{
 "tm": "1RB1RF_1RC0RA_1LD1RE_1LE0LE_0RA0LD_0RB---",
 "rank": 18,
 "class_sheet": "Bell eats counter",
 "class_guess": "Bell eats counter (BMO 7 layout, but a base-3 counter)",
 "confidence": "med",
 "summary": "BMO7-like bell + walking marker; counter is base 3, not binary; the eat step rebuilds it irregularly",
 "mechanism": [
  "Tape = a long 10-sea (the bell) + a short counter at its right end; a marker walks right through the bell.",
  "Each bounce adds 1 to the base-3 counter: a digit = a wall at offset 1/2/3 in its frame; overflow appends a digit.",
  "When the marker hits the counter, the bell eats its low end (grows by g >= 2); g and the new counter depend only on it."
 ],
 "diagrams": [
  {
   "mode": "state:C",
   "png": "img/state_C_one_phase_x2.png",
   "html": null,
   "steps": 158600,
   "caption": "One phase: marker crosses the bell, counter ticks each pass",
   "title": "One phase: entries into state C",
   "why": "One row per entry into C, every 2nd cell. Look at the black/white diagonal and the right-edge strip.",
   "primary": true,
   "start": "mid-run: step 2097352 (a phase start: left record, head at cell -48)",
   "rows": "entries into state C (563 rows, steps 2,097,352..2,255,952; the next phase starts at 2,255,931)",
   "window": "cells -48..544, 1 px = 2 cells: only every 2nd cell is drawn (-48, -46, ..., 544)",
   "legend": "white = 1, black = 0, red = head (it enters C moving right). Because only every 2nd cell is drawn, the bell (10)^M looks solid white where its 1s fall on drawn cells and solid black where the marker has shifted it by one cell: the white/black diagonal is the marker",
   "style": "bbchallenge-default"
  },
  {
   "mode": "state:C",
   "png": "img/counter_from_zero.png",
   "html": null,
   "steps": 21000,
   "caption": "Counter from zero: base-3 ruler, 4-cell digits, staircase = new digit",
   "title": "Counter from zero, from a built tape",
   "why": "From a built tape, counter at 0. Look at the staircase on the right and the pattern within it.",
   "primary": true,
   "start": "constructed: 0^inf A> (10)^100 00101001 0^inf  (bell of 100 blocks, then the counter X(0) = 00101001 = value 0)",
   "rows": "entries into state C (178 rows, steps 0..21,000)",
   "window": "cells 197..224, 1 px = 0.2 cells (each cell is 5x5 px; bell = cells 0..199, counter from cell 200)",
   "legend": "white = 1, black = 0 or never visited, red = head (it enters C moving right)",
   "style": "bbchallenge-default"
  },
  {
   "mode": "records:L",
   "png": "img/phase_starts_counter_1e9.png",
   "html": null,
   "steps": 1000000000,
   "caption": "Counter at every phase start to 1e9: irregular after eats",
   "title": "Counter at every phase start, to 1 billion",
   "why": "Right edge pinned; one row per phase start, to 1e9. Compare the counter strip from row to row.",
   "primary": true,
   "start": "blank",
   "rows": "left records: the start, then one row per new leftmost cell = one row per phase start (313 rows, steps 0..1,000,000,000; rows are not equal time)",
   "window": "cells -109..0, 1 px = 0.33 cells (each cell is 3x3 px), counted from the right edge (each row is shifted so its rightmost visited cell is at 0)",
   "legend": "white = 1, black = 0 or never visited, green = head (at the left edge, just moved left)",
   "style": "bbchallenge-default"
  },
  {
   "mode": "records",
   "png": "img/records_1e8.png",
   "html": "img/viewer.html",
