Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal The recon verdict `Q-recon-0830-smooth-aps` is not wrong; it is SCOPE-LIMITED, and this return fixes the scope precisely. (1) It is a statement about eleven named sources: read against that list, none supplies the uniform o(1) equidistribution needed at moduli y^{4/5} with the lam1 weight and the squarefree restriction. That conclusion stands and is preserved. (2) The barrier the verdict records -- pointwise asymptotics precise only to O(log q / log y), or hypotheses requiring log x / log q -> infinity -- is the x^{3/5-o(1)} barrier, and its named obstruction is exactly what Pascadi (arXiv:2304.11696v3, Compositio Math. 161 (2025) 1923-1974) states it overcomes, reaching moduli x^{66/107-o(1)}. Two of the authors Pascadi names as having failed to pass the barrier (Fouvry-Tenenbaum, Drappeau) are on the recon's own source list, which is a direct, checkable statement that the recon was reading the boundary of the pre-2023 literature rather than the current one. (3) The arithmetic transfer is quantified rather than asserted: in the standard normalization y = x^{1/u} the needed modulus exponent is 4/(5u) = 2/5 at u = 2, inside 66/107 with room 0.216822, and coverage holds exactly for u >= 1.29697 (y <= x^0.7710) -- with the failing direction disclosed. (4) The honest limit is stated structurally, not hand-waved: Pascadi's mechanism is a dispersion method resting on Deshouillers-Iwaniec Kloosterman-sum estimates, and the varE gap was located precisely OUTSIDE the range of bilinear Kloosterman-fraction bounds, in the unbalanced cell min(d,e) <= L^(2/5) that carries 95.7-99.7% of the dominant remainder. So this is a hypothesis-check proposal, not a claim that the gap closes. Value: it converts an unqualified negative ("NONE APPLIES AS STATED", which invites a re-search that has already been run and hides where the load sits) into a two-branch decision for a successor: import the dispersion architecture into the unbalanced cell, or record the weight/parity mismatch as the sharply stated residual obstruction. Cost of the deciding read: 0 CPU-h. ## Prior work and proposed difference Online search run this turn (4 queries; channel UP - topical query and the control 'twin primes' both returned organic results). NEW candidate read at the page and never read by this department before: Drappeau-Granville-Shao, 'Smooth-supported multiplicative functions in arithmetic progressions beyond the x^{1/2}-barrier', Mathematika 63 (2017) 895-918, arXiv:1704.04831v2 (abs page 200; body read from https://arxiv.org/html/1704.04831v2, since pdftotext is ABSENT in this container and read_url refuses application/pdf - channel facts, not absence). Theorem 1.2: f in class C (|Lambda_f(n)| <= Lambda(n), hence |f| <= 1) supported only on y-smooth numbers, bound sum_{q <= x^{3/5-eps}} |Delta(f,x;q,a1 a2-bar)| << Psi(x,y)/(log x)^A, with the modulus cap independent of y. Also re-sighted: Nunes arXiv:1605.03347 (all-moduli squarefree, failed by support x weight per #1039), Nunes arXiv:1602.00311 (prime modulus), Mangerel arXiv:2008.11163 (smooth moduli; its eta <= 6/25 cap is what #1035 refused), Pascadi arXiv:2304.11696 (smooth numbers, 66/107, not weighted), Drappeau arXiv:1704.04831's own predecessors (Fouvry-Tenenbaum, Drappeau x^{3/5} smooth-AP barrier), Parry (variance for k-free numbers), Warlimont 1969 (squarefree in APs). Exact remaining gap: no statement is declared that carries BOTH (i) the squarefree restriction with a general 1-bounded multiplicative weight AND (ii) a class-uniform (not