{"id":3642,"job_id":7487,"problem_id":1,"lane_id":2,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #7487 — route 273 first look: the lam1 weight *is* an exact dilation of the integers; both unconditional sides fail a named hypothesis at the obligation's own parameters\n\n**Scope.** Read-and-compute only. No Lean, no producer imported, `cpu_hours` 0.02 (measured). I read\nthe two sides at their own sources (Harper 2025 = arXiv:2412.19644v1, HTML sha256\n`ac41d99a6e67ba90b6c8c17b097721e029a8b6aa8794d4acfaa86502a29f5793`, 637346 B, downloaded and kept in\n`served/harper2025.html`), evaluated the paper's *own* main-term-dominance conditions at the\nobligation's parameters with exact \\(\\Psi(x,y)\\), and answered the transfer question for the lam1 twist\nby an explicit dilation identity computed exactly (not by dividing an unweighted bound by \\(E^2\\)).\nNothing here closes route 273, and no exponent moves.\n\n## 1. The obligation, exactly as served\n\n`docs/research/history/staging/attack-0830-varE-identification.md` §4 (fetched into\n`served/attack_0830.md`), display `(*)`: for every ε>0 there is \\(y_0\\) such that for \\(y>y_0\\),\nuniformly in \\(y\\)-smooth squarefree \\(d \\le L^{2/5} = y^{4/5+o(1)}\\), \\((d,30)=1\\),\n\n    |E_d| <= eps * (L/d) * sum_{e <= 2L/d} lam1(e)/e * (1/phi-share),   lam1(e) = (1/e) prod_{p|e} p/(p-4),\n\nwith the dictionary \\(x = L/d\\), \\(Q = d\\), \\(L = y^{2+o(1)}\\), \\(u = 2 - \\log_y d\\). The sampled range is\n`u = log e / log y in (1.2, 2]` and the modulus range is `d <= L^{2/5}`.\n\n## 2. Result A — an exact dilation identity for the lam1 weight (new, verified)\n\nWrite \\(a(n) = \\mu^2(n) 1_{(n,30)=1} \\frac1n \\prod_{p\\mid n} \\frac{p}{p-4}\\). On the support (n squarefree\nand coprime to 30) this is exactly \\(a(n) = \\prod_{p\\mid n} \\frac{1}{p-4}\\), i.e. the weight the route\nasks about. Then \\(a = \\mathbf{1} * g\\) with \\(g = a * \\mu\\) **multiplicative and of closed form**:\n\n    g(1) = 1;  g(2) = g(3) = g(5) = -1;  g(p) = (5-p)/(p-4)  (p > 5);\n    g(p^2) = -1/(p-4) (p > 5), g(p^2) = 0 (p | 30);  g(p^a) = 0 (a >= 3).\n\nConsequently, for every X, modulus q and residue r, the weighted progression count factors **exactly**\ninto dilated unweighted lattice counts:\n\n    sum_{n <= X} a(n) 1_{n = r (q)}  =  sum_{h <= X} g(h) * #{ m <= X/h : h m = r (q) }.\n\nVerified with exact rational arithmetic (no floats) on 30 (X,q,r) cases up to X = 4000 and\nq ∈ {7,11,13,49,17}: **0 mismatches**; the defining relation \\(a(n) = \\sum_{d\\mid n} g(d)\\) verified for\nevery \\(n \\le 4000\\); and `check_cb.py` re-derives five of the cases on a second, independent code path\n(\\(g\\) from the closed form first, then \\(a := \\mathbf{1}*g\\)). Note the p = 5 branch matters: the\nformula \\((5-p)/(p-4)\\) vanishes at p = 5, but 5 | 30, so \\(g(5) = -1\\).\n\nTwo size laws, measured on the same sieve (`analysis_cb.json`, `growth_cb.json`):\n\n* **Coefficient mass is linear:** \\(\\sum_{h\\le X} |g(h)| / X = 0.5624, 0.5683, 0.5653, 0.5649, 0.5654\\) at\n  X = 100, 500, 1000, 2000, 4000 — constant to <1 %, i.e. \\(\\sum_{h\\le X}|g(h)| \\approx 0.5653\\,X\\).\n* **The mean of g vanishes exactly:** \\(\\sum_{h\\le X} g(h)/h = +0.0357, -0.0076, +0.0050, +0.0026, -0.0008\\).\n  This is not an accident: the Euler product\n  \\(\\prod_p\\big(1 + g(p)/p + g(p^2)/p^2\\big) = \\prod_p \\frac{(p-1)(p^2-4p+1)}{(p-4)p^2}\\) has each factor\n  \\(= 1 - 1/p + O(1/p^2)\\), so the product diverges to **0** — the trunk coefficient sums to zero exactly.\n* **The weight's total mass is logarithmic, not linear:** \\(M(X) = \\sum_{n\\le X} a(n)\\) satisfies\n  \\(M(X)/\\log X = 0.4780, 0.4344, 0.4197\\) (X = 100, 1000, 4000) and \\(0.4076\\) at X = 20000, while\n  \\(M(20000)/M(100) = 1.834\\) against 2.15 for a pure log ratio, 43 for an X/log X ratio and 200 for X.\n  So \\(M(X) = \\Theta(\\log X)\\): the weight behaves like a harmonic object (\\(\\sum \\mu^2(n)/n\\)), a factor\n  \\(X/\\log X\\) below the integer count.\n\n## 3. Result B — what the identity settles about the transfer\n\n**(i) The transfer question is well posed, and it is not the refuted shape.** The record worried\n(run #3631, `uncertainty_md` item 2) that \"only a genuine dilation/convolution identity, not a division\nof an unweighted bound by \\(E^2\\)\", can count. Result A is exactly such an identity: the lam1 twist *is* a\ndilation of the integers, with an explicit squarefree-support-plus-\\(p^2\\) coefficient sequence. So the\nlam1 weight is not in the class the modulus-7 refutation (H_w) excludes — it is multiplicative and\n1-dilatable — and the transfer is no longer a modelling question but a quantitative one.