{
 "tm": "1RB0LA_0RC1RD_0LD1RE_1LA0RB_0RD1RF_1RC---",
 "rank": 4,
 "class_sheet": "Sync bouncer counter",
 "class_guess": "Sync bouncer counter: TM4 binary counter + 3-block bouncer with moving front",
 "confidence": "high",
 "summary": "TM4's binary counter + 00111/10111 block bouncer; 7 phase rules validated; parity branch, both branches safe",
 "mechanism": [
  "Left: TM4's binary counter, k bits MSB first (1=1010, 0=0010), +1 per bounce.",
  "Right: A blocks 00111, one 111 marker (the front), B blocks 10111. P phase: each bounce adds a B block.",
  "Overflow: counter gains a bit; the front moves right 2 A blocks per visit; on reaching B, all B blocks become A blocks."
 ],
 "diagrams": [
  {
   "mode": "records",
   "png": "img/records_2.7e6.png",
   "html": "img/viewer.html",
   "steps": 2700000,
   "caption": "Left counter strip; diagonal fronts; light B block grows at edge",
   "title": "Left/right records to 2.7M steps",
   "why": "Edge records to 2.7M steps. Look at each horizontal reset and the diagonal that follows it.",
   "primary": true,
   "start": "blank",
   "rows": "left/right records: the start, then one row each time the head reaches a new leftmost or rightmost cell (1,173 rows, steps 0..2,700,000; rows are not equal time)",
   "window": "cells -21..1151, 1 px = 0.5 cells (each cell is 2x2 px)",
   "legend": "white = 1, black = 0 or never visited, red/green = head just moved right/left",
   "style": "bbchallenge-default"
  },
  {
   "mode": "every:60",
   "png": "img/every60_Q8_front_conversion.png",
   "html": null,
   "steps": 110000,
   "caption": "From Q(8,129,1,24,29): front, conversion sweep, then P bounces",
   "title": "One cycle up close, every 60 steps",
   "why": "One cycle of the same run, up close: every 60 steps from step 74,114, the tape Q(8,129,1,24,29).",
   "primary": true,
   "start": "mid-run: step 74114 (the real tape there is exactly Q(8,129,1,24,29); rebuilt from that formula and checked identical)",
   "rows": "every 60 steps (1,834 rows: steps 74,114..184,114), so one zigzag (bounce) spans many rows",
   "window": "cells -17..377, 1 px = 0.5 cells (each cell is 2x2 px)",
   "legend": "white = 1, black = 0 or never visited, red/green = head just moved right/left",
   "style": "bbchallenge-default"
  },
  {
   "mode": "blaze",
   "png": "img/blaze_1e7.png",
   "html": "img/viewer.html",
   "steps": 10000000,
   "caption": "Blaze 1e7: sqrt-growth bouncer with periodic conversion sweeps",
   "title": "Blaze, 10 million steps",
   "why": "The span grows like sqrt(t); the periodic conversion sweeps show as bands.",
   "start": "blank",
   "rows": "Blaze time blocks of 16,384 steps (averaged), 611 blocks stretched to 500 px, so 1 px row ~ 20,000 steps (steps 0..10,000,000)",
   "window": "cells -24..2653, 1 px = 3.35 cells",
   "legend": "white = 0, orange = 1, paler orange = mixed (each pixel averages its cells over its time block); no head drawn",
   "style": "bbchallenge-blaze"
  },
  {
   "mode": "default",
   "png": "img/default_site.png",
   "html": "img/viewer.html",
   "steps": 10000,
   "caption": "bbchallenge default view, first 10k steps",
   "title": "bbchallenge default view, first 10,000 steps",
   "why": "The site's view: a bouncer growing to the right from an almost fixed left end.",
   "start": "blank",
   "rows": "every step 0..10,000, squeezed 20:1 into 500 px as on bbchallenge.org (px row y blends steps 20y+9 and 20y+10)",
   "window": "cells -200..199, 1 px = 1 cell",
   "legend": "white = 1, black = 0 or never visited, red/green = head just moved right/left",
   "style": "bbchallenge-default"
  }
 ],
 "history": {
  "skeleton": "0^inf C_k(n) (00111)^x [111 (00111)^y] (10111)^z 00 C> 0^inf",
  "vars": [
   "phase",
   "k",
   "n",
   "x",
   "y",
   "z"
  ],
  "rows": [
   {
    "step": 1047,
    "values": [
     "P",
     4,
     14,
     4,
     "-",
     0
    ]
   },
   {
    "step": 1222,
    "values": [
     "P",
     4,
     15,
     4,
     "-",
     1
    ]
   },
   {
    "step": 1507,
    "values": [
     "Q",
     5,
     17,
     1,
     2,
     2
    ]
   },
   {
    "step": 1912,
    "values": [
     "Q",
     5,
     20,
     3,
     0,
     3
    ]
   },
   {
    "step": 2826,
    "values": [
     "P",
     5,
