== crossbias, boot=2000, NO LEG RUN, NO BOX TAKEN == ARM 1: the direct coarse-against-fine comparison, paired inside the draw (observation. It mixes the GRID term with ladder position and repeat, which is why arm 2 exists.) -- 4mc-e8 (0xFF0, -t8, create, 4 MiB n=4096) cpu coarse 396.43 (sd 1.7, bracket [384,416] h=32, u=0.388) fine 398.02 (sd 1.1) fine - coarse = +1.59 rows sd 2.0 0.8 sigma SIGN FLIPS IN 24.2% of 2000 resamples wall the fine ladder has NO CROSSOVER IN SHAPE (above_last) - a BOUND, not a point, so no difference is formed. -- 4mc-p4 (0xF, -t4, create, 4 MiB n=4096) cpu coarse 396.40 (sd 0.7, bracket [384,416] h=32, u=0.388) fine 401.23 (sd 0.9) fine - coarse = +4.83 rows sd 1.1 4.3 sigma SIGN FLIPS IN 0.0% of 2000 resamples wall coarse 412.48 (sd 0.5, bracket [384,416] h=32, u=0.890) fine 415.74 (sd 0.1) fine - coarse = +3.25 rows sd 0.5 7.1 sigma SIGN FLIPS IN 0.0% of 2000 resamples -- 4mcb (unpinned, -t16, create, 4 MiB n=4096) REFUSED: the fine ladder's only lower bracket is m=384, whose fold A/A floor is 30.1% CPU / 28.9% wall - crband's own documented ruined first rung. Not graded. A ladder whose bracketing rung is ruined is not a fine reference, and its coarse-to-fine difference measures the ruined rung. == ARM 2: the coarse grid SIMULATED inside each fine ladder Same legs, same sitting, same ladder position - everything but the GRID cancels by construction. -- 4mc-e8-fine [cpu] reference crossover 398.02 on a 4-row grid, legs/rung [4] h rungs u chord bias sd sign flips 8 [392,400] 0.752 396.30 -1.71 1.6 13.2% 8 [396,404] 0.252 398.16 +0.14 1.2 40.5% 12 [392,404] 0.502 395.56 -2.46 2.0 14.5% 12 [396,408] 0.168 399.10 +1.08 1.8 15.4% 16 [392,408] 0.376 396.55 -1.46 2.6 30.0% 16 [396,412] 0.126 398.94 +0.92 1.0 11.7% 20 [392,412] 0.301 396.10 -1.92 2.0 17.1% 20 [396,416] 0.101 399.46 +1.44 1.3 11.5% 24 [392,416] 0.251 396.63 -1.39 2.2 27.7% 24 [396,420] 0.084 399.71 +1.70 1.5 11.0% 28 [392,420] 0.215 396.85 -1.17 2.3 32.6% -- 4mc-p4-fine [cpu] reference crossover 401.23 on a 4-row grid, legs/rung [4], 1 far pair(s) excluded (they bracket a DIFFERENT crossing - the ladder is non-monotone) h rungs u chord bias sd sign flips 8 [396,404] 0.654 399.98 -1.25 1.2 16.7% 8 [400,408] 0.154 400.96 -0.28 0.3 24.6% 12 [392,404] 0.769 402.99 +1.76 0.9 0.0% 12 [396,408] 0.436 398.78 -2.45 1.2 3.0% 16 [392,408] 0.577 404.31 +3.08 1.0 0.0% 16 [400,416] 0.077 401.01 -0.22 0.3 35.8% 20 [396,416] 0.262 398.60 -2.63 1.2 1.8% 24 [392,416] 0.385 406.94 +5.71 1.0 0.0% -- 4mc-p4-fine [wall] reference crossover 415.74 on a 4-row grid, legs/rung [4] h rungs u chord bias sd sign flips 8 [408,416] 0.967 414.40 -1.34 0.4 0.0% 12 [404,416] 0.978 414.57 -1.16 0.3 0.0% 16 [400,416] 0.983 414.49 -1.25 0.4 0.0% 20 [396,416] 0.987 414.50 -1.24 0.4 0.0% 24 [392,416] 0.989 415.15 -0.59 0.2 0.0% -- 1m-res-bracket [cpu] reference crossover 350.21 on a 16-row grid, legs/rung [12] h rungs u chord bias sd sign flips 32 [320,352] 0.944 350.37 +0.16 2.3 46.5% 48 [304,352] 0.963 349.74 -0.47 2.7 17.0% 64 [288,352] 0.972 349.41 -0.80 3.1 13.1% == the first-order law bias = C * h^2 * u(1-u), fitted per ladder C is -(curvature/slope) of log(F/T), in 1/row. ONE SIGN across the corpus would be a correction; MIXED signs are an error term. ladder metric n C (1/row) rms bias rms resid implied bias at h=32,u=0.5 1m-res-bracket cpu 3 -5.389e-03 0.54 0.29 -1.38 rows 4mc-e8-fine cpu 11 -1.060e-02 1.51 1.35 -2.71 rows 4mc-p4-fine cpu 8 +2.266e-02 2.73 2.34 +5.80 rows 4mc-p4-fine wall 5 -2.215e-01 1.15 0.59 -56.70 rows == ARM 3: the correction the algebra implies for EVERY banked 32-row crossover delta = (root of the local CURVE through this ladder's own rungs) - (root of the CHORD). 146 banked 32-row crossovers reduced (38 old-format log(s) skipped by name). delta > 0 (the coarse reading is LOW, so the correction is UP) in 74 of 146. |delta|: median 1.5 rows, 90th pct 6.7, max 23.8 distinguishable from zero (sign flips in under 5% of resamples): 42 of 146, of which 16 positive == ARM 4: arm 3 VALIDATED against arm 1, on the two cells that have both cell metric coarse +delta = corrected fine |err| 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 == ARM 5: what one per-cent of F/T is worth in rows, and the rung-noise term ladder metric crossing bracket 1% of F/T A/A floors (lo | hi) implied rung-noise 4mc-e8 coarse cpu 396.43 [384,416] 3.9 rows 1.74/0.03% | 0.16/1.48% 3.4 rows 4mc-e8 coarse wall 415.27 [384,416] 4.2 rows 1.74/0.58% | 0.18/1.19% 3.6 rows 4mc-p4 coarse cpu 396.40 [384,416] 5.3 rows 0.34/0.48% | 0.07/0.11% 1.3 rows 4mc-p4 coarse wall 412.48 [384,416] 5.1 rows 0.27/0.57% | 0.24/0.13% 1.4 rows 4mc-e8-fine cpu 398.02 [396,400] 2.7 rows 1.10/1.09% | 0.05/0.51% 1.5 rows 4mc-p4-fine cpu 401.23 [400,404] 5.1 rows 0.54/0.29% | 0.08/0.28% 1.4 rows 4mc-p4-fine wall 415.74 [412,416] 0.8 rows 0.60/0.14% | 0.01/0.30% 0.2 rows == ARM 6: is the curvature at 32-row spacing RESOLVED, or noise shaped like one? 172 coarse ladder-metrics carry four or more consecutive 32-row rungs. Adjacent second differences DISAGREE in sign in 391 of 568 pairs = 69%. Pure per-rung noise predicts 73.2%; a resolved curvature predicts well under 50%. => the curvature a 32-row ladder appears to have is, to within a few per cent, its own rung scatter.