   "steps": 100000000,
   "caption": "Records to 1e8: bell grows, counter rides the right edge",
   "title": "Edge records to 100 million steps",
   "why": "The bell grows; the counter rides its right edge.",
   "primary": false,
   "start": "blank",
   "rows": "left/right records: the start, then one row each time the head reaches a new leftmost or rightmost cell (2,058 rows, steps 0..100,000,000; rows are not equal time)",
   "window": "cells -154..1903, 1 px = 1 cell",
   "legend": "white = 1, black = 0 or never visited, red/green = head just moved right/left",
   "style": "bbchallenge-default"
  },
  {
   "mode": "blaze",
   "png": "img/blaze_1e8.png",
   "html": "img/viewer.html",
   "steps": 100000000,
   "caption": "bbchallenge Blaze view, 1e8 steps",
   "title": "Blaze view, 100 million steps",
   "why": "The bell (half 1s) averages to pale orange; growth ~ t^0.34.",
   "primary": false,
   "start": "blank",
   "rows": "Blaze time blocks of 131,072 steps (averaged), 763 blocks stretched to 500 px, so 1 px row ~ 200,000 steps (steps 0..100,000,000)",
   "window": "cells -154..1903, 1 px = 2.57 cells",
   "legend": "white = 0, orange = 1, paler orange = mixed (each pixel averages its cells over its time block); no head drawn; the bell (10)^M is half 1s, so it averages to pale orange",
   "style": "bbchallenge-blaze"
  }
 ],
 "history": {
  "skeleton": "0^inf A> (10)^M W 0^inf",
  "vars": [
   "M = bell blocks at phase start",
   "|W| = counter cells",
   "W as wall gaps (o = 00 hole, # = 11 bump)"
  ],
  "rows": [
   {
    "step": 41,
    "values": [
     2,
     8,
     "1o 1o 5o"
    ]
   },
   {
    "step": 64,
    "values": [
     3,
     7,
     "1o 1o 1o 3o"
    ]
   },
   {
    "step": 163,
    "values": [
     7,
     7,
     "1o 1o 1o 3o"
    ]
   },
   {
    "step": 362,
    "values": [
     8,
     13,
     "1o 1o 1o 3o 2# 4o"
    ]
   },
   {
    "step": 669,
    "values": [
     9,
     19,
     "1o 1o 11o 1o 1o 3o"
    ]
   },
   {
    "step": 1104,
    "values": [
     10,
     25,
     "1o 1o 11o 1o 1o 3o 2# 4o"
    ]
   },
   {
    "step": 1675,
    "values": [
     17,
     19,
     "1o 1o 11o 1o 1o 3o"
    ]
   },
   {
    "step": 2574,
    "values": [
     18,
     25,
     "1o 1o 1o 9o 1o 5o 2# 4o"
    ]
   },
   {
    "step": 10067,
    "values": [
     33,
     50,
     "1o 1o 41o 1o 5o"
    ]
   },
   {
    "step": 128733,
    "values": [
     111,
     19,
     "1o 1o 1o 3o 3o 1o 2# 2o 3o 1o"
    ]
   },
   {
    "step": 529171,
    "values": [
     172,
     29,
     "1o 1o 13o 1o 1o 5o 3o 3o"
    ]
   },
   {
    "step": 1428457,
    "values": [
     241,
     31,
     "1o 1o 11o 1o 1o 1o 3o 5o 3o 3o"
    ]
   },
   {
    "step": 2974083,
    "values": [
     326,
     17,
     "1o 1o 5o 2# 1# 2o 3o 1o"
    ]
   },
   {
    "step": 5636701,
    "values": [
     388,
     33,
     "1o 1o 11o 1o 5o 1o 1o 5o 3o 3o"
    ]
   },
   {
    "step": 9167327,
    "values": [
     444,
     41,
     "1o 1o 25o 1o 1o 5o 3o 3o"
    ]
   },
   {
    "step": 13840601,
    "values": [
     520,
     17,
     "1o 1o 1o 3o 3o 3o 1o 3o"
    ]
   },
   {
    "step": 19709255,
    "values": [
     559,
     67,
     "1o 1o 27o 1o 19o 1o 1o 5o 3o 3o 2# 2o"
    ]
   },
   {
    "step": 27284505,
    "values": [
     666,
     16,
     "1o 1o 3o 1o 3o 1o 5o"
    ]
   },
   {
    "step": 36981311,
    "values": [
     718,
     61,
