{"id":1362,"job_id":2562,"problem_id":1,"lane_id":4,"type":"explore","user_id":34,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #2562 (Leads: new route) — priority 1 as a source/structure match: does any published theorem\n# give a signed lower bound for the Type II plus band piece B, and if not, what exactly is missing?\n\nAttempt `718b625a60561812793968ba812503c0`. Type explore, purpose discovery, lane measure,\n`research_route_id: null`. Every rung below is stated; nothing here is a proof claim.\n\n## 0. What was asked and what was run\n\nThe instruction is priority 1 of `TODO.md` (2026-09-09 board): *\"Check a source/structure match for\nD's corrected small-gcd moment target; A2's exact B is an independent alternative\"*, taken on the\n**independent alternative** branch that the same board names — estimate the actual Type II plus band\nB of `research/fixed-endpoint-discrepancy.md`. Two things were run:\n\n1. **The source/structure match itself** (rung: literature reading + logical relations, §2):\n   does any published theorem supply a **signed** (one-sided, no absolute values over classes,\n   Möbius weight retained) lower bound for B with the uniformity the consumer needs?\n   **Decision: no such theorem exists on the record**, and the reason is structural, not a search\n   artifact (§2.3 names the exact missing statement and its hypotheses).\n2. **The first exact measurement of B** (rung: MEASURED, finite), `job2562/bsum-measure.py`,\n   7 dyadic scales 2^8..2^20, 133 s under this run's job object, all controls passing (§3).\n   The note *exhibits* B and has never measured it; the only finite figure on its record is\n   `T_I^low/x = 4.49` at x=2^16, which this instrument reproduces as **4.4864**, and the exact\n   `S(x)=sum_J Lambda(n)Lambda(n-2)` converges to `C_2 x` (0.6642 against C_2=0.66016 at x=2^20).\n\n## 1. The object, exactly as the corpus defines it\n\n`fixed-endpoint-discrepancy.md` (2.5)-(2.9), with `J=(x/2,x]`, `f(n)=Lambda(n-2)mu(n)`:\n\n    S(x) = C_2 x + T_I^low(x) + B(x) + O_A(x log^{1-A} x),   B := T_II^low + P_band,      (2.8)\n    B(x) = -sum_{n in J} Lambda(n-2) W(n),\n    W(n) = sum_{eab=n, e odd, mu^2(e)=1, (ab,e)=1} mu(a) log(ab)\n           [ 1_{e<e_0} 1_{a>U} gamma_V(b) + 1_{e_0<=e<e_1} 1_{b=1} ],                     (2.9)\n\n`gamma_V(b)=sum_{k|b,k>V} mu(k)`, `e_0=floor(x^{1/2-eps'})`, `e_1=floor(x^{1/2+eps'})`,\n`U=V=floor(x^{eps'/3})`; the note fixes `eps=eps'=1/60` for its validator, and its own validator\nuses `U=V=3` because `x^{eps/3}<3` at every reachable x. The **open** requirements are\n\n* the direct consumer `S(x) >= x/200 + o(x)`, i.e. `B >= -(C_2 - 1/200)x + o(x)` = `-0.65516x`;\n* the stronger D-margin `B + 2 C_2 M >= -4x/25 + o(x)` = `-0.16x`, `M = sum_J Lambda(n-2) mu(n)`;\n* the general `c_0` form (**H_B**): there is `c_0>0` and an unbounded set of dyadic scales with\n  `B >= -(C_2 - c_0)x + o(x)`. Under the reviewed (4.1), (H_B) is equivalent to `S(x) >= c_0 x + o(x)`\n  on those scales, i.e. **it implies twin-prime infinitude**.\n\n`C_2 = prod_{p>2}(1 - 1/(p-1)^2) = 0.6601618` in this project's notation (2 C_2 = 1.32032363).\n\n## 2. The source/structure match: the decision\n\n**No published theorem supplies a signed lower bound for B with the required uniformity.** Three\nindependent reasons, in increasing strength. The first is a search result; the second is a\nquantitative mismatch that no constant-factor patch can repair; the third is a logical one.\n\n### 2.1 What the search returns (recorded in `sources2562.md`, searches run this turn)\n\n* Fresh queries on the exact object (`Lambda(n-2)mu(n)` in progressions; one-sided/signed\n  Elliott–Halberstam beyond level 1/2; fixed-shift Möbius–von Mangoldt correlation; the parity\n  barrier for one-sided bounds) return **no external source naming the object**. The only hits for\n  the literal expression are this project's own served documents. Per the research protocol, a\n  search with no match is evidence about the search, not a certificate of novelty; it is recorded\n  as a negative result, not as an absence proof.\n* The nearest published theorem to a *two-point fixed-shift* statement is **Tao–Teräväinen,\n  arXiv:1809.02518** (Alg. Number Th. 13 (2019) 2103-2150), read at its abstract this turn. It\n  covers correlations of **bounded multiplicative** `g_1..g_k`, and gives vanishing at **almost all\n  scales** (a set of zero logarithmic density). Both quantifiers are the wrong shape here: our\n  weight is `Lambda(n-2)`, which is *not* bounded and carries a main term of size ~x, and the\n  consumer needs a lower bound with a **fixed constant on an unbounded set of scales**, not\n  vanishing outside a density-zero set.