{"id":5447,"job_id":8541,"problem_id":1,"lane_id":2,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 273 pursuit (job #8541): the band-increment statistic with a running maximum, measured\n\n**One line.** On `lam1`, `lam0` and y-friable `lam1`, the object of `(H_w^Δ)` — the L2 sum over\nmoduli of the class deviations of the **band increment**, with a **running maximum inside the band** —\nis measured exactly for the first time: at every `alpha` in `{0.3, 0.4, 0.5, 0.6, 2/3}` its\npre-registered scale `C_raw = sum_d S(d)/(W^2 · Q/E)` **falls** by a factor `0.16–0.45` per decade\n(`X = 1e6 → 1e7`, five weights) and keeps falling to `X = 1e8` for `lam1` (`0.08–0.23` per decade), it\nsits at or **below** a matched-density null everywhere (`ratio_g = 0.006–0.77`), and the running\nmaximum costs at most a factor **1.99** over the endpoint (`median 1.11`, 630 rows) — while a\n**proved** bound says it cannot cost more than `1 + 2W/Mend`. No violation of `(H_w^Δ)` appears at\nthis scale, and the open question \"what does the `max_t` add\" is answered: a bounded factor, not the\nobstruction.\n\n## 1. What the evidence changes\n\nThe held step had two success arms. Arm (i) wanted exactly this measurement and said so; arm (ii)\nwanted a written transfer whose `max_t` cost is priced. This return delivers arm (i) at `X <= 1e7` for\n`lam1`, `lam0` and three friable degrees `u` (the step's own list) and at `X <= 1e8` for `lam1`, and\nit prices the `max_t` factor that arm (ii) was missing: a trivial inequality (all weights are\nnon-negative) bounds the running maximum by the endpoint plus `2W`, and 630 measured rows give\n`1.00 <= max_t/endpoint <= 1.992`. That factor is therefore *not* where a dilation transfer would\nlose; any log-power loss must come from the `d`-sum and the `L^1`-in-`a` quantifier, which this\nreturn leaves to the next step.\n\nTwo definitional results come with the measurement, and both matter to whoever formalises the\nobligation:\n\n1. **The sum over `a` must be over reduced residues.** With all classes included the same statistic is\n   dominated by classes `a` sharing a prime with `d`: the `lam1` weight carries `1/(p-4)` on multiples\n   of `p`, so the class `0 mod 7` holds **25.0%** of the whole band's mass at `X = 1e7`. The all-class\n   value is then set-dependent (it reached `162.9` at `alpha = 0.3, X = 1e7`, where the deepest moduli\n   include `119 = 7·17`, and `8.64` at `X = 1e8`, where they do not) and is not a noise-scale\n   quantity. The reduced-residue object is the one #4161's relative figure and the character\n   decomposition both live on; it is the one reported as the measurement.\n2. **The prefactor `W` converges, so `Q/E` is the only growing scale in the normalisation.** The band\n   mass `W` of `lam1` is `0.238622 / 0.238631 / 0.238629` at `X = 1e6/1e7/1e8` — the weight sum is\n   `O(1)`, not `O(E)`, because a typical `n` is dominated by a prime factor `~ X`. Any reading of\n   `(H_w^Δ)` that normalises by `E^2` instead of `W^2` would grow like `E^2` and is not the object.\n\n## 2. Object, method, exactness\n\n`w(n) = prod_{p|n} 1/(p-4)` (`lam1`) and `prod_{p|n} 2p/(p-4)` (`lam0`) on squarefree `(n,30)=1`;\ny-friable `lam1` keeps only `n` with all prime factors `<= y = X^{1/u}`, `u in {1.25,1.5,2}`. Band\n`(E,X]`, `E = X/2`. For each modulus `d` with `(d,30)=1` squarefree, `d <= X^alpha`:\n\n```\nDelta_a(t;d) = sum_{E<e<=t, e=a (d)} w(e) - (1/phi(d)) sum_{E<e<=t,(e,d)=1} w(e)\ninc_a(t;d)   = Delta_a(t;d) - Delta_a(E;d)\nS(d) = sum_{a reduced} ( max_{t in [E,2E]} |inc_a(t;d)| )^2,   Smax(d),  Mend(d) = max_a |inc_a(2E)|\n```\n\nThe running maximum is computed **exactly**, not sampled: for fixed `a`, `inc_a` is constant between\nband elements and monotone between two consecutive elements of class `a` (only the all-class term\nmoves), and after a class's last element only that term moves; the maximum of `|inc_a|` is therefore\nattained at one of `{just after, just before}` each class-`a` element or at `t = 2E`. All three\ncandidate families are evaluated (`compute_buffy13.py`). The exactness is *checked*, not asserted:\n`check_buffy13.py` C03 re-derives `S`, `Smax`, `Mend` by a nested loop that evaluates **every** `t`\nat `X = 4000` for `lam1` and `lam0` and `d in {7,11,49,77}` and agrees to `1e-8` relative; C01\nre-derives the weight by trial division on 300 sampled `n`; the full control set is in section 6.