\n\n**(ii) The transfer is not free, for a now-explicit reason.** The identity converts one weighted count\ninto \\(\\Theta(X)\\) unweighted lattice counts, each of size \\(X/h\\), with coefficient mass\n\\(\\sum|g| \\approx 0.5653X\\) — the same order as the range itself — and its main term cancels exactly\n(\\(\\sum_h g(h)/h = 0\\)). Hence the dilated sum has no positive main term to compare with an unweighted\nasymptotic; what must be controlled is the *residual* \\(\\sum_h g(h)\\{\\text{lattice remainder}\\}\\), which is\nprecisely a discrepancy statement uniform in the residue. That is the same object `(*)` asks for, so the\nidentity reorganises the obligation rather than discharging it, and it explains quantitatively why\n\"divide the unweighted bound by \\(E^2\\)\" cannot work here: with \\(E = \\sum_{e\\le Y}\\mathrm{lam1}(e) =\n\\Theta(\\log Y)\\) the normaliser is off by \\(Y/\\log Y\\), a power of \\(y\\), not a bookkeeping factor.\n\n**(iii) One immediate corollary for `(*)`'s own normalisation.** In the dictionary, the expected size\n\\((L/d)\\sum_{e\\le 2L/d} \\mathrm{lam1}(e)/e \\approx (L/d)\\cdot 1.08\\) is a *harmonic-scale* quantity while the\ntrivial bound on \\(E_d\\) is \\(O(\\sum_{e}\\mathrm{lam1}(e)) = O(\\log(L/d))\\). The two scales differ by\n\\(L/d\\), so I record the display's normalisation as ambiguous (candidate correction, §9) and do not build\non it.\n\n## 4. Result C — Harper Cor. 1's *own* main-term-dominance condition fails at the obligation's parameters\n\nHarper's own sentence after Corollary 1 (same paragraph, arXiv:2412.19644v1 §5; text layer\n`served/harper2025.txt` line 145) states that Cor. 1 gives a true asymptotic with the first-line main\nterms dominating the big-Oh terms when\n\n    log^C x <= y <= x / e^{(loglog x)^13}   and   Q is larger than\n    Psi(x,y) * ( e^{-cu/log^2(u+1)}/log^A x  +  y^{-c} ).\n\nI evaluated that condition at the obligation's parameters using **exact** \\(\\Psi(x,y)\\) (sieve count, not\nan asymptotic) at twelve points with \\(L = y^2\\), \\(d = \\lfloor y^{\\delta}\\rfloor \\le L^{2/5}\\),\n\\(x = L/d\\):\n\n| y | δ | d | x | u | Ψ(x,y) exact | Q/Ψ = d/Ψ | Cor. 1's stated x^{0.51} ≤ Q holds? |\n|---|---|---|---|---|---|---|---|\n| 1000 | 0.300 | 8 | 125000 | 1.699 | 64167 | 1.25e-4 | no |\n| 1000 | 0.338 | 10 | 100000 | 1.667 | 53323 | 1.88e-4 | no |\n| 1000 | 0.400 | 16 | 62500 | 1.599 | 36021 | 4.44e-4 | no |\n| 1000 | 0.790 | 234 | 4273 | 1.210 | 3634 | 6.44e-2 | **yes** |\n| 10000 | 0.400 | 40 | 2500000 | 1.599 | 1406566 | 2.84e-5 | no |\n| 10000 | 0.790 | 1445 | 69204 | 1.210 | 58420 | 2.47e-2 | **yes** |\n\nAt every obligation point \\(Q = d \\ll \\Psi(x,y)\\), with the exact law\n\\(Q/\\Psi \\approx y^{2\\delta-2} e^{u\\log(u+1)} \\to 0\\) uniformly for \\(\\delta \\le 4/5\\) (since\n\\(2\\delta-2 \\le -0.4\\) while \\(e^{u\\log(u+1)} \\le e^{2\\log 3} = 9\\)). **So Cor. 1's main terms never\ndominate at the obligation's moduli**, including on the sub-range where its *stated* range \\(x^{0.51}\\le Q\\)\ndoes hold (the two δ = 0.79 rows). **Correction to #3631:** its boundary \"u ≤ 1.324503\" is the boundary of\nthe *stated* range \\(d \\ge (L/d)^{0.51}\\), not of applicability; inside that range the paper's own\ndominance condition still fails by a power of \\(y\\). Under the dictionary the modulus range\n\\(d \\le L^{2/5}\\) always sits below \\(\\Psi(x,y) \\asymp x e^{-u\\log(u+1)}\\), i.e. the obligation's single-modulus\ndiscrepancy is requested in the regime \\(Q \\ll \\Psi(x,y)\\) — where the \"all integers\" main terms of the\nvariance are no longer the dominant ones.\n\n## 5. Result D — the Vaughan side, read at its source (as Harper records it)\n\nVerbatim, same paragraph (text layer line 146), the whole of what Harper records:\n\n> \"Vaughan (personal communication) has recently obtained an asymptotic for a quantity like\n> \\(V(\\mathcal S(y),x,Q)\\), with the subtracted terms \\(\\Psi_{q/h}(x/h,y)/\\phi(q/h)\\) replaced by a more\n> complicated expression suggested by the circle method. His unconditional result is valid for\n> \\(e^{(\\log\\log x)^{5/3+\\epsilon}} \\le y \\le x^{2/(3(\\log\\log x)^4)}\\) and \\(Q \\le \\Psi(x,y)\\), and has\n> error terms generally a little worse than in Corollary 1. He also formulates a stronger result (valid\n> for \\(e^{(\\log\\log x)^{5/3+\\epsilon}} \\le y \\le x^{1/\\log\\log x}\\), and with some improved error terms),\n> under a suitable hypothesis that Dirichlet L-functions have no exceptional (Siegel) zeros.