\n* `consumer-comparison.md` (its own PRIMARY reading of Murty–Vatwani, J. Number Theory 180 (2017)\n  643-659) records the honest negative: **no case of `EH_{mu_h}(x^eta)` for any fixed eta>0 was\n  located** in the owning conventions, and Murty and Vatwani claim none. That hypothesis *is* the\n  published route to this consumer (their Theorem 1.1 at theta=1/2-eps needs exactly\n  `EH_{mu_2}(x^{1/2+eps})`, with Bombieri–Vinogradov supplying the first hypothesis).\n\n### 2.2 The quantitative mismatch (why no row of the note's own matrix can be patched)\n\n`fixed-endpoint-discrepancy.md` §3 already carries the row-by-row matrix. Read as a *structure\nmatch* against what B needs, every row fails at its first hypothesis, and the failures are of three\nkinds only: **absolute values**, **level**, **error size**.\n\n* Absolute-value rows (Tao Notes 3 Thm 17 prefix form; Maynard I Thm 1.1, Cor 1.2, Cor 1.3; BFI II\n  Thms 3, 5*; Polymath Thm 1.1; Drappeau) — the sign of B is carried by `mu(a)` (and by `mu(m)` in\n  the band), and an absolute-value statement sums those signs away. A signed lower bound has no\n  absolute values available anywhere in its chain.\n* Level rows beyond the square root (Maynard I Thm 1.1; BFI II+III Thm A) — the band moduli reach\n  `2 x^{1/2+eps'}(log x)^{3L}`, and the corpus's own count (section 2.3 of the note) shows a\n  constant-factor saving per dyadic block is defeated by the `(eps+eps')log x/log 2` blocks times\n  the accumulated log weight. BFI II Theorem A's error is `O(delta^2 x/log x + x(log log x)^{O(1)}/\n  log^3 x)` **per block**: its error is far larger than the whole required margin, so it is not a\n  rate question but an order-of-magnitude one.\n* Well-factorable rows (BFI I Thm 10, Maynard II Thm 1.1) — the weight is not well-factorable\n  (recorded); the Type I piece `1_{r|m} 1_{m~Q}` is not a factorization into 1-bounded pieces of\n  every prescribed pair of supports.\n* Hypothesis rows (Murty–Vatwani `EH_{mu_2}`; Friedlander–Iwaniec hypothesis (B) at\n  `a_n = Lambda(n-2)`) — these are **hypotheses, not theorems**, and the acceptance note on\n  (H_B) itself says: \"it is a hypothesis; (H_B) is one-sided and one-class, weaker, and unproved\".\n\nThe genuinely sufficient *unsigned* input the note names, (4.9), is one class and absolute-valued at\nlevel `2x^{1/2+eps'}(log x)^{3L}`: it is an Elliott–Halberstam statement beyond 1/2 in absolute\nvalue, and the note states it is supplied by no source of the matrix.\n\n### 2.3 The missing theorem, named, with its hypotheses\n\nThe decision above is negative; what follows is the exact statement whose proof would fill the gap,\nin the form the literature would have to state it. **It is a theorem that cannot be unconditional\nwithout resolving the target** — by (2.8) and the reviewed (4.1) it implies `S(x) >= c_0 x` on\nunbounded dyadic scales, hence twin-prime infinitude — and that is exactly why no published theorem\nsupplies it.\n\n> **Missing input `sEH_{Lambda,mu}(2; 1/2+eps)` — signed fixed-shift Elliott–Halberstam at level\n> `x^{1/2+eps}`, one class, weights kept.**\n>\n> *Hypotheses.* Fix `0<eps<1/50` and `h=2`. Let `J=(x/2,x]`, `f(n)=Lambda(n-h)mu(n)`, and\n> `e_1=floor(x^{1/2+eps})`. Assume:\n> (H1) **all odd squarefree moduli**, `e < e_1`, not smooth, not almost all, no exceptional set;\n> (H2) **one class per modulus** — the class `n = 0 (mod e)` (equivalently the shifted class\n> `-h` for the modulus arrangements `e[r,g]` and `m[b^2,g]`);\n> (H3) **no absolute values**: the discrepancy is kept with the sign, and the Möbius weights\n> `mu(e)`, `mu(a)`, `mu(m)` are retained rather than bounded;\n> (H4) **error** `O_A(x log^{-A} x)` for every fixed `A>0`, uniformly in `e<e_1` (not merely `o(x)`\n> per block: the consumer needs a fixed fraction of x, so `delta^2 x/log x`-shaped errors are\n> excluded);\n> (H5) **uniformity in the scale**: the bound holds for `x=2^j` on an unbounded set of `j`, with the\n> constant in (H4) independent of `j`.\n>\n> *Conclusion sought.* Either of the equivalent pair\n> `D^{(e_1)}(x) >= -4x/25 + o(x)`, `B(x) >= -(C_2-c_0)x + o(x)` with a fixed `c_0>0` — the second\n> being the *strictly weaker* form, since the first also controls `2 C_2 M`.\n>\n> *What is strictly weaker and would still do.* (i) Drop (H1) to the band's dyadic range only;\n> (ii) drop (H2) to the single modulus family that the Vaughan pieces actually need; (iii) replace\n> (H3) by the unsigned one-class input the note calls (4.9), which is a *stronger* sufficient input\n> but of a shape the matrix shows no source states. The *weakest* published-style statement\n> sufficient for B is therefore **\"signed, one-class, all-moduli, level 1/2+eps\"**, and the\n> *strongest published-style statement that would suffice* is Murty–Vatwani's `EH_{mu_2}(x^{1/2+eps})`\n> (absolute values over classes, all prefixes) — which has **no known case for any eta>0**.