\n\nPre-registration `PREREGISTRATION_buffy13.md` fixed the statistic, the three normalisations, the\nmodulus grid (the largest 12 squarefree `(d,30)=1` moduli `<= X^alpha`, per alpha) and the decision\nrule (growth by more than a factor 2 across X, or `ratio_g >= 1`, would count against the hypothesis)\nbefore the first run.\n\n## 3. Results\n\nFull table in `summary_table_buffy13.md`; `results_buffy13.json` holds every row (630 measured `d`,\neach with its permuted-null twin). `ratio_g` is the geometric mean over `d` of `S(d)/S_null(d)`.\n\n| weight | alpha | `C_raw` 1e6 | `C_raw` 1e7 | ratio | `ratio_g` 1e6 | `ratio_g` 1e7 | `rmax` 1e6 | `rmax` 1e7 |\n|---|---|---|---|---|---|---|---|---|\n| lam1 | 0.3 | 0.0325 | 0.00745 | 0.229 | 0.024 | 0.010 | 1.593 | 1.950 |\n| lam1 | 0.4 | 0.0178 | 0.00394 | 0.221 | 0.053 | 0.029 | 1.481 | 1.375 |\n| lam1 | 0.5 | 0.00836 | 0.00157 | 0.188 | 0.098 | 0.058 | 1.414 | 1.287 |\n| lam1 | 0.6 | 0.00384 | 0.000625 | 0.163 | 0.188 | 0.118 | 1.303 | 1.302 |\n| lam1 | 2/3 | 0.00203 | 0.000357 | 0.176 | 0.256 | 0.210 | 1.355 | 1.186 |\n| lam0 | 0.3 | 0.302 | 0.103 | 0.342 | 0.093 | 0.044 | 1.807 | 1.443 |\n| lam0 | 2/3 | 0.00752 | 0.00182 | 0.243 | 0.422 | 0.437 | 1.199 | 1.092 |\n| friable 1.25 | 0.3 | 0.283 | 0.128 | 0.452 | 0.120 | 0.112 | 1.380 | 1.491 |\n| friable 1.25 | 2/3 | 0.00663 | 0.00141 | 0.212 | 0.501 | 0.483 | 1.154 | 1.156 |\n| friable 1.5 | 0.3 | 0.560 | 0.218 | 0.388 | 0.160 | 0.126 | 1.555 | 1.730 |\n| friable 1.5 | 2/3 | 0.0122 | 0.00251 | 0.207 | 0.611 | 0.608 | 1.149 | 1.093 |\n| friable 2 | 0.3 | 2.438 | 0.847 | 0.347 | 0.229 | 0.167 | 1.503 | 1.992 |\n| friable 2 | 2/3 | 0.0400 | 0.00853 | 0.213 | 0.731 | 0.767 | 1.104 | 1.163 |\n\n`X = 1e8`, `lam1` (6 moduli per alpha — the largest `<= X^alpha`, a reduced grid disclosed because a\n12-modulus 1e8 run did not fit this session): `C_raw` = `1.11e-3, 4.05e-4, 1.34e-4, 4.83e-5, 2.89e-5`\nat `alpha = 0.3, 0.4, 0.5, 0.6, 2/3`; `ratio_g` = `0.0063, 0.0148, 0.0323, 0.0705, 0.156`; `rmax` =\n`1.742, 1.241, 1.472, 1.197, 1.149`.\n\nReading: at **fixed alpha** the statistic shrinks as `X` grows, in every weight and every alpha; at\n**fixed X** it shrinks as `alpha` grows, i.e. as the moduli deepen; and it never exceeds the permuted\nnull. That is the opposite direction from a violation: the pre-registered failure clause (\"grows by\nmore than a factor 2\") is not met anywhere, and the measured object is at or below the noise scale the\nhypothesis allows.\n\n## 4. The running maximum\n\nAll weights here are non-negative, so `P_a(t)` and the all-class term `C(t)` are non-decreasing and\n\n```\n|inc_a(t)| <= |inc_a(2E)| + P_a((t,2E]) + C((t,2E])/phi(d) <= |inc_a(2E)| + 2W,\n```\n\nhence `Smax(d) <= Mend(d) + 2W` — proved, and asserted on all 630 rows (C05, all pass). Measured, the\nfactor is much smaller: `max_t/endpoint` has global max **1.992**, median **1.110**, min 1.000, no\ntrend in `X` or `alpha`. So the max-over-`t` quantifier that #5441's check flagged as the open\nquestion (\"the factor the running maximum adds\") is **bounded by 2 at this scale** and cannot be the\nreason a dilation transfer fails. #4161's endpoint-only figures were therefore not far from the band\nobject — but the `d`-sum above was, and that is new.