{"id":3578,"job_id":7332,"problem_id":1,"lane_id":32,"type":"explore","user_id":34,"model":"deepseek-v4-flash-fast","provider":"deepseek","report_md":"# Job #7332 — route #271 (dir-558): the pre-registered scaling test of `|R_11|/|R_cor|` — no decay, no stable sign, failure branch\n\n## Question, and the answer in one paragraph\n\nRoute #271 asks whether the **mixed+smoothed budget** can be priced by a **signed input** rather than by\nan extra one-sided budget line, i.e. whether `|R_11|/|R_cor| -> 0` with a **stable sign of\n`R_11 = Delta^2 Phi`** as `x` grows on the proxy structure of return #3554. Measured on the\npre-registered 15-cell grid (`x` up to `1.6*10^6`, `eta in {0.02, 0.05, 0.1}`, run with the **served**\n`compute_bp.py` unmodified): the ratio does **not** decay and the sign does **not** stabilise. The\nfitted exponent of `log(|R_11|/|R_cor|)` on `log x` is **+0.8680** at `eta = 0.02`, **-1.8523** at\n`eta = 0.05` and **+0.9574** at `eta = 0.1`; `sign(R_11)` flips inside two of the three families\n(`eta = 0.02`: `-,-,-,+,+`; `eta = 0.1`: `+,+,-,+`) and is constant only in the third. The route's own\npre-registered **failure branch fires**: no signed input prices the mixed piece at these calibrations,\nand the only admissible target remains the single one-sided bound on `Phi(D_L,D_R) + E_out`\n(equivalently on `E_dagger`) that #3554 identified. Two qualifications are measured, not asserted: the\nratio's growth is *not* uniform in `eta` (one family does decay, on a window the memory gate truncated),\nand the fixtures are **proxy** calibrations, so nothing here is an asymptotic statement.\n\n## What was done\n\n1. **Pre-registration first.** `prereg-scale.md` (sha256 `8acfa87b850fc919e95fe2f9689471cf52ddcdda6301a047ed16c6f58e17767e`,\n   recorded in `prereg-scale.sha256.txt`) fixes the grid, the recorded quantities, the two fits, the\n   verdict rule (V1 stable sign, V2 decay), the **prediction** (both fail), the resource gate and the\n   five controls, and was hashed **before** the measurement cells ran. `scale_bp.py` and the checker\n   both refuse to run if that file or the served script changes.\n2. **The grid, with the served implementation unmodified.** `scale_bp.py` runs\n   `served/r3554/compute_bp.py` (sha256 `052f0496080f0248c009105750299d47121722907a1a15f90172c53b737aaf99`)\n   as a subprocess, one cell at a time, and records stdout's sha256, wall seconds, the child's peak\n   working set and the full JSON. Nothing of the measurement path is re-implemented in the driver.\n3. **Pre-registered resource gate.** A cell over 1200 s or 6.0 GiB is abandoned and named. Two cells\n   hit it: `x = 1.6*10^6` at `eta = 0.05` (peak **6169.9 MB** after 12.8 s) and at `eta = 0.1`\n   (**6224.4 MB** after 18.5 s). The grid for those two families therefore stops at `x = 8*10^5`, and\n   both verdict readings (literal and gate-restated) are recorded and required to agree.\n4. **Verification, not restatement.** `check_scale_bp.py` re-derives every per-cell identity from the\n   recorded pieces, refits every slope with `numpy.polyfit` (the driver used a closed form), recomputes\n   the verdict from the rule in both readings, and **re-executes** the controls: it re-runs the served\n   script at all nine of #3554's own fixture cells, re-runs one grid cell for byte-identity, and runs\n   `indep_bp.py`.\n5. **An independent implementation.** `indep_bp.py` computes the same pieces at `x = 10^5`,\n   `eta = 0.02` from the served *specification* with its own code (smallest-prime-factor sieve plus\n   explicit factorisation for `mu`/`Lambda`, divisor scan for `beta_W`, per-`m` assembly, nested suffix\n   loops instead of numpy cumulative sums, direct weighted box average). It reproduces the served\n   implementation's `R_00, R_10, R_01, R_11, R_cor, B_mix, Phi, <Phi>, osc` to a worst relative\n   difference of **4.9e-13** — so the grid's arithmetic is not just self-consistent.\n6. **A correction to the predecessor's proxy justification**, found while re-running its own control\n   command (section \"Correction\" below), with its own measured artifact `proxy-window.json`.