{ "tm": "1RB0RF_1RC0RB_1LD0RA_1LE0LD_1RB0LC_1LA---", "rank": null, "class_sheet": "Skelet 1-like Class 2", "class_guess": "Skelet 1-like: multi-level counter on phase defects between inert fossil blocks (x3 per epoch)", "confidence": "low", "summary": "Multi-level counter on 00/11 defects between inert fossil blocks; a new fossil 3F+72 cells long each epoch.", "mechanism": [ "The tape is alternating cells with isolated defects (00 or 11). Sweeps flip every pair. A leftward head turns at a 00 in D-phase (E reads 0) or at an 11 in E-phase (C reads 1). A rightward head turns only at a 00 in B-phase (C reads 0), so the phase decides pass or turn.", "Level 1: the head bounces between the right edge and a wall W that moves 1 cell per hit (E-turn 00 at x -> 11 at x-1, C-turn 11 at x -> 00 at x-1; 554/555 single-turn trips to 3e6). W eats the active block X while the right block Y grows by about 1.7 cells per trip.", "Deeper trips stop at the defects between fossil blocks, which cycle through 2 or 3 positions (e.g. 180 -> 179 -> 178 -> 180), and at the left end. This is a mixed-radix counter, and each overflow ends a phase. There are 3 phases per epoch: X grows, then Y grows, then W eats X and Y freezes into a fossil.", "Epoch laws, exact on every epoch from blank to 2e10 steps: fossils F = 2736 -> 8280 -> 24912 -> 74808 cells (F' = 3F + 72); mid blocks X = 248 -> 752 -> 2264 -> 6800 (X' = 3X + 8). There are 3 left records per epoch (3 cells each), and each epoch takes 9.04x longer than the one before.", "Fossil lengths are inert mod 4. Inserting 4j cells into any block (fossils or X) leaves the run unchanged except for those lengths (4/4 starts, to 3.5e9 steps, the new blocks are identical). Inserting 2 mod 4 cells changes the run (3/3; one ran off a 2^25-cell tape).", "Halting needs `C> 111` or `A> 11`, i.e. two adjacent defects in the wrong phase. The run has no halt to 2e10 steps, and neither sibling halts to 2e9." ], "diagrams": [ { "mode": "default", "png": "img/default_site_1e4.png", "html": null, "steps": 10000, "caption": "bbchallenge default view, first 10,000 steps", "title": "bbchallenge default view, 10,000 steps", "why": "The site's view. Fine pair stripes fill a slowly widening bell; the left edge barely moves.", "primary": true, "start": "blank", "rows": "every step: steps 0..9,999 squeezed into 500 px rows as on bbchallenge.org (1 px row = 20 steps)", "window": "cells -200..199, 1 px = 1 cell", "legend": "white = 1, black = 0 or never visited, red/green = head after a right/left move (pale = blended by the squeeze)", "style": "bbchallenge-default" }, { "mode": "every:100", "png": "img/wall_trips_2p7e6.png", "html": null, "steps": 300000, "caption": "Mid-run close-up: wall trips and deep trips", "title": "Wall trips at step 2.7 million", "why": "Rows every 100 steps. Black wedges are left sweeps; their left tips are turns. Compare short and long tips.", "primary": true, "start": "mid-run: step 2,700,000 (tape from the blank run)", "rows": "every 100 steps: steps 2,700,000..3,000,000 (3,001 rows)", "window": "cells -34..3178, 1 px = 2 cells (every 2nd cell sampled; the fossil 970-cell block is at cells 180..1149, the wall near 1270)", "legend": "white = 1, black = 0 (a swept region shows its flipped phase), red/green = head after a right/left move", "style": "bbchallenge-default" }, { "mode": "records:R", "png": "img/rrecords_sampled.png", "html": null, "steps": 7612581, "caption": "Right records, every 8th cell: block phases as bands", "title": "Block phases at right records", "why": "One row per new rightmost cell; sampling every 8th cell turns each alternating block into one flat colour.", "primary": true, "start": "blank", "rows": "right records only: one row per new rightmost cell (5,000 rows, steps 0..7,612,581)", "window": "cells -40..4999, 1 px = 8 cells (every 8th cell sampled, so an alternating block shows its phase as one colour)", "legend": "white = 1, black = 0 or never visited (sampled cells), red = head", "style": "bbchallenge-default" }, { "mode": "records", "png": "img/records_1e8.png", "html": null, "steps": 7546801, "caption": "Left and right records, full resolution", "title": "Records view, 1 px = 1 cell", "why": "Fig 3's span at full resolution: the right edge grows like sqrt(t), the left edge in rare jumps.", "start": "blank", "rows": "left and right records (5,000 rows, steps 0..7,546,801)", "window": "cells -40..4959, 1 px = 1 cell", "legend": "white = 1, black = 0 or never visited, red/green = head after a right/left move", "style": "bbchallenge-default" } ], "history": { "skeleton": "0^inf J F_1 .. F_k X Y C> 0^inf (rows: the 3rd left record of each 9-cell group)", "vars": [ "lo", "hi" ], "rows": [ { "step": 29330, "values": [ -22, 346 ] }, { "step": 91395, "values": [ -25, 586 ] }, { "step": 442986, "values": [ -28, 1152 ] }, { "step": 647767, "values": [ -31, 1394 ] }, { "step": 1212916, "values": [ -34, 2123 ] }, { "step": 