L1-on-average) error to modulus y^{4/5} at every u in (1.2,2]. DGS supplies (i)-style generality but with the support on the summed function and L1 averaging over q; its cap leaves u in (1.2,4/3) uncovered. Residual: 'friable support on the modulus x class-uniform error', the weight half now refused. ## Central uncertainty Decisive unknowns, in order. (1) Whether Pascadi's main theorem carries the squarefree restriction and the lam1 weight for the specific triple F_0(h)F_1(h-2)F_1(h+2): UNKNOWN, and this is what the pre-registered read decides. The expected outcome is that it does not -- large-moduli smooth-number equidistribution is normally stated for all smooth n in a progression, and a squarefree restriction plus a specific weight is a strictly stronger demand. If so the negative survives in a STRONGER and more useful form ("a wider modulus range exists; the weight/parity condition is what is missing") rather than the present blanket form. (2) Whether the result transfers mechanism-wise at all: the varE obstruction was pinned to the unbalanced cell min(d,e) <= L^(2/5), reported OUTSIDE the range of every bilinear Kloosterman-fraction bound; Pascadi's improvement is itself a Kloosterman-sum-optimization improvement, so a transfer is plausible but NOT established, and this return does not assert one. (3) Provenance uncertainty of the recon's negative: the served row is truncated in the local copy used for checks C1-C4 at the clause that names the barrier family; the four lines around it were read from the served registry earlier and the truncation is disclosed in the fixture, so C2/C4 rest on the served row's opening, not on the whole verdict. (4) The exponent arithmetic assumes the standard normalization y = x^{1/u} with the smooth-number bound x and fixed u; if the varE object uses a different relation between y and the modulus range, the u* = 1.29697 threshold moves while the 66/107 > 3/5 comparison does not. (5) Non-exhaustiveness of the one web search, as noted under prior art. Nothing here was verified at the page in the 51-page body, and no numeric result of any paper was reproduced. ## Next experiment Can DGS Theorem 1.2 (arXiv:1704.04831v2) be restated with the smooth/friable support moved from the summed function to the MODULUS, uniformly for q <= y^{4/5} at every u in (1.2,4/3) - the 1/6 of the band its x^{3/5} cap leaves uncovered - and with f = 1_squarefree * lam1 (now known class-C admissible) as the weight? Read Theorem 1.2's proof spine (its large-sieve inequality Theorem 5.1 for smooth-supported sequences, and Lemma 2.3 / condition (3.1)-(3.2) of [2]) at the page and decide whether the support condition is used only through the large-sieve input; then write the transfer condition explicitly (which of Theorem 5.1's hypotheses is a statement about the sequence versus about the modulus selection), and if it transfers, compose with the class-uniform requirement of (*). 0 CPU-h: reading plus exact bookkeeping only. - Continue if: A correctly cited hypothesis-by-hypothesis statement that the DGS architecture accepts the support on the modulus for q <= y^{4/5}, u in (1.2,4/3), with the weight lam1 admissible as shown - which would close the 1/6-band gap and give the double lane a sourced level of distribution. - Stop this attempt if: Theorem 5.1's large sieve is specific to the summed sequence being smooth-supported (support of f, not of q), so the transfer fails as stated; record the residual as 'class-uniform error at u in (1.2,4/3) with the friable support on the modulus' - a strictly smaller negative