     25,
     7,
     "-",
     0
    ]
   },
   {
    "step": 4996,
    "values": [
     "P",
     5,
     31,
     7,
     "-",
     6
    ]
   },
   {
    "step": 5553,
    "values": [
     "Q",
     6,
     33,
     1,
     5,
     7
    ]
   },
   {
    "step": 7003,
    "values": [
     "Q",
     6,
     39,
     5,
     1,
     9
    ]
   },
   {
    "step": 17215,
    "values": [
     "Q",
     7,
     65,
     1,
     16,
     0
    ]
   },
   {
    "step": 27975,
    "values": [
     "Q",
     7,
     89,
     17,
     0,
     8
    ]
   },
   {
    "step": 35105,
    "values": [
     "P",
     7,
     99,
     26,
     "-",
     0
    ]
   },
   {
    "step": 72213,
    "values": [
     "P",
     7,
     127,
     26,
     "-",
     28
    ]
   },
   {
    "step": 74114,
    "values": [
     "Q",
     8,
     129,
     1,
     24,
     29
    ]
   },
   {
    "step": 108318,
    "values": [
     "Q",
     8,
     165,
     25,
     0,
     41
    ]
   },
   {
    "step": 172672,
    "values": [
     "P",
     8,
     208,
     67,
     "-",
     0
    ]
   },
   {
    "step": 310913,
    "values": [
     "P",
     8,
     255,
     67,
     "-",
     47
    ]
   },
   {
    "step": 314750,
    "values": [
     "Q",
     9,
     257,
     1,
     65,
     48
    ]
   },
   {
    "step": 518654,
    "values": [
     "Q",
     9,
     353,
     65,
     1,
     80
    ]
   },
   {
    "step": 1101764,
    "values": [
     "Q",
     10,
     537,
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     131,
     0
    ]
   },
   {
    "step": 1824857,
    "values": [
     "Q",
     10,
     732,
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   },
   {
    "step": 2625871,
    "values": [
     "P",
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   {
    "step": 4076949,
    "values": [
     "P",
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   },
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    "values": [
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   },
   {
    "step": 6289433,
    "values": [
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   },
   {
    "step": 9222995,
    "values": [
     "P",
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   },
   {
    "step": 18983294,
    "values": [
     "P",
     11,
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     437
    ]
   },
   {
    "step": 19012811,
    "values": [
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     12,
     2049,
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     476,
     438
    ]
   },
   {
    "step": 30552161,
    "values": [
     "Q",
     12,
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     676
    ]
   },
   {
    "step": 48261531,
    "values": [
     "P",
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    ]
   },
   {
    "step": 79286557,
    "values": [
     "P",
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     654
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   },
   {
    "step": 79344666,
    "values": [
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     1152,
     655
    ]
   },
   {
    "step": 139306778,
    "values": [
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   },
   {
    "step": 209134028,
    "values": [
     "P",
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    ]
   },
   {
    "step": 316196767,
    "values": [
     "P",
     13,
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     "-",
     1133
    ]
   },
   {
    "step": 316309612,
    "values": [
     "Q",
     14,
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     1,
     2383,
     1134
    ]
   },
   {
    "step": 564062495,
    "values": [
     "Q",
     14,
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     2383,
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   },
   {
    "step": 1093007525,
    "values": [
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     15,
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    ]
   },
   {
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    "values": [
     "P",
     15,
     25768,
     7028,
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     0
    ]
   }
  ]
 },
 "rules": [
  {
   "name": "R1 P-bounce",