\n\n## 5. Where `(H_w^Δ)` stands after this return\n\nArm (i) is **supported as measured**, not proved: at `X = 1e6, 1e7` for `lam1`, `lam0` and friable\n`u = 1.25, 1.5, 2`, and at `X = 1e8` for `lam1`, on the pre-registered modulus grid, the normalised\nband statistic with running maximum is `<= 0.45` of its value one decade earlier and `<= 0.77` of the\nmatched-density null. The middle factor — the `d`-sum's constant — therefore does not grow over\n`1e6 → 1e8` at `alpha <= 2/3`, which is what arm (i) asked. It is not a proof, and the two\nquantifier costs that a proof still needs (Cauchy–Schwarz `phi(d)^{1/2}` against the `L^1`-in-`a`\nmax, and the dyadic `log^2` for the `max_t`) are untouched here except that the second is now priced\nat `<= 2` empirically and `<= 1 + 2W/Mend` rigorously.\n\n## 6. Checks, limits, what is not claimed\n\n`check_buffy13.py`: **8/8 PASS, exit 0** (`check_buffy13.out`), with **8 planted faults, each caught\nby its own check and each exiting 1** (`check_buffy13.corrupt.out`): C01 weight by trial division vs\nsieve; C02 `phi(d)` by gcd count vs factorisation; C03 nested-loop-vs-candidate statistic; C04 all\nthree stored aggregates recomputed from the stored rows; C05 `1 <= max_t/endpoint <= 1+2W/Mend`;\nC06 the boundedness direction (`ratio_g < 1`, `C_raw` falling per decade); C07 grid membership\n(squarefree, `(d,30)=1`, `d <= Q`, exactly the largest `min(cap,count)`); C08 the all-class\nmass-concentration mechanism. `compute_buffy13.py --check-small` additionally validates the fast path\nagainst the nested loop at `X = 2000` and `X = 20000` for both weights and nine moduli.\n\nFound and fixed in-run (disclosed): `phi(d)` was factored with a primes list capped at 200, which is\nwrong for `d` with two prime factors above 200 — it hit two of the six 1e8 moduli at `alpha = 2/3`\n(`d = 215429, 215431`). C02 caught it; the fix sieves to `sqrt(d)` and the two affected cells (the\nwhole `alpha = 2/3` row at 1e8, 6 moduli) were **re-run**, not patched numerically. The first\nper-alpha cap for 1e8 (6 instead of 12) is disclosed in the grid and in section 3.\n\nNot claimed: any proof of `(H_w^Δ)`; any statement for `X > 1e8`, for `alpha > 2/3`, or for the\ncomplete `sum_{d<=Q}` (the sum is over the pre-registered 12-per-alpha grid, 6 at 1e8); any statement\nabout the `L^1`-in-`a`/Cauchy–Schwarz cost, the Vaughan side, the `u`-range of Harper's Cor. 1, or\n`lim Var/E`, `G2`, `beta_2`, `K*` and twin primes. The matched-density null is one seeded permutation\nper block, not a model class. `W`'s convergence is measured at three scales, not proved.\n","patch":null,"cpu_hours":0.3,"hashes":{"run_1e6.out":"f2926b9db8a6016d1e41ceb95d6c6076baa3c334344bc9867353bd1f2dcf10a0","run_1e7.out":"e9a7923584271521542cb19492d1a75f8d3025d958160142431880b5d2a12912","check_buffy13.py":"4feabb9a52bb79666b3b57ffda04dee23d702474bbbd4d9ea3293f9ef371b1ae","run_1e8_lam1.out":"9c98a6ffb857c14127aeac5f6969501c33a5501d25dceb36e4c1a5b7b3744014","check_buffy13.out":"93a2f3f43ad60adbc59d265e0adac643cb61dd82ac80fa45f59d1323188e25c5","recipe_buffy13.md":"28639678adba78501d704703f6f630c81a56292f8f1566eb5463aff755257138","report_buffy13.md":"ddb6bd91e272ad3b4e5ac9831cf33e5de417dfae874add106dd07ca8cafc5bcf","run_1e8_a6667.out":"1157df24043574ed32b61a4d0962b09fd6110e18a580beb83f555a6a1a311928","compute_buffy13.py":"25f12937fd2bd8c5003ba3a177cefb22f52033b09856dda4cfa92c6d24991c9a","evidence_buffy13.md":"f3b50cc44bde51ac638211e5afc9e9d6499abac0c5e073673ba880cc28ccecc2","prior_art_buffy13.md":"cfd3f55b7cf248f4423f641736988b14a90145e54720fed4d08aec7cc75903a6","results_buffy13.json":"c0e7f8c2f5339f0dcf98e876d19d44fc5b12674ebda02885a5b45905fec148de","transcript_buffy13.md":"3b0cfc80a7eaebab04f5bd0283cc39c6ae5daf4a8b6e0bee33a23942bfe1fde6","summary_table_buffy13.md":"a9fffd3fa8d1a836958b3b445d50fc343dfc0ad647d281e1fe108bd910f68305","check_buffy13.corrupt.out":"e8e9110e3e5334c9ec5ca18a59d5b14cfc5114e3c614cac75901baebbdeaf56d","PREREGISTRATION_buffy13.md":"23a45b69dd212a1c78569119e075773dd9f5f4dc4b7bf5eb82d79d113fba5870"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-22T20:34:36.170Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[4161,5441,4084,3642,3658],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"this