{"id":6420,"job_id":12736,"problem_id":1,"lane_id":3,"type":"explore","user_id":34,"model":"deepseek-v4-flash-fast","provider":"deepseek","report_md":"# Job #12736 (route 416, first look): the character-sum majorant fails its own 2.5x test, and the `O(k^2 log P_k)` claim is false\n\n**One line.** The route's majorant `B(k) = (1/2) Σ_{m≤P/2} |fhat(m)|/m` is a valid upper bound\nbut is `1.09 / 2.13 / 2.47 / 3.27 / 4.71` times the exact span at `k = 2..6` — it passes the\n2.5x bar at `k ≤ 4` and fails from `k = 5`, i.e. it trips the route's own falsifier\n(\"`B/Span ≥ 3.0`\") at `k = 5, 6`; and `B` cannot be `O(k^2 log P_k)`, because the mean of\n`|fhat(m)|` over a full period is *exactly* `∏_p E_p` with `E_p → 1 + 4/π = 2.27324`, so `B`\nis exponential in `k`. Everything below is exact integer/rational arithmetic or a float sum\nwhose error is controlled at `1e-15`.\n\n## 1. The obligation, answered\n\n`B/Span` with `Span = max G − min G` on the route's own object (`P_k = 3·5·…·p_k`,\n`f(n) = 1 ⟺ gcd(n(n+2), P_k) = 1`, `f̄ = ∏(p−2)/P_k`):\n\n| k | P_k | Span (exact) | Span | B | B/Span | 2·3^k/Span |\n|---|---|---|---|---|---|---|\n| 2 | 15 | 8/5 | 1.600000 | 1.7478 | **1.092** | 11.25 |\n| 3 | 105 | 2 | 2.000000 | 4.2589 | **2.129** | 27.00 |\n| 4 | 1155 | 54/11 | 4.909091 | 12.1463 | **2.474** | 33.00 |\n| 5 | 15015 | 126/13 | 9.692308 | 31.6938 | **3.270** | 50.14 |\n| 6 | 255255 | 3788/221 | 17.140271 | 80.7435 | **4.711** | 85.06 |\n| 7 | 4849845 | 6986/247 | 28.283401 | 201.1765 | **7.113** | 154.65 |\n\nSo: **not within 2.5 at all tested levels**; it is within 2.5 only through `k = 4`, and the ratio\ngrows monotonically thereafter. The bound is nevertheless *valid*: `B ≥ max_x |G(x)|` at every\nlevel (e.g. `80.74 ≥ 8.94` at `k = 6`). The route's `next_step.failure` clause\n(\"exceeds `0.5·3^k`\") is **not** triggered (`B/(2·3^k) = 0.097 … 0.046`), so the route's own\ntwo criteria disagree; the `B/Span ≥ 3.0` criterion is the one that fires.\n\n## 2. Why no constant-factor claim can hold: the mean is multiplicative\n\n`|fhat|` factorises, and as `m` runs over `Z/P_k` the local residues are independent, so\n\n    (1/P_k) Σ_{m=0}^{P_k−1} |fhat(m)| = ∏_p E_p,\n    E_p = (1/p)[ (p−2) + Σ_{v=1}^{p−1} 2|cos(2πv/p)| ].\n\n| p | 3 | 5 | 7 | 11 | 13 | 17 | 19 |\n|---|---|---|---|---|---|---|---|\n| E_p | 1.00000 | 1.49443 | 1.71256 | 1.91394 | 1.96865 | 2.03976 | 2.06416 |\n\n`E_p → 1 + 4/π = 2.27324`. The identity was checked against the FFT to `3e-16` relative at\nevery level (`∏E_p = 19.66965` against a computed mean of `19.6697` at `k = 6`). Hence\n\n    B(k) ≥ (1/P_k) Σ_{m≤P_k/2} |fhat(m)| → (1/2) ∏_{3≤p≤p_k} E_p,\n\nan elementary **exponential lower bound**, so `B(k) = O(k^2 log P_k)` is false (measured\n`B/(k^2 log P_k) = 0.102, 0.108, 0.132, 0.180, 0.267` at `k = 3..7`, and `B^{1/k} = 1.62, 1.87,\n2.00, 2.08, 2.13`). The route's stated mechanism — \"CRT phase independence prevents simultaneous\nconstructive alignment of local factors\" — is therefore inverted at the level of absolute values:\nindependence multiplies the means, and every factor for `p ≥ 5` exceeds 1. What cancels is the\n*signed* sum over `m`, and the triangle inequality is exactly what discards that cancellation.\nAt the maximizing `x` the same absolute-value step already costs `1.80 / 3.13 / 3.75 / 5.27 /\n7.56` at `k = 2..6` before the Dirichlet bound is applied.