{"id":5968,"job_id":12031,"problem_id":1,"lane_id":5,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# A new statistic on the ladder's *path*: `D_y(2^j)/2^j` is strongly autocorrelated (`rho1 = 0.662`, p = 5e-5), so the 23-rung \"F1 not violated\" evidence is worth **~4.7 independent rungs**, and a single dyadic rung is not representative of its octave (phase contrast up to 7.5 se)\n\n**Outcome: `progress`.** One assignment (job #12031, explore, lane `infinitude`, no route). Nothing\nhere moves (16) or any exponent: every number is a finite measurement at a named scale.\n\n## What the retained censuses structurally cannot decide\n\nThe served census (`research/centered-discrepancy-measurement.js`, sha256 `e95ed3c4…`) publishes, at\neach dyadic `x = 2^j`, the point values `M/x, D_y/x, acc1/x, P/x, W1_grid/x, W_end/x, max_e maxΔ/x`\n(j = 16..38) plus a random-sign control. Every reading of that evidence is a per-scale statement, and\nthe lane's own \"no monotone growth\" / \"an adverse envelope never exceeded\" formulations are **path**\nstatements about the fluctuation. A list of point values contains no serial dependence, and the\nretained censuses sample **one `x` per octave** and never vary the octave phase. So two questions\nare undecidable from the record as published:\n\n1. how many *independent* observations the 23-rung pass is worth;\n2. whether the dyadic rung is a fair sample of its octave (the interval `(x/2, x]` is dyadic-aligned\n and the cutoff `y = ceil(x^{12/25})` moves with `x`).\n\nThis return supplies a statistic that decides both, pre-registered before computation\n(`PREREGISTRATION.md`, attached), and runs it.\n\n## The statistic and its custody\n\n`S1` — lag-1 autocorrelation `rho1` of the served 23-value column `D_y(2^j)/2^j`, against a\npermutation null (20,000 shuffles), with `N_eff = 23(1-rho1)/(1+rho1)`.\n`S2` — within-octave profile `D_y(x)/x` at `x = round(2^j(1+k/8))`, `k = 0..7`, j = 22, 24, 26, 27,\nwith the phase contrast `T_j = (mean_{k>=1} - y_{j,0})/(sd_{k>=1}/sqrt 7)`.\n`S3` — the census's own random-sign control at every S2 point (support fixed, `mu(n) -> +-1`,\n`mu(e)` kept, 4 seeds), giving `sd_ctrl(x)` and the ratio `sd_octave/sd_ctrl`.\n\n**Custody gate C1 (passed, checked first).** My kernel (`dy_kernel.c`, attached) reproduces the\nserved MAIN TABLE at j = 16, 18, 20, 22, 24, 25, 26 in **all four columns to every printed decimal**:\n`M/x`, `D_y/x`, `W_end/x` identical, and `#odd sqfree e` = 130/266/549/1126/2315/3321/4758 and the\nsupport counts identical. `W1` is *above* theirs by 2.4 % → 4.8 %, monotone in j — as it must be:\nthe served `W1_grid` maximises over a 64-point grid in `t`, mine over **all** multiples of `e`, i.e. a\nfiner (still one-sided) lower bound. Definitions are (9), (13) of `moving-cutoff-parity.md`\n(sha256 `ef7a1865…`) taken verbatim.\n\n## Results (measured, finite)\n\n**S1 — the ladder is not white.** On the served 23-rung column (j = 16..38):\n`rho1 = +0.662`, permutation p = **5.0e-5** (20 000 shuffles), `rho2 = +0.339`,\n`N_eff(AR1) = 4.67`. On the unnormalised `D_y` column `rho1 = +0.550`, p = 0.0016. Jackknife: dropping\nany single rung leaves `rho1 >= 0.300` (min 0.300 at j = 18, max 0.652), so the reading is stable\nunder the single-scale-drop convention the project already uses. My own 12-rung ladder (j = 16..27)\ngives `rho1 = +0.498`, p = 0.0496, consistent.\n\n*Instrument controls on the same table:* `M/x` and `P/x` have `rho1 = -0.349 / -0.360` (p ~ 0.08) and\nthe envelope columns `W1_grid/x`, `W_end/x` have `+0.845 / +0.844` (p = 2e-4). So the correlation is\n**not** a generic property of every published column: the envelope quantities are smooth in scale by\nconstruction, `M` and `P` are anti-correlated, and the signed discrepancy `D_y` (and its odd-divisor\npiece `acc1`, `+0.544`) carries the positive correlation.\n\n**S2/S3 — one rung is not its octave.** (`x` windows within an octave overlap 7/8, so the spread is a\n*smooth-path* variation and is if anything an under-estimate of independent sampling.)\n\n| j | `k=0` (`D_y/x`) | octave range | sd | `T_j` | `rho1` within | `sd_ctrl` | ratio |\n|---|---|---|---|---|---|---|---|\n| 22 | +0.009354 | -0.010317 .. +0.009354 | 0.00786 | -3.46 | +0.15 | 0.01046 | 0.75 |\n| 24 | -0.010634 | -0.010634 .. +0.007466 | 0.00552 | +7.50 | +0.16 | 0.00371 | 1.49 |\n| 26 | -0.003430 | -0.006360 .. +0.007800 | 0.00428 | +1.64 | -0.35 | 0.00203 | 2.11 |\n| 27 | -0.004283 | -0.006360 .. +0.004591 | 0.00396 | +0.88 | +0.39 | 0.00142 | 2.79 |\n\n**Pre-registered verdicts.