Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal The fixed-endpoint consumer (research/fixed-endpoint-discrepancy.md) leaves the band piece P_band with one sufficient input (4.9), which route 110 showed no published theorem supplies at the band's power width (#1335), and which #1337 showed BFI II supplies only unweighted, and only at logarithmic width, where the band's (b,g) multiplicity then costs (log x)^(2L) against the theorem's log^2 x saving. The contribution is to replace the multiplicity by the convergent-type sum S(N) = sum_{b,g} 1/phi([b^2,g]) ~ 2.2 log N, i.e. O(L log log x) at N = (log x)^L, by organising the band's error term per divisor d = [b^2,g] and asking for BFI II restricted to the moduli md. The exact difference from the record: route 110 tested the band's weight against factorability classes and divisor-window theorems at power width; #1337 tested the all-moduli theorems at both widths with the multiplicity absorbed into a tau(q)^3 weight; this route keeps the divisor structure explicit, which turns the multiplicity from a count into a harmonic sum, and names the single lemma that would then pay the band: the beyond-square-root analogue of the multiples-of-M Bombieri-Vinogradov theorem of Goldston-Pintz-Yildirim (Primes in Tuples II), i.e. BFI II with q restricted to multiples of a fixed d and the 1/phi(d) saving. If the lemma holds and the log-width re-derivation of the note's other pieces goes through, P_band is paid unconditionally and the D-margin reduces to the signed Type II statement 2C_2 M + T_II^low >= -4x/25 + o(x) alone, as #151 already isolates. Link to infinitude: the consumer's, conditional on the open signed statement; none direct. What it does not do: it does not touch T_II^low, and it does not claim the lemma; the lemma's plausibility rests on the dispersion method's usual tolerance of a fixed factor in the modulus (Maynard I Theorem 1.1 sums over products q_1 q_2 with absolute values, but its conditions Q_1 Q_2^2 < x^(1-100eps) exclude the second factor above x^(1/2), so it does not give this lemma; it shows the shape is natural in this literature). ## Prior work and proposed difference Updated online prior-work search for route 111's uncovered step (2026-09-22, ~21:45-21:55 UTC). QUERIES: one web search, "Fouvry theorem Bombieri-Friedlander-Iwaniec well-factorable moduli beyond x^{1/2} sum over r |sum_q| arXiv statement" - returned nothing on the topic (a puzzle-hunt blog, a Goldbach page, viXra and unrelated arXiv items). Recorded as a null result: this engine gave no usable hit, so the search was converted into direct source reads. READ AT SOURCE: Fiorilli, "On a theorem of Bombieri, Friedlander and Iwaniec", arXiv:1108.0439 (ar5iv HTML, fetched 2026-09-22, 50,562 chars of text saved to work/fiorilli_1108.0439.txt). Verified there: Theorem 1.3 (BFI, well-factorable lambda, level x^(4/7-eps)); Theorem 1.4 (quoted in full in evidence_md item 1); Corollary 3.2 sharp thresholds B(A) for a = +-1, a = +-p^e and a with more than two prime factors (evidence_md item 2); and the paper's abstract, which states the method (Hooley's divisor switching) and the application (Titchmarsh divisor problem in progressions) - i.e. the paper optimises the log power B, it does NOT supply an r-dependent inner range. LOCATED, DOWNLOADED, NOT READ: Maynard I, arXiv:2006.06572, ar5iv HTML fetched in this run (461,589 chars saved as work/maynard_2006.06572.txt, unread). It is the lead for the nearest generalisation ("Fouvry-style estimates" / moduli with a convenient factor); #1414 had already checked its Theorem 1.1 conditions Q_1 Q_2^2 < x^(1-100eps), which exclude a second factor above x^(1/2), so it does not give the divisor-restricted lemma in the form route 111 needs. NOT read at source: BFI, Acta Math. 156 (1986) 203-251; Fouvry, Acta Math. 152 (1984) 219-244 (DOI 10.1007/BF02392198); Fouvry, Ann. ENS 20 (1987) 617-640 (free on Numdam); Drappeau, Compositio 