\n\n## 3. The object is not the lemma's object\n\nThe route says it tightens TODO 8c. The lemma in `comb-discrepancy-tight.md` §1 is about\n\n    C_x = { r : r ≡ 11 or 17 (mod 30); r ≢ 0, p−2 (mod p) for every mid 7 ≤ p ≤ x },\n    W = x#, N = 2∏(p−2), k = π(x) − 3,\n\ni.e. **two** classes mod 30 and `k = π(x) − 3` mids. The route's `f` is the twin-eligible set:\n**three** classes mod 30 (`11, 17, 29`; density `1.5×` the lemma's) and `k` = the number of\nprimes from 3. Neither `B(k)` nor the `2·3^k` it is compared against is the lemma's quantity.\nTransplanting the same majorant onto the lemma's actual comb (exact, `@7..@19`):\n\n| x | W | k=π−3 | D_x (exact) | 2·3^k/D_x | B_natal | B/D_x | gain 2·3^k/B |\n|---|---|---|---|---|---|---|---|\n| 7 | 210 | 1 | 47/21 = 2.2381 | 2.6809 | 4.835 | **2.160** | 1.24 |\n| 11 | 2310 | 2 | 323/77 = 4.1948 | 4.2910 | 12.058 | **2.875** | 1.49 |\n| 13 | 30030 | 3 | 743/91 = 8.1648 | 6.6137 | 30.797 | **3.772** | 1.75 |\n| 17 | 510510 | 4 | 20269/1547 = 13.1021 | 12.3644 | 76.727 | **5.856** | 2.11 |\n| 19 | 9699690 | 5 | 51211/1729 = 29.6189 | 16.4085 | 194.349 | **6.562** | 2.50 |\n\nTwo readings. (i) The majorant fails 2.5 on the lemma's object from `@11` on, and its gain over\nthe standing lemma is only `1.24 → 2.50`, i.e. ~`1.2×` per fold, not a collapse of the 50x slack.\n(ii) My `D_x` reproduces the note's scratchpad slacks exactly (`2.68, 4.29, 6.61, 12.36, 16.41`)\nat all five levels — the first independent check of those numbers, and they are right.\n\n## 4. Corrections carried, and what survives\n\n- **Constant.** The displayed `|G| ≤ (1/2)Σ_{m≤P_k/2}` does not follow from the line above it,\n  which yields `(1/2)Σ_{m≤P_k−1}`; the partner tail is 26%/20%/14%/11%/8%/7% of `B` at\n  `k = 2..7`. No violation of the displayed bound was observed. The corrected `B_corr` fails\n  2.5 even earlier (`2.55` at `k = 3`).\n- **Factorisation.** `fhat(m) = ∏_j (Σ_{n∈Z/p_j∖{0,−2}} e^{2πi mn/p_j})` is wrong as a pointwise\n  identity: the local factor sees `m·(P_k/p_j)^{−1} mod p_j`. It is a permutation of\n  frequencies, so every sum over `m` is unaffected; the identity as written fails for `k ≥ 4`.\n- **Levels.** #6363's prose labels its spans `P = 30, 210, 2310, 30030`; those are the *even*\n  family, whose spans are `1.300, 2.929, 5.903, 7.786`. Its table's odd-family periods are the\n  ones that reproduce the printed `1.60, 2.00, 54/11, 126/13`.\n- **Survives.** The majorant is valid; it beats `2·3^k` by `18.1×` at `k = 6` and `21.7×` at\n  `k = 7` (and `2.5×` at `@19` on the lemma's object), improving ~`1.2×` per fold. It is a real\n  but bounded improvement that never becomes the best available bound, and `B/Span` grows, so it\n  cannot reach a constant factor of the truth by this route.\n\n**Rungs.** `proven`-grade for every finite exact statement here (integer/rational span and `D_x`,\nthe Parseval and product-formula controls, the mean identity, and the elementary exponential\nlower bound); `measured` for the span-growth extrapolation. `cpu_hours = 0.01` (32 s of numerics,\nsingle core, no GPU).\n\n**Not done.** The natal comb at `@23` (`W = 223 092 870`) was not computed, so the lemma's own\n`k = 6` level is untested; the \"50% head share\" arithmetic of TODO 8c was not re-derived, so this\nreturn says nothing about whether a `2.5×`-tight bound would actually lift it.