# Is a 32-row crossover BIASED against a 4-row one? No - and here is what it is instead Lane `crpin-coarse-vs-fine-bias-18sep`, **item 2** of an internal note. **NO LEG WAS RUN AND NO BOX WAS TAKEN.** Every figure here is a re-reduction of logs already committed under `rounds/`. No constant moved; `crates/nzbkit-base/src/par2repair/fastpar.rs` was not touched. **The answer in one paragraph.** The bias is **not systematic**, so there is no correction to apply and the 32-row crossovers in an internal note stand as published. Over the whole banked corpus the implied correction is positive in **74 of 146** 32-row crossings - a coin toss at z = +0.17 - with a median size of 1.5 rows; applied to the three cases where a fine ladder says what the answer is, it makes **all three worse**. Simulating the coarse grid inside each fine ladder shows why: the bias barely moves with the gap (which a curvature bias must do as `h^2`) and flips sign with **which rung** happens to be the lower bracket. And the reason it cannot be corrected is measurable directly - adjacent second differences of `log(F/T)` disagree in sign in **69%** of 568 pairs, against the **73.2%** that pure per-rung noise predicts, so the curvature a 32-row ladder appears to have is its own rung scatter. What replaces the correction is a **grid term of about 1 to 4 rows** to carry beside every 32-row crossover, which is smaller than the 3-13 row replicate spread the campaign already quotes and therefore changes no published claim. Separately, **crpin4m's "all four moved up" is not four observations**: one is a bound, one is a null, and the two that are firm are the same arm's two metrics off the same legs - and that arm is the one both sittings call non-monotone. ## Files | file | what | |---|---| | `crossbias.py` | the whole reduction: six arms plus `--selftest`. Imports `w3winred`, which imports `waskred` - the crossover arithmetic is theirs | | `crossbias-output.txt` | the full run at `--boot 2000`, which is what the campaign-file section quotes | python3 rounds/crossover-bias-2026-09-18/crossbias.py --selftest python3 rounds/crossover-bias-2026-09-18/crossbias.py --boot 2000 Whole run: ~12 s on the dev Mac. `--pairs`, `--sim`, `--census`, `--sensitivity` and `--curvature` select one arm. ## Why this could be done with no box The handoff's item 2 lists three unranked candidates for crpin4m's observation. The first - "log-interpolation bias across a 32-row gap" - is a claim about **arithmetic applied to banked numbers**, and the campaign banks 146 reducible 32-row crossings and four ladders fine enough to be a reference. So the item is answerable from the ledger, and its own handoff says so. ## The instrument, and the one piece of algebra `crossover()` log-interpolates: with `y(m) = log(fold/force)` it takes the two rungs bracketing `y = 0` and returns the root of the **chord** through them. Writing the true curve locally as `y(m) = A(m-x) + B(m-x)^2` with `A > 0`, and putting the true root at fraction `u` of an interval of width `h`: bias = x_chord - x_true = -h*beta*u*(1-u) / (1 + beta*(1-2u)), beta = B*h/A to first order **`-(B/A) * h^2 * u(1-u)`**. Three consequences, each of which is a test this round runs rather than an assumption it makes: 1. **The sign is the curvature's.** Convex `y` puts the chord above the rising curve inside the interval, so it reaches zero early and the coarse ladder reads **low** - which is the direction crpin4m observed. Concave reads high. A campaign-wide correction needs the curvature to have one campaign-wide sign. 2. **It vanishes at both ends.** At `u = 0` or `u = 1` the crossing sits on a rung and the chord is exact, so the correction is not a per-ladder constant. 3. **It is quadratic in the gap.