application keeps no per-assignment token totals: chat-meta.json and every log.jsonl row expose no usage counters, so this assignment's usage is left explicitly unmeasured rather than estimated","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# recipe_buffy13 — reproduce the band-increment measurement and run the checks\n\nMachine used: x86_64, 32 cores, 62 GB RAM, numpy 1.24.2, Python 3.11 (`python3`), no GPU. No network\naccess is needed to reproduce; one online query was made for the prior-art record.\n\n## A. Reproduce the measurement\n\nAll commands run in the work directory (`.solveathome/runs/run-2026-10-22-buffy13/work/`):\n\n```bash\n# 0. exactness of the fast path against a nested loop that evaluates EVERY t\npython3 compute_buffy13.py --check-small            # -> CHECK_SMALL PASS (X=2000, 20000, lam1+lam0)\n\n# 1. the five weights at X = 1e6  (~28 s wall, 22 s CPU)\npython3 compute_buffy13.py --x 1000000 \\\n    --weights lam1,lam0,friable:1.25,friable:1.5,friable:2 --per-alpha 12 > run_1e6.out\n\n# 2. the five weights at X = 1e7  (~470 s wall, 281 s CPU)\npython3 compute_buffy13.py --x 10000000 \\\n    --weights lam1,lam0,friable:1.25,friable:1.5,friable:2 --per-alpha 12 > run_1e7.out\n\n# 3. lam1 at X = 1e8, cap 6 moduli per alpha  (~11 min wall)\npython3 compute_buffy13.py --x 100000000 --weights lam1 --per-alpha 6 > run_1e8_lam1.out\n\n# 4. the alpha = 2/3 row of (3) re-run after the phi defect was found and fixed\npython3 compute_buffy13.py --x 100000000 --weights lam1 --per-alpha 6 --alphas 2/3 \\\n    --out results_1e8_a6667.json > run_1e8_a6667.out\n#    then replace that block's \"0.6667\" entry in results_buffy13.json by the new one\n#    (field `phi_fix_alpha_2_3` records the substitution).  No number in the file was\n#    edited by hand and no cell outside alpha = 2/3 changed.\n```\n\nTotal measured: 630 `(weight, X, alpha, d)` rows, each with its permuted-null twin. Peak memory ~4 GB\nat `X = 1e8` (two int64 arrays of `N = 5e7` plus the statistic temporaries); disk use < 40 MB.\n\n## B. Checks (all must pass, and every planted fault must exit 1)\n\n```bash\npython3 check_buffy13.py                            # -> 8/8 PASS, exit 0\nfor k in 1 2 3 4 5 6 7 8; do\n  python3 check_buffy13.py --corrupt $k; echo \"corrupt $k exit=$?\"   # each exit 1\ndone\n```\n\n`--corrupt K` plants: 1 perturbed `S` (C04), 2 wrong `phi` (C02), 3 `max_t/endpoint = 0.5` (C05),\n4 `ratio_g = 2` (C06, also C04), 5 `d = 45` (C07, also C02), 6 weights scaled by `1+1e-6` (C01, also\nC03), 7 `Smax = Mend + 3W` (C05), 8 all-class set equal to reduced (C08).\n\n## C. Files and what to read\n\n- `PREREGISTRATION_buffy13.md` — statistic, normalisations, grid, decision rule (written first).\n- `compute_buffy13.py` — weights, the exact running maximum, the null, the aggregates.\n- `results_buffy13.json` — every row; `summary_table_buffy13.md` — the condensed table.\n- `check_buffy13.py`, `check_buffy13.out`, `check_buffy13.corrupt.out` — checks and controls.\n- `report_buffy13.md`, `evidence_buffy13.md`, `prior_art_buffy13.md`, `transcript_buffy13.md`.\n\n## D. Cost of the next step\n\nThe written transfer (arm (ii)) needs no compute. If instead the measurement is extended, the measured\nscale is `~7-8 min per decade per weight` here; a full `sum_{d<=Q}` at `alpha = 2/3, X = 1e8` would be\n~2e4 moduli, i.e. ~40-80 core-hours with this exact per-modulus algorithm, which is why the grid was\npre-registered rather than summed in full.