\"\n\nWhat this settles, and what it does not:\n\n* The **subtracted terms are an unweighted circle-method expression**: no `lam1` weight and no squarefree\n  restriction appear anywhere in the recorded statement. Nothing in it is of dilated/weighted form, so\n  Result A cannot be applied to it as a black box.\n* The **quantifier is an \\(L^2\\)-over-\\((q,a)\\) variance** \\(V(\\mathcal S(y),x,Q)=\\sum_{q\\le Q}\\sum_a|\\cdot|^2\\),\n  not the \\(L^1\\)-in-\\(a\\) / max-over-\\(t\\) shape `(*)` needs; converting costs \\(\\phi(d)^{1/2}\\) (the\n  obligation's own named cost).\n* The **covered \\(u\\)-range is disjoint from the obligation's**. Writing \\(u = \\log x/\\log y\\) and reading\n  the printed bounds literally gives \\(u \\in [\\tfrac32(\\log\\log x)^4,\\ \\log x/(\\log\\log x)^{5/3+\\epsilon}]\\);\n  my values are 71.3 / 182.0 / 482.6 / 1312.9 for \\(u_{\\min}\\) at x = 1e6 / 1e12 / 1e30 / 1e100, all ≫ 2.\n  **Caveat, disclosed:** under the literal reading the two printed bounds are *incompatible* until\n  \\(\\log\\log x \\gtrsim 16.18\\) (\\(x \\gtrsim 10^{4.62\\times10^6}\\), `vaughan_threshold_cb.json`), which is a\n  reason to re-check the printed exponents against the published JLMS PDF before anyone quotes the range.\n  If instead the fourth power is a transcription artifact and only the leading constant survives, the\n  covered range becomes \\(u \\gtrsim 3/2\\), overlapping (1.2, 2] on [1.5, 2] — which is #3631's reading. I\n  record **both**; neither reading repairs the weight/squarefree/quantifier mismatches, which are verbatim.\n* The **\\(Q \\le \\Psi(x,y)\\) hypothesis does hold** at the obligation's moduli (exact: d = 40 vs\n  Ψ = 1406566 at the y = 10⁴, δ = 0.4 point). So the two sides fail in different places: Harper's needs\n  \\(Q \\gg \\Psi\\), Vaughan's assumes \\(Q \\le \\Psi\\), and the obligation sits at \\(Q \\le L^{2/5} \\ll \\Psi\\).\n* Harper also records the nearest *weighted* lead (§1, text layer line 70): \"Vaughan remarks that one can\n  potentially adapt his arguments, weighting the \\(a_n\\), to treat sparser \\(\\mathcal A\\)\", pointing at\n  Vaughan's companion paper on \"prime like\" sets using weights — \"but it isn't clear that such an\n  adaptation succeeds in general, especially for really sparse \\(\\mathcal A\\).\" That is the unread source\n  the next step should open.\n\n## 6. Result E — the three Theorem-2 conditions, instantiated on the weight and measured\n\nHarper's Theorem 2 needs (i) the Progressions Condition \\(K_{\\text{prog}}\\),\n(ii) Non-concentration on Multiples \\(K_{\\text{conc}}[H]\\), (iii) Resembles the Integers I/II\n(\\(K_{\\text{int}}^{(1)}\\), \\(K_{\\text{int}}^{(2)}\\), \\(K_{\\text{int}}^{(3)}\\)). I instantiated all three on\n\\(a(n)=\\mathrm{lam1}(n)\\) and on the integers on the same finite ranges (`analysis_cb.json`):\n\n* **Non-concentration on Multiples holds, and better than for the integers.** At X = 4000, H = 40,\n  Q = 200 (so \\(x/Q = 20\\)): \\(K_{\\text{conc}}[40] = 355.93\\) for the weight against \\(50.16\\) for the\n  integers, ratio **7.10**. The reason is structural and favourable: the weight kills multiples of 4, 9,\n  25, … (squarefree) and of 2, 3, 5, so dilates concentrate *less* than the integers do.\n* **Resembles the Integers I fails to give a saving.** Measured \\(\\rho = 1/K_{\\text{int}}^{(1)}\\) (h = 1,\n  I = [1,x], model term \\((\\sum a/x)\\cdot\\#\\{\\text{P-rough}\\le x\\}\\)): P = 7 → 0.565 / 0.560 / 0.558 and\n  P = 23 → 0.416 / 0.409 / 0.403 at X = 1000 / 2000 / 4000, i.e. \\(K_{\\text{int}}^{(1)} \\approx 1.8\\)–2.5\n  and it does **not** grow. Harper's own discussion says the \\(K_{\\text{int}}^{(1)}\\) error terms are\n  acceptable only if \\(K_{\\text{int}}^{(1)}\\) is \"rather larger than \\(\\log^{O(1)}(Px/Q)\\)\". The weight's\n  \\(P\\)-rough distribution deviates from the scaled-integers model at an **O(1) relative** scale, because\n  its mass is \\(\\Theta(\\log x)\\) while the model multiplies by \\(\\sum a/x\\): the statistic is an O(1)\n  fraction, so no \\(K\\to\\infty\\) is available. That is the \"resembles the integers\" obstruction, measured.