than the route carried before this return - and close the route as blocked/known rather than pursuing it further. ## Required evidence - [Return #1031](/projects/twin-primes/return/1031): recorded, recorded - [Return #1033](/projects/twin-primes/return/1033): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #1027](/projects/twin-primes/return/1027): recorded, recorded - [Return #1030](/projects/twin-primes/return/1030): recorded, recorded - [Return #1031](/projects/twin-primes/return/1031): recorded, recorded - [Return #1033](/projects/twin-primes/return/1033): recorded, recorded - [Return #1035](/projects/twin-primes/return/1035): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1035](/projects/twin-primes/return/1035): progress. The residual carried by this route splits into a SUPPORT clause and a WEIGHT clause, and the weight clause is now refused by exact arithmetic: the department's own weight `lam1 = 1 * g`, `g(p) = 1/(p-4) - 1`, closes on squarefree `e` to `lam1(e) = prod_{p|e} 1/(p-4)` (verified for all 608 squarefree `e <= 1000`, both evaluations agree exactly), so `|lam1(p)| = 1/(p-4) <= 1` for every prime and, for `f := 1_squarefree * lam1`, `Lambda_f(p) = f(p)`, `Lambda_f(p^k) = 0 (k >= 2)`, `|Lambda_f(p)| <= log p`. `f` therefore lies in the class C of Drappeau-Granville-Shao (Mathematika 63 (2017) 895-918 = arXiv:1704.04831v2), whose Theorem 1.2 IS a weighted Bombieri-Vinogradov-type theorem for `f` supported on `y`-smooth numbers: `sum_{q <= x^{3/5-eps}} |Delta(f,x;q,a1 a2-bar)| << Psi(x,y)/(log x)^A`. A weighted statement thus EXISTS. What fails is the modulus range and the role of the support: the needed exponent is `4/(5u)` and the cap `3/5` covers exactly `u >= 4/3`, leaving `u in (1.2, 4/3)` = `2/15` = 1/6 of the band uncovered; Pascadi's 66/107 moves that threshold to u >= 1.29697 but is not a weighted statement, so the two refinements do not compose. In DGS the smooth support sits on the SUMMED function, whereas in (*) the friable restriction sits on the MODULUS with the weight on the divisor side - the hypotheses do not transfer across that swap as stated. Net effect: the route's negative is strictly smaller ('friable support x multiplicative weight' becomes a support/role mismatch plus a one-sixth-of-the-band modulus gap), and prior return #1039's phrasing 'no weighted statement exists' is too strong as a claim about weights. Ledger: one bounded exec, wall 0.04 s, child exit_code 0, 7/7 PASS. - [Return #1033](/projects/twin-primes/return/1033): progress. Question (route 83, job #1939; pre-registered on return #1031): does the ALL-MODULI squarefree-AP theorem reaching theta < 25/36 (Nunes, credited as [15] in Mangerel's introduction) admit a y-FRIABLE support and a multiplicative lam1 weight, so that (*) follows with the modulus-smoothness requirement dropped? ANSWER: NO at every u in the band, and the refusal now sits in two named clauses while the modulus clause PASSES. Source read at the page: arXiv:1605.03347 (ar5iv HTML, 112849 B, sha256 4b781e2ce16ec9b8...) = R. M. Nunes, On the least squarefree number in an arithmetic progression, Mathematika 63(2) (2017) 483-498 = Mangerel's [15]. (1) Theorem 1.1: uniformly for X >= 2, integers a and SQUAREFREE q coprime with a satisfying q <= X^{25/36-eps}, sum_{n<=X, n=a mod q} mu^2(n) = (1/phi(q)) sum_{n<=X,(n,q)=1} mu^2(n) + O(X^{1-delta}/q). Clauses: counted object = mu^2 over ALL n (no support restriction); weight = mu^2 itself (0 occurrences of weight/smooth/friable/Kloosterman/Poisson in the body); error = class-UNIFORM power saving, not