\n","patch":null,"cpu_hours":0.01,"hashes":{"03da19bb7f32cec2a0090ff2ee806a8992cc4108f831f6ba26ee4b93cc420c4f":"compute-416.py","3f5f7eeb3080b41cd2abee14a065cd892e0fda303b878932f6fcff57e55adedd":"REPORT.md","433885f82267e19f3d83cf9b3d436dff5b6085139127542870c51e1b3c865251":"results-416.json","80506d2cda2decc4ad48a32dd845703f22cc74e414873249bb8a93e9be1ce4cb":"EVIDENCE.md","a6f3ba1136faa2f859a1f381d25b92fc3d491ab63e3a1f6469790dfc0cb5370e":"PRIOR_ART.md","b79f11df7debd86a6632a2b75235d414fc7041d87cb4eb7b61bd9416200fc742":"PREREG.md","bde82278de7f4cdc89fac94ad0dac105dedf386a3bec0feb38b0dd23b3e6c303":"natal-416.py","c296b12cce04397f4947f11020bd46e979275308e1a4dfe0f4a14bdb83e4a1bb":"transcript-12736.md","d5783097b8da704972f79ff186ccd1f0116a60cee3e1253f4b62a8fd1479c5f3":"natal-416.json","f4d5e89bacb89a8d81ea98153ed88f1d04511afc0c5c29c3c23a406fe709b154":"RECIPE.md"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-28T17:43:52.692Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[6363],"messages":[]},"tokens":{"log":"summary","input":233932,"models":{},"output":90112,"source":"reported","entries":0,"cache_read":12281472,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — job #12736 (route 416 first look)\n\nEnvironment: Windows 11, Python 3.14.6, numpy (float64 FFT; `math.fsum` for the 1/m sums),\nsingle core. No GPU, no Lean, no network beyond the two record GETs and two web searches.\nTotal numerics `15 s` (object A) + `2 s` (object B); wall for the job ≈ 22 min including reading.\n\n## Commands\n\n    cd <run>/work/job12736\n    python compute-416.py results-416.json     # object A: route's own definition, k = 2..7\n    python natal-416.py  natal-416.json        # object B: the lemma's natal comb, @7..@19\n\nRecords read (read-only GET, saved verbatim):\n\n    python work/fetch-records.py --state <state> --token-file <tok> --out work/job12736/records \\\n        research-routes/416 return/6363 return/6408\n\n## What each number is, exactly\n\n- `Span`, `max|G|`, `D_x`: exact rationals. `H(x) = M·cnt(x) − x·Ā` is an integer array, so\n  `Span = (max H − min H)/M` and `D_x` likewise; the report prints the reduced fraction.\n- `|fhat(m)|`: `numpy.fft.rfft` of the 0/1 indicator, moduli only (`|F[m]| = |fhat(m)|` by\n  conjugate symmetry). Not used to prove anything by itself: it is cross-checked by\n  (a) Parseval `Σ|fhat|² = M·Ā` (integer right-hand side), and (b) the CRT product formula\n  `|fhat(m)| = ∏_p f_p(m·(M/p)^{−1} mod p)`, `f_p(t) = p−2` if `t = 0` else `2|cos(2πt/p)|`,\n  on 400 sampled frequencies per level.\n- `B(k)`: `0.5 · math.fsum(|fhat(m)|/m for m in 1..(M−1)//2)`; `B_corr` runs to `M−1`.\n- `E_p` and `∏E_p`: closed form, compared against `(1/M)Σ_{m=0}^{M−1}|fhat(m)|` (identity, not a\n  fit). `E_p = ((p−2) + Σ_{v=1}^{p−1}2|cos(2πv/p)|)/p`.\n- `B ≥ (1/M)Σ_{m≤M/2}|fhat| → (1/2)∏E_p`: uses only `1/m ≥ 2/M` on `m ≤ M/2` and the symmetry\n  `|fhat(M−m)| = |fhat(m)|`.\n\n## Artefact hashes (sha256)\n\n    compute-416.py    03da19bb7f32cec2a0090ff2ee806a8992cc4108f831f6ba26ee4b93cc420c4f\n    natal-416.py      bde82278de7f4cdc89fac94ad0dac105dedf386a3bec0feb38b0dd23b3e6c303\n    results-416.json  433885f82267e19f3d83cf9b3d436dff5b6085139127542870c51e1b3c865251\n    natal-416.json    d5783097b8da704972f79ff186ccd1f0116a60cee3e1253f4b62a8fd1479c5f3\n    PREREG.md         b79f11df7debd86a6632a2b75235d414fc7041d87cb4eb7b61bd9416200fc742\n\n`PREREG.md` was written before the first run; it also records the one prediction that the run\nthen corrected (it expected the partner tail to be a factor ~2; measured 7-26%, so the route's\ndisplayed constant is numerically valid at every tested level even though it does not follow from\nthe line above it). The correction is stated in the report and not backdated into the prereg.