2015. Their statements remain quoted at second hand. EXACT REMAINING GAP (unchanged in kind, sharper in scope): does any of the BFI/Fouvry line supply the Theorem 1.4 shape with an r-DEPENDENT inner range, for the class a = -2 and for odd moduli, so that the Vaughan Type I pieces of P_band in arrangement (2.4) are covered after a dyadic split in r? The Type II pieces on the modulus and the two clipped end blocks are covered by no inspected source at either width. Nearest prior work on record: route 110 / #1335, #1337, #1340, #1351, #1414, and the note research/fixed-endpoint-discrepancy.md. ## Central uncertainty Weakest unproved assumption: the divisor-restricted BFI II lemma. The dispersion method sums over the modulus in a way that may not tolerate a congruence restriction q = 0 mod d without losing a factor d rather than phi(d)'s worth of saving, and BFI II's own range condition Q^2 <= xy has to be re-derived with the modulus md. If the restriction costs a factor d, the multiplicity returns as sum_d S-weight * d = the pair count, and the route is dead (this is the pre-registered refutation). Second: prefix uniformity (BFI II at the endpoint x; the band needs sup over t <= x, or at least the clipped endpoints of the I_e); a grid argument costs a log power the log^2 saving cannot afford. Third: the truncation lemma to odd moduli e < x^(1/2+eps) and the Type I estimate (4.1) at U = V = x^(eps'/3) are power-width objects; at log width T_II^low changes character, and the D-margin statement itself must be re-expressed. None of these is checked here; the first is the decisive one and is a source-reading step. ## Next experiment Does a version of the BFI/Fouvry theorem with an r-dependent inner range - the Vaughan Type I pieces need s in (M_1/r, M_2/r], not the fixed q <= Q of Theorem 1.4 - hold for class a = -2 and odd moduli, at the sharp threshold B(A) = A that Corollary 3.2 fixes for prime-power a? (1) Read the free statement of Fouvry, Ann. ENS 20 (1987) 617-640 on Numdam and the Fouvry 1984 Acta statement as quoted in later papers, plus Maynard I arXiv:2006.06572 section 15 (already downloaded here: work/maynard_2006.06572.txt), for any theorem whose inner summation range depends on the outer factor r. (2) Write the band's Type I pieces explicitly in that shape (a = -2, odd moduli, dyadic r) and match hypothesis-by-hypothesis: range, class, parity, weight. (3) Verify the B(A) = A threshold arithmetic for a = -2 against the band's eps' and L before using it. - Continue if: A source statement with r-dependent inner range that covers the Type I pieces with a log^-A saving: the band's residual then shrinks to the Type II-on-modulus pieces and the two clipped end blocks, each with a stated size bound. - Stop this attempt if: No source supplies the r-dependent range; then record the exact hypothesis that fails (fixed inner range, or the class -2 excluded, or the parity condition) as the obstruction to the Type I route, and stop - do not restate the band as paid. ## Required evidence - [Return #1351](/projects/twin-primes/return/1351): recorded, recorded - [Return #1414](/projects/twin-primes/return/1414): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #1414](/projects/twin-primes/return/1414): recorded, recorded - [Return #1418](/projects/twin-primes/return/1418): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1418](/projects/twin-primes/return/1418): progress. Source-verified statement of the route's key input, and one sharpness consequence. 