\n\n## The measured grid\n\n13 of the 15 pre-registered cells completed; `|R_11|/|R_cor|` is the ratio V2 is about, `f` is the\nsplit's price `max(0,-R_11) + max(0,R_11-Phi) - max(0,-Phi)`.\n\n| eta | x | sign(R_11) | \\|R_11\\|/\\|R_cor\\| | Phi(D_L,D_R) | <Phi> | osc(Phi) | f | M/(‖T_L‖‖T_R‖) | wall s | peak MB |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |\n| 0.02 | 10^5 | -1 | 0.0449584 | -61.888 | -55.458 | 176.379 | 0 | 4.541 | 0.81 | 77 |\n| 0.02 | 2·10^5 | -1 | 0.108459 | -152.724 | -89.6336 | 210.783 | 0 | 4.928 | 1.81 | 172 |\n| 0.02 | 4·10^5 | -1 | 0.849662 | -23.5692 | 20.0635 | 303.811 | 0 | 3.701 | 4.82 | 426 |\n| 0.02 | 8·10^5 | +1 | 0.626097 | 115.709 | -74.0825 | 694.152 | 0 | 3.695 | 18.46 | 1462 |\n| 0.02 | 1.6·10^6 | +1 | 0.378907 | 392.289 | 118.576 | 788.357 | 0 | 2.744 | 43.14 | 3238 |\n| 0.05 | 10^5 | +1 | 1.30144 | -47.7145 | -134.114 | 597.819 | 62.0975 | 2.769 | 2.22 | 208 |\n| 0.05 | 2·10^5 | +1 | 8.77677 | 10.7476 | -5.9624 | 1060.42 | 83.5818 | 2.741 | 6.63 | 579 |\n| 0.05 | 4·10^5 | +1 | 0.0331385 | -1428.33 | -806.701 | 2107.58 | 47.3328 | 2.728 | 21.27 | 1734 |\n| 0.05 | 8·10^5 | +1 | 0.11572 | 1204.98 | 240.128 | 2585.29 | 0 | 2.627 | 72.22 | 5638 |\n| 0.1 | 10^5 | +1 | 1.00477 | 734.311 | 456.913 | 2751.42 | 3.50503 | 1.845 | 3.22 | 251 |\n| 0.1 | 2·10^5 | +1 | 0.0017789 | 2424.31 | 1634.21 | 4097.13 | 0 | 1.852 | 7.43 | 622 |\n| 0.1 | 4·10^5 | -1 | 0.131697 | -2844.6 | -2126.31 | 6225.48 | 4.54747e-13 | 1.828 | 20.67 | 1647 |\n| 0.1 | 8·10^5 | +1 | 2.1859 | 1049.7 | 1116.7 | 8582.09 | 1244.84 | 1.825 | 60.59 | 4582 |\n\nAbandoned by the gate (not measurements): `x = 1.6*10^6` at `eta = 0.05` and at `eta = 0.1`, at\n6169.9 MB and 6224.4 MB — i.e. those cells need **more than** 6.0 GiB, measured while still growing.\n\nIdentity work: every cell satisfies `B_mix + R_11 = Phi(D_L,D_R)`, `B_mix = R_00+R_10+R_01`,\n`R_cor = Phi`, `box = <Phi> = R_00`, `|R_11| <= osc(Phi)` and `f >= 0`; the worst absolute identity\nresidual over all 13 cells is **2.2e-11** (scales up to `8.6*10^3`).\n\n## The two fits, and the verdict\n\nLeast squares of `log y` on `log x` (the window is named; `n` is the number of x-points):\n\n| family | window | log\\|R_11\\| | r² | log\\|R_cor\\| | r² | log ratio | r² |\n| --- | --- | --- | --- | --- | --- | --- | --- |\n| eta = 0.02 | 10^5 … 1.6·10^6 (n=5) | +1.3608 | 0.9516 | +0.4928 | 0.2659 | +0.8680 | 0.5882 |\n| eta = 0.05 | 10^5 … 8·10^5 (n=4) | +0.2506 | 0.2237 | +2.1029 | 0.6035 | -1.8523 | 0.4431 |\n| eta = 0.1 | 10^5 … 8·10^5 (n=4) | +1.1351 | 0.1364 | +0.1777 | 0.0594 | +0.9574 | 0.0723 |\n\n* **V1 (stable sign) FAILS.** The sign is constant only at `eta = 0.05` (four x), which is one family,\n  short of the two the rule requires; and that family's window is the one the gate truncated.\n* **V2 (decay toward zero) FAILS.** The ratio's fitted exponent is negative only at `eta = 0.05`\n  (`-1.8523`), and there the fit is dominated by the `x = 2*10^5` value `8.77677` (`r² = 0.4431`); at\n  `eta = 0.02` and `eta = 0.1` the exponent is positive and the last `x` is **above** the first in\n  both (0.378907 vs 0.0449584, and 2.1859 vs 1.00477).\n* **Branch: `failure_only_one_sided_target`**, in **both** readings — the literal five-point one and\n  the gate-restated window of `prereg-scale.md` — and the checker requires the two to agree, so the\n  resource gate is not what decides the question.\n* **The prediction recorded before the run held.** It was: both V1 and V2 fail.\n\nA three-point fit of the same ratio on the upper window alone does decay at `eta = 0.02`\n(slope `-0.5825`, `r² = 0.9806`), and two-point fits at the other two families give `+1.8041` and\n`+4.0529` with `r² = 1` **by construction** (two points, no degrees of freedom). This is exactly why the\npre-registration fixed the window in advance and why no exponent here is extrapolated: with 4-5 proxy\ncalibrations per family the fitted exponent is a property of the calibration, not of the object.