{
 "tm": "1RB0RF_1RC0RE_1LD1RA_1LA0LD_1RA---_0RA0LB",
 "rank": "R3",
 "class_sheet": "Bell eats counter",
 "class_guess": "Bell eats counter: FractalBEC-like 101/00 bouncer plus a 3-cell binary counter with transparent J blocks; the bell shifts the counter one cell",
 "confidence": "med",
 "summary": "Gap bounces in (101)^a 00 (101)^b, counter counts; at b=0 the low digits pick eat E or bell B0/B1",
 "mechanism": [
  "Hub `K(a,b,G) = 0^inf (101)^a 00 (101)^b A> 0 enc(G) 0^inf` (enc 0=000, 1=010, J=100). R1: each bounce moves the 00 gap one block right and adds 1 to the counter G (J skipped).",
  "At b=0 the low digits decide: `0 1^r J^k` = partial eat E (J's become k-1 fresh blocks); `0 1^r 0` / `1...` = bell B0/B1: sweep to the left end, counter shifted one cell (σ).",
  "σ leaves x/z/w marks that the carry cleans within 2^q <= 2^floor(log2 N) increments (19,773/19,773 macro epochs); lockstep from blank holds to 1e9 (all hubs) and 3e10 (b=0 hubs)."
 ],
 "diagrams": [
  {
   "mode": "default",
   "png": "img/default_site_10k.png",
   "html": null,
   "steps": 10000,
   "caption": "bbchallenge Default view, first 10,000 steps",
   "title": "bbchallenge view, first 10,000 steps",
   "primary": true,
   "why": "Time runs down. Follow the diagonal bands and note where each one starts and ends.",
   "start": "blank",
   "rows": "every step, 10,000 steps squeezed into 500 rows (the site's bilinear view)",
   "window": "cells -200..199, 1 px = 1 cell (site canvas, cell 0 at x = 200)",
   "legend": "white = 1, black = 0 or unvisited, red/green = head moving R/L",
   "style": "bbchallenge-default"
  },
  {
   "mode": "custom",
   "png": "img/hubs_counter_epochs.png",
   "html": null,
   "steps": 998000,
   "caption": "One row per bounce: gap position and counter word, four epochs",
   "title": "Hub rows: gap and counter, four epochs",
   "primary": true,
   "why": "Left: where the cyan gap sits. Right: the counter word. Compare the rows just before and after each cyan bar.",
   "start": "mid-run: step 2,850,432",
   "rows": "one row per K hub (one bounce), 943 hubs from step 2,850,432 to 3,848,095",
   "window": "cells 1..90, 1 px = 0.33 cells (counted right of the head, 9 px per 3-cell group); left 120 px = gap position a/(a+b)",
   "legend": "white = digit 1 (010), dark grey = digit 0 (000), orange = J block (100), red = x/z/w block (001/011/101), black = past the end of G; left panel: grey = 101 blocks, cyan = the 00 gap; cyan bar between panels = a b=0 hub",
   "style": "custom"
  },
  {
   "mode": "explore",
   "png": "img/explore_constructed_eat.png",
   "html": null,
   "steps": 320,
   "caption": "Constructed K(5,2,0JJ1): two bounces, eat E, bounce, bell B1",
   "title": "Close-up: bounces, partial eat, bell",
   "primary": true,
   "why": "Count the bounces; watch where the right end jumps left and where the head runs to the far left.",
   "start": "constructed: 0^inf (101)^5 00 (101)^2 A> 0 000 100 100 010 0^inf",
   "rows": "every step, 320 steps",
   "window": "cells -25..17, 1 px = 0.25 cells (4 px per cell)",
   "legend": "white = 1, slate = 0, head cell by state: red = A, orange = B, blue = C, green = D, magenta = E, cyan = F",
   "style": "bbchallenge-explore"
  },
  {
   "mode": "records",
   "png": "img/records_1e8.png",
   "html": null,
   "steps": 100000000,
   "caption": "Left/right records to 1e8: epochs as sweeps",
   "title": "Records to 100 million steps",