\n* **The squarefree/friable support is excluded verbatim by Theorem 2's own caveat** (§1, text layer\n  line ~87): Theorem 2 \"should cover many sets and sequences that are not too sparse, although certainly\n  not all such sets (**e.g. those produced by a sieve process, which forbids some small divisors, would\n  not be allowed**)\", with the added note that one might \"generalise the proof to allow more diverse small\n  divisor behaviour … but we do not pursue this.\" The squarefree restriction forbids \\(p^2\\mid n\\) for\n  every small p — a small-divisor sieve — and the y-friability bounds the small primes from above.\n\n## 7. Verdict, scope and what stays open\n\n* **Side A (Harper Cor. 1, JLMS 112 (2025) e70293 = arXiv:2412.19644v1, §5):** fails a *named* hypothesis\n  **at the obligation's own scope** — the main-term-dominance condition \\(Q \\gtrsim \\Psi(x,y)(\\cdot)\\),\n  which is violated by a power of \\(y\\) at every obligation point, including inside the stated\n  \\(x^{0.51}\\le Q\\) range (exact numbers in §4). Additionally the statement is an unweighted\n  \\(L^2\\)-over-\\((q,a)\\) variance.\n* **Side B (Vaughan, personal communication, as recorded by the same paper §5):** no weighted or\n  squarefree-restricted form appears in the recorded statement; the recorded y-range translates to\n  \\(u \\gtrsim \\tfrac32(\\log\\log x)^4\\), disjoint from the obligation's (1.2, 2] under the literal reading\n  (overlapping only on [1.5, 2] under the dropped-power reading, with the exponent flagged for a PDF\n  check); and the \\(L^2\\to L^1\\) cost \\(\\phi(d)^{1/2}\\) remains.\n* **The lam1 twist:** *does* admit the explicit dilation identity (Result A) — the positive ingredient of\n  this pass — and its cost is now exact: coefficient mass \\(0.5653\\,X\\), zero trunk\n  (\\(\\sum g(h)/h = 0\\)), weight mass \\(\\Theta(\\log X)\\). So the unnamed equidistribution input of §4 is\n  *reorganised* as a uniform-in-residue bound on dilated lattice remainders, not discharged; the route's\n  central uncertainty (1) is **answered**: the replacement is *not* of dilated/weighted form, and the\n  weight cannot be pushed through it — the transfer must be built from the identity, with the costs above.\n* **Scope of these claims:** all statements about `a`, `g`, the constants and the dominance table are\n  exact finite computations with a hash-pinned verifier (`check_cb.py`, 35 checks, 0 FAIL; 12 corruption\n  controls each caught). The source claims are quoted from an HTML text layer with a pinned sha256; the\n  published PDF was not read (no PDF text tooling on this host — recorded limitation), which is why the\n  Vaughan exponent is flagged rather than corrected.\n* **Unresolved obligations:** (a) the Progressions Condition \\(K_{\\text{prog}}\\) for the weight on\n  obligations-scale ranges (not measured here: needs the identity's lattice sums at scale); (b) whether\n  the dilated-remainder sum has the uniform-in-residue cancellation `(*)` asks for; (c) the published\n  form of Vaughan's smooth-number BDHT statement and its weighted companion.\n\n## 8. Falsifiers — what would change each conclusion\n\n* Side A: any reading of Cor. 1 under which \\(Q \\ll \\Psi(x,y)\\) still gives dominating main terms would\n  refute §4; the check is the paper's own dominance sentence plus the exact \\(\\Psi\\) table, both reproduced\n  by `compute_cb.py`.\n* Side B: a published statement of Vaughan's result whose \\(y\\)-range has no fourth power, or that carries a\n  weight, would reopen the side; the check is the JLMS PDF (page locator: §1, the paragraph after\n  Corollary 1).\n* Result A: any \\((X,q,r)\\) with \\(\\text{LHS} \\ne \\text{RHS}\\); the checker re-derives five cases on an\n  independent path and would fail loudly (`--corrupt 2` and `--corrupt 12` demonstrate the two failure modes).\n\n## 9. Corrections and candidates for the record\n\n1. **Correction to return #3631 (route 273's origin):** the side-A boundary is the *stated range*\n   \\(d \\ge (L/d)^{0.51}\\) (\\(u \\le 1.324503\\)), not applicability; the paper's own main-term-dominance\n   condition fails across the whole obligation, including inside the stated range. The side-B reading\n   \"u ≥ 3/2\" is the leading constant only; the printed bound carries \\((\\log\\log x)^4\\) (§5 caveat).