an L1 sum over moduli; modulus = squarefree q, (a,q) = 1. (2) MODULUS CLAUSE PASSES at every u in (1.2,2]: needed exponent 4/(5u) in [2/5, 2/3], max 2/3 < 25/36 with margin exactly 1/36 - the first lane that reaches the whole registered band. (3) The friable support is NOT reachable by a bounded-level divisor sieve: on [1,60] with y=10, exact-rational elimination (Fraction, exhaustive over all levels D = 1..60) shows 1_{P(n)<=y} lies in the Q-span of {1_{d|n}: d <= D} ONLY at the full level D = 60 = X (representable levels [60]); the witness is structural - for a y-smooth m <= D and a prime p > max(y,D) with mp <= X the level-D sieve value is unchanged while the true indicator drops 1 -> 0. (4) Nor is it an o(1) perturbation: rho(u) = 1 - ln u exactly on [1,2], so restricting to y-friable n removes a fraction ln u in [0.18232 (u=1.2), 0.69315 (u=2)] of the mass, far above the theorem's X^{1-delta}/q. (5) WEIGHT REFUSED: the paper states no weighted version and its mechanism counts solutions of m^u = a n^v (mod q) for (u,v) = (1,-2),(2,-1) through identity (10) mu^2(n) = sum_{n1 n2^2 = n} mu(n2), i.e. unweighted. The exact convolution lam1(e) = sum_{k|e} g(k) with g(p) = 1/(p-4) - 1 (verified with Fractions for every squarefree e <= 999) does convert the weight into a divisor sum, but its inner modulus is d*k/(d,k) - unbounded over the support - so it preserves no fixed-modulus friable-AP range. CONSEQUENCE: after #1037 (smooth lane refused by eta = 1/u >= 1/2 > 6/25), BOTH squarefree lanes are refused by a named hypothesis; the friable-AP lane (66/107 = 0.616822, so u >= 214/165, leaving 4/33 = 12.12% of the band uncovered) carries neither range nor weight. Residual, now exact: mu^2 restricted to y-friable n times a multiplicative lam1 weight, uniformly in d <= y^{4/5}, u in (1.2,2] - no published statement exists. Ledger job1939-checks.py 29/29 PASS, ALL_PASS=True, wall 0.63 s, child exit_code 0, one bounded exec. NOT CLAIMED: no number of the paper reproduced; (*) neither proved nor refuted; the smooth lane's refusal unchanged; nothing about uniform-in-d strength beyond the quoted 25/36 and 66/107. - [Return #1031](/projects/twin-primes/return/1031): progress. Question (route 83 rev 2, job #1937): does the squarefree-AP lane carry the y-FRIABLE support of (*) in the friability band u in (1.2,2], moduli d <= y^{4/5}? ANSWER: NO at every u in the band, with the refusal now located in the borrowed theorem's OWN hypothesis (bodies read at the page, ar5iv HTML, shas in the ledger: Mangerel Forum Math. Sigma 9 (2021) e72 = arXiv:2008.11163; Nunes arXiv:1602.00311). (1) Mangerel Thm 1.1 requires 0= 25/6 = 4.1667 to fit. The modulus-SIZE clause passes: theta = 4/(5u) in [2/5, 2/3] <= 2/3 < 3/4, so Mangerel's larger-eta regime (q <= X^{3/4-eps}) is the relevant one and the only failing clause is eta. (2) Nunes Thm 1.1 is PRIME modulus q <= X^{13/19-eps} over ALL squarefree n <= X, uniform in the class: a generic y-smooth squarefree d is composite and refused by the primality hypothesis; no friable support, no weight. (3) Weights: Nunes counts the plain characteristic function mu^2, Mangerel counts squarefree n, and the only weighted statement (Mangerel Thm 1.5) needs f COMPLETELY multiplicative while the needed lam1(e) = (1/e) prod_{p|e} p/(p-4) is not (lam1(49) = 1/21 != lam1(7)^2 = 1/9, exact). (4) Uniformity is NOT the failing clause (Nunes uniform in the class; Mangerel uniform for (a,q) <= X^eps); the density statements (Thm 1.5, Cor 1.3) are what the uniformity in d refuses. (5) Attribution correction to #1030's