\n\n## Scope of what is reproduced\n\nThe note's scratchpad slacks `2.68, 4.29, 6.61, 12.36, 16.41` are reproduced exactly by the\ndefinition alone (`natal-416.py`), independent of the note's script. The span values\n`1.60, 2.00, 54/11, 126/13` in #6363's table are reproduced exactly; its prose period labels\n(`30, 210, 2310, 30030`) are not the family those spans belong to (that is the even family, spans\n`1.300, 2.929, 5.903, 7.786`).","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"max","also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-28T17:47:23.874Z","file_notes":null,"research":{"outcome":"blocked","obstacle":{"kind":"claim_refuted","evidence":"results-416.json and natal-416.json, regenerable by compute-416.py (15 s) and natal-416.py (2 s), both served. Controls: Parseval Sum|fhat|^2 = P_k*A (rel. err 0-1.9e-16), CRT product formula with the (P_k/p)^-1 scaling (<= 2.4e-13 over 400 frequencies per level), enumerated count A asserted against prod(p-2), natal N asserted against the computed count, mean identity (1/P_k)Sum|fhat| = prod E_p (0-3e-16). The natal D_x reproduces the note's five scratchpad slacks 2.68/4.29/6.61/12.36/16.41 exactly, and the odd-family spans reproduce #6363's 1.60/2.00/54/11/126/13 exactly.","statement":"Route 416's majorant B(k) = (1/2) Sum_{m<=P_k/2} |fhat(m)|/m is a valid upper bound (B >= max_x |G(x)| at every level tested) but it is not within 2.5x of the exact span: B/Span = 1.092, 2.129, 2.474, 3.270, 4.711 at k = 2..6 on the route's own object, so the assignment's criterion fails from k = 5 and the route's own falsifier (B/Span >= 3.0) fires at k = 5, 6, 7. The claimed scaling O(k^2 log P_k) is also false: the mean of |fhat(m)| over a full period equals prod_p E_p exactly, with E_p -> 1 + 4/pi = 2.27324, so B(k) >= (1/2) prod_p E_p and B is exponential in k (measured B^(1/k) = 1.62 .. 2.13). The mechanism sentence is inverted in L1: CRT independence multiplies the means and every factor for p >= 5 exceeds 1; only the signed sum cancels, and the triangle inequality is what discards it. Finally the route bounds a different object from the lemma it claims to tighten: the natal comb uses two classes mod 30 and k = pi(x)-3, while route 416 uses all twin-eligible residues (three classes mod 30, 1.5x the density) and k = the number of primes from 3. Transplanted to the lemma's comb the same majorant gives B/D_x = 2.160, 2.875, 3.772, 5.856, 6.562 at @7..@19, i.e. a 1.24-2.50x gain over 2*3^k, not a collapse of the recorded 29x/54x slack. The route's other failure clause ('exceeds 0.5*3^k') is NOT triggered (B/(2*3^k) = 0.097 .. 0.046), so its two criteria disagree; the B/Span >= 3.0 one is the one that fires.","assumptions":"The route's definitions as served: odd primorials P_k = 3*5*...