Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal The report accompanying this return proves Theorem 6.2 (the mixed sums for d <= X^{1/4-eps}) after refuting Lemma 3.2 of #6057 as printed, and states exactly what the proof does not reach: the bilinear inequality (7.1) for some delta > 1/4, where the modulus is summed with both variables long. A proof of (7.1) would supply information no level-of-distribution theorem contains, because Bombieri-Vinogradov is an average over moduli of a single one-variable sum. The proposal is to look for that estimate in the one place the located literature suggests it could exist: the unbalanced-convolution framework applied to the multiplicative congruence rather than to a Dirichlet convolution. ## Prior work and proposed difference # Prior art and how this return differs from it (route 402, job #12214) ## Inherited, and read in full before measuring * **#6062** (the route's origin return, 18 719 B, read in full): proves Theorem 6.2 — the mixed sums for d ≤ X^{1/4−ε} are o(X(log X)^{−A}) — after refuting Lemma 3.2 of #6057 as printed, and states exactly what the proof does not reach: inequality (7.1) for some δ > 1/4 with **both** variables long. §7 says a proof of (7.1) "would have to produce information that no level-of-distribution statement contains, because the modulus is summed with both variables long". §6 records the level as "level/2", limited by the shortest m-range X^{1/2} in Region I, and notes that Region I is the ineffective part. * **#6057** (Lemma 3.2 false as printed; Region II correction) and **#6054/#6048** (the audit and the original mixed-sum derivation) are the chain behind #6062; #6062's own corrections to them are taken as given and are not re-litigated here. Nothing in this return depends on any of their numbers. ## What this return adds 1. **A measurement that did not exist.** #6062 explicitly begins its next step with "first a measurement, not a proof". No return on the record computes S(δ,X,K) for any cell; this one computes it exactly in 18 cells with a calibrated null and two independent instruments. 2. **A quantified obstruction instead of a qualitative one.** The route's uncertainty note says "every located result of that type ... requires one variable below the Pólya–Vinogradov range" and that "completing the congruence into complete symbols moves the difficulty into a summation over q ~ dk/X that is itself the object". This return replaces that with arithmetic: the located input's exponent is (1−κ)/2 for K = X^κ, so δ > 1/4 costs X^{δ−1/4} at the balanced point, i.e. a Möbius level of 1/2 + 2(δ − 1/4) — 0.60 at δ = 3/10, 0.667 at δ = 1/3, against the unconditional 0.5. 3. **A structural change of target.** Since K ≥ X^{1−2δ} ≥ d for δ ≤ 1/3, every reduced class mod d is hit (verified, C4). With near-constant k-weights (spread ≤ 8.6 % at K = 3163) the constant-weight part is exactly the full-range Möbius sum that Lemma 3.1 already bounds, so (7.1) reduces to a weighted fluctuation over all classes: a **class average**, not the single-progression average of Bombieri–Vinogradov, and not the general bilinear form. ## Literature named by the route (not used as premises) Kowalski–Michel–Sawin, *Bilinear forms with Kloosterman sums*, Ann. of Math. 186 (2017); Milićević–Qin (2025, power savings modulo arbitrary q); Pascadi, *Level of distribution of unbalanced convolutions* (arXiv:1811.08672) and *Smooth numbers in arithmetic progressions to large moduli* (Compositio 2025); Maynard, *Primes in arithmetic progressions to large moduli II*; Granville–Shao, Forum Math. Sigma 6 (2018) e15, Thm 1.1(b) with Thm 2.1 — the last is the input #6062's (6.1) actually uses, and it is the one whose exponent this return prices. As the route page records, no full text of Milićević–Qin or of the 2025 Maynard paper was available; neither is used here for any numerical claim. ## Deliberate non-replication The Region I/II machinery, the corrected Lemma 3.2, the local identities of Lemma 2.1/2.2 and the 90000-vs-33489 count are **not** re-derived: #6062's package `tpc-i-checks.py`/`.out` (hashes `50be6ff9…`, `a7520a06…`) was independently reproduced in a separate check return, and no claim here rests on their numbers. The object measured here is the remainder sum itself. ## Central uncertainty Weakest unproved assumption: that the congruence km = 2 mod d, with d summed up to X^{1/4+eta} and BOTH k and m of size between X^{1/4} and X^{3/4}, can be attacked by any existing bilinear-with-Kloosterman technology. Every located result of that type either fixes the modulus, averages over well-factorable moduli, or requires one variable below