{
 "tm": "1RB0RA_1LC0LF_1LE0LD_1LB---_1RA1LB_0LD0RC",
 "rank": "R2",
 "class_sheet": "Bell eats counter",
 "class_guess": "Bell eats counter: tm108-like 110/010 bell eats a 3-cell-digit binary counter; eat size varies (not BECv1)",
 "confidence": "low",
 "summary": "BECv1 tm108 twin (E1, F0 differ): 110/010 bell eats a binary counter; eat size varies, left language open",
 "mechanism": [
  "Tape = L | bell `1 (110)^e (010)^f 1 0^inf`, e+f = N per epoch; each round e+1, f-1 and L (binary counter, 3-cell digits) gains 1/2.",
  "At f = 0 the bell eats the low 3e'-2 cells of L: f' = N, e' = 1 (odd last round) or 2..47 (even); L then mixes digit kinds.",
  "Exact so far: sweeps R1-R4 and INC2 on pure counters; open: carry/eat on mixed words. tm108 cert: 9/12 hold (HL2, HR2, HE2 fail)."
 ],
 "diagrams": [
  {
   "mode": "blaze",
   "png": "img/blaze_1e8.png",
   "html": null,
   "steps": 100000000,
   "caption": "Blaze 1e8: bell-shaped growth, mostly to the left",
   "title": "Blaze, 100 million steps",
   "why": "Overall shape: the left edge (counter side) versus the right edge (bell side) over time.",
   "primary": true,
   "start": "blank",
   "rows": "Blaze time blocks of 131,072 steps (each averaged), 763 blocks to step 1e8, stretched to 500 px",
   "window": "cells -1929..263, 1 px = 2.74 cells (Blaze image: 1,097 columns of 2 cells = the visited span at 1e8, stretched to 800 px)",
   "legend": "white = 0, orange = 1, paler orange = 1 only part of the time in that pixel's block; no head",
   "style": "bbchallenge-blaze"
  },
  {
   "mode": "custom:rounds",
   "png": "img/rounds_aligned.png",
   "html": null,
   "steps": 469936,
   "caption": "One row per clean round, aligned at the bell's left edge",
   "title": "Rounds aligned at the bell boundary",
   "why": "Left of red: the word L, cell by cell. Right of red: one column per bell block. Compare rows.",
   "primary": true,
   "start": "blank",
   "rows": "one row per clean round start (E0 -> A turn landing on the boundary cell b of `1 (110)^e (010)^f 1`), i.e. every second round of an epoch: 198 rows, steps 250,017..469,936",
   "window": "cells -80..-1, 1 px = 0.33 cells (3x scale), counted from the boundary cell b; then b (red); then 112 columns in block units (1 column = one 3-cell block)",
   "legend": "white = 1, black = 0 (left word L), grey = never visited, red = boundary cell b (a 1), orange = one 110 block, blue = one 010 block, black right of blue = beyond the tape",
   "style": "custom"
  },
  {
   "mode": "every:40",
   "png": "img/reset_every40.png",
   "html": null,
   "steps": 24000,
   "caption": "Close-up: rounds, then the reset at step 300,164",
   "title": "Close-up around one reset",
   "why": "Rows every 40 steps. Follow the white bell region's right border down to where the picture changes.",
   "primary": true,
   "start": "mid-run: step 290000 (configuration from the blank-tape run)",
   "rows": "every 40 steps: 601 rows, steps 290,000..314,000",
   "window": "cells -266..51, 1 px = 0.5 cells (2x scale); absolute positions of the blank-tape run (head at -116 at step 290,000)",
   "legend": "white = 1, black = 0 or unvisited, red/green = head (last move right/left)",
   "style": "bbchallenge-default"
  },
  {
   "mode": "default",
   "png": "img/default_site.png",
   "html": null,
   "steps": 10000,
   "caption": "bbchallenge default view, first 10,000 steps",
   "title": "bbchallenge default view, 10,000 steps",
   "why": "The site's own first view, for comparison with the bbchallenge page.",
   "primary": false,
   "start": "blank",
   "rows": "every step 0..9,999, squeezed into 500 px by the site's bilinear filter (about 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 L A> 1 (110)^e (010)^f 1 0^inf",
  "vars": [
   "N (bell blocks e+f before the reset)",
   "e' (blocks gained = new e)",
   "last round of the epoch (odd = deep turn)",
   "|L| (cells left of the bell)"
  ],
  "rows": [
   {
    "step": 29,
    "values": [
     1,
     2,
     "even",
     1
    ]
   },
   {
    "step": 94,
    "values": [
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    ]
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   {
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    "values": [
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    "values": [
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    "values": [
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    "values": [