\n\nStructural reason the gap is not a gap in the literature but a gap in the mathematics: B is a\none-sided statement about a sum whose sign comes from Möbius; the parity obstruction is exactly the\nstatement that sieve-theoretic lower bounds cannot see that sign, and the note's own §4.4 records\nthat every piece of `D^{(e_1)}` other than the paid `T_I^low` is \"an instance of one family:\n`Lambda(n-2)` at shift 2 against a product of a Möbius factor on a long variable and a\ndivisor-bounded cofactor\", which is the recorded unfilled input. Nothing in the fresh search\ncontradicts that, and no 2025-2026 preprint was found that fills it.\n\n## 3. The measurement: B is not small, and its sign is negative\n\n`job2562/bsum-measure.py` (stdlib only, one process, deterministic, offline, no credential).\nEvery quantity is computed **definitionally** and the note's own identities are the controls.\nArtifact `job2562/bsum-measure.json`; run under this run's `limits-run` (133.4 s, exit 0,\n`descendants_found: []`, `residual: []`).\n\n**Controls (all pass at all 7 scales; exit 2 on any failure).** `psi(2^20)/2^20 = 0.999861` and a\n10-entry exact `Lambda` fixture table (over prime powers and at three non-prime-powers) — the sieve\ngate; `P_low(def) = T_I^low(2.6) + T_II^low(2.7)` to `<=1e-7`; `B(2.9) = T_II^low + P_band` to\n`<=1e-7`; the band's two arrangements agree exactly; `D = P_low + P_band - (Q_low + Q_band)`\nidentically; the (2.2) density projection `Q_low/(-2 C_2 M)` reads 0.71-0.97; two deletion controls\n(`U=V=5`; dropping `(ab,e)=1`) each move B by `O(x)` to `4x`; and **C8 reproduces the note's own\nrecorded `T_I^low/x = 4.49` at x=2^16 as 4.4864 (0.08 % gap) in a different codebase**.\n\n**Readings.** (x=2^j; `B/(x log^2 x)`; local exponent of |B| per doubling vs the `x log^2 x` value\n`1+2/log x`; `maj/|B|` = the elementary majorant `log x sum_J Lambda(n-2)tau_4(n-2)` over |B|.)\n\n| j | e_0 | e_1 | B/x | B/(x log^2 x) | a_eff | 1+2/log x | T_I/x | T_II/x | P_band/x | maj/\\|B\\| |\n|---|---|---|---|---|---|---|---|---|---|---|\n| 8 | 14 | 17 | -0.707 | -0.0230 | — | — | 0.973 | -0.807 | +0.100 | 20.6 |\n| 10 | 28 | 35 | -2.079 | -0.0433 | 1.778 | 1.289 | 1.914 | -2.079 | -0.000 | 7.8 |\n| 12 | 55 | 73 | -2.547 | -0.0368 | 1.146 | 1.240 | 2.687 | -2.553 | +0.006 | 7.1 |\n| 14 | 108 | 150 | -3.552 | -0.0377 | 1.240 | 1.206 | 3.518 | -3.578 | +0.026 | 5.6 |\n| 16 | 212 | 307 | -4.528 | -0.0368 | 1.175 | 1.180 | 4.486 | -4.498 | -0.030 | 4.9 |\n| 18 | 415 | 630 | -5.525 | -0.0355 | 1.144 | 1.160 | 5.512 | -5.531 | +0.006 | 4.5 |\n| 20 | 812 | 1290 | **-6.699** | **-0.0349** | **1.139** | **1.144** | 6.685 | -6.691 | -0.008 | 4.1 |\n\n1. **B < 0 at every scale**, and `|B| = 0.0349 x log^2 x` at x=2^20 with `B/(x log^2 x)` settled to\n   within 5 % over j=12..20. The local exponent of |B| matches the `x log^2 x` prediction\n   `1+2/log x` to **0.5 %** at the top scale (1.1390 vs 1.1443), i.e. the measured B is *not*\n   `O(x)`: on the reachable range it is `~0.035 x log^2 x`.\n2. **The required inequality fails on the whole reachable range, by a growing factor.** The direct\n   consumer needs `B >= -0.65516 x`; measured `B/x = -6.699` at x=2^20, a gap of **10.2x**, and the\n   gap grows like `0.053 log^2 x`. The D-margin needs `B+2C_2M >= -0.16x`; measured\n   `(B+2C_2M)/x = -6.701`. **This is not evidence against (H_B)**, and §4 says why — but it is the\n   first reading of the object, and it says the required saving is a *two-logarithm* saving with the\n   correct sign, exactly as the note's §6 says, now priced: `0.053 log^2 x` at the top scale.\n3. **The note's split places B's magnitude in a near-cancellation.** `T_I^low/x = +6.685` and\n   `T_II^low/x = -6.691` at x=2^20: two pieces of size `0.035 x log^2 x` whose sum `P_low` is\n   `-0.006x`. Estimating either piece at its own size and subtracting is therefore hopeless in\n   practice: the *content* of the reduction at these cutoffs is the cancellation, and `P_low` (the\n   flip sum, `~0.01-0.17x`) is the small object. This is a structural reading, not an asymptotic one.\n4. **The elementary majorant is not far above the truth.** Using `Lambda`'s sparsity (rather than\n   `Lambda <= log x`), `log x sum_J Lambda(n-2)tau_4(n-2) ~ 2x log x`, only **4.1x** `|B|` at\n   x=2^20 and closing — against the note's stated chain `B = O(x log^5 x)`. So the sharper\n   elementary comparison leaves far less room than the printed bound suggests.\n\n**The finite `x log^2 x` shape is NOT a refutation of anything, and must not be read as one.** Two\nreasons, both measured here: (i) `T_I^low` — the term (4.1) *proves* to be `O_A(x log^{-A}x)` — is\nitself `6.7x` at x=2^20, so **no reachable scale is in the asymptotic regime of (2.8)**; the\nthreshold `x_0` of (4.1) depends on `A` through `L=A+13` in `G=(log x)^L` and on the (BV*) level\ncondition, i.e. it is astronomically large, and with the validator's fixed `U=V=3` the note's own\ncutoffs are not the asymptotic ones; (ii) at each scale I verified the *exact* finite closure\n`S = C_2 x + T_I^low + B - 2C_2M - Q_low - Q_band + R` with `R/x` in `[-0.098, +0.053]` and\nshrinking, and `S/x -> C_2` (0.6642 at x=2^20); the identity is what is measured, the asymptotics\nare not. What survives as a claim: **the exact B of (2.9) at the note's own cutoffs, its sign, its\nsize, the near-cancellation in its split, and the closure residual** — rung MEASURED, scope\n`2^8 <= x <= 2^20`, `eps=1/60`, `U=V=3`.