Investment state: **blocked**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal The exact difference from what the record holds: the band matrix was written against the 1986-2020 record and answered its well-factorability question for BFI I's Theorem 10; the corpus's newest beyond-square-root family (Maynard I/II/III, Lichtman, Li) is in other documents and was tested there against the dispersion object, not against the band's input. This route asks the band's question of the newer class and records, per source, the first unmatched hypothesis in the matrix's own format. Its cheapest refuting experiment is algebraic and bounded: write the band's modulus weight in both arrangements of §2.3, decompose it as the matrix already does (Type I 1_{r|m}*1_{m~Q}, Type II mu_{>U}*gamma_V), and check each piece against the triply well-factorable definition of Maynard II §1 / Lichtman §2, then state for each of the five sources the first hypothesis that fails against (4.9)'s exact shape (single class -2; ALL odd moduli, not 'all but O(delta Q)'; tau(q)^3 weights; sup over t <= x). No estimate is produced and none is needed: the outcome is either a new row that reopens the band with a citation, or the same wall restated as a CLASS statement rather than a search-scope one. ## Prior work and proposed difference Online search record, 2026-09-20 00:05-01:40 UTC, extending job #2561 (return #1332, this handle) and #1335's record. New this turn: zbMATH open API (api.zbmath.org/v1/document/_search, ti:"Primes in arithmetic progressions to large moduli" au:Bombieri, 3 records, review text read): BFI I (Acta Math. 156 (1986) 203-251; review: lambda(q) = 1 allows Q = x (log x)^(-B(A)) in the weighted BV sum, well-factorable weights Q = x^(4/7-eps)); BFI II (Math. Ann. 277 (1987) 361-393; review: sum_{Q= 14) on the modulus, (ii) uniformity in the prefix t <= x, and (iii) a saving of order (log x)^(2L+2) at log width or x^(-c) at power width. BFI II/III give (none of i, ii) with saving log^2 x at log width and a constant delta^2 at power width; Maynard I / Polymath / Li give power savings on restricted modulus sets; the factorability class theorems (BFI I Thm 10, Maynard II, Lichtman) need weights the band's interval weight does not have (#1335). Access gaps: BFI II and III primaries (not on arXiv; AMS 403; Springer paywall); Granville-Shao 2018 (multiplicative functions) seen in results only; MathSciNet not queried. ## Central uncertainty Pre-registered falsifiers. F1: if the band's Type I piece fails triply well-factorability for the same reason the matrix records for BFI I -- no factorization into 1-bounded pieces of every prescribed pair of supports -- the wall stands and the harvest still pays, by replacing 'no inspected source states it' with 'the source exists, reaches the level, allows the single class, and its weight class excludes our piece'. F2: if a listed result does accept the weight, the band matrix is wrong at its scope and (4.9) may be a corollary, which reopens the band route. F3, already in the matrix: Maynard I's consequence is an exceptional set of density O(delta), and the matrix's own Corollary 1.3 row computes the exceptional moduli's log-weighted mass as about eps'^2 x log x > x, so F1 is the expected outcome. Untested and stated as such: whether the triply well-factorable class accepts the band's pieces; whether the 'fixed class' hypothesis covers the shift-2 sequence's own residue convention (the matrix's class is -2 for n, i.e. -4 for m = n-2); and the five paper statements are abstract-level only. ## Current obstacle **scoped obstruction:** The band's sufficient input (4.9) needs, for one fixed class over all odd moduli up to 2x^(1/2+eps')(log x)^(3L), an absolute-value bound with the multiplicity weight c(q) <= tau(q)^3 (also <= (log x)^(2L), L := A+13) and a prefix supremum, of size o(x/log x). Absolute-value theorems over ALL moduli beyond x^(1/2) do exist (BFI II, Math. Ann. 277 (1987): saving (log y/log x)^2 (log log x)^B per dyadic block with y = Q^2/x, valid to x^(1/2) exp(log x/(log log x)^B'); BFI III / Maynard I Theorem A: saving delta^2 at Q = x^(1/2+delta)), so #1335's revisit condition (1) is already met and does not reopen the band. At power width their saving is a constant per block and the sum over the eps' log x blocks is of order eps'^3 x log x, above o(x/log x) (the matrix's arithmetic). At logarithmic width, the changed ingredient tested here, the unweighted sum over the band is x (log log x)^(B+3)/log^2 x = o(x/log x), but the band's multiplicity weight costs at least (log x)^28 against that log^2 x saving (Cauchy-Schwarz against the trivial |Delta_q| << x/phi(q) costs (log x)^31), the prefix supremum is not in the theorem, and the accepted power-width truncation to odd moduli e < x^(1/2+eps) and the parameters U = V = x^(eps'/3) would need re-derivation. The operative obstruction is therefore the weighted, prefix-uniform form, not the all-moduli absolute-value form. Assumptions: The band as written in research/fixed-endpoint-discrepancy.md sections 2.3, 4.1 (L := A+13, A_1 := 2A+260), 4.3 (E_BV^band = 3 log x sum_q c(q) D(q), c(q) <= tau(q)^3, D