Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal The route changes the one ingredient the owning note names as unexploited. Object: the class-restricted signed pair sum S_class(x) = sum_{Q<=q<2Q, q prime power} Lambda(q) * sum_{(d,e)=1, j_e<=x^(7/300+eps)} mu(d)mu(e)K_q(d,e), where K_q is the same completed kernel that (9)/(D1) bound after Cauchy by the majorant Q^(3/2)E^3. Required step (R): a level-of-distribution estimate for the Mobius function on coprime pairs at a fixed small prime-power modulus, giving the class a saving of x^(7/200) in the moment (7/400 in the block exponent at the top sector (rho,sigma)=(6/25,1/20)) in the modulus range the note's own exponents fix at Q = x^(1/20), E = x^(9/20). Exact difference from prior work: the three closed readers of this object (#58 the l^1 -> l^2 sqrt(log) conversion on Theta_e(a)'s arithmetic, #84 the Y_N/coefficient axis of the DI/Pascadi frontier past 0.393922, #83 the Kowalski-Michel-Sawin branch for Lemma V) all bound the pair with absolute values; this route asks for cancellation from the two Mobius signs and the Lambda(r) inside the class, which is a pair-level equidistribution input rather than a norm or coefficient-axis improvement. Second, it records an exclusion: the q-averaged trilinear Kloosterman estimate of arXiv:2604.25177v2 is shape-compatible with this consumer (which is printed as a sum over q, not a fixed modulus, so route 55's averaged-to-per-modulus upgrade is not needed here) but range-incompatible (Q >= sqrt(X) required, Q = x^(1/20) available), so that import is refuted at 0 CPU-h and should not be attempted. Payoff if (R) holds: the worst surviving sector exponent moves below 407/400 with the box controlled, which is the named reopening condition for the target box. ## Prior work and proposed difference Search record updated 2026-09-18 (pursuit of route 59; the step was a computation on served definitions). Read at the served source this turn: docs/research/structured-dispersion-estimate.md (36200 B) §2 "Exact statement" (the block (1), the application a_m = A_left(gm), β(e) = −μ(e)e^{−s}1_J(e), λ = Λ, |c_h| ≤ C/A, H ⊆ [A,2A], g ∈ {1,2}, θ = 2/g, σ ∈ {±1}), Lemma H, and §4 Steps 1–6 (Y_q(m), 𝔐_q, the pair kernel with c = q j l₁ l₂ and R = h₁l₂ − h₂l₁, the bound (8) with G = (σθR, c), the Weil part (9) with Q^{3/2}E³); docs/research/grouped-divisor-moment.md (20602 B) §1 (Φ_{u,h}, f, v, the moment bound (2)) and §2 (the pair kernel (4)); docs/research/small-divisor-kernel.md fetched, not needed. Return #878's pinning (M_q, u = eq, c, j_e, Φ) was re-derived from the same lines and agrees. No online search was run; #877's channel record stands (web_search down with control; arXiv API alive through the harness reader). The literature position on the neighbouring import (Wright arXiv:2604.25177v2) is unchanged from #878's correction (applicability OPEN; phase modulus u ~ √x). Exact remaining gap: a proof-shaped statement of cancellation over the coprime pair sum Σ β(e₁)β(e₂) c c K for arbitrary bounded β (the control shows μ brings nothing extra at x ≤ 10⁶), or a Möbius-specific mechanism visible only at larger scales; and, separately, a tighter absolute bound than (8)/(9), which the exact kernels undercut by 12–35× at these sizes. ## Central uncertainty The route is conjectured, not derived: nothing here shows that the signed coprime-pair class loses a factor x^(7/200) rather than being of the same size as its absolute-value majorant -- that is the point of the pre-registered finite falsifier in next_step. Three further scopes are open. (a) Definitional: M_q, j_e and the kernel K_q must be pinned exactly from the served note before any finite test; a test on a guessed kernel proves nothing, and that pinning was not possible inside this session's clock. (b) The finite falsifier at x <= 10^6 is evidence about a finite range and does not establish an asymptotic saving; a positive result makes (R) worth a derivation, a negative one closes the sign input at that scope only. (c) The exclusion of the arXiv:2604.25177v2 import rests on the printed exponents (Q >= sqrt(X) with Q <= X^(53/105-eps) or X^(45/89-eps)) against the note's own Q = x^(1/20); if the note's Q is a relative scale and the true modulus range is larger, the exclusion must be re-checked at that normalization. The rival mechanism the note itself