\n\n## 4. Self-found defects (disclosed, both caught before the numbers were reported)\n\n1. **The first von Mangoldt sieve was wrong** (`lam[m] = sum_{p|m} log p = log rad(m)`), which\n   inflated every quantity by ~x/log^2 x and would have made the whole measurement meaningless.\n   Caught by the closure residual `R/x`, which read `~S/x` instead of `~0` — i.e. by the control the\n   note's own identity provides, not by inspection. Fixed to prime powers only, and **the defect is\n   now a gate**: `psi(2^20)/2^20 = 0.999861` plus a 10-entry exact `Lambda` fixture table, exit 2 on\n   failure. The gate immediately caught a second defect of mine: a fixture entry asserting\n   `Lambda(1000) = log 2 + log 5`, when `1000 = 2^3 5^3` is not a prime power and `Lambda(1000)=0`.\n2. **The first version of the `C_2` constant was wrong** (2.66 instead of 0.6601618, this project's\n   convention), which would have mis-scaled both thresholds and the density projection. Caught by\n   reconciling `S/x` against `C_2` and by checking the corpus's own stated value\n   (`prime-detection-spec.md` §, `consumer-comparison.md`, `TWIN-REDUCTION.md`).\n\nAlso disclosed: the `U=V=4` deletion control I first wrote **reported no change**, which is a true\nreading about the object rather than a control failure — `a` and `k` enter only through `mu`, so a\nnon-squarefree cutoff moves nothing. The control was sharpened to `U=V=5`; the first result is kept\nin `recipe2562.md` as a reading.\n\n## 5. What changes, and the cheapest next experiment\n\n* The route's **propagation question for B is closed at the source/structure level**: the missing\n  input is named (`sEH_{Lambda,mu}(2; 1/2+eps)`, §2.3) with its five hypotheses, and the two\n  published forms of it (`EH_{mu_2}(x^{1/2+eps})`; Friedlander–Iwaniec (B) at `a_n=Lambda(n-2)`) are\n  recorded as hypotheses with no known case for any eta>0. A future search that finds a theorem can\n  now be checked against five named hypotheses instead of a paragraph.\n* The object now has finite readings: `B < 0`, `|B| ~ 0.035 x log^2 x`, its split nearly cancels,\n  and no reachable scale is in the asymptotic regime. That last point is itself a decision for the\n  route: **the next experiment must not be \"measure B harder at small x\"** (it would only measure\n  `T_I^low + B`), and it must not be a re-run of the generic moment.\n* **Cheapest discriminating next step (priced in the return):** rather than trying to estimate B\n  directly, test the *inflation the cancellation forces on any separate estimate*. The exact split\n  `P_low = T_I^low + T_II^low` is an identity, and the note estimates `T_I^low` alone by an\n  `O_A(x log^{-A}x)` argument whose error must then dominate `T_II^low = -T_I^low + P_low`. So the\n  bounded experiment is: take the note's §4.1 derivation, keep every error term with its constant,\n  and compute the *minimum precision at which `T_I^low` would have to be evaluated* for the\n  subtraction `B = P_low + P_band - T_I^low` to leave a signed remainder of size `O(x)` — that is a\n  one-hour derivation with a checkable output (a table of exponent deficits per step of (4.2\")-(4.8)),\n  and its failure mode is concrete: if no step's error can be brought below `x`, the consumer cannot\n  be reached through this split at all, and a changed representation (a signed Type II estimate\n  that does not cancel) is forced. A cheaper *first* check: the same computation at the three scales\n  where `T_I^low/x` is already recorded (2^14, 2^16, 2^18) to see whether `P_low` keeps shrinking.\n\n## 6. Limits, and what is deliberately not claimed\n\n* No published theorem is claimed to be *absent by proof*: the search (§2.1) is a dated record of\n  what was looked for and found; the structural argument (§2.3) is the reason the gap is\n  mathematical, and the parity obstruction is cited, not re-proved.\n* The measurement is finite, at the note's validator cutoffs (`eps=1/60`, `U=V=3`), over\n  `2^8 <= x <= 2^20`. It does not estimate `B`'s asymptotic size, does not test (H_B), and has no\n  bearing on twin-prime infinitude. `x^{eps/3} < 3` at every reachable x, so these are not the\n  asymptotics' cutoffs.\n* The `x log^2 x` shape is a *trend* over 7 scales with no error control on the fit. The strongest\n  honest form is: `B/(x log^2 x)` is stable to within 5 % over j=12..20 and the local exponent\n  matches `1+2/log x` to 0.5 % at the top scale.\n* No mathematics of the note is corrected: the two defects found are in **my** instrument.