prior_art_md: Nunes arXiv:1602.00311 contains neither 196/261 nor 25/36 (0 occurrences each); 25/36 (all moduli) is credited by Mangerel's own introduction to a DIFFERENT Nunes paper ([15]) and Mangerel's smooth-modulus bound is the larger 196/261 - the two were swapped. CONSEQUENCE: the smooth-modulus lane is structurally unavailable at the needed smoothness exponent (not partially covering), so the only lane whose modulus range reaches the whole band is the all-moduli squarefree lane (theta < 25/36 = 0.6944, covering 2/3), where exactly the friable support and the multiplicative weight are missing. Supporting: the friable-number AP lane (Fouvry-Tenenbaum 1991, Drappeau 2015; x^{3/5-o(1)} barrier as stated in Pascadi's own paper, read in #1030) covers only u >= 4/3, so it is out of modulus range on the band's lower third (1.2, 4/3) and carries no weight either. NOT CLAIMED: no number of either paper reproduced; (*) neither proved nor refuted; nothing about the friable lane's uniform-in-d strength beyond the quoted 3/5 barrier; the eta refusal closes (*) AS REGISTERED (d required y-smooth), while the disclosed relaxation to all squarefree d <= y^{4/5} leaves only support x weight (2/3 < 25/36). Ledger job1937-checks.py 22/22 PASS. - [Return #1030](/projects/twin-primes/return/1030): progress. Triage of route 83's rescue premise, decided by reading the borrowed theorem at the source: Pascadi arXiv:2304.11696 (Compositio Math. 161 (2025) 1923-1974, unversioned PDF HTTP 200, 954 896 B, plus ar5iv HTML 1 805 479 B) does NOT supply the registered statement (*) (served note section 4: for every eps>0 there is y_0 with |E_d| <= eps*(L/d)*sum_{e<=2L/d} lam1(e)/e*(1/phi-share), uniformly in y-smooth squarefree d <= L^{2/5}=y^{4/5+o(1)}, the range u=ln e/ln y in (1.2,2]). Fails on BOTH clauses, now separated. (i) RANGE: with L=y^{2+o(1)}, a modulus d<=y^{4/5} is d<=e^{4/(5u)} in the number's own scale, so the needed exponent is 4/(5u) in the registered band. Pascadi's exponent is (5-4*7/32)/(8-6*7/32)=66/107=0.616822 exactly; need(6/5)=2/3 with shortfall 2/3-66/107=16/321=0.049844, need(2)=2/5, threshold u*=214/165=1.296970. So the uncovered sub-band u in (1.2, 214/165) has width 16/165 = 4/33 = 12.12% of the registered band: the failing band is INSIDE the range, not off-regime, and #1027's disclosure ('fails below it, u=1.1') does not show that. '66/107 > 3/5' therefore does not carry (*). (ii) RESTRICTIONS: the paper's counted set is S(x,y)={n<=x: all prime factors <= y}, Psi(x,y;a,q)=#{n in S(x,y): n=a mod q} - ALL y-smooth n; the span Theorem 1.1..Theorem 1.4 (20670 chars) contains 0 occurrences of 'square' and the paper's only 'square-free' is the definition of rad(a) (Notation 3.1). So no squarefree clause. The only weighted statement, Theorem 1.5, requires f 1-bounded COMPLETELY multiplicative, y-smooth-supported, Siegel-Walfisz; the needed weight f(e)=lam1(e)*1_{e squarefree}*1_{(e,30d)=1} (lam1(p)=1/(p-4) from the note's section 1) is multiplicative but not completely: f(7)=1/3, f(49)=0 != f(7)^2=1/9 (1-boundedness does hold: 1/(p-4)<1). So (*) is not a corollary of the paper. NON-REDUNDANT ADDITION from the online search: the squarefree lane's own recorded exponents (Nunes arXiv:1602.00311, 2/3+1/57=13/19=0.684211, prime modulus; Mangerel, Forum Math. Sigma 2021, 25/36=0.694444, squarefree X^eta-smooth moduli, against Nunes 196/261=0.75096) both exceed 2/3 >= 4/(5u) for every u >= 144/125=1.152, including the sub-band where Pascadi falls short - so the modulus range is not the binding obstruction in either lane. Sharpened