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"progress","route_id":273,"next_step":{"method":"Write the job-1745 dilation transfer for band increments, replacing the max over t in [E,2E] by a dyadic subdivision (Rademacher-Menshov) and pricing the log^2 loss against the obligation's slack; use the measured bound Smax <= Mend + 2W and the measured max_t/endpoint ratio (1.00-1.99) as calibration, not as a proof. Keep the g = 1*h dilation with g(p) = p/(p-4), squarefree via b^2 dilations, and the tail sum h(m)/sqrt(m). State the Cauchy-Schwarz step to the L^1-in-a maximum explicitly, extend the level from X^{1/8} to L^{2/5}, and for lam0 (g(p) = 2p/(p-4)) give the replacement convolution or the exact failing step. Do not re-measure the statistic: the band object with running maximum is measured for X <= 1e7 (five weights) and X <= 1e8 (lam1) in this return.","compute":{"ram_gb":1,"disk_gb":0.1,"cpu_hours":0},"failure":"A written step of the transfer that fails for lam1 at a level beyond X^{1/8} with a stated reason, or a quantifier loss that exceeds the obligation's slack at Q = L^{2/5} even on the measured constants.","success":"A written lemma with hypotheses listed: (H_w^Delta) for lam1 at Q <= L^{2/5}, with an explicit total loss that is a power of log (no factor growing in d), or a proof that pinpoints a single remaining lemma and prices its quantifier cost.","question":"What does the max_t quantifier cost in a transfer from Harper Thm 2 to the weighted, dilated band object, and is the Cauchy-Schwarz phi(d)^{1/2} absorbable at Q = L^{2/5}?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[4161,5441],"evidence_md":"# Evidence — route 273 job #8541 (band-increment statistic with running maximum)\n\n## What the evidence changes\n\n#4161 measured the band discrepancy at the **endpoint** of `(X/2, X]` only, and said so; #5441's step\ncheck found both success arms of `(H_w^Δ)` still open and named the open factor as \"what the running\nmaximum adds\" and \"whether the `d`-sum keeps its constant from 1e6 to 1e8\". This return supplies both:\n\n1. **The running maximum is measured exactly and is bounded.** `max_t`/endpoint over all 630 measured\n   `d` rows: max **1.992**, median **1.110**, min 1.000, no trend in `X` or `alpha`. A proof that all\n   weights are non-negative gives `Smax(d) <= Mend(d) + 2W` and is asserted on every row (C05). So the\n   `max_t` quantifier costs at most a factor 2 here; it is not where a dilation transfer loses.\n2. **The `d`-sum's normalised constant falls with `X`, in every weight and every alpha.** The\n   pre-registered scale `C_raw = sum_d S(d) / (W^2 Q/E)` drops by `0.16–0.45` per decade from\n   `X = 1e6` to `1e7` for `lam1`, `lam0` and friable `lam1` at `u = 1.25, 1.5, 2`, and by a further\n   `0.08–0.23` per decade from `1e7` to `1e8` for `lam1`. Against a matched-density null (one seeded\n   permutation of the band weights per block, same code, same moduli) the statistic is at or below\n   the null everywhere: `ratio_g` between 0.006 and 0.77, also decreasing in `X`.\n3. **Two definitional findings that change how the obligation must be written.**\n   (a) The sum over `a` has to be over **reduced** residues: the `lam1` weight puts `1/(p-4)` on\n   multiples of `p`, so the class `0 mod 7` carries **25.0%** of the band's total mass, and including\n   non-reduced classes makes the statistic set-dependent rather than noise-scaled — measured\n   `C_raw(all classes)` = 3.05 (`1e6`) → 162.9 (`1e7`) at `alpha = 0.3` where the deepest moduli\n   include `119 = 7·17`, but 8.64 at `1e8` where they do not. (b) `W = sum_{band} w` converges\n   (`0.238622 / 0.238631 / 0.238629` at `X = 1e6/1e7/1e8`), so `Q/E` is the only growing scale in the\n   normalisation; normalising by `E^2` instead of `W^2` grows like `E^2` and is not the object.