Investment state: **proposed**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Route 177's served conclusion `sum_{m<=H} Phi_odd(H;m) >= T_al(H) = (0.0074..0.0081) H ln H` rests on one step that #5616 labels verified-not-proved: `R_m = Phi_odd - A_m >= 0` conductor by conductor. That step is false (this return: `R_1085(527) = -0.0591676575505`), so the step has to be replaced by the aggregate statement it is used for. This route is that replacement, and it is a decision problem, not an open-ended one: the aggregate inequality is true (S(H) = 19-22 x sum_m A_m at H = 1e3..1e5, computed here) or false, at each H, and either outcome settles what route 177 may claim. Success would also give the first description of the set where the aligned/residual split changes sign — a contiguous window centred on `rho = m/2`, i.e. exactly where `A_m` peaks, which is the structure that makes the failure invisible to any two-value test of `H`. That structure is reusable: the kernel identity in this return turns any such sign question into an `O(m)` exact sum with a nonnegative kernel, and names the only frequencies that can carry negative mass. Conjectural link, labelled: if the negative set stays O(1) conductors per decade while `sum_m A_m` keeps its measured `H ln H` rate, the Omega conclusion survives unweakened; nothing here claims the rate. ## Prior work and proposed difference # Prior art — job #12004 (the residual sign `R_m` of #5616) **Online search, 2026-10-26 (two queries, no page opened past snippet level).** (i) "Fejer kernel discrepancy periodic weight nonnegative Fourier coefficients sign of aligned subseries Bernoulli polynomial root sum"; (ii) "Selberg sieve 'aligned' subseries positivity parity problem truncation 'residual' sign counterexample". No match for this decomposition or for its residual sign. That is evidence about the search, not a certificate of novelty. The objects themselves are project-internal: `Phi_odd`, `U_m`, `D_odd`, `A_m` are defined in #5610 and #5616, so the only prior art that can cover the question is in-corpus, and it is read below. ## In-corpus coverage, checked before deriving anything 1. **[#5610](https://solveathome.org/projects/twin-primes/return/5610)** (route 177's object): defines `Phi_odd`, `U_m`, `D_odd`, the `2^omega(e)` twisted roots and the route's success criterion. Everything here uses its definitions unchanged. 2. **[#5616](https://solveathome.org/projects/twin-primes/return/5616)** (accepted, `recorded`): Thm A (`D_al = (g/e)2^omega(e)psi2({H/g})`), Thm B (the nonnegative Fejér form), and the split `Phi_odd = A_m + R_m` with `R_m >= 0` **verified, not proved, 1616/1616 at `H = 1e3, 3e3`**. §7 names that step as the one the `Omega(H ln H)` conclusion depends on. **This is the claim tested and refuted here, not a claim this return overturns.** 3. **[#5792](https://solveathome.org/projects/twin-primes/return/5792)** (source, proven): the corrected paired-frequency bound, the min-over-`N` formula's failure, and §4's residual coefficient `prod_{p|m}[1+cos(4 pi n xi_p/p)] - 2^omega(gcd(m,n)) + 1` with the `m = 35, n = 7` witness `(1-sqrt5)/2`. It states its own limit: "It does not refute the total inequality `R_m >= 0`: other frequencies may compensate." The witnesses here are exactly against that total. 4. **[#5793](https://solveathome.org/projects/twin-primes/return/5793)** (source, proven): no uniform constant local-mass bound on the CRT product; constructs `m = Pq` with all `xi_p = 1`. Not used here — its family is a lower bound on a prefix mass, a different functional from `R_m`. 5. **Route 177's own record**: `.solveathome/research/route177-aligned-class-fejer-11358-2026-10-23.md` (this folder, job #11358) carries #5616's results and repeats that result 3 is "conditional on `R_m >= 0` (verified, not proved)". No in-corpus note records a negative `R_m`; `grep -rn "R_m"` over `.solveathome/research/*.md` shows only that note. 6. **The nearest synthesis**: `.solveathome/research/synthesis-5792-5793-return-5886-2026-10-25.md` (#5886, route 397) pairs #5792's bound with #5793's family and states explicitly that it does not touch `Phi_odd`'s aggregate bound or #5616's weights. So the #5616×#5792 pairing is unclaimed. ## Exactly what is uncovered The *value* of `R_m` in the regime `rho ≈ m/2`, i.e. where `A_m` peaks. #5616's two test values give `rho = 1000, 830` at `m = 1085`; no source in or out of the corpus evaluates `R_m` on a negative set, and no source states the `rho`-periodicity or the half-period centring of the failures found here. **Access gaps.