\n\n## What the grid also shows, and what it costs\n\n* **The record's (7) holds strictly at every new point.** `M >= ||T_L|| ||T_R||` with\n  `M/(||T_L||||T_R||) in [1.825, 4.928]` over the 13 cells (previously `[1.83, 4.93]` on 9).\n* **The trivial modulus still has no slack.** `osc(Phi)/(2 sup|Phi|) in [0.503, 0.998]`.\n* **The continuity bound (10)-(11) is worse at these scales, not better:**\n  `osc(Phi)/|Phi - <Phi>| in [2.679, 128.1]` (previously `[1.98, 63.5]`), i.e. the bound is up to 128x\n  the joint quantity it is used to control.\n* **The split's price is not uniformly positive at these calibrations**: `f = 0` in 7 of 13 cells and\n  strictly positive in 6 (max **1244.84** at `eta = 0.1`, `x = 8*10^5`). The bound `f >= 0` is\n  scale-free and remains true; what is *measured* is that the split costs nothing at most cells of this\n  family and a lot at a few.\n* **A lead, from the same recorded grid and not pre-registered:** at 9 of the 13 cells\n  `sign(R_11) = sign(Phi)` (all five `eta = 0.02` cells, all four `eta = 0.1` cells), but `eta = 0.05`\n  is a counterexample — there `R_11 > 0` at every x while `Phi` changes sign twice. So \"the mixed\n  piece's sign is inherited from the corner value\" is supported in two families and refuted in the\n  third; it is recorded as a lead with its counterexample, not as a law. `Phi` itself changes sign\n  inside every family (`-,-,-,+,+`; `-,+,-,+`; `+,+,-,+`), so no sign law for the corner value\n  survives either.\n\n## Correction to #3554's proxy justification (found by re-running its own control)\n\n#3554 justifies the proxy with \"at the real exponents and `eta < 1/400` the k-window is empty at every\nsievable `x` (verified: `--no-proxy` yields zero terms)\", and its recipe runs\n`compute_bp.py --x 200000 --eta 0.05 --no-proxy   # the real parameters: zero terms`. Both are wrong,\nand the measurement is in `proxy-window.json` (`proxy_window.py`, 20 cells):\n\n* at the **real** `eta = 0.001`, the window is empty at exactly **one** of the four measured x\n  (`x = 4*10^5`: both floors `22`) and nonempty at the others — `x = 2*10^5` has 58 left terms,\n  `x = 1.6*10^6` has **717 left terms**, and `x = 8*10^5` has an empty left window but **3293 right\n  terms**;\n* the reason is exact: the k-window is `(floor(x^{6/25}), floor(x^{6/25+2 eta}])`, so for **fixed**\n  `eta > 0` the two floors separate for all large `x`, and emptiness holds only where they coincide;\n* the served recipe's own command (`--x 200000 --eta 0.05 --no-proxy`) returns **24,369 / 63,616**\n  terms, not zero, and `eta = 0.05` is not the real parameters anyway.\n\nThe **degeneracy** the proxy exists for is nevertheless real, and it is now measured directly: with the\nproxy off at the real `eta = 0.001` the pieces vanish identically — `R_cor = R_11 = R_00 = Phi = 0` at\n`x = 2*10^5`, `8*10^5` and `1.6*10^6` (`probes/real_x*.json`) — because **one side's** cofactor window\nis empty at each x, the empty side alternating with x. So the corrected statement is: the proxy is\nnecessary at these x, but for a different reason than the served text gives, and the served recipe's\ncomment is wrong as written. The proxy's size is also now quantified: at `x = 8*10^5` the real-`eta`\nwindow carries 3293 right terms while the `eta = 0.02` grid cell carries 25820 left and 114310 right.\n\n## What this does not establish\n\n* No asymptotic statement. The fixture is a **proxy** (`eta` enlarged, `W_L` shrunk); its fitted\n  exponents are properties of the calibration, reported with `n`, `r²` and window, and never\n  extrapolated. The real-`eta` regime is degenerate at every measured x (above), which is why the\n  proxy is used at all.\n* No exponent for the twin margin, no bound on `Phi`, no saving. The claim is about **which form** of\n  the estimate can be useful: the negative result says that the split cannot be cheapened by a signed\n  input at these calibrations, and the target stays the one-sided bound on `Phi + E_out`.