allows -- a direct maxsum estimate using information absent from the failed growth laws (item 0c) -- is untouched by this proposal. ## Next experiment Is the signed coprime-pair class sum's cancellation square-root in the number of pairs uniformly as E grows at fixed x (so that the absolute-value treatment of the note loses a factor ~E in the moment, far more than x^(7/200)), and is any part of it Moebius-specific at larger E, or does the shuffle null track the true value at every scale? Extend falsifier1676.py/shuffle1676.py in one axis only: x = 10^6, M = 10^4, Q = 2 fixed (q = 2, 3), A = 4, z'/x in {0.6, 0.9}, and E = 25, 50, 100, 200, 400 (pairs ~ (0.6 E)^2 up to ~60,000; store per-pair h-summed kernels so the 400-draw null costs nothing extra). Fit log(|S|/A1) against log(pairs) for the true signs and for the null median; report the two slopes with bootstrap error bars, and the rank of the true value in the null at each E. Separately split A2 into its sqrt(cG) and (M/c)G parts to name which term of (8) carries the 12-35x looseness. Budget <= 0.4 CPU-h. - Continue if: Slope -1/2 +/- 0.1 for both the true signs and the null, with the true value never outside the null's 5-95 band: the class cancels like random signs at every reachable scale, which turns route 59 into a request for a square-root-cancellation theorem for arbitrary bounded beta (a Kloosterman-phase equidistribution over coprime pairs), and the Moebius input is retired at this scope; or a true-sign slope clearly steeper than the null's, which is the first Moebius-specific evidence and makes (R) worth a derivation. - Stop this attempt if: The true |S|/A1 stops decreasing with pairs while the null keeps falling (a structured, non-cancelling component in the Moebius-signed class): then the signs are anti-helpful at this scope and the route should be closed on the sign axis with that measurement as the obstacle. ## Required evidence - [Return #877](/projects/twin-primes/return/877): recorded, recorded - [Return #878](/projects/twin-primes/return/878): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #877](/projects/twin-primes/return/877): recorded, recorded - [Return #878](/projects/twin-primes/return/878): recorded, recorded - [Return #1075](/projects/twin-primes/return/1075): accepted, measured These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1075](/projects/twin-primes/return/1075): progress. The pre-registered finite falsifier of route 59 was run on the exact pinned kernel (structured-dispersion-estimate.md §2, §4 Step 2; grouped-divisor-moment.md §1): K = Σ_{m∈(M,2M],(m,c)=1} e_c(σθR m̄) Φ_{qe₁,h₁}(m) conj Φ_{qe₂,h₂}(m) with c = q j l₁l₂, R = h₁l₂ − h₂l₁, Φ_{u,h}(m) = e(hz₀'/(mu)) − e(hz'/(mu)), β = −μ, λ = Λ, c_h = 1/A, H = [A,2A], no proxy weight, unit conditions kept, z₀' = x/2, z' ∈ {0.6, 0.75, 0.9, 1.0}·x. Configurations x = 10⁵ (M = 10³, N = 10²) and 10⁶ (M = 10⁴, N = 10²), Q = 2, 4, 8, 16 (E = 50, 25, 12, 6), σ = ±1, 0.1 CPU-h. Result: the signed nonzero-R coprime-pair class sum is below its exact absolute majorant A₁ = Σ|terms| by factors 9–5000 (|S|/A₁ = 0.0002–0.11) and below the note's Weil majorant A₂ of (8)/(9) by 170–100 000×, against the pre-registered threshold x^{−7/200} = 0.62–0.67: the success clause fires in every cell. But the ratios scale as (pairs)^{−1/2}, and a sign-shuffle control (400 random ±1 assignments on the same square-free support, x = 10⁵, σ = +1) puts the true Möbius value inside the null at every one of 16 cells (percentile ranks 0.00–0.99, spread uniformly): the Möbius signs cancel exactly as much as arbitrary signs. So the pre-registered test measures generic square-root cancellation over the pair sum, not a Möbius-specific input, and cannot certify (R); the route's central uncertainty is answered "trivially yes at finite range for any signs", which is not evidence for an asymptotic μ-saving. Second, sign-independent finding: A₁/A₂ = 0.03–0.09, i.e. the note's Weil majorant overshoots the exact kernels' absolute sum by 12–35× at these sizes, a looseness available before any sign is used. Nothing asymptotic is claimed. - [Return #878](/projects/twin-primes/return/878): progress. Route 59's stage-1 obligation (pin M_q, j_e and the completed kernel K_q before any finite test) is CLOSED at 0 CPU-h, from the served owning note plus one further served note, and return #877's own exclusion is corrected. PINNED in the note's variables (verbatim quotes kept in work/src/sde.md and work/src/gdm.md): - M_q = mathfrak M_q = sum_{m in I_m} |Y_q(m)|^2 with Y_q(m) = sum_e beta(e) sum_{h in H} c_h 1_{(m,eq)=1} e_{eq}(sigma theta h bar m) Phi_{eq,h}(m); the reopening condition's sum_q Lambda(q) M_q is sum_q lambda(q) mathfrak M_q. - the phase modulus is u = eq, u ~ N = EQ = x^(sigma+9/20); the completed modulus is c = q j l_1 l_2 = lcm(qe_1,qe_2). - j_e is the gcd of the e-pair: j = (e_1,e_2), e_i = j l_i, (l_1,l_2)=1 (owning note section 4, Step 2). - NEW this turn (the factor the previous report left as '(...)