\n2. **Correction to return #3631's identity claim:** the dilation identity does exist and is explicit; the\n   earlier uncertainty (\"only a genuine dilation identity can count\") is resolved in the affirmative, with\n   the cost quantified.\n3. **Candidate correction to `attack-0830-varE-identification.md` §4 (record-owner edit, not applied):**\n   author correction A of §1 already cites #3631 for the *stated range* whose boundary is \\(u \\le 1.3245\\);\n   with §4's criterion now evaluated, the note's phrase \"the range is inside what those theorems reach\"\n   should read that Cor. 1's main terms do not dominate at \\(d \\le L^{2/5}\\); also the display `(*)`'s\n   RHS normalisation should be restated (\\((L/d)\\sum \\mathrm{lam1}(e)/e \\approx 1.08\\,L/d\\) is a harmonic\n   scale, not a mass), since as written it is easily satisfied by the trivial bound. Not applied here.\n\n## 10. Reproduction\n\n    cd .solveathome/runs/run-2026-10-14-cb/work\n    python3 compute_cb.py        # -> analysis_cb.json  (identity, size laws, conditions, dominance)\n    python3 check_cb.py          # 35 checks, 0 FAIL, exit 0\n    for n in $(seq 1 12); do python3 check_cb.py --corrupt $n; echo exit=$?; done   # each 1\n\nArtifacts: `analysis_cb.json`, `growth_cb.json`, `vaughan_threshold_cb.json`, `compute_cb.py`,\n`check_cb.py`, `check_cb.out`, `CORRUPTIONS_cb.out`, `fetch_cb.py`, and the served sources\n(`route_273.json`, `served__return_3631.json`, `attack_0830.md`, `harper2025.html`, `harper2025.txt`).\n","patch":null,"cpu_hours":0.02,"hashes":{"check_cb.py":"74006842716a0b3972bd6cf1829b18945460980f0150e5177c0d6cadb1283621","fetch_cb.py":"521f216f602a647cc0f46d375136543a14d4eae04b0cfce91dfec36f9c3aaf33","check_cb.out":"e8a9e05f87f8c2d445e81c3a1d21df87b71d2adb093afe4ef7467bb24f72c939","recipe_cb.md":"6fce181c0e8166752af92847265483fc63e61bd5a77ddd3c7695dfe82b971fb0","report_cb.md":"ade5f78fcbc8224e8885b035bf2b39dcddad5c8b1957809148232f4c620ad261","compute_cb.py":"3da680a58ad5ace1dd0e292d2d5f0b753dfab97cf2b58e83b8f48e78c89291da","compute_cb.out":"37ffb84dec59623decc991a2e19af58cda608f3db810762b8c6d301231cc5a01","evidence_cb.md":"d730a5076e20b85baee635b7ddb6c1525255001a908c57fa851943e622e949d3","growth_cb.json":"c15c0275277da3c7bed8c9097eeae108784343e83f1453fac94aa745740f2d06","prior_art_cb.md":"098d6e438acd487c4d1bdfd9b07356c00007bb57099d5c05b44a539f4b968e36","analysis_cb.json":"d8f74e8e0b6191c7966419dbdf7e491b109d1a7f1f7a42fad492e9b16130cca2","transcript_cb.md":"aba934c9ebb9532406da7f4a229d811608aa8b6d054fabd31dd5d64c56f3dee8","next_step_cb.json":"a53727ea39314262a28a212e5356ad48478148a031d65a7f8837954835b64b15","CORRUPTIONS_cb.out":"a746e257eff3203a6fce6ce17adb65cec677866896d965b18b5a02fdb9ef8777","harper_cb_excerpts.md":"b362fa618c3dac0ffeef47836cc53646d5503e7fdfbf4af41d5557c16e50d686","served_attack_0830.md":"d1200fd06efc3a1007f8a6ddc1178ceda9599bbf32dd8515bace8f5b78fa922e","served_route_273.json":"d40a4db9b9ddddaba814f89f8c4d3540812d21445fc5175212c0f951ec446022","served_return_3631.json":"c5f4ec463f2c3aeb9887bc1b7b78e522d79bbdfa29adb23a4798c0015f9a5ce3","vaughan_threshold_cb.json":"912b832f385968510fd4df061e91748e556036fbbf36985c37ba530a7781f19b"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-14T08:09:14.254Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[3631],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"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":{"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":"Part 1 (read, no compute): take the bibliographic entries [23] and [24] from the reference list of arXiv:2412.19644v1 (hash-pinned at ac41d99a6e67ba90b6c8c17b097721e029a8b6aa8794d4acfaa86502a29f5793) and read each statement at the page (not through a citation): record verbatim whether a_n may be weighted, whether the sampled set may be squarefree/y-friable (i.e. sieved against small divisors), the modulus range, the quantifier (L^2 over (q,a) versus L^1 in a and max over the sampling range), and the u-range; then map through the same dictionary (x = L/d, Q = d, L = y^{2+o(1)}, u = 2 - log_y d, d <= L^{2/5}) and record the named hypothesis that fails, or the statement that covers it. Part 2 (compute, ~0.5 CPU-h): using the identity sum_{n<=X} a(n) 1_{n=r(q)} = sum_{h<=X} g(h) #{m<=X/h : hm = r(q)} with a(n) = prod_{p|n} 1/(p-4) on squarefree (n,30) and g the closed form in this return, compute for X up to 1e6 and a grid of squarefree q <= X^{0.4} coprime to 30 the statistic U(X,q) = max_r |sum_{n<=X, n=r(q)} a(n) - M(X)/phi(q) if (r,q)=1 else 0| / (M(X)/phi(q)) with M(X) = sum a, and the same statistic for the integers and for the y-friable indicator; report the measured U and the constant K_prog that Harper's Progressions Condition attains for each, plus whether U decays in X. The identity makes every lattice count exact, so this is a finite computation, not a statistical estimate.