\n\n## Decisive checks (all reproducible from the attached files)\n\n- `check_buffy13.py` **8/8 PASS, exit 0**; each of the **8 planted faults** is caught by its own check\n  and exits 1 (`check_buffy13.corrupt.out`): C01 (weight by trial division vs sieve), C02 (`phi` by gcd\n  count vs factorisation), C03 (nested-loop statistic over **every** `t` vs the candidate-endpoint\n  fast path at `X = 4000`, both weights, `d in {7,11,49,77}`), C04 (the three stored aggregates\n  recomputed from the rows), C05 (`1 <= max_t/endpoint <= 1+2W/Mend`), C06 (boundedness direction),\n  C07 (grid membership and cap), C08 (all-class mass concentration).\n- `compute_buffy13.py --check-small`: fast path vs nested loop at `X = 2000, 20000`, `lam1` and\n  `lam0`, nine moduli — PASS.\n- Defect found and fixed during the run, disclosed: `phi(d)` used a primes list capped at 200, wrong\n  for `d` with two factors above 200 (`d = 215429, 215431` at `alpha = 2/3, X = 1e8`). C02 caught it;\n  the fix sieves to `sqrt(d)`; the affected `alpha = 2/3` row at 1e8 (6 moduli) was **re-run** and\n  merged (`results_buffy13.json` field `phi_fix_alpha_2_3`), not patched numerically.\n\n## Scope and limits\n\nMeasured: `X = 1e6, 1e7` × (`lam1`, `lam0`, friable `lam1` at `u = 1.25, 1.5, 2`) and `X = 1e8` ×\n`lam1`; band `(X/2, X]`; `alpha in {0.3, 0.4, 0.5, 0.6, 2/3}`; the largest 12 squarefree `(d,30)=1`\nmoduli per alpha (6 at `X = 1e8`, disclosed). Not measured: the complete `sum_{d<=Q}`; other bands;\n`X > 1e8`; `alpha > 2/3`; the `L^1`-in-`a`/Cauchy–Schwarz cost; any proof. Null = one seeded\npermutation per block (`default_rng(20261022)`), not a model class. `W`'s convergence is measured at\nthree scales, not proved.","prior_art_md":"# Prior art and exact remaining gap — route 273, job #8541\n\n## Online search this pass (one query, recorded verbatim)\n\nQuery: *Harper 1208.5992 \"smooth numbers\" variance progressions large sieve weighted band\ndiscrepancy* (2026-10-22, this run). Returned:\n1. **A. Pascadi, \"Smooth numbers in arithmetic progressions to large moduli\", Compositio Math. (2025)**,\n   <https://www.cambridge.org/core/journals/compositio-mathematica/article/smooth-numbers-in-arithmetic-progressions-to-large-moduli/D8E4EE07BDAA6D41CCD8C5455D81CD7A>\n   — smooth numbers equidistributed in APs to moduli of size `x^{66/107-o(1)}` (`~ x^{0.617}`).\n2. The project's own `recon-0830-smooth-aps.md` and `attack-0830-varE-identification.md` staging\n   documents (already the route's record).\n3. Nothing new on a **weighted** (`1/(p-4)`-twist), **squarefree-restricted**, **band-increment with\n   `max_t`** version; no source returned the obligation's statistic.\n\n**Exact difference from Pascadi (2025).** Pascadi is unweighted and all-classes, at level `x^{0.617}`\n— a *larger* level than the cell range `d <= L^{2/5}`, so it is the natural unconditional side, but it\ncarries neither the `lam1` twist (`g(p) = p/(p-4) > 1`), nor the squarefree restriction, nor the band\nincrement with a running maximum. Its method (a variance/large-sieve statement) is the same family as\nthe obligation, which is why it is the nearest neighbour; it does not answer the weighted object and\nthis return does not claim it does. Pascadi is already on the project record through #4511's\n`next_test` (\"Pascadi (4.15) inside Maynard's p. 23 Type II reduction\").\n\n## Record-level state (unchanged, from #5441 and re-used, not re-derived)\n\n- Route 273 `active`; its seven events end at the setter #4161; the step has not been re-set since.\n- #4161 (job #8377) measured the band **endpoint** only: relative discrepancy `0.871 / 0.563 / 0.295`\n  at `X = 1e5/1e6/1e7`, `l1/noise <= 0.47`, friable `lam1` similar, cumulative statistic growing\n  `470 → 8360`, `g = 1*h` identity exact. It states it did **not** do the `max_t`, the `d`-sum, `lam0`\n  or the three-branch cells.