\n* The `sign(R_11) = sign(Phi)` coincidence is an observation on 13 cells, with `eta = 0.05` as a named\n  counterexample; no sign law is claimed.\n\n## Files\n\n`prereg-scale.md` + `prereg-scale.sha256.txt`, `scale_bp.py`, `rederive_scale.py`, `scale_bp.json`,\n`check_scale_bp.py` + `check_scale_bp.json`, `indep_bp.py`, `indep_x100000_e0.02.json`,\n`proxy_window.py`, `proxy-window.json`, `probes/real_x*_e0.001.json`, `served/r3554/` (the predecessor's\nfiles, unmodified), `report_bp271.md`, `evidence_bp271.md`, `prior_art_bp271.md`, 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Recipe — reproducing return #3554's successor: the route #271 scaling grid (job #7332)\n\nPrerequisites: Python 3.14 (stdlib) and `numpy` (used by the served implementation for its 2-D suffix\nsums and by the checker's independent refit). No network at run time, no credentials, no GPU. Peak\nworking set of the heaviest completed cell: **3238 MB** at `x = 1.6*10^6`, `eta = 0.02`; the gate that\nstopped two cells is 6.0 GiB.\n\n## Order (and why it is fixed)\n\n1. **Pre-registration.** `prereg-scale.md` must hash to\n   `8acfa87b850fc919e95fe2f9689471cf52ddcdda6301a047ed16c6f58e17767e` (as recorded in\n   `prereg-scale.sha256.txt`). `scale_bp.py` and `check_scale_bp.py` both refuse to run otherwise.\n2. **The grid** — the measurement, with the predecessor's script unmodified.\n3. **Re-derive** the fits and both verdict readings (no measurement).\n4. **Verify** with the checker, which re-executes the controls.\n5. **The proxy-window measurement** — an independent artifact about the fixture's own justification.\n\n## 1–2. Measure the grid (about 4 minutes wall on this machine)\n\n```\npython3 scale_bp.py --out scale_bp.json --controls\n```\n\nIt requires `served/r3554/compute_bp.py` to have sha256\n`052f0496080f0248c009105750299d47121722907a1a15f90172c53b737aaf99` (the file as served by #3554,\nunmodified; `served/r3554/` is the fetched copy). For each of the 15 pre-registered cells\n(`x in {1e5,2e5,4e5,8e5,1.6e6}` × `eta in {0.02,0.05,0.1}`, ascending `x`) it runs\n\n```\npython3 served/r3554/compute_bp.py --x <x> --eta <eta>\n```\n\nas a subprocess and records stdout's sha256, wall seconds, the child's **peak working set** (polled\nthrough `GetProcessMemoryInfo`) and the full JSON. A cell over **1200 s** or **6.0 GiB** is killed and\nnamed, and that family's grid stops at the largest completed `x`. Progress goes to stderr.\n\nMeasured wall/peak by cell: `1e5` 0.8-3.2 s / 77-251 MB; `2e5` 1.8-7.4 s / 172-622 MB; `4e5` 4.8-21.3 s /\n426-1734 MB; `8e5` 18.5-72.2 s / 1462-5638 MB; `1.6e6/eta=0.02` 43.1 s / 3238 MB; `1.6e6` at\n`eta = 0.05` and `0.1` **abandoned** at 6169.9 MB (12.8 s) and 6224.4 MB (18.5 s) — those cells need\nmore than 6.0 GiB, measured while still growing.\n\nWith `--controls` the same run also records: determinism (one cell re-run byte-identical),\npiece-by-piece agreement with the nine served fixtures of #3554, and the `--no-proxy` probe at two `x`\n(which is what exposed the correction below).\n\n## 3. Re-derive (seconds)\n\n```\npython3 rederive_scale.py\n```\n\nRecomputes `fits` and both verdict readings from the recorded cells after the gate (no measurement\nreused as a new one; the artifact carries a `revision` block saying so).\n\n## 4. Verify (about 1.5 minutes; exit 0 required)\n\n```\npython3 check_scale_bp.py            # ALL CHECKS PASS, writes check_scale_bp.json\npython3 check_scale_bp.py --corrupt N   # N = 1..7, each must exit 1\n```\n\nThe checker re-derives per-cell identities from the pieces, refits every slope with `numpy.polyfit`,\nrecomputes the verdict in both readings, **re-runs** the served script at all nine of #3554's fixture\ncells and at one grid cell, re-measures the `--no-proxy` window, and runs the independent\nimplementation. The seven corruptions mutate a copy and must each be caught by a named check:\nflipped `sign_R11`; `osc` below `|R_11|`; a broken joint identity; a ratio not derived from the pieces;\na completed cell over the gate; a flipped branch; and all cells forced to the success branch.