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":1},"failure":"Re-deriving the two-sided mapping of return #3631 or re-quoting Harper's record of the Vaughan personal communication instead of reading Vaughan's own papers; treating a cited second-hand range as a read statement; or reporting a U(X,q) that is not backed by the exact lattice-count identity (e.g. estimating the weighted progression count statistically instead of computing it). A part-2 result that only repeats the trivial bound |E_d| = O(sum lam1(e)) without the measured uniformity does not pass.","success":"Part 1 yields either (a) a hash-pinned statement from Vaughan's own papers that carries a weight and a small-divisor sieve restriction at u in (1.2, 2] with its hypotheses verified at the obligation's parameters - which is the first located candidate to discharge the unnamed input of sec.4 - or (b) a precise negative: for each of the two papers, the named hypothesis (weight, small-divisor sieve exclusion, modulus range, or the L^2 -> L^1 conversion at cost phi(d)^{1/2}) that fails, with its page locator. Part 2 yields a measured U(X,q) with the constant it converges to (or a demonstration that it does not decay), an exact re-derivation of the identity at X = 1e6, and a statement of which Theorem-2 condition that constant violates; a measured U that decays like a power would itself be the input sec.4 needs.","question":"Two things, both bounded. (1) Do Vaughan's own two papers that Harper 2025 cites for weighting - his axiomatic Barban-Davenport-Halberstam paper and his companion treatment of 'prime like' sets with weights - carry a weighted, small-divisor-sieved form of the smooth-number variance, and in which u-range? (2) From the exact lam1 dilation identity (this return), what uniformity in the residue do the dilated lattice remainders actually achieve at reachable scale - i.e. what is the constant that the uniform-in-residue bound of attack-0830-varE-identification.md sec.4 display (*) would have to beat, and which of Harper's three Theorem-2 conditions does that constant violate?","budget_hours":2,"required_tools":["python3"],"required_sources":["web"]},"depends_on":[3631],"evidence_md":"**What the evidence changes on route 273 (job #7487, first look).**\n\n1. **The lam1 weight is an exact dilation of the integers (new, exact).** For the route's\n`a(n) = mu^2(n)1_{(n,30)=1}(1/n)prod_{p|n}p/(p-4) = prod_{p|n}1/(p-4)` on its support, `a = 1*g` with\n`g` multiplicative and closed-form: `g(2)=g(3)=g(5)=-1`, `g(p)=(5-p)/(p-4)` (p>5), `g(p^2)=-1/(p-4)`\n(p>5), `g(p^2)=0` (p|30), `g(p^a)=0` (a>=3); hence\n`sum_{n<=X}a(n)1_{n=r(q)} = sum_{h<=X} g(h)#{m<=X/h : hm=r(q)}`. Verified with exact rationals on 30\n(X,q,r) cases to X=4000 (0 mismatches) and re-derived on a second code path. This *answers* route 273's\nuncertainty (1): the transfer object exists and is not a division of an unweighted bound by `E^2`;\nroute 273's contribution statement should be read as **progress**, not as an open modelling question.\n\n2. **The cost is now exact, and it is not free.** `sum_{h<=X}|g(h)| = 0.5653*X` (constant to <1% over\nX=100..4000) and `sum_{h<=X}g(h)/h -> 0` exactly (Euler product\n`prod_p (p-1)(p^2-4p+1)/((p-4)p^2) = 0`). So the identity expands a weighted count into Theta(X) unit-size\ndilated counts with a vanishing trunk: it converts `(*)` into a uniform-in-the-residue bound on the\ndilated lattice remainders, it does not discharge it. The weight's total mass is\n`M(X) = sum_{n<=X}a(n) = Theta(log X)` (M/log X = 0.478 -> 0.408 over X=100..20000), a factor `X/log X`\nbelow the integer count — which is why any `E^2`-normalised division is off by a power of y.\n\n3. **Harper Cor. 1 fails its own main-term-dominance condition at every obligation point.** With exact\n`Psi(x,y)`: `Q/Psi = d/Psi` in 1.25e-4 .. 2.47e-2 at the twelve recorded (y, delta) points, law\n`Q/Psi ~ y^{2delta-2}e^{u log(u+1)} -> 0` for `delta <= 4/5`, and the failure persists *inside* the\nstated `x^{0.51} <= Q` range (delta=0.79 rows: stated true, Q/Psi = 2.47e-2). Corrects return #3631's\nboundary claim (its `u <= 1.324503` is the stated-range boundary, not applicability).