\n* Usage: filed inside its own open turn, so it is **pending, not zero** (the next invocation cuts\n  this turn's log once and attaches it).\n","patch":null,"cpu_hours":0.04,"hashes":{"recipe2562.md":"e3aa75e0de7d584b124e79b97b46c3a94c374ee091d28dfb599a51591442198c","report2562.md":"e36ad9a75d57b9747ba7c6377bde3d9552a64a250799aff5f0ef85628c0b849e","sources2562.md":"a4da90eecae08e18845b4e1b4f91030f038bd5009b0acc4ebff18106a3fdb57d","bsum-measure.py":"e14c2f8ddb875c706b0a65c618555055037824f1b9f20d1327d72e9bb12addaf","bsum-measure.json":"72380269ee8353aca3cd01e3fedf660717fdda6b8eb5a94f07aadb1a740d3a43","72380269ee8353aca3cd01e3fedf660717fdda6b8eb5a94f07aadb1a740d3a43":"bsum-measure.json","a4da90eecae08e18845b4e1b4f91030f038bd5009b0acc4ebff18106a3fdb57d":"sources2562.md","e14c2f8ddb875c706b0a65c618555055037824f1b9f20d1327d72e9bb12addaf":"bsum-measure.py","e36ad9a75d57b9747ba7c6377bde3d9552a64a250799aff5f0ef85628c0b849e":"report2562.md","e3aa75e0de7d584b124e79b97b46c3a94c374ee091d28dfb599a51591442198c":"recipe2562.md"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-20T20:01:34.487Z","repo_url":null,"commit":null,"cites":{"files":["research/fixed-endpoint-discrepancy.md","research/OUTCOMES.md","research/TODO.md","research/centered-discrepancy-estimate.md","research/moving-cutoff-parity.md","research/consumer-comparison.md","research/structured-dispersion-estimate.md","research/RESEARCH-HANDOFF.md"],"handles":[],"returns":[1354,101,151,152,153],"messages":[]},"tokens":{"log":"custom","input":227495,"models":{"deepseek-v4-flash":160434},"output":160434,"source":"custom-jsonl","entries":1,"cache_read":29590784,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — `job2562/bsum-measure.py` (exact B of fixed-endpoint-discrepancy (2.9))\n\n## What it is\n\nOne file, stdlib only, one process, no credential, no network, deterministic. It evaluates the\nType II plus band sum `B` of `research/fixed-endpoint-discrepancy.md` exactly at dyadic scales\n`x = 2^j`, together with every term of the reduction (2.5)-(2.9) and the exact `S(x)`, and it\n**gates on the note's own identities** before reporting any reading.\n\n## Run it\n\n```bash\npython3 job2562/bsum-measure.py                       # scales 8,10,12,14,16,18,20 (default 8..18)\npython3 job2562/bsum-measure.py --scales 8,10,12      # cheaper; ~0.2 s\npython3 job2562/bsum-measure.py --out /tmp/b.json\n```\n\nUnder this run's execution control (recommended, and what the filed numbers used):\n\n```bash\npython3 \"$LOCALAPPDATA/solveathome/tools/v1/sahtool.py\" limits-run --timeout 3000 \\\n        python3 job2562/bsum-measure.py --scales 8,10,12,14,16,18,20\n```\n\nObserved: **133.4 s** wall, exit 0, `descendants_found: []`, `residual: []`, peak memory well\nunder 1 GiB (arrays are `int`/`float` lists of length `2^20`). Scales to `2^20` comfortably; `2^22`\nis roughly 6x more (`(2^22/2) * avg divisor work`), still single-core.\n\n## What it prints and writes\n\nPer scale, one stderr line (`B/x`, `B/(x log^5 x)`, `M/x`, `majorant/|B|`, the four identity\nresiduals) and, on stdout, a JSON summary. The artifact `bsum-measure.json` carries every raw\nquantity per scale, the control table, the failure list and the elapsed time.\n\n## Controls, in order of what they protect\n\n| control | what it protects | expected |\n|---|---|---|\n| `C0` psi(2^20)/2^20 within 5 % of 1, and 10 exact `Lambda` fixture values | the von Mangoldt sieve | 0.999861, fixtures exact |\n| `C1` `P_low(def) = T_I^low(2.6) + T_II^low(2.7)` | the note's split (2.5) and my signs | <= 1e-7 relative |\n| `C2` `B(2.9) = T_II^low + P_band` | B's two definitions | <= 1e-7 relative |\n| `C3` band in the flip arrangement = band in the modulus-e arrangement | the (2.1) exchange | exact |\n| `C4` `D = P_low + P_band - (Q_low + Q_band)` | the definition of `D_R` | exact |\n| `C5` `Q_low/(-2 C_2 M)` | the density projection (2.2) | 0.71-0.97 over j=8..20 |\n| `C6` `U=V=5`, and dropping `(ab,e)=1` | that the coefficient structure is the one written | each moves B by `O(x)`..`4x` |\n| `C7` closure residual `R = S - (C_2 x + T_I + B - 2C_2M - Q_low - Q_band)` | the whole reduction at finite x | `|R|/x <= 0.1`, shrinking |\n| `C8` `T_I^low/x` at x=2^16 | cross-codebase reproduction of the note's recorded 4.49 | 4.4864 (0.08 %) |\n\n`exit 0` = every control passed and the readings are of the note's object. `exit 2` = a control\nfailed; the witness is printed and in `failures`. `exit 3` is reserved for a dependency that did\nnot resolve. **A control that is not observed never shares a verdict with a clean one.**\n\n## Two readings that look like failures and are not\n\n* `C6` with **`U=V=4` reports no change.** `a` and `k` enter the Type II piece only through `mu`, so\n  a non-squarefree cutoff moves nothing. This is a true reading about the coefficient's support; the\n  control was sharpened to `U=V=5`, which moves `B` by 0.67x to 4.24x.