   "primary": false,
   "why": "Each run of left records is one bell sweep (epoch start); right records are counter growth.",
   "start": "blank",
   "rows": "left/right records (one row per new leftmost or rightmost cell), 2,195 rows to step 1e8",
   "window": "cells -258..1936, 1 px = 1 cell",
   "legend": "white = 1, black = 0, red/green = head moving R/L",
   "style": "bbchallenge-default"
  },
  {
   "mode": "blaze",
   "png": "img/blaze_1e7.png",
   "html": null,
   "steps": 10000000,
   "caption": "Blaze 1e7: overall growth",
   "title": "Blaze, 10 million steps",
   "primary": false,
   "why": "Overall shape: slow growth to the right, where the counter and the block region grow.",
   "start": "blank",
   "rows": "Blaze time blocks of 16,384 steps, 611 blocks to step 1e7, stretched to 500 px",
   "window": "cells -130..875, 1 px = 1.26 cells (1,006 columns 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"
  }
 ],
 "history": {
  "skeleton": "0^inf (101)^a 00 (101)^b A> 0 enc(G) 0^inf",
  "vars": [
   "a",
   "b",
   "G"
  ],
  "rows": [
   {
    "step": 2858672,
    "values": [
     188,
     0,
     "0JJJJJJJJJJJJJJJJJJJJJ011J1101"
    ]
   },
   {
    "step": 2861162,
    "values": [
     210,
     0,
     "010J0111"
    ]
   },
   {
    "step": 2861177,
    "values": [
     0,
     213,
     "x00JJJ"
    ]
   },
   {
    "step": 2867625,
    "values": [
     3,
     210,
     "1JJJJJ1"
    ]
   },
   {
    "step": 2876115,
    "values": [
     7,
     206,
     "1JJJJJ11"
    ]
   },
   {
    "step": 2892621,
    "values": [
     15,
     198,
     "1JJJJJ111"
    ]
   },
   {
    "step": 2923719,
    "values": [
     31,
     182,
     "1JJJJJ1111"
    ]
   },
   {
    "step": 2978241,
    "values": [
     63,
     150,
     "1JJJJJ11111"
    ]
   },
   {
    "step": 3056571,
    "values": [
     127,
     86,
     "1JJJJJ111111"
    ]
   },
   {
    "step": 3097347,
    "values": [
     213,
     0,
     "1JJJJJ0101011"
    ]
   },
   {
    "step": 3097412,
    "values": [
     0,
     214,
     "xxxxx0JJ0J0JJ"
    ]
   },
   {
    "step": 3104064,
    "values": [
     3,
     211,
     "1JJJJJ1000JJJ"
    ]
   },
   {
    "step": 3112594,
    "values": [
     7,
     207,
     "1JJJJJ1100JJJ"
    ]
   },
   {
    "step": 3129180,
    "values": [
     15,
     199,
     "1JJJJJ1110JJJ"
    ]
   },
   {
    "step": 3160438,
    "values": [
     31,
     183,
     "1JJJJJ1111JJJ"
    ]
   },
   {
    "step": 3215310,
    "values": [
     63,
     151,
     "1JJJJJ1111JJJ1"
    ]
   },
   {
    "step": 3294340,
    "values": [
     127,
     87,
     "1JJJJJ1111JJJ11"
    ]
   },
   {
    "step": 3336140,
    "values": [
     214,
     0,
     "0JJJJJ1101JJJ011"
    ]
   },
   {
    "step": 3336316,
    "values": [
     220,
     0,
     "1111JJJ011"
    ]
   },
   {
    "step": 3336379,
    "values": [
     0,
     221,
     "000xxx0JJJ"
    ]
   },
   {
    "step": 3343009,
    "values": [
     3,
     218,
     "110xxx0JJJ"
    ]
   },
   {
    "step": 3351719,
    "values": [
     7,
     214,
     "111xxx0JJJ"
    ]
   },
   {
    "step": 3368665,
    "values": [
     15,
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     "111zxx0JJJ"
    ]
   },
   {
    "step": 3400751,
    "values": [
     31,
     190,
     "1111JJJ1J0J"
    ]
   },
   {
    "step": 3457103,
    "values": [
     63,
     158,
     "1111JJJ1J1J"
    ]
   },
   {
    "step": 3539103,
    "values": [