     "1o 1o 33o 1o 5o 1o 1o 3o 2# 4o 2# 2o 2# 2o"
    ]
   },
   {
    "step": 48739545,
    "values": [
     799,
     50,
     "1o 1o 27o 1o 13o 1o 5o"
    ]
   },
   {
    "step": 62765319,
    "values": [
     871,
     23,
     "1o 1o 1o 3o 1o 5o 2# 4o 3o 1o"
    ]
   },
   {
    "step": 79028529,
    "values": [
     928,
     49,
     "1o 1o 25o 1o 9o 1o 1o 1o 2# 2o 3o 1o"
    ]
   },
   {
    "step": 97840939,
    "values": [
     1002,
     40,
     "1o 1o 17o 1o 5o 1o 7o 1o 5o"
    ]
   },
   {
    "step": 119322197,
    "values": [
     1083,
     31,
     "1o 1o 13o 1o 1o 1o 2# 2o 1o 3o 1o 3o"
    ]
   },
   {
    "step": 144732311,
    "values": [
     1168,
     45,
     "1o 1o 21o 1o 5o 1o 1o 1o 2# 2o 1o 3o 1o 3o"
    ]
   },
   {
    "step": 174085265,
    "values": [
     1252,
     52,
     "1o 1o 9o 1o 11o 1o 5o 1o 3o 1o 11o 1o 5o"
    ]
   },
   {
    "step": 207484435,
    "values": [
     1326,
     49,
     "1o 1o 15o 1o 5o 1o 5o 1o 1o 3o 2# 4o 2# 2o 1o 3o"
    ]
   },
   {
    "step": 244435833,
    "values": [
     1401,
     31,
     "1o 1o 1o 3o 1o 3o 2# 2o 3o 1o 1o 3o 2# 2o 1o 3o"
    ]
   },
   {
    "step": 285517799,
    "values": [
     1469,
     51,
     "1o 1o 1o 3o 3o 19o 2# 6o 3o 3o 2# 2o 1o 3o"
    ]
   },
   {
    "step": 331301633,
    "values": [
     1556,
     37,
     "1o 1o 1o 3o 3o 3o 3o 3o 3o 3o 3o 1o 2# 2o 1o 3o"
    ]
   },
   {
    "step": 382417731,
    "values": [
     1642,
     41,
     "1o 1o 5o 1o 5o 1o 9o 1o 1o 3o 2# 2o 3o 1o 1o 3o"
    ]
   },
   {
    "step": 438884121,
    "values": [
     1705,
     71,
     "1o 1o 41o 1o 13o 1o 1o 5o 3o 3o"
    ]
   },
   {
    "step": 501203395,
    "values": [
     1791,
     55,
     "1o 1o 5o 1o 19o 1o 5o 1o 1o 1o 2# 2o 2# 6o 3o 3o"
    ]
   },
   {
    "step": 568808345,
    "values": [
     1887,
     31,
     "1o 1o 1o 1o 2# 10o 1o 5o 1o 3o 2# 2o"
    ]
   },
   {
    "step": 643095903,
    "values": [
     1957,
     51,
     "1o 1o 17o 1o 1o 3o 1o 5o 1o 5o 2# 4o 1o 3o 2# 2o"
    ]
   },
   {
    "step": 723172281,
    "values": [
     2025,
     75,
     "1o 1o 27o 1o 27o 1o 1o 3o 2# 2o 2# 2o 2# 2o"
    ]
   },
   {
    "step": 809229495,
    "values": [
     2106,
     69,
     "1o 1o 1o 11o 3o 19o 3o 11o 3o 3o 1o 3o 2# 2o 2# 2o"
    ]
   },
   {
    "step": 902342053,
    "values": [
     2188,
     57,
     "1o 1o 17o 1o 1o 3o 3o 5o 1o 5o 1o 7o 3o 3o 2# 2o"
    ]
   },
   {
    "step": 991609360,
    "values": [
     2263,
     47,
     "1o 1o 5o 1o 1o 3o 1o 15o 3o 1o 1o 5o 3o 1o 2# 2o"
    ]
   }
  ]
 },
 "rules": [
  {
   "name": "sweep-in",
   "from": "A> (10)^n",
   "to": "10 (01)^(n-1) B>",
   "steps": "2n",
   "cases": "n>=1",
   "validated": {
    "instances": 60,
    "passed": 60
   },
   "source": "manual"
  },
  {
   "name": "sweep-R",
   "from": "B> (10)^n",
   "to": "(01)^n B>",
   "steps": "2n",
   "cases": "n>=0",
   "validated": {
    "instances": 60,
    "passed": 60
   },
   "source": "manual"
  },
  {
   "name": "sweep-L",
   "from": "(10)^n <D",
   "to": "<D (01)^n",
   "steps": "2n",
   "cases": "n>=0",
   "validated": {
    "instances": 60,
    "passed": 60
   },
   "source": "manual"
  },
  {
   "name": "shift-A",
   "from": "A> (01)^n",
   "to": "(10)^n A>",
   "steps": "2n",
   "cases": "n>=0 (used inside the counter)",
   "validated": {
    "instances": 60,
    "passed": 60
   },
   "source": "manual"
  },
  {
   "name": "Inc (one bounce, X form)",
