imported are recorded so the next attempt does not spend its budget re-running them.