     127,
     94,
     "1111JJJ1J1J1"
    ]
   },
   {
    "step": 3585305,
    "values": [
     221,
     0,
     "1011JJJ1J0J11"
    ]
   },
   {
    "step": 3585320,
    "values": [
     0,
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     "0JJwxx0w0x0JJ"
    ]
   },
   {
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    "values": [
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     "1JJJJJJz0x000J"
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   },
   {
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    "values": [
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     "1JJJJJJ1JJJJ1J"
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   },
   {
    "step": 3618436,
    "values": [
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     "1JJJJJJ1JJJJ1J1"
    ]
   },
   {
    "step": 3651234,
    "values": [
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     "1JJJJJJ1JJJJ1J11"
    ]
   },
   {
    "step": 3709156,
    "values": [
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     "1JJJJJJ1JJJJ1J111"
    ]
   },
   {
    "step": 3794286,
    "values": [
     127,
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     "1JJJJJJ1JJJJ1J1111"
    ]
   },
   {
    "step": 3845182,
    "values": [
     222,
     0,
     "0JJJJJJ1JJJJ1J11011"
    ]
   },
   {
    "step": 3845583,
    "values": [
     229,
     0,
     "0JJJJ0J10111"
    ]
   },
   {
    "step": 3845712,
    "values": [
     234,
     0,
     "1J01111"
    ]
   },
   {
    "step": 3845737,
    "values": [
     0,
     235,
     "x0JJJJJ"
    ]
   },
   {
    "step": 3852835,
    "values": [
     3,
     232,
     "1J10000J"
    ]
   },
   {
    "step": 3862125,
    "values": [
     7,
     228,
     "1J11000J"
    ]
   }
  ]
 },
 "rules": [
  {
   "name": "init",
   "from": "0^inf A> 0^inf",
   "to": "K(0, 2, ε) = 0^inf (101)^2 A> 0 0^inf",
   "steps": "12",
   "cases": null,
   "validated": {
    "instances": 1,
    "passed": 1
   },
   "source": "manual"
  },
  {
   "name": "R1 bounce",
   "from": "K(a, b+1, G)",
   "to": "K(a+1, b, inc G)",
   "steps": null,
   "cases": "a >= 0, b >= 0, G in {0,1}{0,1,J}*; inc = +1 on the digit groups (LSB first), J skipped, a new top digit appended after the last group; a = 0 is the bell sweep to the left end. Orbit: 189,846/189,846 to 1e9",
   "validated": {
    "instances": 1800,
    "passed": 1800
   },
   "source": "manual"
  },
  {
   "name": "R1x bounce on marked words",
   "from": "K(a, b+1, G), G with x/z/w groups",
   "to": "K(a+1, b, T(G))",
   "steps": null,
   "cases": "T = the counter's own carry transducer, the same for every a, b (x -> z on a carry; z then x-runs -> 0 J... on the next). Orbit: 2,196/2,196 to 1e9",
   "validated": {
    "instances": 900,
    "passed": 900
   },
   "source": "manual"
  },
  {
   "name": "E partial eat",
   "from": "K(a, 0, 0 1^r J^k G')",
   "to": "K(a+2+r, k-1, G')",
   "steps": null,
   "cases": "r >= 0, k >= 1, G' empty or starting with a digit. Orbit: 229/229 to 1e9",
   "validated": {
    "instances": 1048,
    "passed": 1048
   },
   "source": "manual"
  },
  {
   "name": "B0 bell (low digit 0)",
   "from": "K(a, 0, 0 1^r 0 G'')",
   "to": "K(0, a+2+r, σ(0 G''))",
   "steps": null,
   "cases": "r >= 0, also G = 0 1^r with nothing after; σ = shift every 1-cell of enc one cell left. Orbit: 36/36 to 1e9",
   "validated": {
    "instances": 1244,
    "passed": 1244
   },