** 32 against 4 is a factor of 64, which is what makes a 4-row ladder usable as ground truth for an 8-to-28-row simulation. `--selftest` pins the closed form against the numeric interpolation in sign, magnitude and its vanishing at both ends, and pins the seven crpin4m crossovers this round quotes. Three perturbations were checked to redden it by name: a wrong published crossover, a disarmed `0 <= u <= 1` guard, and an inverted sign convention. ## The population, and the one cell that is REFUSED Inventorying every banked ladder's rung set gives **three** cells carrying both a coarse and a fine ladder on one box, binary, fixture and pool. Two are usable: | cell | coarse | fine | note | |---|---|---|---| | `4mc-e8` `0xFF0` `-t8` | `crpin4m` `cpe8` 32-row | `cpe8f` 4-row | one sitting | | `4mc-p4` `0xF` `-t4` | `crpin4m` `cpp4` 32-row | `cpp4f` 4-row | one sitting; **both ladders non-monotone** | | `4mcb` unpinned `-t16` | `crband` `csolo` 32-row | `cband` 8-row | **REFUSED** | **The `cband` refusal is a finding, not an omission.** `crband`'s own README records that it put its first rung inside the region it was measuring. That rung, m = 384, carries a **fold A/A floor of 30.1% CPU and 28.9% wall** against 0.5-4.2% at its other four, and it is the only lower bracket every crossing in that ladder has. So `cband` is not a fine reference, and its coarse-to-fine difference - which is **negative** (-3.7 rows in wall, sign flipping in 0.0%) - measures the ruined rung rather than the grid. The refusal is pinned in `--selftest`, which re-measures the floor rather than trusting this sentence. **That leaves the corpus with no clean second arm**, which is the single biggest limit on this round and is what is owed. ## Arm by arm ### Arm 1 - the direct comparison, paired inside the draw | cell | metric | coarse | fine | fine - coarse | sd | sign flips | |---|---|---:|---:|---:|---:|---:| | `4mc-e8` | CPU | 396.43 | 398.02 | **+1.59** | 2.0 | **24.2%** | | `4mc-e8` | wall | 415.27 | **bound** (`above_last`) | - | - | - | | `4mc-p4` | CPU | 396.40 | 401.23 | **+4.83** | 1.1 | 0.0% | | `4mc-p4` | wall | 412.48 | 415.74 | **+3.25** | 0.5 | 0.0% | | `4mcb` | both | - | - | **REFUSED** | - | - | **"All four moved up" is not four observations.** The e8 wall comparison is a bound and cannot be formed at all. The e8 CPU comparison is a **null** - its sign flips in a quarter of resamples. The two that are firm are the **same arm's two metrics, computed from the same legs**, so they are one observation with two read-outs, not two; and that arm is the one crpool4m and crpin4m **both** report as non-monotone, whose `m = 412` cell reverses direction between 408 and 416 at A/A floors of 0.4% and 0.6%. A tight floor is agreement, not correctness, and this is the case that rule exists for. ### Arm 2 - the grid simulated INSIDE each fine ladder Within one fine ladder the pair of rungs `h` apart that brackets the crossing **is** a simulated coarse grid: same legs, same sitting, same ladder position, so everything but the grid cancels by construction. Grouping the 27 simulated biases by which rung was the lower bracket is the whole result: | ladder | metric | lower rung | n | mean bias | across h | |---|---|---:|---:|---:|---| | `4mc-e8-fine` | CPU | 392 | 6 | **-1.69** | h 8 -> 28: -1.71 -> -1.17 | | `4mc-e8-fine` | CPU | 396 | 5 | **+1.06** | h 8 -> 24: +0.14 -> +1.70 | | `4mc-p4-fine` | CPU | 392 | 3 | **+3.52** | h 12 -> 24: +1.76 -> +5.71 | | `4mc-p4-fine` | CPU | 396 | 3 | **-2.11** | h 8 -> 20: -1.25 -> -2.63 | | `4mc-p4-fine` | CPU | 400 | 2 | -0.25 | h 8 -> 16: -0.28 -> -0.22 | | `4mc-p4-fine` | wall | 392..408 | 5 | -1.12 | flat across h 8 -> 24 | | `1m-res-bracket` | CPU | 288..320 | 3 | -0.37 | h 32 -> 64: +0.16 -> -0.80 | **The gap does almost nothing and the rung choice does everything.