{
 "tm": "1RB1RF_1LC1RE_---0LD_1LE0LD_1RA0RF_0RE0LB",
 "rank": 15,
 "class_sheet": "Bell eats counter",
 "class_guess": "Bell eats counter: unary left pair + binary right counter; reset absorbs counter digits",
 "confidence": "med",
 "summary": "Left pair (110)^a 0 (011)^n trades 1 block/trip; right binary counter +1 per 2 trips; reset at n<=2 grows L",
 "mechanism": [
  "Tape = 0^inf (110)^a 0 (011)^n, then a binary counter Q at the right end (3-cell digits, LSB nearest the head).",
  "One trip = zig-zag from the right end to the a|n boundary and back: a+1, n-1. Q gets +1 every 2nd trip.",
  "At n = 1 or 2: reset (two full sweeps), a restarts at 2 and L = a+n grows. Trips ignore Q; the reset ignores a."
 ],
 "diagrams": [
  {
   "mode": "state:B:R",
   "png": "img/phase_trips.png",
   "html": null,
   "steps": 40000,
   "caption": "Boundary (black dots) walks right one block per trip; counter ticks; reset",
   "title": "One row per trip, across one reset",
   "why": "One row per trip. Black dots mark the a|n boundary; the counter sits at the right edge.",
   "primary": true,
   "start": "mid-run: step 152677",
   "rows": "every entry into state B moving right: one per trip, plus 1-4 extra rows a few steps apart while the counter carries; 196 rows over steps 152,677..192,677; the reset falls between rows 162 and 163 (steps 181,068 and 181,780)",
   "window": "cells -46..240, 1 px = 0.5 cells (each cell 2x2 px)",
   "legend": "white = 1, black = 0 or unvisited, red/green = head (last move right/left); the black diagonal dots are the 0 between (110)^a and (011)^n",
   "style": "bbchallenge-default"
  },
  {
   "mode": "every:2",
   "png": "img/reset_zoom.png",
   "html": null,
   "steps": 1700,
   "caption": "Reset: last short trips, then two full-width sweeps",
   "title": "Zoom on the reset, every 2 steps",
   "why": "Every 2 steps around the reset: compare the sweeps at the top with those below them.",
   "primary": true,
   "start": "mid-run: step 180953",
   "rows": "every 2 steps, steps 180,953..182,653 (851 rows, 2 px each; no head drawn on row 0). The head moves at most 2 cells between rows, so each sweep is a clean diagonal",
   "window": "cells -46..230, 1 px = 0.5 cells (each cell 2x2 px)",
   "legend": "white = 1, black = 0 or unvisited, red/green = head (last move right/left)",
   "style": "bbchallenge-default"
  },
  {
   "mode": "state:B:R",
   "png": "img/state_auto_1e6.png",
   "html": "img/viewer.html",
   "steps": 1000000,
   "caption": "B-entries moving right (≈ trips); each diagonal is one phase",
   "title": "One row per trip, to 1 million steps",
   "why": "One row per trip to 1M steps; each long diagonal is one phase. Compare the phases with each other.",
   "primary": true,
   "start": "blank",
   "rows": "every entry into state B moving right: one per trip, plus extra rows a few steps apart while the counter carries (2,299 of 4,871 gaps are under 20 steps); 4,872 rows to step 1,000,000",
   "window": "cells -72..430, 1 px = 1 cell",
   "legend": "white = 1, black = 0 or unvisited, red/green = head (last move right/left)",
   "style": "bbchallenge-default"
  },
  {
   "mode": "blaze",
   "png": "img/blaze_1e7.png",
   "html": "img/viewer.html",
   "steps": 10000000,
   "caption": "Blaze 1e7: slow t^0.35 growth, no chaos",
   "title": "Blaze view, 10 million steps",
   "why": "Smooth, slow growth (span ~ t^0.35) with no chaotic patches.",
   "primary": false,
   "start": "blank",
   "rows": "Blaze time blocks of 16,384 steps (each averaged), 611 blocks to step 1e7, stretched to 500 px (1 px = about 20,000 steps)",
   "window": "cells -138..958, 1 px = 1.37 cells (Blaze image: 1 column per cell, 1,097 columns stretched to 800 px)",
   "legend": "white = 0, orange = 1, paler orange = 1 only part of the time (or in part of the cells) of that pixel's block; no head",
   "style": "bbchallenge-blaze"
  },
  {
   "mode": "default",
   "png": "img/default_1e4.png",
   "html": "img/viewer.html",
   "steps": 10000,
   "caption": "bbchallenge default view, first 10k steps",
   "title": "bbchallenge Default view, first 10,000 steps",
   "why": "For orientation: early zig-zag trips as bbchallenge.org shows them.",
   "primary": false,
   "start": "blank",