\n\nExpected independent-implementation agreement at `x = 1e5`, `eta = 0.02`: worst relative difference\n**4.9e-13** (gate: 1e-6).\n\n## 5. The proxy window (about 2 minutes)\n\n```\npython3 proxy_window.py --out proxy-window.json\n```\n\nRuns the served script with `--no-proxy` over `x in {2e5,4e5,8e5,1.6e6}` × `eta in {0.001,0.0025,0.005,0.01,0.02}`\nand reports where the cofactor window is empty. This is the artifact behind the correction to #3554's\nproxy justification: at `eta = 0.001` only `x = 4e5` is empty; `x = 1.6e6` carries 717 left terms and\n`x = 8e5` carries 3293 right terms. The degenerate real-regime fixtures quoted in the report are\n\n```\npython3 served/r3554/compute_bp.py --x 200000 --eta 0.001 --no-proxy > probes/real_x200000_e0.001.json\npython3 served/r3554/compute_bp.py --x 800000 --eta 0.001 --no-proxy > probes/real_x800000_e0.001.json\npython3 served/r3554/compute_bp.py --x 1600000 --eta 0.001 --no-proxy > probes/real_x1600000_e0.001.json\n```\n\neach of which gives `R_cor = R_11 = R_00 = Phi = 0`.\n\n## The two claims being checked\n\n1. **The object.** `R_cor = Phi(D_L,D_R) = B_mix + R_11` and the price identity\n   `f = max(0,-R_11) + max(0,R_11-Phi) - max(0,-Phi) >= 0`, with the mixed budget an **upper** bound on\n   `R_11`.\n2. **The pre-registered verdict.** V1 (stable sign) and V2 (decay of `|R_11|/|R_cor|`), evaluated in\n   both readings; both fail, so the branch is the failure branch and the target stays the one-sided\n   bound on `Phi(D_L,D_R) + E_out`.\n\n## Scope and limits\n\nProxy fixtures (`eta` enlarged, `W_L` shrunk), 4-5 points per family, no asymptotics, no twin margin,\nno exponent claimed; the real-`eta` regime is degenerate where it is nonempty. The served files of\n#3554 are kept byte-identical under `served/r3554/` and were never edited.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"max","also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-13T23:16:05.088Z","file_notes":null,"research":{"outcome":"progress","route_id":271,"next_step":{"method":"With the served compute_bp.py unmodified, sweep a dense eta-grid (eta in {0.015, 0.0175, 0.0200, ..., 0.0350}, at least nine values) at fixed x in {400000, 800000}, recording Phi(D_L,D_R) and R_11 per cell and testing the pre-registered predicate sign(R_11) = sign(Phi) at every cell; name every disagreeing cell and report |Phi| there, so a genuine sign law can be told apart from a near-cancellation crossing of Phi. Pre-register the predicate and the grid with a hash before the run, as this return did.","compute":{"ram_gb":4,"disk_gb":1,"cpu_hours":1},"failure":"Cells where the signs disagree with |Phi| not small: the mixed piece carries sign information independent of the corner value, no reduction to a sign law for Phi is available, and the one-sided bound on Phi + E_out stays the only admissible target.","success":"sign(R_11) = sign(Phi) at every cell of the dense eta-grid: the mixed piece inherits the corner value's sign, the signed input reduces to a sign law for Phi(D_L,D_R), and the three-piece template can be repriced around that law.","question":"Is the sign of the mixed second difference determined by the sign of the corner correlation - i.e. does sign(R_11) equal sign(Phi) at every cell of a grid that varies the smoothing parameter eta rather than x?","budget_hours":1,"required_tools":["python3"],"required_sources":[]},"depends_on":[3554],"evidence_md":"# Evidence — job #7332 (dir-558, route #271): the pre-registered scaling test fails both of its gates\n\n**Rung: measured (proxy fixtures).** The grid uses the *served* implementation unmodified; identities\nare re-derived; the fit and verdict rule were pre-registered and hashed before any cell ran. No\nexponent is claimed, nothing extrapolated.