     12,
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    ]
   },
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    "values": [
     13,
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     22
    ]
   },
   {
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    "values": [
     15,
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    ]
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    "values": [
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    ]
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   },
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    "values": [
     35,
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    ]
   },
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     473,
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    "values": [
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    "values": [
     1389,
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   },
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    "values": [
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    "values": [
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    "values": [
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    "values": [
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    "values": [
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    "values": [
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    "values": [
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   },
   {
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    "values": [
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    "values": [
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    ]
   },
   {
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    "values": [
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    ]
   },
   {
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    "values": [
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   },
   {
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    "values": [
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   },
   {
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    "values": [
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    ]
   },
   {
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    "values": [
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    ]
   },
   {
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    "values": [
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   },
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   },
   {
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    "values": [
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   {
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    "values": [
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   }
  ]
 },
 "rules": [
  {
   "name": "R1 A-sweep over 110 (as BECv1 HR1 shape)",
   "from": "l A> (110)^n 1 r",
   "to": "l (001)^n A> 1 r",
   "steps": "7n",
   "cases": "any l, r",
   "validated": {
    "instances": 200,
    "passed": 200
   },
   "source": "manual"
  },
  {
   "name": "R4 A-sweep over 101 (= tm108 HR1)",
   "from": "l A> (101)^n r",
   "to": "l (010)^n A> r",
   "steps": "7n",
   "cases": "any l, r",
   "validated": {
    "instances": 200,
    "passed": 200
   },
   "source": "manual"
  },
  {
   "name": "R2 B-sweep back over 010 (= tm108 HR3)",
   "from": "l (010)^n <B r",
   "to": "l <B (101)^n r",
   "steps": "3n",
   "cases": "any l, r",
   "validated": {
    "instances": 200,
    "passed": 200
   },
   "source": "manual"
  },
  {
   "name": "R3 B-sweep back over 001 (uses F0, ours only)",
   "from": "l (001)^n <B r",
   "to": "l <B (100)^n r",
   "steps": "3n",
   "cases": "any l, r",
   "validated": {
    "instances": 200,
    "passed": 200
   },
   "source": "manual"
  },
  {
   "name": "INC2 pure counter, suffix 001 (two rounds)",
   "from": "0^inf 1 W(v) 001 A> 1 (110)^e (010)^(f+2) 1 0^inf",
   "to": "0^inf 1 W(v+1) 001 A> 1 (110)^(e+2) (010)^f 1 0^inf",
   "steps": null,
   "cases": "W(v) = v in k blocks 000/001, MSB first, 0<=k<=7; v = 2^k-1 overflows to 1 (000)^(k+1) 001; e>=1, f>=0",
   "validated": {
    "instances": 200,
    "passed": 200
   },
   "source": "manual"
  },
  {
   "name": "INC2 pure counter, suffix 000 100 101",
   "from": "0^inf 1 W(v) 000100101 A> 1 (110)^e (010)^(f+2) 1 0^inf",