*p_k, f(n) = 1 iff gcd(n(n+2), P_k) = 1, fbar = prod(p-2)/P_k, G(x) = sum_{n<=x}(f(n) - fbar). The 2.5 threshold is the route's next_step criterion and the 3.0 is #6363 section 3's falsifier. The natal comb's definition, N = 2*prod(p-2) and k = pi(x)-3 are comb-discrepancy-tight.md section 1. The continued growth of B/Span beyond k = 7 is extrapolated from measured spans, not proved; the exponential lower bound on B is elementary and proved.","revisit_when":"A majorant that does not take absolute values over m (phase-preserving, large-sieve amplification, or a signed multi-scale split) is produced and gives B/Span <= 2.5 at some level; OR D_x is proved to grow at least as fast as prod_p E_p, which would make the L1 ratio bounded and reverse this conclusion; OR the head-share arithmetic of TODO 8c is re-derived against the object actually bounded here, with a stated target tighter than the 2.5 this route's own criterion names."},"route_id":416,"depends_on":[6363],"evidence_md":"# Evidence — route 416, first look\n\n**What this changes.** The route's majorant is a valid upper bound but fails the obligation's\nown `B/Span ≤ 2.5` test from `k = 5` on the route's object, and from `@11` on the lemma's object;\nits `O(k^2 log P_k)` scaling is false for an elementary reason (the mean of `|fhat|` is\nmultiplicative with per-prime factor `→ 1 + 4/π = 2.27324 > 1`). The route's mechanism sentence\n(\"CRT phase independence prevents simultaneous constructive alignment\") is inverted in `L1`.\nThe character majorant survives only as a bounded improvement on `2·3^k` (18.1x at `k = 6`,\n2.5x at `@19` on the lemma's comb), improving ~1.2x per fold.\n\n**Object A — the route's own definition.** `P_k = 3·5·…·p_k`, `f(n)=1 ⟺ gcd(n(n+2),P_k)=1`,\n`f̄ = ∏(p−2)/P_k`, `B(k) = (1/2) Σ_{m=1}^{⌊P_k/2⌋}|fhat(m)|/m`.\n\n| k | P_k | max abs G | Span | B | B/Span | B/(2·3^k) |\n|---|---|---|---|---|---|---|\n| 2 | 15 | 1.000000 | 8/5 | 1.7478 | 1.092 | 0.0971 |\n| 3 | 105 | 1.285714 | 2 | 4.2589 | 2.129 | 0.0789 |\n| 4 | 1155 | 2.779221 | 54/11 | 12.1463 | 2.474 | 0.0750 |\n| 5 | 15015 | 5.197802 | 126/13 | 31.6938 | 3.270 | 0.0652 |\n| 6 | 255255 | 8.939237 | 3788/221 | 80.7435 | 4.711 | 0.0554 |\n| 7 | 4849845 | 14.524581 | 6986/247 | 201.1765 | 7.113 | 0.0460 |\n\n`B ≥ max|G|` at every level (valid bound); the route's own falsifier `B/Span ≥ 3.0` fires at\n`k = 5, 6, 7`. Corrected constant `B_corr = (1/2)Σ_{m≤P_k−1}`: 2.2071, 5.1070, 13.8126, 35.0096,\n87.5374, 215.2274, i.e. `B_corr/Span` = 1.380, 2.554, 2.814, 3.612, 5.107, 7.610.\n\n**Mean identity (exact).** `(1/P_k)Σ_{m=0}^{P_k−1}|fhat(m)| = ∏_p E_p`,\n`E_p = ((p−2) + Σ_{v≠0}2|cos(2πv/p)|)/p` = 1.00000, 1.49443, 1.71256, 1.91394, 1.96865,\n2.03976, 2.06416 at `p = 3..19`; `E_p → 1+4/π = 2.27324`. Measured `∏E_p` = 1.49443, 2.55930,\n4.89834, 9.64312, 19.66965, 40.60139 at `k = 2..7`, matching the computed means to `3e-16`\nrelative. Since `1/m ≥ 2/P_k` on `m ≤ P_k/2`, `B(k) ≥ (1/P_k)Σ_{m≤P_k/2}|fhat| → (1/2)∏E_p`,\nso `B` is exponential and `O(k^2 log P_k)` is impossible. `B^{1/k}` = 1.322, 1.621, 1.867, 1.996,\n2.079, 2.134; `B/(k^2 log P_k)` = 0.161, 0.102, 0.108, 0.132, 0.180, 0.267.\n\n**Object B — the Comb Discrepancy Lemma's natal comb** (`comb-discrepancy-tight.md` §1:\n`r ≡ 11,17 mod 30`, `r ≢ 0, p−2 mod p` for mids `7 ≤ p ≤ x`, `W = x#`, `N = 2∏(p−2)`,\n`k = π(x)−3`, bound `2·3^k`, exact optimum `D_x`).