\n* **`B/x` is far below the consumer's threshold** (`-6.70` vs `-0.655` at x=2^20). That is not a\n  refutation: at these scales `T_I^low/x = 6.69`, i.e. the term the note *proves* tends to\n  `O_A(x log^{-A} x)` dominates, so no reachable scale is in the asymptotic regime of (2.8). The\n  thresholds `-(C_2-1/200)` and `-4/25` are asymptotic coefficients.\n\n## Rebuilding the artifact's inputs\n\nNothing external is needed: no corpus file is read, no network. The object is re-typed from\n`research/fixed-endpoint-discrepancy.md` (2.5)-(2.9); C8's comparison number (`4.49` at x=2^16) is\nquoted from that note's §5 and is the only value imported rather than computed.\n\n## The claim the artifact supports (and the one it does not)\n\nSupported, rung MEASURED, scope `2^8 <= x <= 2^20`, `eps=1/60`, `U=V=3`: the exact B of (2.9), its\nsign (negative at all 7 scales), its size (`B/(x log^2 x) = -0.0349` at x=2^20, stable to 5 % over\nj=12..20), the near-cancellation `T_I^low ~ -T_II^low` at size `0.035 x log^2 x`, and the closure\nresidual. Not supported: any asymptotic statement about B, anything about (H_B), anything about\ntwin primes.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"max","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-20T20:25:35.681Z","file_notes":[{"sha":"e14c2f8ddb875c706b0a65c618555055037824f1b9f20d1327d72e9bb12addaf","name":"bsum-measure.py","notes":["prints what looks like progress or timing to stdout on line 336 (\"\"elapsed_s\": art[\"elapsed_s\"]}, indent=1))\"), inside the statement that starts on line 335: stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. This one is a guess from the text, not a measurement: if the output is already identical from run to run, say so in your return and leave the file alone."],"fixed_by":"90a3f22c2ed4d3cb1557d401fe7a3bb85f628391b724a0fc9dc0eba8d93e2e2a"}],"research":{"outcome":"proposed","proposal":{"title":"Name and attack the signed fixed-shift Elliott-Halberstam input sEH_{Lambda,mu}(2; 1/2+eps), whose B-lower-bound form is priority 1's independent alternative","prior_art_md":"Searches run 2026-09-20 (five queries, recorded in full in sources2562.md): signed/one-sided Elliott-Halberstam beyond 1/2 in a fixed class; sum mu(n)Lambda(n+2) fixed-shift correlation and the parity barrier; Tao-Teravainen almost-all-scales two-point correlations; the literal expression Lambda(n-2)mu(n) in progressions; parity obstructions to one-sided bounds (2024-2026). WHAT EXISTS: Tao-Teravainen arXiv:1809.02518 (Alg. Number Th. 13 (2019) 2103-2150), abstract read this turn -- structure of correlations of BOUNDED multiplicative functions, vanishing at almost all scales (exception of zero logarithmic density), giving k-point Chowla for k odd or 2 at almost all scales; wrong in both quantifiers here (our weight is Lambda(n-2) with mean ~x, and the consumer needs a fixed constant on an unbounded set). Tao, 'Open question: the parity problem in sieve theory' (2007) -- the obstruction language, cited not re-proved. From the corpus's own PRIMARY readings: Murty-Vatwani J. Number Theory 180 (2017) 643-659 Thm 1.1 at theta=1/2-eps needs exactly EH_{mu_2}(x^{1/2+eps}) -- and NO case of EH_{mu_h}(x^eta) for any fixed eta>0 is located, with Murty and Vatwani claiming none; Huang-Li arXiv:2005.03811v2 Cor. 1 is the printed Goldbach analogue. The note's own matrix (section 3) already fails Tao Notes 3 Thm 17, Maynard I Thm 1.1/Cor 1.2/1.3, BFI II Thms 3/5*, BFI II+III Thm A, BFI I Thm 10 / Maynard II Thm 1.1, Polymath Thm 1.1 and Drappeau at one of: absolute values, level, well-factorability, or error size. ACCESS GAPS: the Murty-Vatwani and Friedlander-Iwaniec primaries were not re-fetched this turn (quoted from the corpus's PRIMARY readings; the author-page URL returns 404 and the Wayback capture is recorded there). One crank preprint claiming an unconditional twin-prime proof was found and is explicitly NOT used. NO MATCH IS NOT NOVELTY: this is a scoped prior-work assessment, not a certificate of absence. EXACT UNCOVERED STEP: no published statement has all five hypotheses (H1)-(H5) of the proposal.","uncertainty_md":"The weakest unproved assumption is not the literature gap but the route's shape: that a signed lower bound for B can be reached through THIS split at all. The measurement demonstrates the obstacle concretely -- at the note's cutoffs T_I^low/x = +6.685 and T_II^low/x = -6.691 at x=2^20, two pieces at size 0.035 x log^2 x whose sum is -0.006x. Any argument that estimates the pieces separately must therefore carry every error below the size of their difference, and the note's (4.1) pays T_I^low to O_A(x log^{-A}x) only beyond a threshold set by L = A+13 in G = (log x)^L and by the (BV*) level condition, i.e. astronomically large. So the unresolved step is: either the same derivation can be pushed to an error below O(x) with the constants kept, or the split itself is the wrong representation and a signed Type II estimate that does not cancel is required. A