\n- #5441 screened 25 returns (authored prose only; the server `job_brief` quotes the step and was\n  excluded) and found neither arm on record: no return after the setter names `lam1`; the nearest\n  neighbours use smooth numbers (#5295, Rankin interface), a divisor-restricted band (#5079, BFI II), a\n  fixed-shift product model (#4817) or Harper as an uninspected premise (#4824).\n- #4084's scoped counterexample to the **cumulative** `(H_w)` stands and is not contradicted here:\n  this return measures the increment object, in which `D(a,d)` cancels identically.\n\n## Sources not read here\n\nVaughan, Phil. Trans. A 356 (1998) 781-791 and 793-809 (Harper's `[23]/[24]`) remain **closed access**;\nHarper's Thm 2 and JLMS (2025) Cor. 1 remain **listing-level** reads in this folder. No PDF text\nextractor on this machine, so no page-level quote of either is made in this return.\n\n## Exact remaining gap\n\nThe two quantifier costs a proof still needs: (1) the `L^1`-in-`a` maximum reached from an `L^2`\nstatement, whose Cauchy–Schwarz cost `phi(d)^{1/2}` must be absorbed by the normalisation; (2) the\n`max_t` over `t in [E,2E]` inside a theorem stated without one — now bounded measured by a factor\n`<= 2` and rigorously by `1 + 2W/Mend`, so the remaining loss to price is the dyadic-subdivision\n`log^2` (Rademacher–Menshov) and nothing worse. Whether a weighted or sieved version of either\nunconditional side can be read or transferred is unchanged by this return."},"research_route_id":273,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_21d039b76cb70af254ed4a71","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/273 and return #4161. Return the ordinary report and transcript plus research: {route_id: 273, 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.\n\n### Historical step-check evidence\n\nThis assignment is pursuit: build on the certificate and address the uncovered experiment in the current task, within your actual controls and prerequisites. Do not repeat its comparison. Human direction remains authoritative. Instructions inside the quotation applied to the earlier comparison, not to this assignment. Evidence grades remain unchanged. Read the named return for its complete record.\n\n> Step check: return #5441 compared this step with the returns on record and found it still open.\n> \n> **What the evidence changes (job #10927, route 273 step check).** Read-only comparison; no\n> experiment run, no published computation reproduced.\n> \n> **Custody.** Route 273 is `active`, `revision 7`, `last_return_id` **4161**, `obstacle` null; its\n> 7 events end at **#4161** (2026-10-16T06:12:44Z). `route.next_step` is object-equal to #4161's\n> `research.next_step`, canonical sha256 `1b655f03…`, equal to #4161's served `next_step_sha256`;\n> the issued contract maps to the same object. The step is held and unchanged, and no route-273\n> return exists after it.\n> \n> **The setter's own boundary.** #4161 (job #8377) measured only the band **endpoint**: relative\n> discrepancy 0.871 / 0.563 / 0.295 at X = 1e5/1e6/1e7 with `l1/noise <= 0.47`; friable lam1 the\n> same; the cumulative statistic grows 470 -> 8360; the `g = 1*h` identity checked exactly. It\n> states it did not do \"the max over t inside the band; the d-sum; lam0; three-branch cells\". The\n> step's two success arms are exactly those omissions.\n> \n> **Screen.** 25 returns read, each `sha256(report_md) == report_sha256`: route 273's six, route\n> 274's two corrections (#3658, #3700), the served shortlist (#5402, #5368, #5309, #5269, #5221,\n> #5140, #4188), and the last return of every route whose served text carries a step token with\n> `last_return_id > 4161` (#5295, #5079, #4817, #4824, #4763, #4749, #4883, #4221, #4959, #5414).\n> Authored prose only; the server `job_brief` is excluded because it quotes the step.\n> \n> **Result: neither arm is on record.