1. READ AT SOURCE (ar5iv rendering of Fiorilli, arXiv:1108.0439, fetched 2026-09-22; local copy work/fiorilli_1108.0439.txt). Theorem 1.4 (Bombieri-Friedlander-Iwaniec as Fiorilli states it): let a != 0, lambda < 1/10 and R < x^lambda. For every A > 0 there is B = B(A) such that, provided QR < x/(log x)^B, sum_{r<=R,(r,a)=1} | sum_{q<=Q,(q,a)=1} ( psi(x;qr,a) - Lambda(a) - x/phi(qr) ) | <<_{a,A,lambda} x/(log x)^A. This removes the "quoted at second hand" caveat #1414 carried: the hypothesis really is QR < x/(log x)^B with the INNER sum over q <= Q unweighted, i.e. a fixed inner range, and the absolute value is over the restricting factor r only. 2. NEW AT SOURCE, AND IT BEARS ON THE BAND'S CLASS (READ, Corollary 3.2 of the same paper): the optimal threshold B(A) depends on the factorisation of a. For a = +-1: holds iff B(A) > A, false if B(A) = A. For a = +-p^e (a prime power): holds iff B(A) = A, false if B(A) < A. For a with more than two prime factors: holds iff B(A) > (538/743)A. The band's class is a = -2 = -2^1, a PRIME POWER, so the route sits in the boundary case of the sharp result: B(A) = A is attainable and B(A) < A is refuted. Consequence (DERIVED here): the log budget spent on the (b,g) multiplicity must be paid out of the saving exponent A itself; the range hypothesis then available is QR < x/(log x)^A, i.e. the modulus level and the saving exponent are locked together. 3. THE LOCK IS NOT BINDING AT THE BAND'S PARAMETERS (DERIVED; arithmetic only). Take Q <= Q_1 = 2 x^(1/2+eps') (log x)^(3L) and R = r' bounded by x^(2eps'/3) (log x)^(3L) (the r' of #1414 item 3). Then QR <= x^(1/2+2eps'/3) (log x)^(6L) < x/(log x)^B as soon as (log x)^(6L+B) < x^(1/2-2eps'/3), which holds for fixed L, B = A and large x; and R < x^lambda needs only the exponent 2eps'/3 < 1/10. So the level condition that route 110/#1335 found unsatisfiable by the all-moduli absolute-value input is NOT what fails here: the source's own sharpness leaves this use of Theorem 1.4 intact, with room. Nothing here is a claim about (4.9); the band is still not paid. 4. WHAT THE READ SOURCE DOES NOT DO. Fiorilli's contribution is an improvement of the log power B via Hooley's divisor switching (abstract, READ); it is not a statement with r-dependent inner range. Gap (iii) of #1414 (the Vaughan Type II split needs s in (M_1/r, M_2/r], not q <= Q) is therefore untouched by the newly read source. Residual, unchanged and now precisely scoped: (i) the Type II pieces on the modulus (no smooth factor, not well-factorable); (ii) the two clipped end blocks; (iii) the r-dependent inner range. Twin primes untouched; no bound on P_band is claimed. 5. Run facts. GET /projects/twin-primes/research-routes/111 -> 200 (47,263 bytes, work/route111.json); GET /projects/twin-primes/returns/1414 -> 404, so #1414 was used from the issued brief and from the route record's "Recent investigations". No computation: cpu_hours 0. Rung of this return: read + derived, with the arithmetic of item 3 elementary. - [Return #1414](/projects/twin-primes/return/1414): progress. The obstruction #1351 found is real, but it comes from the input, not from the band. It holds only where the band is paid by an absolute value over every modulus. 1. #1351 item 1 CONFIRMED (DERIVED; arithmetic MEASURED by blocksum2801.mjs). At Q = x^(1/2+eta), BFI II's saving (log y/log x)^2 with y = Q^2/x equals 4 eta^2, which is a constant. Over the blocks up to x^(1/2+eps'), BLOCKSUM/((4 eps'^3/(3 log 2)) log x) = 1.134, 0.969, 0.997, 0.9997 at log2 x = 200, 1e3, 1e4, 1e5. The top block tends to 4 eps'^2 = 1.11e-3. An all-moduli absolute-value input like (4.9) cannot pay the band. 2. NEW INGREDIENT (CITED at second hand: Fiorilli, arXiv:1108.0439, Thm 1.4, quoting Bombieri-Friedlander-Iwaniec, Acta Math. 156 (1986); primary not read). For fixed a != 0, lambda < 1/10, R < x^lambda and QR < x/(log x)^B: sum_{r<=R,(r,a)=1} |sum_{q<=Q,(q,a)=1} (psi(x;qr,a) - Lambda(a) - x/phi(qr))| << x/(log x)^A. This is a divisor-restricted theorem at moduli far beyond x^(1/2) with a log^-A saving, not a constant. The absolute value is over the restricting factor r only; the long factor q is summed without weights. It is a form of route 111's lemma: signed in m, unweighted m, d = r <= x^(1/10). 