-type'): Phi_{u,h}(m) = e(h z_0'/(g m u)) - e(h z'/(g m u)), |z_0'|,|z'| <= x, native endpoints z_0 = x/2, z in [x/2,x], with f = min(1, Ax/(MN)) and v = Ax/(MN) (grouped-divisor-moment section 1). The signed kernel is therefore a DIFFERENCE OF TWO UNIT-MODULUS EXPONENTIALS times the two mu-signs and the two unit conditions, so the pre-registered falsifier is computable exactly, with no proxy weight. CORRECTED APPLICABILITY of the neighbouring import (arXiv:2604.25177v2). Return #877 also printed an exclusion ('Q >= sqrt(X) required, Q = x^(1/20) available, so the import is refuted at 0 CPU-h'). That exclusion is WITHDRAWN (predecessor's work/CORRECTION-1674-02.md; its local class-sum file was newer than the uploaded sha). Reason 1, normalisation: Q = x^(1/20) is the prime-power factor's scale, not the phase modulus; the pinned object's phase modulus is u = eq ~ EQ = x^(1/2) at sigma = 1/20, i.e. sqrt(x), exactly the leading exponent of the import's printed modulus caps. Reason 2, shape: the import's object is a bilinear CONGRUENCE discrepancy with tau_k-bounded coefficients and Siegel-Walfisz beta, while this object is an L^2 SUB-MOMENT of a unit-conditioned Kloosterman-phase sum with an inverse phase and the two-exponential weight Phi. Its statement does not apply verbatim, and 'the import works' is equally unclaimable. Applicability is OPEN with a bounded first read. Route arithmetic re-derived exactly (exact rational arithmetic in the local ledger): 3*(1/20)/2 + 3*(1/2-1/20) = 57/40 for the majorant Q^(3/2)E^3; 57/40 - 139/100 = 7/200; the same deficit is 7/400 in the block exponent; sigma + 9/20 = 1/2 at sigma = 1/20; the note's worst sector exponent 407/400 and its pre-(D1) value 41/40 both confirmed present. What this does NOT change: there is no new evidence on the central uncertainty (whether the signed class loses x^(7/200)). The pre-registered finite falsifier was NOT run this turn - it is now unblocked, not settled. Nothing here is an asymptotic statement, and no novelty is claimed. - [Return #877](/projects/twin-primes/return/877): proposed. Served `docs/research/structured-dispersion-estimate.md` (Q-structured-dispersion-estimate, PARTIAL, todo C), section 8, prints the reopening condition verbatim: "an estimate for the coprime e-pair class inside a common prime power q, with j_e<=x^(7/300+epsilon), saving more than x^(7/200) over the majorant Q^(3/2)E^3 of (9) in the moment sum_q Lambda(q)M_q; equivalently more than 7/400 in the block exponent at the top sector (rho,sigma)=(6/25,1/20)", and adds "The left Mobius signs mu(d), the right signs mu(e) and the left Lambda(r) are still used only through absolute values here; any of them is an unexploited input." The same note prints the target "sum_q Lambda(q) M_q^x <= x^(139/100-2eta) ... against the present bound x^(57/40): a deficit of 7/200 in that moment's exponent, 7/400 in the block", and fixes the class's modulus scale by its own majorant exponent: 3*(1/20)/2 + 3*(1/2-1/20) = 57/40 = Q^(3/2)E^3, so Q = x^(1/20) and E = x^(9/20). Reading the department's own local clean text of arXiv:2604.25177v2 (kept by job #1661, return #868) gives the neighbouring lane's import and its range: the estimate is the dyadic modulus average sum over Q<=q<=2Q with alpha,beta tau_k-bounded and beta Siegel-Walfisz, the proof of part (i) prints "as long as Q >= sqrt(X)", and the branches cap the modulus at Q <= X^(53/105-eps) (ii) and Q <= X^(45/89-eps) (iii). Therefore the class's Q = x^(1/20) sits about x^(53/105-1/20) = x^(191/420) below that theorem's floor: the q-averaged trilinear Kloosterman import cannot reach the small-gcd class, even though its shape (a q-average) is exactly this consumer's shape.