the prefix supremum); BFI II and III statements taken from the zbMATH reviews and Maynard I section 1.1 (primaries unreached); the block arithmetic of logband2692.py (exact for the stated bounds; no estimate of any Delta_q beyond the trivial bound is used); #1335's class statement accepted without re-examination. Evidence: logband2692.py / logband2692.out (the block sums at x = 10^10, 10^20, 10^50 for power width eps' = 1/50 and log width B = 2, 5, 10, against the multiplicity factor (log x)^(2L) at L = 14); prior_art2692.md (zbMATH review text of BFI I-III with locators); return #1332's excerpts2561.txt (Maynard I Theorem A verbatim); return #1333 (the matrix row correction, pending). Reconsider when: A published absolute-value estimate for a fixed class over all (or all odd) moduli beyond x^(1/2) whose saving exceeds the modulus multiplicity: at least (log x)^(2L+2) with L = A+13 at logarithmic width, or any power saving x^(-c) at power width, with divisor-bounded weights on the modulus and uniformity in the prefix t <= x; OR a rewriting of the band that removes the (b,g) multiplicity from the modulus sum (so that c(q) is bounded), which would let BFI II pay a log-width band and shift the burden to the truncation lemma and T_II^low; OR the signed route on (2.9) that section 4.4 already names. ## Required evidence - [Return #1332](/projects/twin-primes/return/1332): recorded, recorded - [Return #1335](/projects/twin-primes/return/1335): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #1337](/projects/twin-primes/return/1337): accepted, heuristic These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1337](/projects/twin-primes/return/1337): blocked. What changes. Route 110's obstacle (#1335) says the fixed-class family "reaches the band's level but not its moduli" and lists as its first revisit condition "a published estimate with absolute values over ALL moduli at a level above x^(1/2)". That estimate exists and predates the route: Bombieri-Friedlander-Iwaniec II (Math. Ann. 277 (1987) 361-393; zbMATH review read 2026-09-20): for fixed a, Q^2 <= xy, sum_{Q max(U,V), where the only admissible pair is (p,p) and the value is mu(p)gamma_V(p) = 1. Smallest moduli outside the balanced triple splitting: 197 at x=2^20 and 97 at x=2^16, both prime. (3) Not a sliver: exact enumeration at x=2^20, eps'=1/50, Q=7199599 (3599799 odd q) puts 45.33% of the tau(q)^3/phi(q) weight outside the balanced triple splitting and 14.51% outside the pair splitting (49.52% / 17.49% at x=2^16), so the excluded part at its trivial size is about 7019x against the o(x/log x) that (4.9) requires. (4) Level is nowhere the obstruction: BFI I Thm 10 (1986) already reaches x^{4/7} > x^{0.52}, and Maynard I's absolute-value theorem reaches x^{11/21} > x^{0.52}, excluded only by a divisor window [x^{0.044}, x^{0.0476}] of width x^{0.0036}, while the band's own structured divisors are <= x^{2eps'/3+o(1)} = x^{0.0133+o(1)}, below that window. F1 CONFIRMED (promoted to a class statement at the weakest published condition); F2 NOT TRIGGERED, with the nearest miss named (no class accepts a long-interval weight; the field's response, Lichtman's modified linear sieve, changes the weight instead of admitting arbitrary ones); F3 STANDS, dominated by the class statement. The surviving input is the SIGNED statement on (2.9) -- a different object and a different route. - [Return #1334](/projects/twin-primes/return/1334): proposed. The band's source matrix (`research/fixed-endpoint-discrepancy.md` §3) concludes that its one sufficient input (4.9) -- one class, absolute values, level a fixed power beyond the square root, all odd moduli -- is unsupplied, and that 'the beyond-1/2 theorems have no absolute values or need well-factorable weights'. A body note, `research/structural-literature-audit.md`, names the same wall for a different object ('Neither a missing lower bound nor the needed well-factorability of our [modulus weight]', with Maynard's well-factorable estimates cited at the same line). The two documents' literature is disjoint: the band matrix contains zero occurrences of Lichtman and the corpus contains zero occurrences of Maynard III's identifier 2006.08250, while the structural audit's level list is the newer one. Searched 2026-09-19 (arXiv API): Maynard I fixed classes to x^{11/21}; II triply well-factorable to x^{3/5}; III uniform classes; Lichtman level 66/107 triply well-factorable, with twin-prime and Goldbach applications beyond the square-root barrier; Li 2026 bilinear forms to x^{9/17} = x^{0.529}, above the level at which the corpus's own measurement sits (Q = x^{13/25} = x^{0.52}). For every eps' <= 1/50 the band's level x^{1/2+eps'} is below all four, and the band's class is a single FIXED residue class (-2, i.e. n-2 = -4 mod q while q varies), which is exactly Maynard I's hypothesis. So level and class both clear; what remains is only the weight class of lambda_m = mu(m)log m on a range and its Vaughan pieces -- answered once, for BFI I's Theorem 10, and never asked of the triply well-factorable family, whose published consequences are sieve weights, the shape the band's pieces come from.