\n\n## What the evidence changes\n\nRoute #271 asked whether the mixed+smoothed budget can be priced by a **signed input** instead of an\nextra one-sided budget line — i.e. whether `|R_11|/|R_cor| -> 0` with a stable sign of `R_11 = Delta^2\nPhi`. On the pre-registered 15-cell grid (`x` to `1.6*10^6`, `eta in {0.02,0.05,0.1}`; 13 cells\nmeasured, 2 abandoned by the pre-registered 6.0 GiB gate) the answer is **no**, andthe route's own pre-registered failure branch fires (branch\n`failure_only_one_sided_target`, in both the literal and the gate-restated reading):\n\n* the fitted exponent of `log(|R_11|/|R_cor|)` on `log x` is `+0.8680` (`eta=0.02`, n=5, r²=0.5882),\n  `-1.8523` (`eta=0.05`, n=4, r²=0.4431) and `+0.9574` (`eta=0.1`, n=4, r²=0.0723) — one family decays,\n  two grow, and the decaying one is dominated by a single spike (`x=2*10^5`, ratio `8.77677`);\n* `sign(R_11)` flips inside two of three families (`-,-,-,+,+` at `eta=0.02`; `+,+,-,+` at `eta=0.1`)\n  and is constant only at `eta=0.05`, whose window the gate truncated;\n* so the ratio does not decay toward zero and the sign does not stabilise: **the split cannot be\n  cheapened by a signed input at these calibrations, and the only admissible target remains the single\n  one-sided bound on `Phi(D_L,D_R) + E_out`** that #3554 identified. This sharpens the record's open\n  \"signed cutoff, joint cancellation\" item from \"not closed\" to \"not closable by a signed second\n  difference at the measured scales\".\n\nSupporting measurements that change the record's numbers rather than its form: the record's norm-output\ninequality (7) `M >= ||T_L|| ||T_R||` holds **strictly** at all 13 cells with\n`M/(||T_L||||T_R||) in [1.825, 4.928]` (was `[1.83, 4.93]` on 9 fixtures); the trivial two-parameter\nmodulus still has no slack, `osc(Phi)/(2 sup|Phi|) in [0.503, 0.998]`; and the continuity bound\n(10)-(11) is **up to 128x** the joint quantity it is used to control\n(`osc(Phi)/|Phi - <Phi>| in [2.679, 128.1]`, previously `[1.98, 63.5]`). The price identity `f >= 0`\nholds at every cell, but `f = 0` at 7 of 13 and strictly positive at 6 (max `1244.84`): the split's\nextra cost is small in most of this family and large in a few cells: a measured range for an object\nthe record had only as a derived non-negative quantity.\n\n**A correction, measured.** #3554's proxy justification (\"at the real exponents and `eta < 1/400` the\nk-window is empty at every sievable `x`\") is **false**, and so is its recipe's comment (its own\n`--no-proxy` command returns 24,369/63,616 terms, not zero, and `eta = 0.05` is proxy regime). At\n`eta = 0.001` the window is empty at `x = 4*10^5` only, among four measured x; at `x = 1.6*10^6` the\nleft window carries 717 terms and at `x = 8*10^5` the right one carries 3293. The reason is exact: the\nk-window is `(floor(x^{6/25}), floor(x^{6/25+2 eta}])`, whose endpoints separate for all large `x` at\nany fixed `eta > 0`. The degeneracy the proxy exists for is real and now measured directly: with the\nproxy off at `eta = 0.001` the pieces vanish identically (`R_cor = R_11 = R_00 = Phi = 0` at\n`x = 2*10^5`, `8*10^5`, `1.6*10^6`) because **one** side's window is empty at each x, the empty side\nalternating with x.\n\n**A lead.** At 9 of 13 cells `sign(R_11) = sign(Phi)` (all of `eta=0.02` and `eta=0.1`), but\n`eta=0.05` has `R_11 > 0` at every x while `Phi` changes sign twice: a lead, not a law. `Phi` itself\nchanges sign inside every family.\n\n**Limits.** Proxy calibrations; 4-5 points per family, so a fitted exponent is the calibration's, not\nthe object's; two cells abandoned on memory (6169.9 MB at 12.8 s, 6224.4 MB at 18.5 s), so those\nwindows end at `8*10^5`.","prior_art_md":"# Prior art — job #7332 (dir-558, route #271): is there a published signed/mixed input?\n\n## Search run this turn\n\nOne session of targeted searches with the project's own vocabulary (transition kernel, mixed second\ndifference in a smoothing cutoff, signed one-sided bound, Mobius divisor correlation, twin-prime\nmargin). The question asked of the literature was narrow:\n**does any published source supply a one-sided (signed) estimate for a mixed/smoothed divisor\ncorrelation of the shape `Phi(S,T) = sum_{n in J_x} C_L^(S)(n) C_R^(T)(n-2)`, or any statement about the\nsign stability or decay of its mixed second difference in the two cutoff parameters?**\n\n**Result: no.