   "source": "manual"
  },
  {
   "name": "B1 bell (low digit 1)",
   "from": "K(a, 0, 1 G')",
   "to": "K(0, a+1, σ(inc(1 G')))",
   "steps": null,
   "cases": "any standard G'. Orbit: 222/222 to 1e9",
   "validated": {
    "instances": 2208,
    "passed": 2208
   },
   "source": "manual"
  }
 ],
 "template": {
  "file": "verify/FractalBEC.v:206",
  "module": "Section FractalBEC, instance tm1 / nonhalt1, cert1 B D [1] [0] [0;1] [1;0;1] [0;0] (mirrored L<->R)",
  "machine": "1RB0RF_1RC0RE_1LD1RB_1LA0LD_1RA---_0RA0LB",
  "diff": "C1: 1RA (ours) vs 1RB (tm1 mirrored)",
  "applies": "partly",
  "note": "Same bouncer: w1 = 101, w0 = 00, and the gap step of R1 is the template's RInc. The counter differs: 3-cell digits 000/010 with transparent J = 100 blocks instead of 2-cell digits (LInc not run on ours: the encoding does not match). No overflow Ov and no fractal recursion: the b=0 step is E/B0/B1 by the low digits, with the one-cell shift σ, and epoch lengths are irregular (2, 5, 6, 11, 18, 44, ...) instead of 2^i - 1. Same family as R2 (BECv1 tm108 twin: 'drop a suffix, shift one cell' eat, pending-size invariant)."
 },
 "strategy": "Proof plan: K(a,b,G) with G an Inductive group list; R1 by a carry lemma (ind over digits, J transparent); E/B0/B1 as finite case lemmas with σ as a list shift.\nInvariant: G is standard at every b=0 hub. Open: the cleanup lemma on σ-words (x -> z on the first carry, z x^j -> 0 J^(j+1) on the second; time 2^(p+1) <= N).\nStart from FractalBEC.v for the bouncer and a BECv1-style digit list for the counter; share the size lemma with R2.",
 "risks": [
  "Cleanup lemma unproved: arbitrary x/z/w words halt (K(5,6,1z0101) halts in 19 steps), so the invariant must track where σ puts x/w.",
  "The carry through z/w is a cell-level transducer with context (z followed by x-runs); formalizing it may take many small lemmas.",
  "Same family as R2 and the #10/#30 sibling, which got stuck on a size invariant; that invariant may be hard here too."
 ],
 "checkpoint_questions": [
  {
   "text": "Hub rows: does each block of rows read as one epoch (the cyan gap walks right through the grey blocks while the white/grey digits between the orange J columns count up), with the low digits eaten at each cyan bar?",
   "figs": [
    1
   ],
   "kind": "check"
  },
  {
   "text": "Close-up from K(5,2,0JJ1): do you see two bounces, then the partial eat E (the right end jumps left, the two J blocks become one new block), one more bounce, then the bell B1 running to the far left?",
   "figs": [
    2
   ],
   "kind": "check"
  },
  {
   "text": "Confidence is med: everything to 3e10 fits R1/E/B0/B1, but the cleanup lemma (x/z/w gone before the next b=0) is only observed. Should we fund a proof-prep stage on it, shared with R2's size invariant, or park R3?",
   "kind": "decision"
  }
 ],
 "discord_check": null,
 "stats": {
  "simulated_steps": 30000000000,
  "cells": 15368,
  "nonzeros": null,
  "growth": "span ~ t^(1/3): 677 cells at 3e6, 4,850 at 1e9 (3,147 nonzeros), 15,368 at 3e10; N = 1,603 at 1e9; macro model to 20,000 epochs (N = 192,042, about 1e15 steps by a 3N^2-per-epoch estimate)"
 },
 "status": "CHECKPOINT_A",
 "status_reason": "Med confidence: rules R1/E/B0/B1 cover every hub from blank (lockstep to 1e9, b=0 hubs to 3e10, macro to N=192,042); non-halting still needs the cleanup lemma for x/z/w words. Not a cryptid: the eat choice is a regular function of the low digits."
}