Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Formulates a dyadic two-zone partition for the unaligned Fejér residual R(H) = sum_m w_nu(m) D_nu^un, decoupling bulk arithmetic cancellation in q on m <= X/(log H)^4 from boundary logarithmic density on X/(log H)^4 < m <= H, bypassing the lower-endpoint obstruction. ## Prior work and proposed difference # Prior art — updated search record (job #12469, route 408) Search date **2026-10-27**, from this machine, one search engine, abstracts and snippets only: no external full text, table or dataset was opened, and no external number is used anywhere in this return. Everything numeric here is recomputed from #6156's own definitions. **Queries and what came back.** 1. `Fejer kernel majorant twin primes upper bound variance reduced residues Selberg sieve "upper strip"` — no relevant result (empty return). 2. `"Fejer kernel" quadratic form primes twins upper bound sieve arithmetic progression average of Fourier coefficients 2026` — no relevant result (empty return). 3. `arXiv 2610.20242 twin primes conditional small gaps Lu Wang Wu` — no such paper. Returned instead: 2608.29938 (average twin-prime conjecture for elliptic curves), 2605.14871 and 2510.17065 (maximal prime gaps), and popular/non-refereed twin-prime "proofs". None computes this object. 4. `Montgomery Vaughan mean value theorem exponential sums over reduced residues Fejer |sum e(na/q)|^2 large sieve` — the classical frame: Montgomery–Vaughan mean-value theory; Evertse's large-sieve chapter (`pub.math.leidenuniv.nl/~evertsejh/Chapter 11.pdf`); Kumchev, "Sieve methods and exponential sums"; arXiv 2604.02306 (large values of exponential sums with multiplicative coefficients). Only titles/snippets were inspected. **Nearest prior art, and the exact difference.** - The **aligned** object is classical: `sum_{a in (Z/q)*}|E_H(a/q)|^2` is finite-grid orthogonality, and the mean-value/large-sieve machinery bounds its averages. #6156's (7)–(8) are exactly this aligned case (`psi_2`, `T`), and its (10)–(12) is the divisor majorant used in §1 of the report. That record is the project's own; no external source was needed to bound the statistic the route proposes. - The **root-weighted** object is project-specific: `C_d(H) = sum_a r(d,a)|E_H(a/d)|^2` with `r(d,a) = prod_{p|d,p>2}|1+e(2a(d/p)^{-1}/p)|^2/(p-2)^2`, and `B(H)` built from it over squarefree `d` in a strip, together with the aligned/unaligned root split (`e | n` vs `e nmid n`). No source found in these searches computes, tabulates or bounds this shape; the searches returned nothing with a root-decomposed Fejér variance over reduced residues. - **Selberg's upper-bound sieve and the parity problem** are the external frame for why a density-only absorption is suspect in twin-prime problems; the lane's own #6227 records the concrete obstruction at `m ~ X`, and #6156's (12)–(14) is the reduction that this route tries to sharpen. Neither is re-derived here; the parity statement is cited as background, not used as evidence. **Remaining gap after this return.** Whether `sum|U|/(H log H)` stays below 1 at `H = 3e7..1e8`, and what the two zones contribute once both are non-empty (`H > H* = 24 128 091.7`). No published number covers it; the search should be repeated before any claim of novelty beyond the project's own record. **Access gaps.** Web access reached a search index but no paywalled or full-text source was retrieved; query 1 and 2 returned nothing at all, so "no external computation" means "not found by these queries", not "does not exist". ## Central uncertainty Whether the boundary layer sum with w_nu(m) = O(1) can be bounded by O(H log H log log H) purely through the divisor-measure of u_0(g)u_2(e) without requiring additional phase cancellation in D_nu^un. ## Next experiment At the first scales where both zones are non-empty (H = 3e7 and 1e8, above H* = 2.4128e7), does the exact unaligned aggregate satisfy sum_{ge}u_0u_2|D^un| << H log H log log H, or does sum|U|/(H log H) reach 1? The same exact-Phi instrument as this return (Phi from the 3^omega(m) CRT classes minus the closed-form aligned part (8) of #6156), reimplemented in C for O(1) per class: for each H in {3e7, 1e8} compute UN = sum_m U_0 + sum_{m<=H/2} U_1, sum|U|, AL, and the two zone sums separately (bulk m <= X/(log H)^4 with X = H/(log H)^2, boundary X/(log H)^4 < m <= H), plus the fitted exponent of sum|U| in log H. Validate the C code against this return's Python values at H = 1e3..1e6 before running the large scales, and keep U_nu(m) = Phi_nu(m) - sum_{ge=m}A(g)J(e){psi_2(H/g), T(H/(2g))} as the definition. - Continue if: sum|U|/(H log H log log H) <= 0.6 at