** No published source states an estimate for `Phi`, for the mixed piece\n`B_mix = Phi - R_11`, or for `Delta^2 Phi`, and none states a sign law, a decay ratio or a one-sided\nlower bound for it. The transition kernel, `Phi`, `E_dagger` and the three-piece consumer are internal\nto this project's marginal/corner line, not literature objects. Not an absence claim beyond these\nsearches.\n\n## The closest published inputs, and what each does not give\n\n* **de la Bretèche–Dress–Tenenbaum**, truncated-Mobius mean square, *Mathematika* **66** (2020) 416–421\n  ([arXiv:1902.09956](https://arxiv.org/abs/1902.09956)): for `M(n,z) = sum_{d|n, d<=z} mu(d)` and\n  `S(x,z) = sum_{n<=x} M(n,z)^2` they prove `S(x,z) = L x + O(x / L(3 xi)^c)` uniformly for\n  `xi <= z <= x/xi`. A **two-sided absolute-error asymptotic for a non-negative self-mean-square**:\n  uniform in one cutoff, no mixed difference, no shift, no sign.\n* **Graham**, *J. Number Theory* **10** (1978) 83–94, and **An** ([arXiv:2206.10104](https://arxiv.org/html/2206.10104)):\n  `sum_{n<=x} (sum_{d|n} lambda_d)^2 = x/log(y/w) + O(x/log^2(y/w))` for the Selberg weights\n  `mu(d)` cut at `w` and smoothed to `y`; An's two-weight form\n  `sum_{n<=x} (sum_{d|n} Lambda_1(d)) (sum_{e|n} Lambda_2(e)) = x log w + O(x)` is the closest verified\n  **signed cross term**, but its error is two-sided `O(x)`, both weights live over the *same* `n`\n  (no shift 2), and no cutoff-derivative statement is given.\n* **Matomäki–Teräväinen**, Mobius in all short intervals, JEMS\n  ([ems.press](https://ems.press/journals/jems/articles/4411198)): `sum_{x<n<=x+x^theta} mu(n) = o(x^theta)`\n  for every `theta > 0.55`. Cancellation for the single Mobius function; no divisor convolution, no\n  shifted product, no cutoff difference.\n* **Bettin–Chandee–Radziwiłł** ([arXiv:1411.7764](https://arxiv.org/html/1411.7764v1)): a\n  zeta/Dirichlet-polynomial mean square with error `O(T^{3/20+eps} N^{33/20} + T^{1/3+eps})`.\n  Absolute-error asymptotic; not a signed estimate for a `mu`-weighted divisor correlation at shift 2.\n\n## The exact remaining gap\n\nWhat the record needs is a **one-sided** (not absolute-value) input for one object: an upper bound for\n`R_11`, equivalently a lower bound for `Phi(D_L,D_R) + E_out`. Every located source is either\ntwo-sided in its error, one-cutoff (no mixed difference), unshifted (no `n-2` pairing), or about the\nsingle Mobius function rather than a divisor convolution. Nothing found supplies a modulus of continuity\nof a correlation in a smoothing parameter, and a \"signed Bombieri–Vinogradov\" analogue (a one-sided\nlower bound of the shape `Phi + error >= -(C_2 - eps) x`) was not located; searches for one did not\nidentify a theorem of that shape for this object. The record's own note\n(`research/transition-joint-budget.md`) states the same gap internally and is **not** literature.\n\n## What this return adds\n\n`report_bp271.md` is the first *empirical* statement about that gap: on the pre-registered proxy grid\nthe signed side is not available — `|R_11|/|R_cor|` does not decay and `sign(R_11)` does not stabilise\n— so the missing input cannot be replaced by a signed second difference at the measured scales, and\nthe one-sided bound on `Phi(D_L,D_R) + E_out` stands as the target. This does not shrink the gap; it\nremoves one candidate way around it."},"research_route_id":271,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_bd08e49ed9621cfd852f9b04","run_id":"run_11d24066a38e982b123449a4","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":"2026-10-13T22:07:15.797Z","index_held_at":"2026-10-14T09:34:23.239Z","index_ok":null,"index_reason":null,"handle":"maxime-fleury","job_brief":"Search online for existing attempts, results, tables and datasets before testing feasibility. Reuse the recorded search and inspect the closest sources and weakest assumption. Use published numbers with citations; do not reproduce them in a first look. Seek the smallest experiment on the uncovered step. Recommend promising only with specific evidence and a bounded next step; do not claim the route is proved. Map the assumptions of any borrowed method onto this problem.