   "rows": "every step 0..9,999, squeezed into 500 px by the site's bilinear filter: px row y blends steps 20y+9 and 20y+10, so it is effectively one sample every 20 steps",
   "window": "cells -200..199, 1 px = 1 cell",
   "legend": "white = 1, black = 0 or unvisited, red/green = head (last move right/left)",
   "style": "bbchallenge-default"
  }
 ],
 "history": {
  "skeleton": "0^inf (110)^a 0 (011)^n 1 W B> 0^inf",
  "vars": [
   "a",
   "n",
   "L=a+n",
   "W (x=110, y=111, |=00)"
  ],
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    "values": [
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 },
 "rules": [
  {
   "name": "T: trip pair (clean counter)",
   "from": "0^inf (110)^(2c) 0 (011)^(n+3) <C 1010010 D(c) 0^inf",
   "to": "0^inf (110)^(2c+2) 0 (011)^(n+1) <C 1010010 D(c+1) 0^inf",
   "steps": "16n + 6*nu(c) + 62, nu = trailing 1-bits of c (exact fit)",
   "cases": "c>=1, n>=0; D(c) = c LSB-first, digit 1=001, 0=000; fails for n+3<3 (reset)",
   "validated": {
    "instances": 131,
    "passed": 131
   },
   "source": "manual"
  },
  {
   "name": "T': trip pair, any natural counter word",
   "from": "0^inf (110)^a 0 (011)^(n+3) <C Q 0^inf",
   "to": "0^inf (110)^(a+2) 0 (011)^(n+1) <C inc(Q) 0^inf",
   "steps": "16n + K(Q)",
   "cases": "Q, inc(Q) = 15 consecutive natural pairs at 3e8/6e8/9e8; a, n random",
   "validated": {
    "instances": 60,
    "passed": 60
   },
   "source": "manual"
  },
  {
   "name": "R: reset (clean counter)",
   "from": "0^inf (110)^(2c) 0 (011)^n <C 1010010 D(c) 0^inf, n in {1,2}",
   "to": "0^inf (110)^2 0 (011)^(2c+n+dL-2) <C Q' 0^inf",
   "steps": "about 16L (not fitted)",
   "cases": "dL=1 if n=2; n=1: dL=4 (c even), 5 (c=3 mod 4), 6+nu(c>>2) (c=1 mod 4)",
   "validated": {
    "instances": 48,
    "passed": 48
   },
   "source": "manual"
  },
  {
   "name": "R': reset ignores a",
   "from": "0^inf (110)^a 0 (011)^n <C Q 0^inf, natural (n,Q) at a reset, a replaced",
   "to": "0^inf (110)^2 0 (011)^(a+n+dL-2) <C Q' 0^inf, same dL and Q' for every a",
   "steps": null,
   "cases": "natural resets after 3e8 and 6e8 (both n=2); 6 values of a each (incl. 0, 2, 4, large)",
   "validated": {
    "instances": 12,
    "passed": 12
   },
   "source": "manual"
  }
 ],
 "template": {
  "file": "vendor/busycoq/verify/IndBECv1.v",
  "module": "Others (nonhalt7)",
  "machine": "1LB1LF_1RC1LB_---0RD_1RE0RD_1LA0LF_0LE0RB",
  "diff": "1 transition up to mirror: B1 = 1RE (ours) vs 1RB (template)",
  "applies": "partly",
  "note": "Its config_BEC ex-rule (r 1 B> 1^n 0 l -> r <D 0^(n+1) 1 l) holds 24/24 on it, 2/24 on ours (n=0 only). Method fits (binary inc), certificate doesn't; BECv7 TM1 shape also fits."
 },
 "strategy": "Rules on G(a,n,Q): T by ind over trips with a binary-inc side lemma on Q (3-cell digits); reset lemmas split on n in {1,2} and Q's low digits; nonhalt via progress_nonhalt_cond with 'Q in a regular language'. Cheap first try: IndBECv1 solve_hlin_nonhalt on the mirrored TM with config_BEC/config_BEC_Pos, block 3, d1=[1;0;0]; fallback BECv7 TM1 Longitudinal/tail style.",
 "risks": [
  "Q carries a growing low-digit stack (13 to 27 blocks by 1e9); need a closed regular language for it",
  "Reset branches on Q's low digits: natural dL=1 if L even (98/98), 2-25 if L odd; all branches must be safe",
  "Template certificate does not transfer; decider may need new ex-rule params or fail"
 ],
 "checkpoint_questions": [
  {
   "text": "Fig 1: does the black dotted diagonal move right row by row while the counter at the right edge ticks?",
   "figs": [
    0
   ]
  },
  {
   "text": "Fig 2: after the short trips at the top, do the sweeps switch to full width (the reset) and stay regular after it?",
   "figs": [
    1
   ]
  },
  {
   "text": "Fig 3: does every phase look alike (one long diagonal, then a reset), only getting longer?",
   "figs": [
    2
   ]
  }
 ],
 "discord_check": null,
 "stats": {
  "simulated_steps": 1000000000,
  "cells": 5210,
  "nonzeros": 2940,
  "growth": "L=a+n: 83 at 1.8e5, 1717 at 1e9 (~t^0.35); ~276 phases by 1e9"
 },
 "status": "CHECKPOINT_A",
 "status_reason": "Mechanism found and core rules validated; not a cryptid: values grow polynomially and every probed reset branch is safe"
}