{
 "tm": "1RB0RA_1LC0LD_1LE1LD_0LB0RE_1RF0LA_---0RA",
 "rank": 7,
 "class_sheet": "Bell eats counter",
 "class_guess": "Bell eats counter: BECv1 tm10 with a broken carry (spaced-digit binary counter + unary marker)",
 "confidence": "med",
 "summary": "BECv1 tm10 twin (E1 differs): spaced-digit counter eats a unary run; 7 rules validated, EatDigit open",
 "mechanism": [
  "Tape: a counter on the left, then a long run of 110 blocks holding a 2-cell 00 marker.",
  "Each round: A sweeps right to the marker, moves it 1 block left, B sweeps back, counter +1 (same as BECv1 tm10).",
  "Unlike tm10 (E1), digits are 111110 blocks spread through the 110 run; the marker eats the lowest digit on arrival."
 ],
 "diagrams": [
  {
   "mode": "state:E:L",
   "png": "img/stateEL_3e5.png",
   "html": "img/viewer.html",
   "steps": 300000,
   "caption": "One row per entry into E moving left; dark diagonal = marker",
   "title": "One row per left entry into E",
   "why": "One row each time E is entered moving left. Follow one dark diagonal (the 00 marker) down the rows.",
   "primary": true,
   "start": "blank",
   "rows": "every entry into state E moving left: 1,456 rows to step 300,000, 7 to 984 steps apart (median 45), so several rows per round",
   "window": "cells 0..435, 1 px = 0.5 cells (each cell 2x2 px); counted from each row's leftmost visited cell, i.e. the left edge is pinned at x = 0 (frame = left)",
   "legend": "white = 1, black = 0 or unvisited, red/green = head (last move right/left)",
   "style": "bbchallenge-default"
  },
  {
   "mode": "records:L",
   "png": "img/recordsL_left_1e7.png",
   "html": "img/viewer.html",
   "steps": 10000000,
   "caption": "Left edge fixed: white streaks = digits, born at edge, eaten later",
   "title": "Left records, left edge pinned",
   "why": "Left edge pinned. Follow the white streaks: where each one starts and where it ends.",
   "primary": true,
   "start": "blank",
   "rows": "left records only: the start, then one row per new leftmost cell (1,424 rows to step 10,000,000; not evenly spaced in time)",
   "window": "cells 0..149, 1 px = 0.5 cells (each cell 2x2 px); counted from each row's leftmost visited cell (frame = left), so only the 150 leftmost cells are shown and the rest of the tape is cut off",
   "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": "Bell-shaped left edge; horizontal bands = epochs (digit eaten)",
   "title": "Blaze, 10 million steps",
   "why": "Bell-shaped left edge; each horizontal band is an epoch, ending when a digit is eaten.",
   "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 -1423..77, 1 px = 1.88 cells (Blaze image: 1 column per cell, 1,501 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_site.png",
   "html": "img/viewer.html",
   "steps": 10000,
   "caption": "bbchallenge view: first rounds, sweeps and marker",
   "title": "bbchallenge default view, first 10,000 steps",
   "why": "The site's view of the first rounds: sweeps and the marker.",
   "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"
  },
  {
   "mode": "every:1",
   "png": "img/closure_halt_L4word.png",
   "html": null,
   "steps": 197,
   "caption": "A 4-gram-legal word whose stale-1 cascade halts",
   "title": "Closure escape: an L4 word halts",
   "why": "Follow the head's zig-zag left through the counter: the cascade bounces, then F reads 0 at the bottom.",
   "primary": false,
   "start": "constructed: 0^inf 1 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 1 <E 1 1 0 1 1 0 0 0 1 1 0 0^inf",
   "rows": "every step: 198 rows, step 0 to the halt at step 197",
   "window": "cells -30..11, 1 px = 0.33 cells (each cell 3x3 px)",
   "legend": "white = 1, black = 0 or unvisited, red/green = head (last move right/left)",
   "style": "bbchallenge-default"
  }
 ],
 "history": {
  "skeleton": "0^inf <E 110 W (110)^G 00 (110)^M 0^inf",
  "vars": [
   "W (counter word; 2=110 4=11110 5=111110 3=1110 z3=extra 00 marker)",
   "G",
   "M"
  ],
  "rows": [
   {
    "step": 20053019,