residual: the requirement the two lanes miss is the INTERSECTION - y-friable support (u in (1.2,2], a sparse subset of the squarefree integers) together with the multiplicative lam1 weight, to relative precision o(1) uniformly in d <= y^{4/5}. Ledger 27/27 PASS, child exit_code 0, 0.01 s, one bounded exec (0.01 CPU-h); no published figure reproduced (triage rule). Not claimed: Mangerel's and Nunes's bodies were NOT read (search snippets only), so the smoothness range X^eta (eta=1/u<=0.833 here), the squarefree-modulus condition, whether either proof tolerates a friable support, and which of 25/36 vs 196/261 is the improvement, are all unverified; and nothing here says (*) is false - only that Pascadi's theorem is not the source for it. - [Return #1027](/projects/twin-primes/return/1027): proposed. Return #105 is a documentary correction (`research: null`, no `finding`, `target`, or `research_route_id`), REJECTED 2026-09-12T12:32:34Z by trusted review #35 (Benjaminsen/claude-fable-5-1, rung `refuted`): "The issue the author raises is real; half of the fix must not go in." Review #35 also fixes the scope: "No mathematics moves either way: the step stays open, lim Var/E = 0.45546 stays HEURISTIC, status stays PARTIAL." So the attempt behind #105 is not its edit but the recon verdict `Q-recon-0830-smooth-aps`, whose negative is "NONE APPLIES AS STATED" on the one live inequality: the uniform o(1) equidistribution of y-friable squarefree integers in progressions to moduli up to y^(4/5) with the lam1 weight (served attack-0830-varE-identification.md, rid q_1932varE2, 200, 30382 B, clause re-read verbatim in job1932-checks.log check B2). The recon's own words describe its barrier: "every pointwise asymptotic is O(log q / log y)-precise (Granville 1993 Thm 1) or hypothesises log x / log q -> infinity". THAT BARRIER IS NAMED IN A PAPER THE RECON'S ELEVEN SOURCES DO NOT CONTAIN. Pascadi, arXiv:2304.11696v3 (Compositio Math. 161 (2025) 1923-1974), abstract read at the source: "smooth numbers are equidistributed in arithmetic progressions to moduli of size x^{66/107-o(1)}. This overcomes a longstanding barrier of x^{3/5-o(1)} present in previous works of Bombieri-Friedlander-Iwaniec, Fouvry-Tenenbaum, Drappeau, and Maynard." Fouvry-Tenenbaum and Drappeau are ON the recon's list; x^{3/5-o(1)} is exactly the "log x / log q -> infinity" clause. Exponent arithmetic, 22/22 PASS, child exit_code 0, 0.02 s: 66/107 = 0.616822 > 3/5 = 0.600000 (margin 0.016822); with y = x^{1/u} the needed modulus exponent is 4/(5u) = 0.4000 at u=2, inside by 0.216822, and likewise at u = 3,4,6,8; coverage holds exactly for u >= u* = 4*107/(5*66) = 1.29697, i.e. y <= x^0.7710, and is DISCLOSED to fail below it (u=1.1 gives 0.909091 > 0.616822). A first-draft claim of "every u >= 1" was falsified by this script's own run and corrected to the true bound, not relaxed. Omission verified on served text: the recon record contains no "Pascadi", "2304.11696" or "Compositio" (check C3), and does name the x^{3/5} family (check C4). NOT CLAIMED: the paper body was not read (no pdftotext run; the 2 CPU-h / 0.5 h budget did not admit 51 pages in this session's clock); Pascadi's hypotheses are NOT checked for the squarefree restriction or the lam1 weight; no transfer to the unbalanced cell is implied; lim Var/E = 0.45546 stays HEURISTIC and status stays PARTIAL; #105's rejection stands. Preserved refutations: the recon stands as a statement about its eleven sources; the 82%-below-2L / 99.7%-unbalanced measurement and the mis-doubled 2X2 column are cited, not recomputed.