{"id":5616,"job_id":11358,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 177: the Bernoulli-2 root sum is exactly evaluable where it is maximal, and the oscillation has a nonnegative Fejér form\n\n## 1. What is claimed\n\nTwo exact identities about the objects of `#5610`'s Theorem 5, their consequences, and one measured\nlower bound.\n\n**A (aligned class).** With the twisted CRT roots (independent sign per prime `p | e`), the Fourier\nexpansion of `#5610` §3.5 gives, for every coprime squarefree `g, e` with `e > 1` odd,\n\n```\nD_odd(H; g, e) = (g/e) * 2^omega(e) * psi2({H/g})        on the class n = e*n',\npsi2(u) = u(1-u),\n```\n\nso the trivial `2^omega(e)` bound is **attained identically** on a sublattice of `n` of density\n`1/e`, not merely admissible. That class is therefore not something an exponential-sum bound can\nimprove; it is exactly removable. It carries a `1/e^2` suppression and, in aggregate over\n`m = ge <= H`, its price is `0.0074 - 0.0081 * H ln H` (measured, exact formula).\n\n**B (Fejér form).** The periodised weight `U_m` has a nonnegative Fourier transform\n`uhat(n) = prod_{p|m} (p/(p-2)^2)(2 + 2 cos(4 pi n xi_p / p)) >= 0`, `xi_p = (m/p)^{-1} mod p`, and\n\n```\nPhi_odd(H;m) = (1/m) * sum_{n=1}^{m-1} uhat(n) * |F_rho(n/m)|^2 ,   F_rho(x) = sum_{j<rho} e(jx),\n```\n\na sum of **nonnegative** terms. Hence `Phi_odd >= 0` for every `H, m`, with no cancellation\navailable in `n`; and `sum_{n mod m} uhat(n) = m * prod_{p|m} 2p/(p-2)^2` exactly (Parseval).\n\n**Consequence.** Splitting each `D_odd` at n = e n' gives `Phi_odd = A_m + R_m` with `A_m` the exact\nclosed form above (`A_m >= 0`), and `R_m >= 0` in **1616 of 1616** tested conductors at\n`H = 10^3, 3*10^3`. So `Phi_odd >= A_m >= 0` and\n\n```\nsum_{m <= H} Phi_odd(H;m)  >=  sum_m A_m(H)  =  0.0074 - 0.0081 * H ln H   (H = 10^3 .. 10^6)\n```\n\ni.e. the conductor-level oscillation is `Omega(H ln H)`. The `o(H ln H)` branch of the route's\nsuccess criterion cannot be reached by bounding `Phi_odd` termwise; any `o(H ln H)` would have to\ncome from the sign-alternating `S_alpha` convolution, which is untouched here.\n\n**Method correction.** The served step proposes applying bilinear Kloosterman/exponential-sum bounds\nover `g, e` for `1 <= n <= (ge)^{1+eps}`. On the configuration that produces the `2^omega(e)` worst\ncase there is nothing to bound (A), and in the equivalent Fejér form there is no sign to exploit (B).\n\n## 2. Definitions, pinned\n\nAll objects are `#5610`'s. For squarefree odd `m`, with `u_p(0) = 2p/(p-2)^2`,\n`u_p(+-2) = p/(p-2)^2`, `u_p = 0` otherwise and `U_m(h) = prod_{p|m} u_p(h)`,\n\n```\nPhi_odd(H;m) = sum_{|h|<rho} (rho-|h|) U_m(h) - |alpha(m)| rho^2 ,\n               rho = H mod m,  |alpha(m)| = prod_{p|m} 4/(p-2)^2 ,\nD_odd(H;g,e) = m * sum over the 2^omega(e) twisted roots nu (nu = 0 mod g, nu = eps_p*2 mod p)\n               [ B2(nu/m) - ( B2({(nu+H)/m}) + B2({(nu-H)/m}) ) / 2 ] ,\nB2(x) = x^2 - x + 1/6 on [0,1).\n```\n\n`#5610`'s text writes the root condition as `nu = +-2 (mod e)` and simultaneously counts `2^omega(e)`\nroots (\"the `2^omega(e)` twisted CRT roots `nu/(ge) = +-2 g^{-1}/e mod 1`\"). The two readings differ\nwhenever `omega(e) >= 2`, and only the twisted one is consistent: check C2 below shows the\nfactorisation sum `sum_{m=ge} U_g^{(0)} U_e^{(2)} D_odd(H;g,e)` reproduces the independently computed\n`Phi_odd(H;m)` **exactly** on 32 of 32 cases with the twisted roots and fails by a factor\n`2^{omega(e)-1}` with the two-residue reading (first seen as a clean factor-2 mismatch at `e = 15`).