   "from": "0^inf (10)^a A> (10)^(b+1) X(v) 0^inf",
   "to": "0^inf (10)^(a+1) A> (10)^b X(v+1) 0^inf",
   "steps": null,
   "cases": "b>=1. X(v) = D(v mod 3) 1 T(v div 3) 001; D = 0010|0110|0100; T = bijective base-3 digits LSB first, 1/2/3 = 0001/0011/0010",
   "validated": {
    "instances": 450,
    "passed": 450
   },
   "source": "manual"
  },
  {
   "name": "Inc (one bounce, any frames)",
   "from": "0^inf (10)^a A> (10)^(b+1) F 0^inf",
   "to": "0^inf (10)^(a+1) A> (10)^b inc(F) 0^inf",
   "steps": null,
   "cases": "b>=1. F = frames (d_i in 1..3, even width w_i>=4), wall gaps 1,d_0,w_0-d_0,...; inc = base 3 on d (3->1 carries), overflow appends (1,4); widths unchanged",
   "validated": {
    "instances": 400,
    "passed": 400
   },
   "source": "manual"
  },
  {
   "name": "Phase (M-1 bounces)",
   "from": "0^inf A> (10)^(M) X(v) 0^inf",
   "to": "0^inf (10)^(M-1) A> 10 X(v+M-1) 0^inf",
   "steps": null,
   "cases": "M>=2; Inc applied M-1 times",
   "validated": {
    "instances": 40,
    "passed": 40
   },
   "source": "manual"
  },
  {
   "name": "Eat (bell meets counter)",
   "from": "0^inf (10)^N A> 10 F 0^inf",
   "to": "0^inf <E (10)^(N+g(F)) F' 0^inf",
   "steps": null,
   "cases": "g and F' depend on F only (N vs N+k, 60/60). Examples: [(1,6)]->+7,[(1,6)]; [(3,6)]->+2,[(1,10),(1,6)]; [(1,10)]->+7,[(1,4),(1,6)]. Closed form: OPEN",
   "validated": {
    "instances": 60,
    "passed": 60
   },
   "source": "manual"
  }
 ],
 "template": {
  "file": "vendor/busycoq/verify/BMO7.v",
  "module": null,
  "machine": "1RB1RF_1RC0RA_1LD1RC_1LE0LE_0RA0LD_0RB---",
  "diff": "C1: 1RC (BMO 7) -> 1RE (ours); all other 11 transitions equal",
  "applies": "partly",
  "note": "Left bell bounce (BMO7 LIncs) holds here 36/36; binary RInc fails on every odd counter (16/40), BigStep shape 3/8 with non-binary right side. Same layout, new counter."
 },
 "strategy": "Keep BMO7's layout: Longitudinal sideRLs for bell bounces (reuse LIncs), a new ternary Inc lemma over frames (ind on digits, DivModCases mod 3). Next stage: close the Eat step by enumerating reachable counter shapes; if finite-parametric, prove it with a WF invariant + progress_nonhalt_cond (BMO7 pattern), else a verified checker over frame lists.",
 "risks": [
  "Eat step not closed: phase-start counters irregular (dN 1 in ~half of phases, else 4..38).",
  "If eat outcomes depend on counter digits pseudo-randomly, this becomes a cryptid-like case.",
  "Larger than BMO7 (441 lines): two-level counter + eat lemma family; likely >15 KB."
 ],
 "checkpoint_questions": [
  {
   "text": "Fig 1: is the black/white diagonal the marker crossing the bell, with the counter in the strip at the right edge?",
   "figs": [
    0
   ]
  },
  {
   "text": "Fig 2: do you read the ruler pattern as a base-3 counter, each new staircase step to the right a new digit?",
   "figs": [
    1
   ]
  },
  {
   "text": "Fig 3: does the counter on the right look irregular from one phase start to the next?",
   "figs": [
    2
   ]
  }
 ],
 "discord_check": null,
 "stats": {
  "simulated_steps": 1000000000,
  "cells": 4585,
  "nonzeros": 2285,
  "growth": "bell M ~ t^0.342; 312 phases by 1e9; span -312..4272"
 },
 "status": "CHECKPOINT_A",
 "status_reason": "Bell/marker and base-3 counter rules validated; eat step is N-independent but its closed form is open. No Collatz-style rule found (values grow polynomially), so not flagged cryptid yet."
}