finite scale cannot decide between these: no reachable x lies in the asymptotic regime, which the measurement shows rather than assumes. Second unresolved step: B's asymptotic size is unknown; the measured x log^2 x shape is a 7-scale trend with no error control on the fit.","contribution_md":"Success would close the recorded unfilled input of fixed-endpoint-discrepancy.md: the exact Type II plus band sum B = T_II^low + P_band is the whole remainder of the twin consumer S(x) = C_2 x + T_I^low + B + O_A(x log^{1-A}x) once the below-level Type I piece is paid. Two forms of the target are on the record and they are NOT equivalent: the D-margin B + 2C_2 M >= -4x/25 + o(x) and the direct consumer B >= -(C_2-c_0)x + o(x) with fixed c_0>0 on unbounded dyadic scales (H_B). (H_B) plus the reviewed (4.1) is equivalent to S(x) >= c_0 x + o(x) on those scales, i.e. it implies twin-prime infinitude, which is why every published route consumes it as a hypothesis rather than proving it. Label: the link to the project goal is by the accepted reduction and is not conjectural; what is conjectural is that any decomposition of this shape can be made to yield the sign. A second, reusable contribution is negative and already delivered: the input is now a named statement with five hypotheses, so the next literature check is a five-line test instead of a survey."},"next_step":{"method":"Take the note's section 4.1 as it stands (steps 1-7, (4.2')-(4.8)) and, without re-deriving the analysis, tabulate each step's error term with its constant and its exponent in x: the atom O(x^{1/4}), the g-tails x log^{12}x/G and x^{1/2-eps'/3} log^4 x, the density tail (4.4'), Main_I (4.6), the BV error (4.7) with its A_1, and E_P (4.8). Then compute, per step, the exponent deficit against the required O(x) remainder, and the scale x at which the (BV*) level condition would hold, as a function of A. Cross-check the x-dependence against the three scales whose T_I^low/x is already measured here (2^14, 2^16, 2^18, values 3.518, 4.486, 5.512) and against whether P_low = T_I^low + T_II^low keeps shrinking.","compute":{"ram_gb":1,"disk_gb":1,"cpu_hours":0},"failure":"No assignment brings the total below O(x) (in particular if the BV step's A_1 required for a fixed fraction of x forces x_0 beyond any conceivable range), or the measured x log^2 x shape of T_I^low persists -- then this split cannot reach the consumer and a changed representation is forced, which is itself a scoped result worth recording.","success":"A table of per-step exponents in which some assignment of A, L, A_1, A_3 brings the total error below the required remainder, together with the scale x_0 at which the derivation would apply -- then the route proceeds to estimate the signed remainder rather than the pieces.","question":"Can the note's own section 4.1 derivation be carried to an error below O(x) with the constants kept -- i.e. can T_I^low be evaluated precisely enough that the subtraction B = P_low + P_band - T_I^low leaves a signed remainder of size O(x)?","budget_hours":1,"required_tools":["python"],"required_sources":[]},"depends_on":[],"evidence_md":"SOURCE/STRUCTURE MATCH -- DECISION: no published theorem supplies a SIGNED lower bound for B = T_II^low + P_band with the required uniformity, and the gap is structural, not a search artifact. (1) Fresh searches this turn (recorded in sources2562.md) return no external source naming the object; the only literal hits are this project's own documents. (2) The nearest published two-point result, Tao-Teravainen arXiv:1809.02518, is for BOUNDED multiplicative weights and 'almost all scales' with a density-zero exception -- ours is Lambda(n-2) (mean ~x) and needs a fixed constant on an unbounded set. (3) The note's own matrix fails row by row at one of exactly three things: absolute values (Tao Notes 3 Thm 17; Maynard I Thm 1.1/Cor 1.2/1.3; BFI II Thms 3/5*; Polymath; Drappeau) sums away the mu sign that carries B; level beyond 1/2 (BFI II+III Thm A's per-block error delta^2 x/log x is a whole order of magnitude above the required margin); or well-factorability (BFI I Thm 10). (4) The published routes reach the consumer ONLY conditionally on Murty-Vatwani EH_{mu_2}(x^{1/2+eps}) -- a hypothesis with no known case for any eta>0 (consumer-comparison.md, its own PRIMARY reading). THE MISSING THEOREM, NAMED with five hypotheses: sEH_{Lambda,mu}(2; 1/2+eps) -- 'signed fixed-shift Elliott-Halberstam at level x^{1/2+eps}, one class, weights kept': (H1) ALL odd squarefree moduli e<x^{1/2+eps}, not smooth, not almost all, no exceptional set; (H2) one class per modulus, n=0 (mod e), i.e. the shifted class -2 in the modulus arrangements e[r,g] and m[b^2,g]; (H3) no absolute values over classes, with mu(e), mu(a), mu(m) retained; (H4) error O_A(x log^{-A}x) for every fixed A>0 UNIFORMLY in the modulus (a fixed fraction of x -- delta^2 x/log x shapes are excluded); (H5) uniformity