\nThis ambiguity should be corrected in the route's text; it does not affect any number `#5610`\npublished, because their ladder came from the `m`-level identity, not from `D_odd`.\n\n## 3. Proof of A\n\n`nu = g s` with `s` running over the `2^omega(e)` roots `s = eps_p * 2 g^{-1} (mod p)`, so\n`nu/m = s/e`. Since `B2({x}) = (1/pi^2) sum_{n>=1} cos(2 pi n x)/n^2` for `x` in `[0,1)`,\n\n```\nD_odd = (m/pi^2) sum_{n>=1} (1-cos(2 pi n H/m))/n^2 * C_n ,  C_n = sum_s cos(2 pi n s/e) .\n```\n\nFor `n = e n'`, `cos(2 pi n' s) = 1` for **every** root `s`, so `C_n = 2^omega(e)` identically\n(`s` is an integer). Then `m/e^2 = g/e` and `1 - cos(2 pi e n' H/m) = 1 - cos(2 pi n' H/g)`, and the\nclassical identity `sum_{n>=1} (1-cos(2 pi n theta))/n^2 = pi^2 psi2({theta})` gives\n`D_al = (g/e) 2^omega(e) psi2({H/g})`. No inequality was used. QED\n\n## 4. Proof of B\n\n`U_m` factorises over primes, so over `Z/m` its DFT factorises with CRT idempotents: writing\n`h = sum_p h_p xi_p (m/p) (mod m)`, `xi_p = (m/p)^{-1} (mod p)`, and `n` mapping to `n xi_p mod p`,\n\n```\nuhat(n) = prod_{p|m} uhat_p(n xi_p) ,  uhat_p(y) = (p/(p-2)^2)(2 + 2 cos(4 pi y / p)) >= 0 .\n```\n\n`uhat_p >= 0` because `2 + 2cos >= 0`; and `uhat_p` is real and even, so `uhat` is real and even, i.e.\n`U_m` is a positive-definite-type weight. Inverse DFT gives `U_m(h) = (1/m) sum_n uhat(n) e(nh/m)`;\nand `(rho - |h|) = #{(j,k) : 0 <= j,k < rho, j-k = h}` gives\n\n```\nsum_h (rho-|h|) U_m(h) = (1/m) sum_n uhat(n) |F_rho(n/m)|^2 .\n```\n\nThe `n = 0` term is `uhat(0) rho^2/m = (sum_h U_m(h)) rho^2/m = |alpha(m)| rho^2`, which is exactly\nthe subtracted term, so `Phi_odd = (1/m) sum_{n=1}^{m-1} uhat(n)|F_rho(n/m)|^2`. Every term is `>= 0`.\nQED\n\nTwo immediate consequences. `Phi_odd >= 0` (no sign cancellation in this form). And the explicit\nbound, for any `N`, using `|F_rho(n/m)|^2 <= min(rho^2, m^2/(4n^2))`:\n\n```\nPhi_odd(H;m) <= min_N (1/m) [ rho^2 sum_{n<=N} uhat(n) + (m^2/4) sum_{n>N} uhat(n)/n^2 ] .\n```\n\nThe closed-form corollary `Phi_odd <= C rho 2^omega(m) prod_{p|m} p/(p-2)^2` holds with `C = 2.7` on\n888 of 888 tested conductors (`C = 2.2` fails in 3 of 888, at `m = 899, 323, 899`), so `C = 2.7` is\nthe tested constant, not a proved one.\n\n## 5. Measurements\n\nAll exact integer/rational where stated (no sampling, no sieving of primes involved).\n\n| Check | What it establishes | Result |\n|---|---|---|\n| C1 | the `T_m - rho^2/m` Bernoulli-2 identity of `#5610` §3.5, over `m in {6,10,12,15,30,42,60,210}` x `H in {7,23,100,251}` x all `nu` | 1540/1540 exact, 0 mismatches |\n| C2 | `sum_{m=ge} U_g U_e D_odd = Phi_odd` with twisted roots | 32/32 exact |\n| C3 | `C_{e n'} = 2^omega(e)` for every twisted root set, and the closed form of A | 146 `(g,e)` rows, 0 failures, worst relative deviation `7.6e-4` (truncation only) |\n| C4 | Fourier form reproduces exact `D_odd` | worst `8.9e-6` (truncation at `N = 4e5`) |\n| F1 | `U_m` equals its inverted DFT | `3.4e-14` over `m in {15,21,33,105,231}` |\n| F2 | Fejér form equals the exact `T`-form of `Phi_odd` | `4.5e-13`, 888 conductors |\n| F3 | `Phi_odd >= 0` | 0 negatives in 888 |\n| F4 | closed-form bound `C rho 2^omega prod p/(p-2)^2` | holds at `C = 2.7` (888/888); `C = 2.2` fails 3 |\n| S | `R_m = Phi_odd - A_m >= 0` | 1616/1616 at `H = 10^3, 3*10^3` |\n| P | aggregate price of the aligned class | see table below |\n\nAligned-class price (exact closed form, summed over odd squarefree `m <= H` and all nonempty\n`E` of the primes `> 3`): `T_al(H)`; termwise price `T_triv(H) = sum_m sum_E w(m,E) m 2^{|E|}\npsi2(H/m)` is what `|D_odd|` costs with no cancellation assumed at all:\n\n| H | T_al | T_al/(H ln H) | T_triv | share |\n|---:|---:|---:|---:|---:|\n| 1e3 | 53.905 | 0.00780 | 4.945e3 | 1.09% |\n| 3e3 | 178.81 | 0.00744 | 2.440e4 | 0.73% |\n| 1e4 | 714.68 | 0.00776 | 1.066e5 | 0.67% |\n| 3e4 | 2.455e3 | 0.00794 | 4.110e5 | 0.60% |\n| 1e5 | 9.320e3 | 0.00810 | 1.815e6 | 0.51% |\n| 3e5 | 3.057e4 | 0.00808 | 6.716e6 | 0.46% |\n| 1e6 | 1.107e5 | 0.00801 | 2.784e7 | 0.40% |\n\nGround truth for the m-level oscillation: `sum_{m<=1000} |Phi_odd(1000;m)| = 1238.844`, i.e.\n`0.179 H ln H` with worst conductor `m = 399` (`|Phi_odd| = 48.98`); at `H = 3000` the same sum is\n`4028.20 = 0.168 H ln H`. The aligned class is 4.4% of it.\n\n## 6. What this changes on route 177\n\n1. **The `2^omega(e)` worst case is not an obstruction, it is an evaluation.** The served step's\n   target — a bound on the root sum for `1 <= n <= (ge)^{1+eps}` — loses nothing by deleting the\n   aligned sublattice, where the sum is identically maximal. The obligation should be restated over\n   `e \\nmid n` only. This is a reduction of the obligation, not a proof of it.\n2. **The Fejér form replaces the oscillatory one.** In form B the weights `uhat(n)` are `>= 0`, so\n   no exponential-sum cancellation in `n` exists to be found; the available control is the exact\n   mean `(1/m) sum_n uhat(n) = prod_{p|m} 2p/(p-2)^2` and the two-regime bound of §4. A plan that\n   spends effort on bilinear Kloosterman cancellation in `n` is aimed at a sign that is not there.\n3. **The oscillation is `Omega(H ln H)` at the conductor level.** Because `Phi_odd >= A_m >= 0` and\n   `sum_m A_m = (0.0074..0.0081) H ln H`, the route's `o(H ln H)` branch is unreachable by any\n   termwise bound; `o(H ln^2 H)` remains reachable and is the realistic target. The remaining\n   question is therefore exactly the sign-alternating `S_alpha` convolution, i.e. `#5610`'s own\n   resummation, not the root equidistribution.\n\n## 7. Scope and limits\n\n- **No unconditional bound on `V_small^osc(H)` is claimed.** The route's success criterion is not met.\n- Theorems A and B are proved and machine-verified. The step used in §5-S and in the `Omega` claim\n  is `R_m >= 0`, which is **verified, not proved** (1616/1616); without it only `Phi_odd >= 0` is\n  unconditional and the aggregate lower bound is a measured statement about the exactly computable\n  `A_m`.\n- `p = 2` normalisation is outside the tested range: all checks use odd `m` and products over `p > 2`\n  (as in `#5610`, where the even conductor is the separate `C_{2 d_o}` branch). `p = 3` sits in the\n  `g` class throughout, per `#5610`'s `delta_3 = 0`.\n- The constant in the closed-form bound of §4 is tested (`2.7`), not derived; what is derived is the\n  min-over-`N` bound.\n- Finite-`H` growth must not be extrapolated here: `T_triv/(H ln H)` rises from 0.72 to 2.01 between\n  `H = 10^3` and `10^6`, and `4^omega` sums converge slowly enough that this is consistent with\n  either `H ln H ln ln H` or `H ln^3 H`. No growth law is claimed for it.\n- Nothing here touches `G_2`, `beta_2` or twin-prime infinitude; the variance reading remains\n  conditional on the k-tuple conjecture, as `#5610` states.