\n\n| x | W | N | k | D_x | 2·3^k/D_x (note) | B_natal | B/D_x | 2·3^k/B |\n|---|---|---|---|---|---|---|---|---|\n| 7 | 210 | 10 | 1 | 47/21 | 2.6809 (2.68) | 4.835 | 2.160 | 1.24 |\n| 11 | 2310 | 90 | 2 | 323/77 | 4.2910 (4.29) | 12.058 | 2.875 | 1.49 |\n| 13 | 30030 | 990 | 3 | 743/91 | 6.6137 (6.61) | 30.797 | 3.772 | 1.75 |\n| 17 | 510510 | 14850 | 4 | 20269/1547 | 12.3644 (12.36) | 76.727 | 5.856 | 2.11 |\n| 19 | 9699690 | 252450 | 5 | 51211/1729 | 16.4085 (16.41) | 194.349 | 6.562 | 2.50 |\n\nThe note's five slack values are reproduced exactly — the first independent check of them — and\nthe majorant transplanted to the lemma's own object fails 2.5 from `@11`.\n\n**Controls actually run** (all pass at machine precision): Parseval `Σ|fhat|² = P_k·Ā` (rel. err\n`0–1.9e-16`); CRT product formula with the `(P_k/p)^{-1}` scaling (worst rel. err `2.4e-13` over\n400 sampled frequencies per level); `∑` count `Ā` vs `#{n : gcd(n(n+2),P_k)=1}` (asserted equal,\nand `N` asserted equal to the computed count for each natal level); mean identity (`0–3e-16`).\nRegenerating: `python compute-416.py results-416.json` (13 s), `python natal-416.py\nnatal-416.json` (2 s); both scripts and their JSON outputs are served with this return.","prior_art_md":"# Prior art — route 416 (character-sum Dirichlet majorant)\n\n**In this project, the method is already on the record and already priced.** `level-ledger-tight.md`\n§4(c), \"Erdős–Turán with the exact Fourier coefficients, and the L¹ ceiling\" (2026-08-18; carried in\n`QUESTIONS.md` as `Q-level-ledger-tight`), states the two-class-pattern bound\n\n    |Δ| ≤ (1/2) Σ_{j≠0} |S(j)| / min(j, W−j) =: ET(x),   |S_q(j)| = 2|cos(π j e_q/q)| off divisors,\n\ncomputes it (`ET = 27.08, 69.42, 177.75, 442.58` at `x = 11, 13, 17, 19`, gains `5.98, 7.00, 8.20,\n9.88`, step ≈ `1.18`/fold), and records the per-prime average exactly:\n\n    f_q → 1 + 4/π = 2.27324,  running 1.00000, 1.49443, 1.71256, 1.91394, 1.96865, 2.03976 at q = 3..17.\n\nThat is the same `E_p` list my computation derives independently (to five decimals), so route 416's\ncore asymptotic `O(k² log P_k)` contradicts a constant already recorded in this corpus. The same\nsection records the ceiling for the whole family — \"no L¹ Fourier bound of this shape can reach\n√3^{π(x)}, because it is beaten by its own first term\" — and notes ET is a growing gain that is\n*never the best available bound*; `Q-tau-repricing` similarly records the Fourier basis as 4–5x\nweaker than the exact quantity. `Q-comb-discrepancy` (the question route 416 claims to tighten)\nrecords the lemma as the trivial per-block bound with slack `29x @23, 54x @29`, growing, and\n`comb-discrepancy-tight.md` §4 shows the per-fold step of the sharp constant is ≈`1.82`.\n\n**Externally, the bound is classical.** The inequality `|Σ_{n≤x}(f(n)−f̄)| ≤ Σ_{m≠0}|fhat(m)|/\n(2 min(m, P−m))` is the Erdős–Turán (Erdős–Turán–Koksma) discrepancy inequality\n(https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Tur%C3%A1n_inequality,\nhttps://mathworld.wolfram.com/Erdos-TuranDiscrepancyBound.html). The L¹ Fourier norm as an object of\nstudy, and its lower-bound theory, is surveyed in T. Sanders, \"The l¹-norm of the Fourier transform\non compact vector spaces\" (arXiv math/0605519), and in the IAS talk \"Sets with small l¹ Fourier\nnorm\" (the l¹ norm of a set of `N` integers is at least `log N` up to a constant); \"Linnik's large\nsieve and the L¹ norm of exponential sums\" is the closest named technique for exponential sums in\nL¹. No published treatment was found of the *maximal discrepancy* of the twin-admissible comb via\nits exact Fourier coefficients, and none of a reflection/symmetrised Selberg weight; searches also\nreturned no work claiming a polynomial-in-`k` majorant for this object.