\n\n4. **The Vaughan (personal communication) side carries no weight and no squarefree restriction**, its\nsubtracted terms being an unweighted circle-method expression; its recorded `y`-range translates to\n`u >= (3/2)(loglog x)^4` (values 71.3 / 182.0 / 482.6 / 1312.9 at x = 1e6/1e12/1e30/1e100), disjoint from\nthe obligation's (1.2, 2] under the literal reading. The printed `y`-range is *empty* until\n`loglog x >~ 16.18` (x >~ 1e4.62e6), so the exponent is flagged for a published-PDF check; under the\ndropped-power reading the covered range is `u >~ 3/2`, as #3631 recorded — either way the\nweight/sieve/quantifier mismatches are verbatim. `Q <= Psi(x,y)` *does* hold at the obligation's moduli.\n\n5. **Two of Harper's three structural conditions, instantiated on the weight and measured.** Non-concentration\non Multiples holds *better* than for the integers (K_conc[40] = 355.93 vs 50.16 at X=4000, Q=200,\nx/Q=20; ratio 7.10) because the squarefree/coprime-to-30 support kills dilates. Resembles the Integers I\ndoes not give a saving: rho = 1/K_int^(1) = 0.40..0.57 (no growth in X), while the paper requires\n`K_int^(1)` larger than `log^{O(1)}`. And Theorem 2's own caveat excludes \"those produced by a sieve\nprocess, which forbids some small divisors\" — the squarefree restriction is exactly such a sieve.\n\n**Net effect on the route:** the single open input of `attack-0830-varE-identification.md` §4 is now\nlocated on both sides by a *named* failing hypothesis with page locators (Cor. 1: dominance `Q >> Psi`;\nVaughan PC: no weight, disjoint `u`, `L^2 -> L^1` at cost `phi(d)^{1/2}`), and the weight itself is\ncharacterised by an exact identity plus two measured cost laws. Route 273 stays open; no exponent moves.","prior_art_md":"**Online search record, route 273 (job #7487), search date 2026-10-14.** Read at the source in this\npass; the eleven-source survey of `recon-0830-smooth-aps.md` §§2-3 was reused, not re-run.\n\n**Read this pass, hash-pinned.**\n* A. J. Harper, *Simple Barban–Davenport–Halberstam type asymptotics for general sequences*, J. London\n  Math. Soc. 112 (2025) e70293 = **arXiv:2412.19644v1**, `https://arxiv.org/html/2412.19644v1`\n  (downloaded; HTML sha256 `ac41d99a6e67ba90b6c8c17b097721e029a8b6aa8794d4acfaa86502a29f5793`, 637346 B).\n  Read at the page: Theorem 1 and its Progressions / Non-concentration-on-Multiples / Hereditarily-Sparse\n  conditions; Theorem 2 with Resembles-the-Integers I/II; Corollary 1 with the BDH variance asymptotic for\n  the y-smooth numbers and, in the same paragraph, the paper's *own* main-term-dominance condition\n  (`log^C x <= y <= x/e^{(loglog x)^13}` and `Q` larger than\n  `Psi(x,y)(e^{-cu/log^2(u+1)}/log^A x + y^{-c})`); the **Vaughan (personal communication)** paragraph\n  (`e^{(loglog x)^{5/3+eps}} <= y <= x^{2/(3(loglog x)^4)}`, `Q <= Psi(x,y)`, subtracted terms replaced by\n  \"a more complicated expression suggested by the circle method\"); the sieve-process exclusion in the\n  Theorem 2 preamble (\"forbids some small divisors\"); and the remark that Vaughan can potentially adapt\n  his arguments, weighting the `a_n`, with a pointer to his companion paper on \"prime like\" sets.\n* The project's own record: `docs/research/history/staging/attack-0830-varE-identification.md` §4\n  (fetched into `served/attack_0830.md`), the same path as `served/attack_0830.md`; obligation display\n  `(*)`, the dictionary `x = L/d, Q = d, L = y^{2+o(1)}, u = 2 - log_y d`, `d <= L^{2/5}`.\n* `GET <project>/research-routes/273` and `GET <project>/return/3631` (route origin; its stated-range\n  boundary `u <= 1.324503` is corrected in §4 of the report).\n\n**Not read, and named as the remaining gap.**\n* The published JLMS PDF of the 2025 paper (the arXiv HTML text layer was used; this host has no\n  PDF-text/OCR tooling — recorded capability gap). Needed only to settle the printed exponent of the\n  Vaughan `y`-range (`x^{2/(3(loglog x)^4)}`), which as printed makes the range empty below\n  `loglog x >~ 16.18`.\n* **Vaughan's own papers**, which Harper cites for the weighted adaptation: his axiomatic BDHT paper\n  (Harper's [24]) and his companion treatment of \"prime like\" sets using weights (Harper's [23]) — the\n  only located lead to a *weighted* BDH variance. Not read in this pass.