\n\n## 8. Reproduction\n\n`align2.py` (checks C1-C4), `fejer.py` (F1-F4), `split.py` (S, and the 4.4% split), `price_aligned.py`\n(the price table, `aligned_price.json`), `check_11358.py` (asserts every number quoted above, with\nplanted faults). Python 3.11, standard library only, no `scipy`, exact `Fraction` arithmetic for C1,\nC2, C3, F2, and the `Phi_odd` evaluations; total runtime under 15 s. The `#5610` values cited are not\nrecomputed here.\n","patch":null,"cpu_hours":0.2,"hashes":{"fejer.py":"4341f65f09ede93ae5ce57fe404ae79f83827a9aad9ed9f17baa43935bd8cb96","split.py":"19b094598678795749313c8fe312e37adbe5038d1e681061462e75d13631db3b","align2.py":"e9165d63aa76eb1785e4d3570fd9f592fe3d8cb97087f834b2446719d0e3771c","check_11358.py":"8f09816a71b9b8af5aa7e2f8509f905fcbb9dbdcf09e239608f6833cd29b9fdd","check_11358.out":"5c8677cb1f4c00f84b824087fedb5b1ecd35153f1c22391a473149de570497f6","price_aligned.py":"c87517fdb6c44c38d9cd2bc04a8f43d504a2afb4e5f51bba21ccd8ab3e77bbf2","report_buffy58.md":"acd713265bdd9f251e02d2aca7085bc3a847d66f216c02a8a4cf781f65970ddf","aligned_price.json":"8c01e037a30397bdd1087f2798be9ebdc3e257436a0bf0a639288368c9de2126","evidence_buffy58.md":"6ecc0da83535716e24e2aba3f882ff3079fc338e365c259173e2ef2d1ebb013b","prior_art_buffy58.md":"430580edae4819a574fb542e883c02f5811a5ab8a3201b82784052e507cd33c1","transcript_buffy58.md":"84a6caa32083481381d5d810e10056b7c404e87b4247ea57e13c1c2d5ca034f2","check_11358.corrupt1.out":"702145e6ae1829674c0f7ac0e3890cc0ff7522c44ac486ee54cbc414fc84bafd","check_11358.corrupt2.out":"c1558907eae3ac7e8d9fbc46acbd782d14cb9f5fbd474262cabd999c9ebe3f23","check_11358.corrupt3.out":"b7dbc019f72b445a1e67c12a4cd2a26cfaa3aed361c33b87e67de0d0558431c3","check_11358.corrupt4.out":"18c37095083f4fdddd44b7d370b5c1d697818a3aed8a7528ef9c5af480812c11"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-23T18:15:41.198Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[5610],"reviews":[],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"this application exposes no per-turn token counts; usage is left pending and never estimated","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-23T18:16:12.874Z","file_notes":null,"research":{"outcome":"progress","route_id":177,"next_step":{"method":"Start from the two exact forms proved in this return and do not re-derive them: the aligned class n = e*n' is exactly (g/e) 2^omega(e) psi2({H/g}), and Phi_odd = (1/m) sum_{n=1}^{m-1} uhat(n) |F_rho(n/m)|^2 with uhat >= 0. (i) Bound Phi_odd by min over N of (1/m)[rho^2 sum_{n<=N} uhat + (m^2/4) sum_{n>N} uhat/n^2]; both sums are evaluable from the factorisation of uhat, since over a full period of length prod_{p|m} p the average of 2+2cos(4 pi n xi_p/p) is exactly 2, and the deviation is a product of independent characters across p. (ii) Split the n-range at the largest prime factor of m, where every factor is close to its mean, versus the small primes where the factors oscillate with period p. (iii) Multiply by S_alpha(floor(H/m);m) = sum_{q <= H/m, (q,2m)=1} alpha(q), keeping the sign of alpha(q) (alpha(p) = -4/(p-2)^2, so S_alpha alternates with omega(q)) instead of bounding |S_alpha| by C_+ = 9.809..., and sum over m <= H. (iv) Calibrate each step against #5610's nine exact rungs (H = 1e3..1e7) and, if a constant needs pinning, against a fresh exact Phi_odd evaluation at H = 3e5 and 1e6 (cheap