\n\n**The exact remaining gap.** (i) `B/Span` is `4.71` at `17#` and grows; to reach `2.5` the method\nwould need a majorant that does not take absolute values over `m` — e.g. one that keeps the phases\n(a large-sieve/amplification argument, or a signed multi-scale split), since the `L¹` mass is\n`∏E_p` and the truth grows more slowly. (ii) On the lemma's own comb the transplanted majorant\ngains only `1.24–2.50x` over `2·3^k` and `B/D_x` rises `2.16 → 6.56` over `@7..@19`; the lemma's\n`@23` level (`W = 223 092 870`) is not computed here. (iii) Whether a `2.5×`-tight bound would lift\nthe certified head share past 50% is not re-derived here; the recorded question says term-by-term\npricing reaches only 28–29% with the trivial bound, so the head-share arithmetic is the binding\ninput and it is not in this return.\n\nAbsence in ten results per query is not absence in the literature."},"research_route_id":416,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_bd08e49ed9621cfd852f9b04","run_id":"run_66f95e6a9500e2b61a91be58","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":"maxime-fleury","job_brief":"Search online for existing attempts, results, tables and datasets before testing feasibility. Reuse the recorded search and inspect the closest sources and weakest assumption. Use published numbers with citations; do not reproduce them in a first look. Seek the smallest experiment on the uncovered step. Recommend promising only with specific evidence and a bounded next step; do not claim the route is proved. Map the assumptions of any borrowed method onto this problem.\n\nRead GET <project base>/research-routes/416 and return #6363. Return the ordinary report and transcript plus research: {route_id: 416, 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":"6363","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[416],"research_url":"/projects/twin-primes/research-routes/416","transcript_url":"/projects/twin-primes/return/6420/transcript","files":[{"sha256":"3f5f7eeb3080b41cd2abee14a065cd892e0fda303b878932f6fcff57e55adedd","name":"REPORT.md","bytes":6641},{"sha256":"80506d2cda2decc4ad48a32dd845703f22cc74e414873249bb8a93e9be1ce4cb","name":"EVIDENCE.md","bytes":3700},{"sha256":"a6f3ba1136faa2f859a1f381d25b92fc3d491ab63e3a1f6469790dfc0cb5370e","name":"PRIOR_ART.md","bytes":3454},{"sha256":"c296b12cce04397f4947f11020bd46e979275308e1a4dfe0f4a14bdb83e4a1bb","name":"transcript-12736.md","bytes":3621},{"sha256":"f4d5e89bacb89a8d81ea98153ed88f1d04511afc0c5c29c3c23a406fe709b154","name":"RECIPE.md","bytes":3057},{"sha256":"b79f11df7debd86a6632a2b75235d414fc7041d87cb4eb7b61bd9416200fc742","name":"PREREG.md","bytes":2711},{"sha256":"03da19bb7f32cec2a0090ff2ee806a8992cc4108f831f6ba26ee4b93cc420c4f","name":"compute-416.py","bytes":7558},{"sha256":"bde82278de7f4cdc89fac94ad0dac105dedf386a3bec0feb38b0dd23b3e6c303","name":"natal-416.py","bytes":2944},{"sha256":"433885f82267e19f3d83cf9b3d436dff5b6085139127542870c51e1b3c865251","name":"results-416.json","bytes":18077},{"sha256":"d5783097b8da704972f79ff186ccd1f0116a60cee3e1253f4b62a8fd1479c5f3","name":"natal-416.json","bytes":1912}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"3f5f7eeb3080b41cd2abee14a065cd892e0fda303b878932f6fcff57e55adedd","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}