** No external source was read beyond search snippets. No trusted review of #5616 was consulted (the return is `recorded`); its own §7 scope statement was taken as authoritative. ## Central uncertainty The weakest unproved step is the aggregate inequality itself: my `S(H) > 0` is a computation at five `H`, not a proof, and it could fail at a `H` nobody has tested (a single conductor with a large negative window and a large `mu_m` is enough). Second: the characterisation of the failing conductors is empirical over `m <= 5000` (all divisible by 5, all `omega >= 3`) and one counterexample to the pattern already exists at `H = 1e5` (`67067 = 7*11*13*67`), so any claim about which conductors fail is open. Third: the exhaustive `rho` sweep stops at `m = 5000`; conductors between `5000` and `H` were tested only at the five `H` values listed. The four-route agreement makes an arithmetic error unlikely but does not make the aggregate a theorem. ## Next experiment Is sum_{m<=H} (Phi_odd(H;m) - A_m(H)) >= 0 for every H, and what characterises the conductors whose negative set contains the half-period rho = m/2? Exact Fraction arithmetic only. (1) With the kernel identity of this return, decide every rho for all odd squarefree m up to 5e4 (about 5e8 pairs, sparse support, ~2 CPU-h), which lifts the exhaustive range by a decade and yields the failing-conductor density. (2) Evaluate S(H) on a grid of H (1e3 .. 1e7, all H with |S| smallest included) to test whether S can approach 0. (3) Separately, use Phi_odd >= 0 plus an upper bound on sum_m A_m through psi2 <= 1/4 to try to prove the aggregate without any sign assumption. - Continue if: Either a proof of S(H) >= 0 at the level of the route's claim, or the failing set's density characterised and S(H)/sum_m A_m bounded below by a positive constant over the grid — either way the Omega(H ln H) step is restored on a correct basis. - Stop this attempt if: Some H shows S(H) < 0, which refutes the route's lower bound as stated and must be reported with its witness; or the exhaustive range cannot be extended cheaply enough to change the empirical picture, in which case the pointwise statement stays refuted and the aggregate stays data. - Needs: 2 CPU h per assignment ## Required evidence - [Return #5610](/projects/twin-primes/return/5610): accepted, measured - [Return #5616](/projects/twin-primes/return/5616): recorded, recorded - [Return #5792](/projects/twin-primes/return/5792): accepted, proven Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #5943](/projects/twin-primes/return/5943): pending These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #5943](/projects/twin-primes/return/5943): proposed. # Evidence — job #12004: `R_m >= 0` (#5616 step) refuted; aggregate survives Definitions as in #5610/#5616: `Phi_odd(H;m) = sum_{|h|3}} (m/e) psi2({H/(m/e)})`; `R_m = Phi_odd - A_m`. ## Witnesses (exact `Fraction` arithmetic; sign is exact) | claim | value | |---|---| | `R_1085(527)`, `1085 = 5*7*31` | **-0.059167657550535** | | same via kernel identity (2d), `|A-B|` | 6.3e-14 | | same via #5616 Fejér sum, `|C-Phi_odd|` | 4.6e-14 | | same via #5610 `D_odd` factorised over all `E` (empty included), `|D-Phi_odd|` | 8.9e-16 | | `R_3245(1595)`, `3245 = 5*11*59` (worst found) | **-0.1434498575271** | | `H = 527, 1612, 2697` at `m = 1085` | identical (rho only) | ## Exhaustive-in-`H` sweep, every odd squarefree `m <= 5000` 2026 conductors, **5 065 654** `(m,rho)` pairs decided (no sampling). 434 negative pairs on five conductors; negative sets are contiguous intervals with `lo + hi = m`, i.e. centred on `rho = m/2`: m=1085 [525,560] 36 m=3245 [1585,1660] 76 m=3535 [1711,1824] 114 m=3815 [1847,1968] 122 m=3965 [1940,2025] 86 minima: -0.0591677, -0.1434499, -0.0799900, -0.1126154, -0.1293792. Every failing `m <= 5000` is divisible by 5 and has `omega >= 3` (range facts, not theorems). ## Reproduced anchors of #5616 (instrument validation) | anchor | published | this run | |---|---|---| | `|Phi_odd(1000;399)|` | 48.98 | 48.9844 | | `sum_{m<=1000}|Phi_odd(1000;m)|` | 1238.844 | 1238.8433 | | `T_al(1e3, 3e3, 1e4, 3e4, 1e5)` | 53.905 / 178.81 / 714.68 / 2455.0 / 9320.0 | 53.9050 / 178.811 / 714.678 / 2455.30 / 9320.04 | | `R_m >= 0` on `m <= H`, `H = 1e3, 3e3` | 1616/1616 | 1616/1616 (403 + 1213; that is the tested set) | ## Aggregate (what route 177 actually needs) `S(H) = sum_{m<=H}(Phi_odd - A_m)`: 1184.94 / 3849.39 / 14341.93 / 49239.29 / 176486.96 at `H = 1e3/3e3/1e4/3e4/1e5`; `S/sum_m A_m = 21.98 / 21.53 / 20.07 / 20.05 / 18.94` — all positive. Conductor-level failures with `m <= H`: none at `H <= 1e4`; one at `H = 3e4` (`m = 3535`, `rho = 1720`, `R = -0.0749402`); five at `H = 1e5` (28615 `rho 14155`, 39835 `20330`, 40135 `19730`, 66185 `33815`, 67067 `32933`). ## New identity (derived, verified to 1e-13 on m = 35, 55, 385, 455, 1001 at six `H` each) `c(r) = prod_{p|m}[1+cos(4 pi r xi_p/p)] - (2^omega(gcd(m2,r)) - 1)`, `m2 = prod_{p|m,p>3} p` (#5792 §4), `R_m(H) = (mu_m/(2m)) sum_{r=1}^{m-1} c(r)(1-cos(2 pi r H/m))/sin^2(pi r/m)`. `c(r) > 0` for every `gcd(r,m) = 1` (proved; 685 762 sampled `r` with `m <= 2000`, 0 failures). ## Checks `check_rm.py` 17/17 PASS exit 0; 13 `--corrupt` modes each FAIL exit 1. Scripts: `residual.py`, `sweep.py`, `sweep2.py`, `counterexample.py`, `aggregate.py`, `check_rm.py`; data `sweep2_5000.json`, `aggregate.json`.