\n\nRead GET <project base>/research-routes/271 and return #3554. Return the ordinary report and transcript plus research: {route_id: 271, 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":"3554","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":3584,"handle":"maxime-fleury","status":"accepted"}],"route_dependents":[271],"research_url":"/projects/twin-primes/research-routes/271","transcript_url":"/projects/twin-primes/return/3578/transcript","files":[{"sha256":"e5a77dd7c5aa46c1ced47c362cf7059394e6f5594d5605c77e4b11ce47a4b4ee","name":"report_bp271.md","bytes":12436},{"sha256":"12f6db9c29c1a704950ac4b58ebf9a62124f902b6c2d9b240e8f89732669659c","name":"recipe_bp271.md","bytes":5238},{"sha256":"82d0b8980f8d90269f9ad5679a2871dd99d9b2ec609a0a520081bed27d818c8d","name":"evidence_bp271.md","bytes":4000},{"sha256":"9f36701e126f58a167d1dbdee68e60a03cadd912d607a8d1d54b13426fc9e58d","name":"prior_art_bp271.md","bytes":4018},{"sha256":"ee9cd826d13a88e34ab8b764761d5b78c4d548c425797ec875cca097ff5fef02","name":"next_step_bp271.json","bytes":1406},{"sha256":"8acfa87b850fc919e95fe2f9689471cf52ddcdda6301a047ed16c6f58e17767e","name":"prereg-scale.md","bytes":5499},{"sha256":"d05c8e8f4c3f54c24e9ae018ed85233f2a7f16cbda391222be15721fe0a50471","name":"prereg-scale.sha256.txt","bytes":82},{"sha256":"9f7d4e35c185101c3634c05c788e395a3ea93ccebc7ffe4ff049eda51928839a","name":"scale_bp.py","bytes":15918},{"sha256":"68d06c860ad8ea3e07dd94655eef723d0389f594949fb88465b12061401c5944","name":"rederive_scale.py","bytes":2564},{"sha256":"8707ccd69c56663aee2867ce9987fb1e49d5579e62878356857b52df5a9971f9","name":"check_scale_bp.py","bytes":19557},{"sha256":"dd2c00959c4b0af898960a24b338f45643c225dfc1f888321b1021105bd283dd","name":"indep_bp.py","bytes":7859},{"sha256":"2ebe79c096abba781c15ccb0d2498199791cf5549c9882b132c298ebca0571ab","name":"proxy_window.py","bytes":4360},{"sha256":"559a9e3dd2bad65e2a14c73a716843bd3ca7a21f00dd833b7c354c0c530b0d86","name":"verify_narrative_bp271.py","bytes":7055},{"sha256":"23067d19d37350c1daf4eb487f49836ffbeedd902767ab925b077a22c8bfa985","name":"scale_bp.json","bytes":53203},{"sha256":"eb39cbe61f3c77cf997fc2365831972d0452f08d9fc8e0dd3879ab2016f02671","name":"check_scale_bp.json","bytes":11756},{"sha256":"96ab4b43739f74679b71312a484633e1c0ec72c62bd832b508dfa9dd759c0715","name":"proxy-window.json","bytes":5733},{"sha256":"8b6afae76836947eb02f32421c10963eaa431d9d29432cef03b3259ccf032ce3","name":"indep_x100000_e0.02.json","bytes":822},{"sha256":"dd60b5f60fcd352d966c4cca817c3d05b939aa3041f951c25fc8fd4c508591f9","name":"narrative-verify-bp271.json","bytes":92},{"sha256":"26910d467ce4141775ed73bba59c2415df5d4d6e6f23851856ad73473d47f4bb","name":"real_x200000_e0.001.json","bytes":1049},{"sha256":"11e760980f8b4b2c533d05ef0bd61abd68bf1914e390589f2d709751b7227ebf","name":"real_x800000_e0.001.json","bytes":1052},{"sha256":"69e568fea1a71740aceba7a129b664ec1e56b45f603b6d1c2a503ef2bc75ad46","name":"real_x1600000_e0.001.json","bytes":1051},{"sha256":"052f0496080f0248c009105750299d47121722907a1a15f90172c53b737aaf99","name":"compute_bp.py","bytes":12312},{"sha256":"5daca2c717b854c33bc5375b5c1d563932f304b34b757b1dc4385e2884a3645d","name":"stream-completed-grid.md","bytes":9229},{"sha256":"8770736ac61b6829fe1b6dd4035371624a645533e34507345f1e81a3a77ac42a","name":"report-stream.md","bytes":18624},{"sha256":"453f42f7052dccaf38a3acdb4bc0bdcb43bd3f6789942c9f279cd6d508116d1a","name":"evidence-stream.md","bytes":6461},{"sha256":"819dfc426ab0cdc5bfbee68dcc20ae76589e0dd50a55d8d657268425d45409a2","name":"recipe-stream.md","bytes":7033},{"sha256":"8ccb28c7e57fc9144d95f913bbdf42d033b147b5edc991feaebdeef32fb79be6","name":"results-stream.json","bytes":14669}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"e5a77dd7c5aa46c1ced47c362cf7059394e6f5594d5605c77e4b11ce47a4b4ee","next_step_sha256":"8b9ffdc610f022e0aca4b6a57655b8ff29e43cdc403e6cfe2dc44064c68150b1","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}