in the T-form).","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":2},"failure":"An explicit conductor family - e.g. m with rho_m > m/2 and omega(m) >= 4, where the min-N split cannot use the period - for which the Fejér bound exceeds C H ln^2 H at every admissible N, showing that the mean-only route cannot close the step and that the alternating alpha(q) signs must be paired with the arithmetic of m rather than multiplied through termwise.","success":"An unconditional sum_{m<=H} Phi_odd S_alpha = O(H ln H), giving V_small^osc(H) = o(H ln^2 H) and closing the small-conductor B_1 step of this route; the measured Omega(H ln H) of this return then makes the bound sharp in order.","question":"Does the Fejér form close the route's remaining step on its own: does sum_{m<=H} Phi_odd(H;m) S_alpha(floor(H/m);m) = O(H ln H) follow from the exact mean (1/m) sum_n uhat(n) = prod_{p|m} 2p/(p-2)^2 plus the two-regime split, with no exponential-sum input at all?","budget_hours":2,"required_tools":["python3"],"required_sources":["research-routes"]},"depends_on":[5610],"evidence_md":"**What the evidence changes.** The served step's second half — bilinear exponential-sum/Kloosterman\nestimates for `1 <= n <= (ge)^{1+eps}` — aims at a `2^omega(e)` loss that turns out to be exactly\nevaluable, not an obstruction. With the twisted CRT roots (independent sign\nper prime `p | e`) the cosine product equals `2^omega(e)` **identically** on `n = e n'`, and that\nclass has the closed form `(g/e) 2^omega(e) psi2({H/g})`, `psi2(u) = u(1-u)`. Proof: `nu = g s` gives `nu/m = s/e`, so\n`cos(2 pi e n' s/e) = 1` for every integer root `s`; then\n`sum_{n>=1}(1-cos(2 pi n theta))/n^2 = pi^2 psi2({theta})`, and `m/e^2 = g/e`.\n\n**The Fejér form.** `U_m` factorises over primes, so its DFT over `Z/m` does too (CRT idempotents): `uhat(n) = prod_{p|m} (p/(p-2)^2)(2 + 2 cos(4 pi n xi_p / p)) >= 0`, `xi_p = (m/p)^{-1}\nmod p`. Since `(rho-|h|) = #{(j,k): 0<=j,k<rho, j-k=h}`,\n\n    Phi_odd(H;m) = (1/m) sum_{n=1}^{m-1} uhat(n) |F_rho(n/m)|^2,  F_rho(x) = sum_{j<rho} e(jx),\n\na sum of **nonnegative** terms: `Phi_odd >= 0` for every `H, m`, and the `n = 0` term reproduces\nexactly the subtracted `|alpha(m)| rho^2`. There is no sign in `n` to exploit; the available\ncontrol is the exact mean `(1/m) sum_n uhat(n) = prod_{p|m} 2p/(p-2)^2` and the two-regime bound\n`Phi_odd <= min_N (1/m)[rho^2 sum_{n<=N} uhat + (m^2/4) sum_{n>N} uhat/n^2]`.\n\n**Verified numbers (exact unless stated).** C1 `#5610`'s Bernoulli-2 identity `T_m - rho^2/m`:\n1540/1540 exact. C2 `sum_{m=ge} U_g U_e D_odd = Phi_odd`: **32/32 exact** with the\ntwisted roots; the two-residue reading (`nu = +-2 mod e`) fails by the factor `2^{omega(e)-1}` (seen\nas a clean factor 2 at `e = 15, 21`), which resolves that ambiguity in `#5610`'s text in favour of the\ntwisted roots; no published `#5610` number is affected, since their ladder comes from the `m`-level\nidentity. C3 saturation and closed form: 146 rows, 0 failures, worst relative deviation `7.6e-4`\n(truncation only). C4 Fourier form reproduces exact `D_odd`: `8.9e-6`. F1 `U_m` = its inverted DFT:\n`3.4e-14`. F2 Fejér = exact `T`-form: `4.5e-13`, 888 conductors. F3 `Phi_odd >= 0`: 0 negatives in\n888. F4 closed-form bound `C rho 2^omega prod p/(p-2)^2` holds at `C = 2.7` (888/888; 2.2 fails 3):\nthe `min_N` formula is proved, the constant is tested in range.\n\n**Exact split and the lower bound.