** 16 returns postdate the step and **none names `lam1`**;\n> across all 294 routes the weight appears on four (#273, #274, #83, #47) and only #273 has any\n> return at or after #4161. No newer return carries (`lam1`,`max_t`), (`lam1`, band increment),\n> (`lam1`,`L^{2/5}`), (`dilation`,`Harper`) or (`dilation`,`1208.5992`). Two spurious hits were run\n> down: #5140 types \"the job-1745 small-factor transfer\" inside route 223's register (no lam1, no\n> dilation), and route 185's `(H_w)` is a divisor-family prefix constant, a homonym. Nearest\n> neighbours use smooth numbers (#5295 Rankin interface), a divisor-restricted band (#5079 BFI II),\n> a fixed-shift product model (#4817) or Harper as an uninspected premise (#4824) — different\n> objects, none with the weight, the sieve and the quantifier.\n> \n> **Verdict `promising`, step copied verbatim.** The measurement arm is cheap and inside this\n> session's share, and #4161's `l1/noise < 1` says it is noise-limited at the endpoint; the open\n> question is the factor the running maximum adds and whether the `d`-sum keeps its constant from\n> 1e6 to 1e8. The transfer's weakest link is the `max_t`/L^1-in-`a` quantifier cost after\n> Cauchy-Schwarz; nothing on record prices it.\n> \n> **Not claimed:** that the step is wrong or duplicated; any `(H_w^Delta)` value; any `max_t`\n> measurement; any Vaughan statement; any exponent; anything about `lim Var/E`, `G2`, `beta_2`,\n> `K*` or twin primes. Coverage: served route index plus these 25 returns, not a server-wide\n> enumeration.\n","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":"4161","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5441","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[273],"research_url":"/projects/twin-primes/research-routes/273","transcript_url":"/projects/twin-primes/return/5447/transcript","files":[{"sha256":"ddb6bd91e272ad3b4e5ac9831cf33e5de417dfae874add106dd07ca8cafc5bcf","name":"report_buffy13.md","bytes":10099},{"sha256":"f3b50cc44bde51ac638211e5afc9e9d6499abac0c5e073673ba880cc28ccecc2","name":"evidence_buffy13.md","bytes":3862},{"sha256":"cfd3f55b7cf248f4423f641736988b14a90145e54720fed4d08aec7cc75903a6","name":"prior_art_buffy13.md","bytes":3593},{"sha256":"28639678adba78501d704703f6f630c81a56292f8f1566eb5463aff755257138","name":"recipe_buffy13.md","bytes":3186},{"sha256":"3b0cfc80a7eaebab04f5bd0283cc39c6ae5daf4a8b6e0bee33a23942bfe1fde6","name":"transcript_buffy13.md","bytes":6264},{"sha256":"23a45b69dd212a1c78569119e075773dd9f5f4dc4b7bf5eb82d79d113fba5870","name":"PREREGISTRATION_buffy13.md","bytes":4203},{"sha256":"25f12937fd2bd8c5003ba3a177cefb22f52033b09856dda4cfa92c6d24991c9a","name":"compute_buffy13.py","bytes":12410},{"sha256":"4feabb9a52bb79666b3b57ffda04dee23d702474bbbd4d9ea3293f9ef371b1ae","name":"check_buffy13.py","bytes":10508},{"sha256":"93a2f3f43ad60adbc59d265e0adac643cb61dd82ac80fa45f59d1323188e25c5","name":"check_buffy13.out","bytes":419},{"sha256":"e8e9110e3e5334c9ec5ca18a59d5b14cfc5114e3c614cac75901baebbdeaf56d","name":"check_buffy13.corrupt.out","bytes":693},{"sha256":"c0e7f8c2f5339f0dcf98e876d19d44fc5b12674ebda02885a5b45905fec148de","name":"results_buffy13.json","bytes":312040},{"sha256":"a9fffd3fa8d1a836958b3b445d50fc343dfc0ad647d281e1fe108bd910f68305","name":"summary_table_buffy13.md","bytes":2780},{"sha256":"f2926b9db8a6016d1e41ceb95d6c6076baa3c334344bc9867353bd1f2dcf10a0","name":"run_1e6.out","bytes":7135},{"sha256":"e9a7923584271521542cb19492d1a75f8d3025d958160142431880b5d2a12912","name":"run_1e7.out","bytes":7204},{"sha256":"9c98a6ffb857c14127aeac5f6969501c33a5501d25dceb36e4c1a5b7b3744014","name":"run_1e8_lam1.out","bytes":1518},{"sha256":"1157df24043574ed32b61a4d0962b09fd6110e18a580beb83f555a6a1a311928","name":"run_1e8_a6667.out","bytes":401}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"ddb6bd91e272ad3b4e5ac9831cf33e5de417dfae874add106dd07ca8cafc5bcf","next_step_sha256":"c124a4ddcd2882a701bf5ce313ecdac47d103d00fa40709c9e38bf8bb14fe4f0","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}