    "values": [
     "5 2^3 5 2^7 5",
     549,
     87
    ]
   },
   {
    "step": 20088414,
    "values": [
     "4 5 2^3 5 2^7 5",
     541,
     95
    ]
   },
   {
    "step": 20224880,
    "values": [
     "5 2 5 2^3 5 2^7 5",
     509,
     127
    ]
   },
   {
    "step": 20290067,
    "values": [
     "4 5 2 5 2^3 5 2^7 5",
     493,
     143
    ]
   },
   {
    "step": 20530341,
    "values": [
     "5 2 5 2 5 2^3 5 2^7 5",
     429,
     207
    ]
   },
   {
    "step": 20638216,
    "values": [
     "4 5 2 5 2 5 2^3 5 2^7 5",
     397,
     239
    ]
   },
   {
    "step": 20987802,
    "values": [
     "5 2 5 2 5 2 5 2^3 5 2^7 5",
     269,
     367
    ]
   },
   {
    "step": 21113469,
    "values": [
     "4 5 2 5 2 5 2 5 2^3 5 2^7 5",
     205,
     431
    ]
   },
   {
    "step": 21342621,
    "values": [
     "5 2 5 2 5 z3 2^4 5",
     641,
     9
    ]
   },
   {
    "step": 21474698,
    "values": [
     "4 z3 5 2^10 5",
     615,
     35
    ]
   },
   {
    "step": 21474724,
    "values": [
     "4 2 5 2^10 5",
     615,
     35
    ]
   },
   {
    "step": 21592226,
    "values": [
     "5 2^3 5 2^10 5",
     591,
     59
    ]
   },
   {
    "step": 21630405,
    "values": [
     "4 5 2^3 5 2^10 5",
     583,
     67
    ]
   },
   {
    "step": 21778007,
    "values": [
     "5 2 5 2^3 5 2^10 5",
     551,
     99
    ]
   },
   {
    "step": 21848762,
    "values": [
     "4 5 2 5 2^3 5 2^10 5",
     535,
     115
    ]
   },
   {
    "step": 22111308,
    "values": [
     "5 2 5 2 5 2^3 5 2^10 5",
     471,
     179
    ]
   },
   {
    "step": 22230319,
    "values": [
     "4 5 2 5 2 5 2^3 5 2^10 5",
     439,
     211
    ]
   },
   {
    "step": 22624449,
    "values": [
     "5 2 5 2 5 2 5 2^3 5 2^10 5",
     311,
     339
    ]
   },
   {
    "step": 22772388,
    "values": [
     "4 5 2 5 2 5 2 5 2^3 5 2^10 5",
     247,
     403
    ]
   },
   {
    "step": 23036592,
    "values": [
     "5 2 5 2 5 2 5 2 5 2^3 5",
     6,
     656
    ]
   },
   {
    "step": 23084955,
    "values": [
     "4 5 2 z3 2^9 5",
     658,
     9
    ]
   },
   {
    "step": 23085024,
    "values": [
     "4 3 2^13 5",
     658,
     9
    ]
   },
   {
    "step": 23095714,
    "values": [
     "4 4 2^13 5",
     656,
     11
    ]
   },
   {
    "step": 23138087,
    "values": [
     "4 2 z3 4 2^14 5",
     648,
     19
    ]
   },
   {
    "step": 23138134,
    "values": [
     "4 3 2^2 4 2^14 5",
     648,
     19
    ]
   },
   {
    "step": 23282700,
    "values": [
     "5 2 5 2^17 5",
     620,
     47
    ]
   },
   {
    "step": 23322927,
    "values": [
     "4 5 2 5 2^17 5",
     612,
     55
    ]
   },
   {
    "step": 23478721,
    "values": [
     "5 2 5 2 5 2^17 5",
     580,
     87
    ]
   },
   {
    "step": 23553572,
    "values": [
     "4 5 2 5 2 5 2^17 5",
     564,
     103
    ]
   },
   {
    "step": 23832502,
    "values": [
     "5 2 5 2 5 2 5 2^17 5",
     500,
     167
    ]
   },
   {
    "step": 23959705,
    "values": [
     "4 5 2 5 2 5 2 5 2^17 5",
     468,
     199
    ]
   },
   {
    "step": 24386603,
    "values": [
     "5 2 5 2 5 2 5 2 5 2^17 5",
     340,
     327
    ]
   },
   {
    "step": 24550926,
    "values": [
     "4 5 2 5 2 5 2 5 2 5 2^17 5",
     276,
     391
    ]
   },
   {
    "step": 24880544,
    "values": [
     "5 2 5 2 5 2 5 2 5 2 5 2^17 5",
     20,
     647
    ]
   },
   {
    "step": 25203475,
    "values": [
     "4 5 2 z3 2^3 5",
     634,
     61
    ]
   },
   {
    "step": 25203544,
    "values": [
     "4 3 2^7 5",
     634,
     61
    ]
   },
   {
    "step": 25213802,
    "values": [
     "4 4 2^7 5",
     632,
     63
    ]
   },
   {
    "step": 25254447,
    "values": [
     "4 2 z3 4 2^8 5",
     624,
     71
    ]
   },
   {
    "step": 25254494,
    "values": [
     "4 3 2^2 4 2^8 5",
     624,
     71
    ]
   },
   {
    "step": 25393012,
    "values": [
     "5 2 5 2^11 5",
     596,
     99
    ]
   }
  ]
 },
 "rules": [