   "to": "0^inf 1 W(v+1) 000100101 A> 1 (110)^(e+2) (010)^f 1 0^inf",
   "steps": null,
   "cases": "as above (overflow 1 (000)^(k+1) 000100101)",
   "validated": {
    "instances": 200,
    "passed": 200
   },
   "source": "manual"
  },
  {
   "name": "ROUND (blank run): bell e+1, f-1",
   "from": "0^inf L A> 1 (110)^e (010)^(f+1) 1 0^inf",
   "to": "0^inf L' A> 1 (110)^(e+1) (010)^f 1 0^inf (next round)",
   "steps": null,
   "cases": "every round from blank to step 100,000,000,000; L' = L with the counter's half-step (carry depth = turn offset)",
   "validated": {
    "instances": 3935408,
    "passed": 3935408
   },
   "source": "manual"
  },
  {
   "name": "RESET-odd: epoch ends on a deep-turn round",
   "from": "0^inf L A> 1 (110)^N 1 0^inf (f = 0, last round odd)",
   "to": "0^inf L* A> 1 (110)^1 (010)^N 1 0^inf; boundary 1 cell left",
   "steps": null,
   "cases": "all 480 odd resets from blank to step 100,000,000,000",
   "validated": {
    "instances": 480,
    "passed": 480
   },
   "source": "manual"
  },
  {
   "name": "RESET-even: bell eats 3e'-2 cells of L",
   "from": "0^inf L A> 1 (110)^N 1 0^inf (f = 0, last round clean)",
   "to": "0^inf L* A> 1 (110)^e' (010)^N 1 0^inf; boundary 3e'-2 cells left",
   "steps": null,
   "cases": "e' in 2..47, not yet predicted from L; all 602 even resets from blank to step 100,000,000,000 (boundary shift 2-3e' checked on all 1081/1081 resets)",
   "validated": {
    "instances": 602,
    "passed": 602
   },
   "source": "manual"
  }
 ],
 "template": {
  "file": "verify/BECv1_6x2_solved.v:872",
  "module": "BECv1 (Section in verify/BECv1.v); tm108 / nonhalt108, cert Build_BEC_cert_v1 B A E A ... 1 0 0",
  "machine": "1RB0RA_1LC0LF_1LE0LD_1LB---_1RA1LE_0LB0RC",
  "diff": "E1: 1LB (ours) vs 1LE (tm108); F0: 0LD (ours) vs 0LB (tm108)",
  "applies": "partly",
  "note": "scripts/template_check.py runs tm108's 12 cert hypotheses on both machines: tm108 12/12 (sanity), ours 9/12 (25/25 each). Fail 0/25: HL2 (binary carry 1 <E -> <E 1, needs E1=1LE), HR2 (00 marker step, uses F0) and HE2 (marker birth). So the sweeps and the eat pieces HE0/HE1/HE3 carry over; the counter (3-cell digits), the marker (a 110|010 border, not 00) and the eat size (3e'-2 cells, e' = 1..47, not a = 1) do not."
 },
 "strategy": "Deepen before any proof: (1) derive the carry automaton on mixed digit words and the eat automaton (predict e' from L);\n(2) find a language for L closed under INC2/ROUND and RESET, with a size bound N >= c|L|, and closure-fuzz it;\n(3) then a BECv1-style file with an Inductive digit list, or a CTL/RWL certificate if the language is regular.",
 "risks": [
  "The eat size e' depends on the counter's digits at the reset (arithmetic feedback); if the reachable L words are not regular, a hand proof is hard.",
  "Real left words with a too-small bell (N < |L|) do halt (2/300 replay starts), so the invariant needs a quantitative N vs |L| bound.",
  "Sibling #29 (F0 = 0LB) looks similar but its trajectory differs (1e8: span -1917..265 vs ours -1929..263); a shared proof is not guaranteed."
 ],
 "checkpoint_questions": [
  {
   "text": "rounds_aligned: does each triangle read as one epoch (orange e grows, blue f shrinks), and does the left word get shorter at each new triangle, i.e. the bell eats the counter?",
   "figs": [
    1
   ],
   "kind": "check"
  },
  {
   "text": "Close-up: around step 300,164 the bell's white region gives way and a new pattern starts further left. Does that match 'the bell eats the counter' to you?",
   "figs": [
    2
   ],
   "kind": "check"
  },
  {
   "text": "Confidence is low: the left-word language and the eat size e' are open. Should we spend a deepen stage on them (carry/eat automaton + closure fuzz), or park this machine behind the others?",
   "kind": "decision"
  }
 ],
 "discord_check": null,
 "stats": {
  "simulated_steps": 100000000000,
  "cells": 23535,
  "nonzeros": null,
  "growth": "span ~ t^(1/3): 453 cells at 1e6, 2,193 at 1e8, 10,590 at 1e10 (4,082 nonzeros), 23,535 at 1e11; bell N = 7822 after 1082 epochs"
 },
 "status": "CHECKPOINT_A",
 "status_reason": "Low confidence. ROUND/RESET structure holds on every round from blank to 100,000,000,000 steps (0 exceptions) and INC2 is exact on pure counters, but the left-word language and the eat size are not derived. Not a cryptid candidate yet: no halt depends on the eat sizes in the tests."
}