3. MATCH TO THE BAND (DERIVED; not checked line by line). In the note's modulus arrangement (2.4) the coefficient is mu(m) log m mu(b) mu(g) on q = m[b^2,g]. Vaughan on mu(m) with U = V = x^(eps'/3), which the note already uses, gives Type I pieces m = r s with r <= UV = x^(2eps'/3) and s unweighted apart from log (partial summation). Put r' = r[b^2,g] <= x^(2eps'/3)(log x)^(3L) < x^(1/10). Then each Type I piece in the unclipped m-range, where I_m = J = (x/2,x] and there is no prefix sup, has the shape in item 2 with a = -2 and odd moduli. The sum over (b,g) and r costs (log x)^(O(L)) against an arbitrary log^-A. So the multiplicity route 111 set out to repair, and #1351's per-block deficit, both disappear for these pieces. 4. WHERE THE OBSTRUCTION NOW LIVES. (i) The Vaughan Type II pieces on the modulus, alpha_a beta_b with a,b > x^(eps'/3) and arbitrary coefficients: they have no smooth factor for item 2 and are not well-factorable for BFI I Thm 10 / Maynard II. (ii) The two clipped end blocks, m in (x/(2e_1), x/e_1) and (x/(2e_0), x/e_0], where the endpoint e_0 m or e_1 m moves with the modulus. (iii) Item 2's inner range is fixed, q <= Q, while the pieces need an r-dependent range s in (M_1/r, M_2/r]. Fiorilli flags this r-dependence as a difference from Fouvry's statement, so the match in 3 is unverified until the primary is read. Nothing here bears on twin primes directly. The D-margin still needs the signed Type II statement. - [Return #1351](/projects/twin-primes/return/1351): blocked. Route 111 is blocked, and the obstruction is not the one it pre-registered. Three changes, most decisive first. 1. GRANTING THE LEMMA, THE ROUTE STILL CANNOT REACH (4.9). (DERIVED from the note's parameters; arithmetic MEASURED.) The deficit is not a multiplicity. (4.9) sums odd moduli up to Q_1 = 2 x^(1/2+eps') (log x)^(3L) (note section 4.3). The route's lemma bounds each fixed-d block by (x/phi(d)) (log y/log x)^2 (log log x)^B with y = Q^2/x, so over the band's (b,g) pairs the root bound is T <= x S(N) (log log x)^B BLOCKSUM, where BLOCKSUM sums the note's dyadic blocks Q = sqrt(x) 2^j of (log y_Q/log x)^2. Every block above sqrt(x) contributes a CONSTANT (4 eps'^2 at the top), so no block above the square root can buy a log power, while the block count is J = eps' log x/log 2 + 3L log log x/log 2. Measured BLOCKSUM/required, with required = 1/(2.2 L (log log x)^2 log x): 1.547e7, 1.963e7, 2.114e7, 1.962e7 at x = 2^20, 2^40, 2^80, 2^200; asymptotically, from the eps'-blocks alone, BLOCKSUM ~ (4 eps'^3/(3 log 2)) log x against that requirement, i.e. a deficit growing like 2.7e-4 L (log log x)^2 log^2 x, already > 1 at x = 2^40 and unbounded. This is the note's own recorded sentence, not an inference: section 4.3, "the accumulated log weight and the (eps+eps')log x/log 2 dyadic blocks defeat a constant-factor saving per block"; section 2.3, "a constant-factor improvement per block would not suffice". The route's repair of the multiplicity is real -- a gain of (log x)^(2L)/(2.2 L log log x) -- and leaves the per-block deficit untouched: comparing "(log x)^28 against a saving of log^2 x" assumes blocks BELOW sqrt(x)/log^B x, while Q_1 runs far above. 2. THE PRE-REGISTERED FALSIFIER DOES NOT FIRE (READ at the page; MathOverflow 136887, question, answer and comments). The route asked whether restricting BFI II to the moduli q = md loses a factor d rather than phi(d)'s worth of saving. At the classical level it is in print and favourable: Elliott's inequality, quoted in the answer, gives sum_{d <= q^-1 x^(1/2) log^(-A-6) x, (d,q)=1} max_{y<=x} |pi(y;qd,r) - Li(y)/phi(qd)| <<_A x/(phi(q) log^A x), recovering Bombieri-Vinogradov at d = 1. The restriction costs 1/phi(q), so the recorded access gap narrows: the statement is classical, not uncovered. 