** An `h^2` law says the bias must grow **12-fold** from h = 8 to h = 28; the largest observed growth within a group is **1.6-fold**, and one group shrinks. Between groups - the same ladder, the same crossing, one rung moved by four rows - the bias **flips sign** and moves 2.75 rows on e8 and 5.6 rows on p4. Fitting `bias = C h^2 u(1-u)` per ladder leaves an rms residual of 1.35 against an rms bias of 1.51 on e8 (the law explains about a tenth of it), and returns C values of opposite sign on the two arms of one sitting. The `4mc-p4-fine` wall fit is the honest absurdity that shows the fit is not measuring curvature: every point sits at `u > 0.96`, where `u(1-u)` is nearly zero, so a real and constant -1.2 row offset is fitted as C = -0.22/row and would imply -57 rows at mid-interval. ### Arm 3 - the correction the algebra implies, over the whole corpus For every banked 32-row crossing, `delta` is the root of the local **curve** through that ladder's own rungs minus the root of the **chord** - both recomputed inside every bootstrap draw so the difference is paired. - **146 crossings reduced** (38 old-format logs skipped by name, the silent- garbage parse class the round-2 error-bar file documents). - `delta > 0` - the coarse reading low, the correction up - in **74 of 146**. **z = +0.17.** A coin toss. - `|delta|`: median **1.5** rows, 90th percentile 6.7, max 23.8. - Distinguishable from zero (sign flips under 5%): 42 of 146, of which **16 positive** - if anything leaning negative (z = -1.54), the **opposite** sign to the one crpin4m's observation would need. - Per round, the positive fraction runs from 1/8 (`g4win`) to 14/19 (`rowgate-09-15`) with no ordering by class, pool, block size or date. ### Arm 4 - arm 3 validated where the answer is known | cell | metric | coarse | + delta | corrected | fine | error before -> after | |---|---|---:|---:|---:|---:|---| | `4mc-e8` | CPU | 396.43 | -0.60 | 395.83 | 398.02 | 1.59 -> 2.19 **WORSE** | | `4mc-p4` | CPU | 396.40 | -0.85 | 395.55 | 401.23 | 4.83 -> 5.68 **WORSE** | | `4mc-p4` | wall | 412.48 | -2.85 | 409.63 | 415.74 | 3.25 -> 6.11 **WORSE** | Three of three, and **in the wrong direction every time**: the measured curvature on those arms is concave over their own bracketing interval, so the correction it implies points **down** while the fine ladder reads **up**. Applying it would be worse than doing nothing, which is the whole verdict. ### Arm 5 - the rival explanation, priced A crossover is a root of `log(F/T)`, so a cell wrong by `e` per cent moves it by `e / (100 dy/dm)` rows. On these ladders one per cent of `F/T` is worth **3.9-5.3 rows** at the coarse crossings and 0.8-5.1 at the fine ones, and the bracketing rungs' own A/A floors are 0.03-1.74%. That prices "one bracketing rung drew badly" at **1.3 to 3.6 rows** - the same size as every coarse-to-fine gap in arm 1, present at **every** spacing, and not removable by refining the grid. ### Arm 6 - why no correction is recoverable, measured directly The chord bias is proportional to the second derivative of `log(F/T)`. A real curvature changes slowly along a ladder, so adjacent second differences should mostly **agree** in sign. Per-rung noise does the opposite: adjacent second differences share two rungs with opposite weights, giving a correlation of exactly **-2/3** and a Gaussian sign-**disagreement** rate of `1/2 + arcsin(2/3)/pi` = **73.2%**. Over 172 coarse ladder-metrics carrying four or more consecutive 32-row rungs: **391 of 568 adjacent pairs disagree, 69%.** Three points below the pure-noise prediction and twenty above what a resolved shape would give. **The curvature a 32-row ladder appears to have is, to within a few per cent, its own rung scatter** - so there is nothing there to build a correction out of, and arm 3's failure in arm 4 is the expected outcome rather than a surprise. ## What to do instead of a correction **Carry a grid term of about 1 to 4 rows beside every 32-row crossover**, added in quadrature to its bootstrap bar, and read it as an order of magnitude the way both error-bar files read theirs. Arm 3's `|delta|` distribution (median 1.5, 90th percentile 6.7) and arm 5's rung-noise term (1.3-3.6) are two routes to the same scale. **Arm 5's arithmetic now lives in `w3winred.rung_noise` and is printed beside every graded crossover** (18 Sep 2026, item 1 of an internal note, which this file's own bullet owed). `w3winred.py --grade` prints the bracketing pair, `u`, rows per one per cent of `F/T`, the two bracketing rungs' A/A floors and the implied rung-noise term under its bootstrap table, for any crossover in the campaign rather than the seven quoted here. **One copy, called from both**: this file's arm 5 is the printing and the ladder roster, the numbers come from there, and every figure in the table above is unchanged by the move - both selftests pin it from their own side, so a copy dragged back reddens. The two errors are printed **side by side** and are not combined: how to fold a sampling bar into a systematic is a judgement neither script makes for the reader. **It changes no published claim.