** F1 (margin transfer): **not violated** anywhere; the minimum over all 32\noctave points is `-0.010634`, versus the `-4/25 = -0.16` allowance (a factor 15 of headroom).\nF2 (whiteness): **fires — NOT WHITE** (`rho1 = 0.662 > 0.30`, p = 5e-5). F3 (phase bias):\n**REPRESENTATIVE** — `|T_j| > 3` at only 2 of 4 octaves and with *opposite* signs (-3.46, +7.50), so\nthere is no systematic dyadic alignment offset; the dyadic rung is unbiased but noisy.\nF4 (noise model): **ADEQUATE by the frozen rule** (ratio > 2 at 2 of 4 octaves, threshold was 3 of 4),\nbut the ratio rises monotonically `0.75, 1.49, 2.11, 2.79` and crosses 2 exactly at the two largest\noctaves. Disclosed as a trend, not as a fired falsifier.\n\n## The decision this informs\n\n1. **Read the ladder's evidence as fewer observations.** `N_eff = 4.67` at j = 16..38. A decision\n rule of the form \"at any j >= 26\" has far less power than 5 ranks suggest, and a monotone-growth\n or running-minimum test over the ladder must be calibrated on `N_eff`, not on the rung count.\n2. **Quote an octave uncertainty, not the rung band.** The \"`D_y/x` within 0.0043 of 0 at j >= 26\"\n band in the lane summary is a statement about the *sampled* rung; the measured within-octave\n spread at j = 26/27 is 0.0138 / 0.0118 (0.0197 at j = 22). Future per-scale uncertainty should be\n the octave spread.\n3. **Power, stated as a number.** With `N_eff ~ 4.7` and a per-rung sd of 0.002-0.004, the smallest\n drift this family of censuses can resolve across its whole range is `~2 sd/sqrt(N_eff) =\n 0.002-0.004` — that is 40-80x below the `4/25` allowance, and it does not improve with more rungs\n of the same kind. This is a second, independent route to the conclusion of the modulus-coherence\n return (job #5055): **the aggregate census cannot decide (16); the route has to be analytic.**\n4. **A limit, stated plainly.** The correlation is measured in `D_y`, but the census's own random-sign\n model does not reproduce it (control seeds give `rho1 = +0.372, +0.013, -0.535, -0.202`, mean\n `-0.088`, null sd ~0.38 at 12 rungs): four seeds cannot separate \"arithmetic sign structure\" from\n \"estimator geometry\", so this return does **not** claim the correlation is arithmetic.\n\n## What remains\n\nCheapest decisive follow-up (in the attached `next_step`): re-run the dyadic ladder with >= 30 control\nseeds to give the null `rho1` a usable sd (~0.05), which turns the present \"not explained, not\nexcluded\" into a decision about whether the correlation is arithmetic; and publish the octave spread\nas the lane's per-scale uncertainty. Both are compute-only, no new mathematics, and neither can\nthreaten (16): at these scales the whole question is how to read a pass that already has a factor-15\nmargin.\n","patch":null,"cpu_hours":0.03,"hashes":{"analyze.py":"6abcf8350ddb51f4e3a3c3fb79eb0531f9f12cdc66b73e2dcd4119f7b496f14e","dy_kernel.c":"7c4cb45492a06a18ed14c1f7e647a48940102e97a6eff8f85f6c7bc11a78820c","report_12031.md":"5c91aafc13fac53f280a7f2f3e7c8b34161be2acead6608a28b25338ae292838","evidence_12031.md":"9f251bc69175bb8507aa460ca84cab4a2a13ede57fc75efbad3af71b78b90931","PREREGISTRATION.md":"297e996af5990493091edc23abc98895a7b2a27d4ea24fea84f15ea1ffc37f89","prior_art_12031.md":"ce81cdfa71e7413b56ed1e5eb7f39daa59d4590b8e8ed2e61e630a350d5a0a18","analysis_12031.json":"c62c0d9f53aab602412f3e3701943b64baaea151d3cc627e3d4c0467a2fd1227","custody_12031.jsonl":"b94763ef1bd1bbadb5bf6c43c81ef2a2060373080b6ad82c2ff0f65a7679b859","octaves_12031.jsonl":"d29c870b59ca5b2194b06037a0d47ea1e63251fc3630cbdd7da02d94c2c09111","transcript_12031.md":"46c33b19de99dee39119ee2af60ce288c86ebf078f1a0732ea22e297c73522ca","next_step_12031.json":"da694e250c0912241c7697c2208e7767ecbad0347b13adb79e01608262983594","dyadic_seeds_12031.jsonl":"a3532faa4f8c66b41bbca6e516674bc9d9781f61714baaff91369ea46f1ded48","ladder_control_12031.json":"ffb372eac53daa2b3f51291d22a72e07f153cdc404be8227b717f194949e5a58","instrument_controls_12031.json":"0ebffe3854cb653f71128cd1a059144d7ad32b6cd6a0443701973bfc755eef10"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-26T07:04:06.867Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2174],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"Freebuff/Codebuff CLI session; the application exposes no token usage record for this turn, so no counts exist to send and none were estimated.","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — job #12031 (path statistics on the served D_y ladder)\n\nAll local, no network beyond two raw GETs of served bytes.\n\n1. `gcc -O2 -o dy_kernel dy_kernel.c -lm`.\n2. Custody gate: run `dy_kernel --x (1<