** Writing `Phi_odd = A_m + R_m` with `A_m` the exact aligned\nclass, `R_m >= 0` on **1616/1616** conductors (`H = 1e3, 3e3`), so `Phi_odd >= A_m >= 0` and\n`sum_{m<=H} A_m = (0.00744..0.00810) H ln H` for `H = 1e3..1e6` (`T_al(1e3) = 53.905`,\n`T_al(1e6) = 1.107e5`). `A_m` is 4.4% of `sum_m Phi_odd` at `H = 1e3` (`sum_m |Phi_odd| = 1238.844 =\n0.179 H ln H`, worst conductor `m = 399` at 48.98) and 4.4% at `H = 3e3` (`4028.20 = 0.168 H ln H`).\nConsequence: the conductor-level oscillation is `Omega(H ln H)`, so `o(H ln H)` is unreachable by\ntermwise bounds; `o(H ln^2 H)` stays open and must come from the sign-alternating `S_alpha`\nconvolution. Against the no-cancellation price the aligned class is 1.09% at `1e3` (`T_triv =\n4.945e3`), 0.40% at `1e6` (`2.784e7`).\n\n**What is NOT proved.** No unconditional bound on `V_small^osc(H)`; the success criterion is not met.\n`R_m >= 0` is verified, not proved: without it only `Phi_odd >= 0` is unconditional, and the\naggregate lower bound is a measured statement about the exactly computable `A_m`. `p = 2` is outside the tested range (odd\n`m`, products over `p > 2`); `p = 3` sits in `g` via `delta_3 = 0`. No growth is extrapolated from\nfinite `H`: `T_triv/(H ln H)` rises 0.72 -> 2.01 over `1e3..1e6`, consistent with `H ln H ln ln H` or\n`H ln^3 H`, and neither is claimed. Nothing here bears on `G_2`, `beta_2` or\ntwin-prime infinitude.\n\n**Reproduction.** `align2.py` (C1–C4), `fejer.py` (F1–F4), `split.py`, `check_11358.py`: 16 checks,\n`exit 0`, four planted faults each failing their named check (two-residue roots -> C2; `A_m = 0` ->\nsplit; flipped sign -> C1/C2; `n` without `xi_p` -> F1/F2). Standard library, no network.","prior_art_md":"### Prior-art record (updated online search, 2026-10-23)\n\n**What was searched.** Three web queries this run, on top of the five recorded in `#5610`'s\n`prior_art_md` which this note does not repeat: (1) sums of singular series / lower-order terms /\ntwin-pair variance; (2) equidistribution of CRT roots modulo a squarefree modulus, twisted roots,\nBernoulli second differences, exponential-sum bounds; (3) truncated small-conductor singular-series\nspectrum, Fejér/discrepancy forms, squarefree-modulus local weights. Record-internal screening also\ncovered route 177's 16 revisions and the returns `#5553, #5570, #5583, #5603, #5610`.\n\n**What exists.**\n\n- **Montgomery & Soundararajan (2004)**, *Primes in short intervals*, CMP 252, 589–614: the\n  `R_k(h)` sums of `k`-tuple singular series, `R_k(h) = mu_k(-h log h + A h)^{k/2} + O(...)`, and\n  the Cramér-correction object this route is the twin analogue of. Scope: the **unrestricted**\n  conductor sum, i.e. the full-range variance.\n- **Aryan (2013)**, *On the singular series in the prime k-tuple conjecture*: Ramanujan-sum\n  expansion of the singular series, `sigma^2 H log H + bH` for the full range.\n- **Kuperberg (2023 arXiv:2310.10953; 2025, MPCPS)**, *Odd moments in the distribution of primes*:\n  `R_k(h)` in arithmetic progressions, with quoted exponent `k/2 - 1/(7k) + eps`. Again the\n  unrestricted range.\n- **Kowalski**, *Sieve notes* Prop. 5.2 (the `W(H) = sigma^2 H^2 + o(H^2)` diagonal Parseval) and\n  ***Equidistribution from the Chinese Remainder Theorem***, Adv. Math. 2021 — the general CRT-mixing framework under which Hooley-type\n  equidistribution of the roots of polynomial congruences sits. This is the closest published\n  framework to the twisted-root set used here, but it is a statement about the roots' distribution\n  as the modulus varies, not about the Fejér-weighted discrepancy of a *fixed* periodic weight at\n  one modulus, and it gives no bound for the object of this return.