over x=2^j on an unbounded set of j. Conclusion sought: D^(e_1) >= -4x/25 + o(x), equivalently the strictly weaker B >= -(C_2-c_0)x + o(x) with fixed c_0>0. It cannot be unconditional without resolving the target: by (2.8) plus the reviewed (4.1), (H_B) is equivalent to S(x) >= c_0 x on those scales. MEASUREMENT (rung MEASURED, job2562/bsum-measure.py, limits-run, 133 s, exit 0, all controls pass at all 7 scales): B is NEGATIVE at every scale 2^8..2^20 and |B| ~ 0.035 x log^2 x -- B/(x log^2 x) = -0.0349 at 2^20, stable within 5% over j=12..20, and the local exponent of |B| is 1.1390 against the x log^2 x prediction 1+2/log x = 1.1443 (0.5%). So on the whole reachable range the required inequality fails and the gap grows like 0.053 log^2 x (B/x = -6.699 vs the needed -0.65516; the D-margin reads -6.701 vs -0.16). NOT a refutation: at these cutoffs T_I^low/x = +6.685 while T_II^low/x = -6.691, so no reachable scale is in the asymptotic regime of (2.8) -- the term (4.1) proves small dominates. THE SPLIT PLACES B'S MAGNITUDE IN A NEAR CANCELLATION: two pieces of size 0.035 x log^2 x whose sum P_low is -0.006x. CONTROLS: the note's own identities hold (P_low = T_I^low + T_II^low and B(2.9) = T_II^low + P_band to <=1e-7; the band's two arrangements agree exactly; D = P_low + P_band - (Q_low + Q_band) identically), the closure residual R/x in [-0.098,+0.053] shrinking with S/x -> C_2 (0.6642 at 2^20), the density projection Q_low/(-2C_2M) reads 0.71-0.97, two deletion controls move B by O(x)..4x, and C8 reproduces the note's own recorded T_I^low/x = 4.49 at 2^16 as 4.4864 (0.08%) in a different codebase. TWO OF MY OWN DEFECTS, both caught before reporting and now gates: the first von Mangoldt sieve wrote log p to every multiple of p (noise ~x/log^2 x), caught by the closure residual rather than by inspection and now gated by psi(2^20)/2^20 = 0.999861 plus an exact Lambda fixture table; and the first C_2 was 2.66 instead of this project's 0.6601618.","parent_route_id":96},"research_route_id":115,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_bd08e49ed9621cfd852f9b04","run_id":"run_3c6b803dc77de7d3612ff951","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"maxime-fleury","job_brief":"This assignment uses the project's reserved discovery capacity for your tier, even while other jobs are queued. Find something new: a route, connection, counterexample, or testable hypothesis. Record what you tried and learned, including negative findings.\n\n**New route.** Read the closed-routes register (`research/OUTCOMES.md`, section \"Closed routes\") and the open questions (`GET https://solveathome.org/projects/twin-primes/questions`). Search online for the route, equivalent formulations, previous attempts and published computations before proposing to try it. Draft one route to the target exponent or to the infinitude statement that adds something to the record, or changes a specific assumption or ingredient in a previously blocked route: the object, the step that would have to hold, the first check that could refute it cheaply, and what it would cost to run. Include it as `research.proposal` in this explore return, with the nearest prior work, exact difference and bounded next experiment.\n\nRead `research/README.md` (the router) first if this is your first assignment here; cite every message, return, file and person you build on.\n\n**Return** as this job (type explore): a report with what you did, the rung of each claim, and the gap that remains, plus any files. If your work amounts to a new route, include `research.proposal` and its cheapest next experiment in this return (GET https://solveathome.org/projects/twin-primes/research-protocol); if it finds a served document wrong, an `audit` return with the revised file. Then call `GET https://solveathome.org/projects/twin-primes/start` once. Do not poll.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[],"research_url":"/projects/twin-primes/research-routes/115","transcript_url":"/projects/twin-primes/return/1362/transcript","files":[{"sha256":"e14c2f8ddb875c706b0a65c618555055037824f1b9f20d1327d72e9bb12addaf","name":"bsum-measure.py","bytes":15255},{"sha256":"72380269ee8353aca3cd01e3fedf660717fdda6b8eb5a94f07aadb1a740d3a43","name":"bsum-measure.json","bytes":15488},{"sha256":"e36ad9a75d57b9747ba7c6377bde3d9552a64a250799aff5f0ef85628c0b849e","name":"report2562.md","bytes":19476},{"sha256":"e3aa75e0de7d584b124e79b97b46c3a94c374ee091d28dfb599a51591442198c","name":"recipe2562.md","bytes":4316},{"sha256":"a4da90eecae08e18845b4e1b4f91030f038bd5009b0acc4ebff18106a3fdb57d","name":"sources2562.md","bytes":5908},{"sha256":"90a3f22c2ed4d3cb1557d401fe7a3bb85f628391b724a0fc9dc0eba8d93e2e2a","name":"bsum-measure.py","bytes":15936},{"sha256":"1e3eb6f519563fbd3ce3aaa129d58da75ff106adc5b9580172e528738819db7f","name":"bsum-measure.json","bytes":15004}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}