4664921, "values": [ -37, 3819 ] }, { "step": 6952350, "values": [ -40, 4608 ] }, { "step": 12289411, "values": [ -43, 6792 ] }, { "step": 43669198, "values": [ -46, 11918 ] }, { "step": 65532919, "values": [ -49, 14320 ] }, { "step": 114511040, "values": [ -52, 20881 ] }, { "step": 398088769, "values": [ -55, 36257 ] }, { "step": 597235426, "values": [ -58, 43526 ] }, { "step": 1041302739, "values": [ -61, 63206 ] }, { "step": 3599625458, "values": [ -64, 109372 ] }, { "step": 5399920255, "values": [ -67, 131214 ] }, { "step": 9400201940, "values": [ -70, 190263 ] } ] }, "rules": [ { "name": "S1 B-sweep", "from": "B> 01^(2n)", "to": "10^(2n) B>", "steps": "4n", "cases": "odd count: B> 01^(2n+1) -> 10^(2n+1) A> in 4n+2 steps (60/60 too)", "validated": { "instances": 60, "passed": 60 }, "source": "manual" }, { "name": "S2 C-sweep", "from": "C> 10^(2n)", "to": "01^(2n) C>", "steps": "4n", "cases": null, "validated": { "instances": 60, "passed": 60 }, "source": "manual" }, { "name": "S3/S4 D-sweeps left", "from": "01^(2n) 10", "to": "10 B>", "steps": "4", "cases": "A> 11 halts (2 steps); C> 111 halts (3 steps)", "validated": { "instances": 62, "passed": 62 }, "source": "manual" }, { "name": "W defect turn (single-turn trips)", "from": "E-turn: 00 at x | C-turn: 11 at x", "to": "11 at x-1 | 00 at x-1; some passed 00-defects reset +2 cells", "steps": null, "cases": "every single-turn trip from step 2e4 to 3e6; the 1 miss (step 446,473) is at an epoch change; which passed defects reset is not yet derived", "validated": { "instances": 555, "passed": 554 }, "source": "manual" }, { "name": "Fossils inert mod 4", "from": "run from step 597,279,037 with 4j cells inserted in blocks", "to": "same run; only those block lengths change", "steps": "+0.2..4.1 M steps over 2.4e9", "cases": "4 random insertions (all blocks / first fossil / X only), compared at the next 2 left records; inserting 2 mod 4 cells changes the run (3/3 negative controls)", "validated": { "instances": 4, "passed": 4 }, "source": "manual" }, { "name": "Epoch laws", "from": "fossil F_k, mid block X_k", "to": "F_(k+1) = 3 F_k + 72, X_(k+1) = 3 X_k + 8", "steps": "epoch time x9.04", "cases": "F: 2736 -> 8280 -> 24912 -> 74808 (to 1e10); X: 248 -> 752 -> 2264 -> 6800; observed, not derived", "validated": { "instances": 6, "passed": 6 }, "source": "manual" } ], "template": { "file": null, "module": null, "machine": "1RB0LE_1RC0RC_1LD0RB_1LA0LE_1LD1LF_1LD---", "diff": "different table (4-5 transitions from #11/#33 under the best renaming); no shared rules", "applies": "partly", "note": "Nearest proved neighbour: our kill #11 (Sk1C2, SBCv6 TM21 architecture). Its .v has one counter level: shift lemmas S1-S6, window lemmas T1-T14, digit lemmas Lb/Lf/Rb/Ro, the rules M_spec / Mdrop_spec / O_spec, and an abstract INV over one digit list. W4-3 matches only the bottom layer (sweep and turn lemmas). Its counter has unboundedly many levels: fossil defects with 2-3 states, the 3-phase epoch and the left-end counter, and #11 has no analogue of these. The closest open cases are #32/R4 (multi-level counter, parked)." }, "strategy": "Build an exact abstract model at the defect level: turn/pass/reset rules by phase, the wall trip, and fossils kept only mod 4. Check that it reproduces every trip to 1e9, then look for a recursive invariant over the fossil list (epoch = 3 counter overflows). Siblings 512/588 would then need only a new init.", "risks": [ "The phase rule is not derived yet: which defects turn, pass or reset depends on the head's phase. The left-end region (a small counter that grows 3 cells per phase) is not ruled.", "A proof needs a recursive multi-level invariant with no template (the same shape as #32/R4, parked). Expect several stages.", "Residues mod 4 matter: a block of the wrong residue changes the run (one perturbed start ran off the tape), so the invariant must pin every fossil residue." ], "checkpoint_questions": [ { "text": "Fig 2: does the head bounce between the right edge and a wall that creeps left (the short wedge tips near the middle), with a longer excursion to the left every few trips?", "figs": [ 1 ], "kind": "check" }, { "text": "Fig 3: do the sampled columns show flat bands (old blocks) that keep their edges, while a new, wider band opens at the right?", "figs": [ 2 ], "kind": "check" }, { "text": "Fund one ~2 h stage to derive the phase rule and an exact defect-level model (recommended), or park W4-3 with #32/R4 as a multi-level counter?", "kind": "decision" } ], "discord_check": null, "stats": { "simulated_steps": 20000000000, "cells": 269297, "nonzeros": 134642, "growth": "span ~ t^0.5 (x3 per epoch, epoch time x9.04); left edge -9 cells per epoch (3 left records of 3 cells); lo = -70, hi = 269,226 at 2e10" }, "status": "CHECKPOINT_A", "status_reason": "Mechanism partly ruled (sweeps, turns, wall trip, epoch laws, mod-4 inertness); level-2 phase rule, left end and invariant open: confidence low." }