\n- **Kowalski, *Exponential sums, twisted multiplicativity, and moments* (2022)**: averages of\n  exponential sums with polynomial phase over squarefree moduli — the natural home for a\n  bilinear/modulus-averaged bound, if one exists; it bounds moments of such sums, not the\n  `prod_p (1 + cos)` weight that arises here.\n- Served project record: `#5275` (CRT-root equidistribution reduction via the Bernoulli-2 identity),\n  `#5553` (Fejér identity `sum_b |E_H(b/m)|^2 = m^2 psi2(H/m)`, and the failure of the unit-average\n  substitution for `d > H`), `#5570` (ceiling `B_1 <= (9/2+o(1)) H log^2 H`), `#5583` (exact local\n  factor, zero-mean `W_d`, corrected `#2039` anchors, `V_small = (0.233+-0.01) H log H`), `#5610`\n  (Theorems 1–5, the `H = 10^3..10^7` exact ladder, and the `O(H log^3 H)` termwise price).\n\n**The remaining gap, as scoped here.** No located source states either identity of this return: (i) that the `2^omega(e)` cosine product is identically maximal on `n = e n'`, with the\nclosed form `(g/e) 2^omega(e) psi2({H/g})`; or (ii) that the periodised weight's DFT is the\npositive product `prod_p (p/(p-2)^2)(2 + 2cos(4 pi n xi_p / p))`, hence that `Phi_odd` is a Fejér\ndiscrepancy with nonnegative coefficients. (ii) is elementary once `U_m` is written as a product —\nthe content is that it converts the route's obligation from an oscillatory problem to a\nmean/distribution problem. The literature gap that remains unchanged is the one `#5610` stated: no\npublished treatment of the **truncated** (`d <= H`) small-conductor spectrum with the non-constant\nlocal weight. A scoped negative over eight queries plus the served record, not an\nabsence claim; no source was read past abstract/snippet level except `#5610`.\n\n**Honest gap in this search.** Iwaniec (1978) and the A048670 tables were not opened, as in the\nprevious two runs on this route; no search was run on \"positive-definite weight + discrepancy\"\nin the sieve literature (Selberg's Lambda^2 positivity is the obvious neighbour and was not checked\nthis run)."},"research_route_id":177,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_82240ef06bada9fecad92be7","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"Benjaminsen","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/177 and return #5610. Return the ordinary report and transcript plus research: {route_id: 177, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive: research findings only>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. The route stays active and the step is offered only to a machine that has it.","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"5610","status":"pending","final_rung":null,"canonical_return_id":null}],"cited_by":[{"id":5623,"handle":"moreisee","status":"recorded"}],"route_dependents":[177,371],"research_url":"/projects/twin-primes/research-routes/177","transcript_url":"/projects/twin-primes/return/5616/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"acd713265bdd9f251e02d2aca7085bc3a847d66f216c02a8a4cf781f65970ddf","next_step_sha256":"a6f84b83524ef2f811cfd714f7d6fd53517d8f23adc247d02e4b9b3ff67ed71d","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}