** The campaign's own external replicate evidence is 3-13 rows (3.6 on the resident anchor across three nights, 13.6 on the windowed cell, 3-7 on the pinned sub-box envelope, 2-6 on crpin4m's own create-path replication), so a 1-4 row grid term is **inside** the scatter the campaign already quotes and does not move a single row of either graded table. ## The corpus has grown since this sitting Re-running `crossbias.py --boot 2000` on origin/main of 18 Sep 2026 prints **76 of 150** where the table above reads 74 of 146, and **401 of 584 = 69%** where arm 6 reads 391 of 568 = 69%. That is four more banked ladder logs reaching the two corpus-wide arms, not a change in any arithmetic: the verdicts (not systematic, z near zero; the curvature is the rungs' own scatter) and every per-cell figure are identical, and the banked `crossbias-output.txt` is this lane's own sitting and is left as it was run. Checked against the tip before and after the rung-noise move, which changes neither count. ## Stated limits - **A fine ladder is a reference, not the truth.** Its own crossover carries the same chord bias at its own spacing; at 4 rows against 32 that residual is 1/64 of the term being measured, which is what makes it usable. - **No arm reaches h = 32 on a 4-row ladder.** They span 28 rows. The only simulated h = 32 in the corpus is `cband`'s, and `cband` is refused. Every statement about h = 32 here is an extrapolation in h - which is safe only because the measured h-dependence is near zero, and would not be if it were not. - **Two usable cells, one box, one part, one payload, one block size, one phase.** Core Ultra 9 386H, GFNI-256, random bytes, 4 MiB, n = 4,096, the CREATE path. Arms 3, 5 and 6 reach the whole corpus; arms 1, 2 and 4 do not. - **The one arm with a firm coarse-to-fine gap is non-monotone in both its ladders and in both sittings.** Its tight A/A floors are agreement, not correctness. - **A bootstrap resamples the same legs.** A rung wrong in both its copies is invisible to it - the round-2 error-bar file's section 2 has the worked case, and `cband`'s m = 384 would be one if it were not caught by its floor. - **Arm 3's `delta` is one instrument's** (a local quadratic through four rungs). It is not trusted on its own; it is validated in arm 4 and independently corroborated by arms 2 and 6, which share none of its arithmetic. - **Nothing here is a statement about the WINDOWED or REPAIR paths' shape** in any direction other than arms 3, 5 and 6's corpus-wide counts. - **NO CONSTANT MOVED.** The 384 / 416 / `create_ntt_min_rows` decision is the maintainer's and remains open. This lane is the seventh to measure an input and the seventh to move nothing. ## Owed after this lane - **A clean second coarse/fine cell.** The corpus has exactly one usable arm with a firm gap and it is the non-monotone one. A `-t8` `0xFF0` pair on a MONOTONE arm - or a repeat of `cband` with its first rung spent below the region, which crpin4m has already shown works - would settle whether arm 1's +4.8 is the arm or the grid. **Needs intel-core-ultra-9-386h, so do not pre-mint a claim for it** (the handoff's own rule). - **The rung-noise term belongs in the reducer, not in prose.** Arm 5 computes "rows per one per cent of `F/T`" from any ladder in four lines, and every crossover in this campaign would be better quoted with it beside the bootstrap bar. Nobody has put it there. - **`4mc-p4`'s non-monotonicity is still item 3 of the same handoff** and is untaken. This round makes it more load-bearing than it was: that arm is now also the only firm evidence for the coarse-to-fine shift.