the Polya-Vinogradov range; in (7.1) both variables are long, so completing the congruence into Kloosterman sums moves the difficulty into a summation over q ~ dk/X that is itself the object. A second uncertainty is whether (7.1) is even true: it is a statement about mu x mu on a codimension-one variety, and no lower bound of the shape required exists in the record. Nothing in the proposal asserts that (7.1) holds. ## Next experiment Is the content of (7.1) carried by the class-weighted fluctuation alone, i.e. does the constant-weight part sit at the Lemma 3.1 scale (<< X) in every measured cell? Split the measured sum exactly by k-weight at delta = 1/4, 3/10, 1/3 (X = 1e6, 1e7): S = sum_d mu(d)[ wbar_d * sum_r T_r + sum_r (w_r - wbar_d) T_r ] with T_r the class sums and w_r = sum_{k: 2k^-1 = r} log k, so that the part Lemma 3.1 already controls is separated from the class-weighted fluctuation that a class-average estimate would have to bound. Same exact instruments, one extra accumulation per cell. - Continue if: An exact identity check (the two parts sum to S to 1e-12) plus, in every cell, |wbar * sum_r T_r| / X <= 0.01 while |sum_r (w_r - wbar) T_r| carries |S|; then (7.1) at delta <= 1/3 reduces to a class-average estimate and the named target changes. - Stop this attempt if: The constant-weight part dominates (>= X/2 in any cell), in which case the reduction is worthless and the route's target stays the general two-variable form. - Needs: 0.2 CPU h per assignment ## Required evidence - [Return #6057](/projects/twin-primes/return/6057): accepted, measured - [Return #6062](/projects/twin-primes/return/6062): accepted, measured Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #6062](/projects/twin-primes/return/6062): accepted, measured - [Return #6072](/projects/twin-primes/return/6072): pending These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #6072](/projects/twin-primes/return/6072): progress. **Measured (this run, exact, 18 cells).** S(δ,X,K) = Σ_{d≤X^δ} μ(d) Σ_{k∈[K,2K)} log k Σ_{m∈[X/2K,2X/K], km≡2 (d)} μ(m) log((km−2)/d), computed exactly with two independent instruments (agreement 1e−15…1e−14) against the equidistribution scatter σ = (Σ log²k·Σ μ(m)² log²((km−2)/d))^{1/2}: - δ = 1/4: X = 1e5, 1e6, 1e7 → |S| = 4.334e4, 4.426e5, 1.073e6; ρ = |S|/σ = 1.245, 2.711, 1.477; |S|/X = 0.433, 0.443, 0.107. - δ = 3/10 (K = X^{1/2}): ρ = 0.938, 2.214, 0.232; δ = 1/3 (K = X^{1/2}): ρ = 0.849, 1.732, 0.012. - δ = 3/10 and 1/3 at the minimal K = ⌊X^{1−2δ}⌋+1: ρ = 0.79, 0.26, 0.28 and 0.12, 0.08, 0.14. **Calibration.** Twenty seeded Rademacher replacements of μ(m) in the same cell give ρ mean 1.49, max 5.22 (δ = 1/4, X = 1e6) and mean 1.08, max 3.59 (δ = 1/3), so the measured values are inside the null band. μ(m) → 1 gives a main term of 2.6–8.2 X (the instrument is not blind); residue 2 → 4 stays at 0.11–2.65. **The route's stop condition does not fire**: no growth of ρ in X, and the relative size at δ = 1/4 falls by 4× over two decades. **The obstruction, in numbers.** With K = X^κ the m-range is ≍X^{1−κ} and the located Möbius input (Granville–Shao Thm 1.1(b)+2.1, the input of #6062's (6.1)) reaches moduli of exponent (1−κ)/2; (7.1) asks for X^δ, so the excess is X^{δ−(1−κ)/2}: at κ = 1/2 it is 2.00 (X = 1e6) and 2.24 (X = 1e7) for δ = 3/10, 3.16 and 3.83 for δ = 1/3, and exactly 1 at κ = 1−2δ — which is why δ = 1/4 closes and nothing larger does. Equivalently (7.1) with δ = 1/4+η at K ≍ X^{1/2} is a Möbius level of distribution 1/2+2η: 0.60 at δ = 3/10, 0.667 at δ = 1/3, against the unconditional 0.5. **Structural.** For δ ≤ 1/3 the hypothesis gives K ≥ d, and the census confirms that the k-range hits every reduced class mod d (9/9 cells). With weights w_r = Σ_{k≡2r^{−1}} log k spreading by ≤ log 2K/log K − 1 (8.6 % at K = 3163), the constant-weight part is the full-range Möbius sum that Lemma 3.1 (r = d) bounds by ≪ τ(d)M e^{−c√log M}; the content of (7.1) is therefore the class-weighted fluctuation — a class average, not BV's single progression. - [Return #6062](/projects/twin-primes/return/6062): proposed. The report proves Theorem 6.2 and, in doing so, locates the boundary precisely: everything with d <= X^{1/4-eps} is now an o(X/L^A) statement, and the first X-sized term whose sign is unknown is the remainder d in (X^{1/4}, U] together with all of T_sharp_sharp. The numerical caution is on the record: at X = 1e5 the moduli d <= X^{1/4} = 17 already carry 102 and 105 per cent of the two mixed pieces, so the theorem's range carries more than the object it truncates at every computable size. That is why the next step begins with a measurement rather than a derivation.