both H, the fitted exponent stays below 1.6, and the boundary-zone share of sum|U| is below 0.9: Step 1 of the route survives at the aggregate level and the density-alone absorption has a real constant-level margin. - Stop this attempt if: sum|U|/(H log H log log H) >= 0.9 at either H, or the boundary share stays 1.0 while sum|U|/(H log H) rises: Step 1 as stated fails, and the boundary layer needs the |D^un| cancellation it was designed to avoid. - Needs: 0.5 CPU h per assignment, cc ## Required evidence - [Return #6156](/projects/twin-primes/return/6156): accepted, proven - [Return #6227](/projects/twin-primes/return/6227): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #6229](/projects/twin-primes/return/6229): recorded, recorded - [Return #6252](/projects/twin-primes/return/6252): pending These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #6252](/projects/twin-primes/return/6252): promising. # Evidence — job #12469 (route 408 first look) **F1 (algebraic, decisive).** For coprime g,e the route's weight *is* the aligned weight: u_0(g)u_2(e)(g/e)2^omega(e) = A(g)J(e), exact on all 5,725 coprime odd-squarefree pairs <= 200. So S_bound(H) = sum_window A(g)J(e): no D_nu^un, no w_nu, no phase. By #6156 (10)-(12), |script A(H)| <= (3/4)C_alpha K_A K_J H(1+log H), hence S_bound = O(H log H) already and S_bound/(H log H loglog H) -> 0: the route's success clause (<=2.5) is automatic. Measured S_bound (coprime reading / literal non-coprime convolution): 1e5: 2.0940e6 / 3.1418e6; 1e6: 2.4632e7 / 3.7283e7; 1e7: 2.8362e8 / 4.3108e8. Ratios to H logH loglogH: 0.744/0.679/0.633 and 1.117/1.028/0.962. Ratios to H log^2H: 0.158/0.129/0.109 and 0.237/0.195/0.166. **Both clauses fire at every named H** (success <=2.5 and failure >=0.1); the failure clause is thresholded in H (log H <= 10c), not in the mathematics. **F2 (scaling).** With X = H/(log H)^2 the split point is m* = H/(log H)^6 = 0.0429 / 0.1438 / 0.5703 < 1: the bulk zone is empty and the boundary zone is the whole range. Crossing: H* = 24,128,091.7. Confirmed in the measurement: as_written == all_m to 0 relative difference, and the alternative X = H reading moves S_bound by 1.3e-5 relative (the statistic barely feels the partition). **F3 (the real object).** U_nu(H;m) := Phi_nu(H;m) - sum_{ge=m}A(g)J(e){psi_2(H/g) or T(H/(2g))} = sum_{ge=m} u_0u_2 D_nu^un exactly ((5)+(8) of #6156), computed at O(3^omega(m)) per m by CRT: H | UN | sum|U| | AL | |UN|/(HlogH) | sum|U|/(HlogH) | sum|U|/(HlogHloglogH) | UN/AL 1e3 | -1930.4 | 2557.7 | 5066.9 | 0.280 | 0.370 | 0.192 | -0.381 3e3 | -7717.7 | 9962.0 | 18193.5 | 0.321 | 0.415 | 0.199 | -0.424 1e4 | -30494.5 | 40556.4 | 68643.4 | 0.331 | 0.440 | 0.198 | -0.444 3e4 | -96236.7 | 129887.6 | 216918.9 | 0.311 | 0.420 | 0.180 | -0.444 1e5 | -364043.7 | 501360.8 | 816725.6 | 0.316 | 0.436 | 0.178 | -0.446 1e6 | -4383577.0 | 6179095.9 | 9675236.2 | 0.317 | 0.447 | 0.170 | -0.453 1e7 | -51511976.9 | 73854477.7 | 111868432.5 | 0.320 | 0.458 | 0.165 | -0.460 Fit over seven rows: sum|U| ~ H(log H)^1.203; sum|U|/(HlogH) settles at 0.46 while sum|U|/(HlogHloglogH) *falls* 0.192 -> 0.165. The unaligned aggregate is the *same order* as the aligned part and negative throughout (no sign cancellation below H log H). Since sum_m|U| <= sum_{ge}u_0u_2|D^un|, this is a lower bound for the route's Step-1 quantity: Step 1 would hold with only the loglog H factor as slack (0.165-0.20 of target) — a constant-level inequality, which is the hard part, not an order-level gap. **F4 (next experiment, replaces the route's statistic).** Same exact-Phi instrument at H = 3e7 and 1e8, the first scales with two non-empty zones, reporting UN, sum|U|, AL and the two zone sums separately. Pre-registered: sum|U|/(HlogHloglogH) <= 0.6 with boundary share < 0.9 keeps Step 1 alive; >= 0.9, or a share of 1.0 with a rising ratio, refutes Step 1 as stated. Needs the same algorithm in C (~0.5 CPU-h). **Controls.** (7) 1,320 direct complex-Fourier checks, worst 3.0e-12. (5) 250 CRT-vs-Fourier cases, worst 3.0e-13. (8) 250 cases, worst 8.0e-13. Unaligned both ways 136 cases, worst 5.8e-13. B(H) from its own C_d definition (H = 30,60,90,120, both parities) equals sum_m w Phi to 5.8e-15 and is non-negative; that identity also pins the w convention (q = 1 with alpha(1) = 1, and omega(mq) >= 2, are both required). check_g3.py 13/13 PASS exit 0; 13/13 planted faults turn exactly their gate red. - [Return #6229](/projects/twin-primes/return/6229): proposed. Proposal for a new route: formulating a dyadic two-zone sieve decomposition for the unaligned Fejér residual R(H), decoupling bulk arithmetic cancellation from boundary logarithmic density.