  {
   "name": "R1 A-sweep (=BECv1 HR1)",
   "from": "l A> (011)^n r",
   "to": "l (010)^n A> r",
   "steps": "5n",
   "cases": null,
   "validated": {
    "instances": 60,
    "passed": 60
   },
   "source": "manual"
  },
  {
   "name": "R2 B-sweep back (=HR3)",
   "from": "l (010)^n <B r",
   "to": "l <B (011)^n r",
   "steps": "3n",
   "cases": null,
   "validated": {
    "instances": 60,
    "passed": 60
   },
   "source": "manual"
  },
  {
   "name": "R3 marker step (=HR2)",
   "from": "l A> 011 00 011 r",
   "to": "l <B 00 011 011 r",
   "steps": "11",
   "cases": null,
   "validated": {
    "instances": 60,
    "passed": 60
   },
   "source": "manual"
  },
  {
   "name": "R4 marker birth at right end (=HE2)",
   "from": "l 010 A> 0^inf",
   "to": "l <B 00 011 0^inf",
   "steps": "6",
   "cases": null,
   "validated": {
    "instances": 60,
    "passed": 60
   },
   "source": "manual"
  },
  {
   "name": "L0 round, low digit 0->1",
   "from": "l 0 <E 11 (011)^(a+1) 00 (011)^b 0^inf",
   "to": "l 1 <E 11 (011)^a 00 (011)^(b+1) 0^inf",
   "steps": "8a+16",
   "cases": "any left context l",
   "validated": {
    "instances": 60,
    "passed": 60
   },
   "source": "manual"
  },
  {
   "name": "OP overflow 2 5 X -> 2 4 5 X",
   "from": "0^inf <E 110 111110 X (110)^(G+2^f) 00 (110)^M 0^inf",
   "to": "0^inf <E 110 11110 111110 X (110)^G 00 (110)^(M+2^f) 0^inf",
   "steps": null,
   "cases": "X = (110)^g1 111110 ... (110)^gk 111110, gi>=1, k>=0; f=k+1; G,M>=0",
   "validated": {
    "instances": 120,
    "passed": 120
   },
   "source": "manual"
  },
  {
   "name": "OQ overflow 2 4 5 X -> 2 5 2 5 X",
   "from": "0^inf <E 110 11110 111110 X (110)^(G+2^(f+2)) 00 (110)^M 0^inf",
   "to": "0^inf <E 110 111110 110 111110 X (110)^G 00 (110)^(M+2^(f+2)) 0^inf",
   "steps": null,
   "cases": "X as in OP; f=k+1; G,M>=0",
   "validated": {
    "instances": 120,
    "passed": 120
   },
   "source": "manual"
  }
 ],
 "template": {
  "file": "vendor/busycoq/verify/BECv1_6x2_solved.v (+ BECv1.v Section)",
  "module": "tm10 / nonhalt10 (line 88), cert Build_BEC_cert_v1 B A E A ... 1 0 0",
  "machine": "1RB0RA_1LC0LD_1LE1LD_0LB0RE_1RF1LE_---0RA",
  "diff": "E1: 0LA (ours) vs 1LE (tm10); all other 11 transitions equal",
  "applies": "partly",
  "note": "Simulated our TM on tm10's cert: 10/11 hypotheses hold (HR1-3, HE0-3, HL1,3,4,5; 25/25 each); HL2 = binary carry 1<E1 -> <E11 fails 0/25.\nSo the unary half carries over verbatim; LeftBinaryCounter must become a spaced-digit counter (5-blocks + 110 filler)."
 },
 "strategy": "Keep BECv1's unary half (HR1-3, HE0-3 as es-lemmas). Replace LeftBinaryCounter by an Inductive digit list (5-digit + filler gaps, 4-mode at the MSB), BECv7.v style;\nprove OP/OQ by ind over sweeps + carries, then a finite EatDigit case table; close with progress_nonhalt_cond on a shape predicate (any gaps G,M), as BECv7's canonical.",
 "risks": [
  "EatDigit is messy: it makes 3/4/6 blocks and a transient second 00 marker; the case table may be large.",
  "Digit gaps evolve irregularly (3,7 -> 10 -> 17 -> 11 ...): proof must quantify over all gaps (shape invariant), not track values.",
  "mxdys's BECv1/IndBECv1 search failed here; expect new counter lemmas (BECv7-size file, 30-50 KB)."
 ],
 "checkpoint_questions": [
  {
   "text": "Fig 1: does each dark diagonal (the 00 marker) move steadily left, 1 block per round, like the BECv1 unary counter?",
   "figs": [
    0
   ]
  },
  {
   "text": "Fig 2: do the white streaks read as counter digits, born at the left edge and eaten each epoch?",
   "figs": [
    1
   ]
  }
 ],
 "discord_check": null,
 "stats": {
  "simulated_steps": 1000000000,
  "cells": 7783,
  "nonzeros": 4191,
  "growth": "left edge ~t^0.36 (bell), right edge ~t^0.27 (+2 cells per epoch, ~135 epochs by 1e9)"
 },
 "status": "CHECKPOINT_A",
 "status_reason": "Mechanism found, 7 rules validated (540/540 sims); not cryptid-like: 0 halts in 1560 random shapes + 251 perturbed real configs; EatDigit table open."
}