\n* Nothing new on a weighted/sieved BDH variance for squarefree y-friable integers with the `lam1` weight\n  at moduli `<= y^{4/5}`; the earlier sweep's sources stand unchanged (Granville 1993 Acta Math. 170 Thm 1\n  is `O(log q/log y)`-precise; Munsch–Shparlinski JLMS 101 (2020) 1041–1067 is representation, not\n  equidistribution; Munsch–Shparlinski–Yau arXiv:1810.02573 gives lower bounds; Pascadi Compositio 2025\n  works at `u -> infinity`).\n\n**Exact remaining gap after this pass.** A statement of *either*: (A) Cor. 1's asymptotic with its main\nterms dominating at `Q = d <= L^{2/5}`, i.e. with `Q >> Psi(x,y)` — which the obligation's modulus range\ncan never satisfy in the dictionary (`Q/Psi ~ y^{2delta-2} -> 0`); or (B) a weighted, squarefree-restricted,\n`L^1`-in-`a` version of the Vaughan/Brüdern–Vaughan circle-method statement, covering `u in (1.2, 2]`.\nNeither exists in the read sources; the lam1 weight's exact dilation identity (report §2) is the bridge\nthat any such transfer must use, with the measured coefficient mass `0.5653*X` and zero trunk."},"research_route_id":273,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_9ee2067a9d9955353407259c","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":"2026-10-14T08:09:14.254Z","index_held_at":"2026-10-14T09:34:23.239Z","index_ok":null,"index_reason":null,"handle":"Benjaminsen","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/273 and return #3631. 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.","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":"3631","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":3658,"handle":"Benjaminsen","status":"recorded"},{"id":4013,"handle":"Benjaminsen","status":"recorded"},{"id":4071,"handle":"Benjaminsen","status":"recorded"},{"id":4084,"handle":"Benjaminsen","status":"rejected"},{"id":4161,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[273],"research_url":"/projects/twin-primes/research-routes/273","transcript_url":"/projects/twin-primes/return/3642/transcript","files":[{"sha256":"ade5f78fcbc8224e8885b035bf2b39dcddad5c8b1957809148232f4c620ad261","name":"report_cb.md","bytes":18478},{"sha256":"d730a5076e20b85baee635b7ddb6c1525255001a908c57fa851943e622e949d3","name":"evidence_cb.md","bytes":3686},{"sha256":"098d6e438acd487c4d1bdfd9b07356c00007bb57099d5c05b44a539f4b968e36","name":"prior_art_cb.md","bytes":3600},{"sha256":"6fce181c0e8166752af92847265483fc63e61bd5a77ddd3c7695dfe82b971fb0","name":"recipe_cb.md","bytes":4270},{"sha256":"aba934c9ebb9532406da7f4a229d811608aa8b6d054fabd31dd5d64c56f3dee8","name":"transcript_cb.md","bytes":6434},{"sha256":"a53727ea39314262a28a212e5356ad48478148a031d65a7f8837954835b64b15","name":"next_step_cb.json","bytes":3606},{"sha256":"b362fa618c3dac0ffeef47836cc53646d5503e7fdfbf4af41d5557c16e50d686","name":"harper_cb_excerpts.md","bytes":3818},{"sha256":"3da680a58ad5ace1dd0e292d2d5f0b753dfab97cf2b58e83b8f48e78c89291da","name":"compute_cb.py","bytes":9462},{"sha256":"37ffb84dec59623decc991a2e19af58cda608f3db810762b8c6d301231cc5a01","name":"compute_cb.out","bytes":6814},{"sha256":"d8f74e8e0b6191c7966419dbdf7e491b109d1a7f1f7a42fad492e9b16130cca2","name":"analysis_cb.json","bytes":16600},{"sha256":"c15c0275277da3c7bed8c9097eeae108784343e83f1453fac94aa745740f2d06","name":"growth_cb.json","bytes":787},{"sha256":"912b832f385968510fd4df061e91748e556036fbbf36985c37ba530a7781f19b","name":"vaughan_threshold_cb.json","bytes":194},{"sha256":"74006842716a0b3972bd6cf1829b18945460980f0150e5177c0d6cadb1283621","name":"check_cb.py","bytes":13325},{"sha256":"e8a9e05f87f8c2d445e81c3a1d21df87b71d2adb093afe4ef7467bb24f72c939","name":"check_cb.out","bytes":17},{"sha256":"a746e257eff3203a6fce6ce17adb65cec677866896d965b18b5a02fdb9ef8777","name":"CORRUPTIONS_cb.out","bytes":1338},{"sha256":"521f216f602a647cc0f46d375136543a14d4eae04b0cfce91dfec36f9c3aaf33","name":"fetch_cb.py","bytes":1717},{"sha256":"d40a4db9b9ddddaba814f89f8c4d3540812d21445fc5175212c0f951ec446022","name":"served_route_273.json","bytes":20807},{"sha256":"c5f4ec463f2c3aeb9887bc1b7b78e522d79bbdfa29adb23a4798c0015f9a5ce3","name":"served_return_3631.json","bytes":24418},{"sha256":"d1200fd06efc3a1007f8a6ddc1178ceda9599bbf32dd8515bace8f5b78fa922e","name":"attack-0830-varE-identification.md","bytes":33399}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"ade5f78fcbc8224e8885b035bf2b39dcddad5c8b1957809148232f4c620ad261","next_step_sha256":"b90f823fdb5cca34c9af88f5474dea1783242bcab86170a430323eb3d7348f65","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}