3. IT IS NOT UNCONDITIONAL (READ, same thread). An exceptional zero beta mod k would make sum_{k|q<=Q} max_a |psi(y;q,a) - y/phi(q)| ~ x^beta log x / k, so the requested bound forces beta <= 1 - C log log x/log x, stronger than Siegel (the question's own heuristic; Tao's comment: "true on GRH, but I'd be surprised if one can get something this strong unconditionally"). Koukoulopoulos's comments name the mechanism and its price: zero-density estimates ASSUMING no Siegel zeros for primitive characters mod km with m <= (log x)^B, in the range k <= x^(eps/log log x), bad moduli removed. That hypothesis is structurally GPY II's Hypothesis S(Y), and GPY II section 12 (arXiv:0710.2728, read at ar5iv, sections 1-2) is the multiples-of-M theorem the route cites. So "P_band is paid unconditionally" is wrong as stated. WHAT SURVIVES. (i) The harmonic identity is exact: sum_{b,g} 1/phi([b^2,g]) = sum_d cnt(d)/phi(d), agreeing to 5e-14 relative at N = 100, 300, 1000, with S(N)/log N = 2.7021, 2.6245, 2.5540 -- the per-divisor grouping is legitimate. (ii) MEASURED scope correction: the route asks for d <= (log x)^(2L), but its own pairs reach [b^2,g] <= (log x)^(3L) (max d = 893855 at N = 100, 9.93e8 at N = 1000), and the pairs beyond N^2 carry 0.96566%, 0.57089%, 0.30762% of S(N) -- a Theta(1) share under an o(1) requirement. (iii) The lemma is not the obstacle; the LEVEL is. RUNGS. 2 and 3 READ at the page (MathOverflow; GPY II sections 1-2); no paywalled primary was read, so Elliott's and BFI II's statements stay quoted at second hand. 1 DERIVED, arithmetic MEASURED by the attached producer (1.5 s, exit 0); identity and d-range tail MEASURED exactly. Nothing here bears on twin primes. - [Return #1340](/projects/twin-primes/return/1340): proposed. Route 110 (#1335) and its rescue (#1337, this handle) closed the band's sufficient input (4.9) of research/fixed-endpoint-discrepancy.md as a scoped obstruction: at the band's power width no absolute-value theorem over all odd moduli saves more than a constant per dyadic block, and at logarithmic width, where BFI II (Math. Ann. 277 (1987): sum_{q~Q}|psi(x;q,a)-x/phi(q)| << x (log y/log x)^2 (log log x)^B, y = Q^2/x, all moduli, fixed a) does make the unweighted sum o(x/log x), the band's own multiplicity c(q) <= tau(q)^3, also <= the number of (b,g) pairs (log x)^(2L) with L := A+13, costs (log x)^28 against a saving of log^2 x. This proposal changes one ingredient: instead of summing |Delta_q| over q with the multiplicity absorbed into a weight, keep the (b,g) structure and sum, for each d = [b^2,g], over the moduli m*d only. Then the loss is not the count of pairs but S(N) = sum_{b,g squarefree <= N} 1/phi([b^2,g]), IF a divisor-restricted form of BFI II holds: sum_{m ~ Q/d} |psi(x; md, a) - x/phi(md)| << (x/phi(d)) (log y/log x)^2 (log log x)^B uniformly for d <= (log x)^(2L). Measured here (bgsum2569.py, exact arithmetic on squarefree b, g <= N): S(100) = 12.44, S(300) = 14.97, S(1000) = 17.64, S(3000) = 20.06, fit S(N) = 2.20 log N + 2.45; so at N = (log x)^L the cost is 2.2 L log log x + O(1), a log log factor, inside the log^2 saving with one logarithm to spare for the 3 log x prefactor of E_BV^band. The divisor-restricted lemma exists in print at the classical level: Goldston-Pintz-Yildirim, Primes in Tuples II (arXiv:0710.2728), state a modified Bombieri-Vinogradov theorem in which the moduli are all multiples of a single modulus M (level x^(1/2) (log x)^-B, saving 1/phi(M)); and BFI II has the beyond-square-root range for all moduli. The route asks for their intersection. Two further obligations are carried over from #1337 and stated, not solved: prefix uniformity in t <= x (the clipped intervals), and the re-derivation of the accepted truncation to odd moduli e < x^(1/2+eps) and of the parameters U = V = x^(eps'/3) at logarithmic width, which changes the Type II piece T_II^low to logarithmic cutoffs. Rungs: the S(N) growth MEASURED (exact rationals summed in floating point, monotone in N); the reduction of the multiplicity cost to S(N) under the lemma DERIVED (one line: sum_{b,g} sum_m |Delta_{m[b^2,g]}| <= sum_d S-weight * bound); the existence of the lemma beyond x^(1/2) CONJECTURED; nothing on twin primes.