   "from": "0^inf C_k(n) (00111)^X (10111)^z 00 C> 0^inf",
   "to": "0^inf C_k(n+1) (00111)^X (10111)^(z+1) 00 C> 0^inf",
   "steps": "32X+32z+16t(n)+47, t(n) = trailing 1-bits of n",
   "cases": "n+1 <= 2^k-1, X >= 1",
   "validated": {
    "instances": 65,
    "passed": 65
   },
   "source": "manual"
  },
  {
   "name": "R2 overflow",
   "from": "0^inf C_k(2^k-1) (00111)^X (10111)^z 00 C> 0^inf",
   "to": "0^inf C_(k+1)(2^k+1) 00111 111 (00111)^(X-2) (10111)^(z+1) 00 C> 0^inf",
   "steps": null,
   "cases": "X >= 2",
   "validated": {
    "instances": 25,
    "passed": 25
   },
   "source": "manual"
  },
  {
   "name": "R3 Q-bounce (front +2)",
   "from": "0^inf C_k(n) (00111)^x 111 (00111)^(y+2) (10111)^z 00 C> 0^inf",
   "to": "0^inf C_k(n+3) (00111)^(x+2) 111 (00111)^y (10111)^(z+1) 00 C> 0^inf",
   "steps": "96x+32y+32z+16(t(n)+t(n+1)+t(n+2))+197",
   "cases": "n+3 <= 2^k-1",
   "validated": {
    "instances": 65,
    "passed": 65
   },
   "source": "manual"
  },
  {
   "name": "R4 even end (y=0)",
   "from": "0^inf C_k(n) (00111)^x 111 (10111)^z 00 C> 0^inf",
   "to": "0^inf C_k(n+z+2) (00111)^(x+z+1) 00 C> 0^inf",
   "steps": null,
   "cases": "n+z+2 <= 2^k-1",
   "validated": {
    "instances": 25,
    "passed": 25
   },
   "source": "manual"
  },
  {
   "name": "R5 odd end (y=1)",
   "from": "0^inf C_k(n) (00111)^x 111 00111 (10111)^z 00 C> 0^inf",
   "to": "0^inf C_k(n+2z+7) (00111)^(x+z+4) 00 C> 0^inf",
   "steps": null,
   "cases": "n+2z+7 <= 2^k-1",
   "validated": {
    "instances": 25,
    "passed": 25
   },
   "source": "manual"
  },
  {
   "name": "R4o even end + overflow",
   "from": "0^inf C_k(n) (00111)^x 111 (10111)^z 00 C> 0^inf",
   "to": "0^inf C_(k+1)(2^k+1+3m) (00111)^(1+2m) 111 (00111)^(x+z-1-2m) 00 C> 0^inf",
   "steps": null,
   "cases": "m = n+z+2-2^k >= 0, x+z <= 2^(k-1); outside that regime 2/25 fail",
   "validated": {
    "instances": 25,
    "passed": 25
   },
   "source": "manual"
  },
  {
   "name": "R5o odd end + overflow",
   "from": "0^inf C_k(n) (00111)^x 111 00111 (10111)^z 00 C> 0^inf",
   "to": "0^inf C_(k+1)(2^k+1+3m) (00111)^(1+2m) 111 (00111)^(x+z+2-2m) 00 C> 0^inf",
   "steps": null,
   "cases": "m = n+2z+7-2^k >= 0, x+z <= 2^(k-1); outside that regime 4/25 fail",
   "validated": {
    "instances": 50,
    "passed": 50
   },
   "source": "manual"
  }
 ],
 "template": {
  "file": "vendor/busycoq/verify/SBCv7.v",
  "module": "TM4 (line 610); also TM37 (line 7021)",
  "machine": "1RB0LA_0RC1RD_0LD1RE_1LA0RB_0RD1RF_0RA---",
  "diff": "F0 only: 0RA (TM4) / 1RB (TM37) -> 1RC; A-E identical",
  "applies": "partly",
  "note": "Left counter carries over (same d1/d0 digits; TM37's overflow form reached 5/5). Right side doesn't: TM4's BigStep holds 2/8, only when F0 is never read."
 },
 "strategy": "Longitudinal proof in TM4 style: reuse the counter lemma (BC.Mul2/BC.Incs on d0/d1), add new segRLs lemmas for 00111/10111/111 words, prove R1-R5o with es/ind/follow.\nThen progress_nonhalt_cond with a linear invariant on (n, X, z)/2^k that holds for both parity branches.\ninit: c0 -->* Q(5,17,1,2,2) at step 1507 by esx.",
 "risks": [
  "Branch = parity of X (X grows ~3X/2: Collatz-like); proof needs an invariant valid for both branches (holds numerically under adversarial branching to k=600).",
  "R5o (odd end + overflow) has front and conversion running at once: hardest low-level lemma.",
  "Outside the regime (x+z >> 2^k) the head breaks through the counter and can halt; the invariant must exclude this."
 ],
 "checkpoint_questions": [
  {
   "text": "Fig 1: after each horizontal reset (an overflow), does the diagonal read as one moving front, the 111 marker? Fig 2 shows one cycle up close.",
   "figs": [
    0,
    1
   ]
  },
  {
   "text": "Fig 1: about two-thirds down, does one diagonal follow straight on from the previous one, with no diagonal-free stretch between them (the odd-X branch)?",
   "figs": [
    0
   ]
  }
 ],
 "discord_check": null,
 "stats": {
  "simulated_steps": 3000000000,
  "cells": 48013,
  "nonzeros": 22929,
  "growth": "span ~ t^0.5 (cycle k: ~2^k blocks in ~4^k steps); left edge -2 cells per cycle"
 },
 "status": "CHECKPOINT_A",
 "status_reason": "Mechanism found; 7 rules validated by simulation; the phase map built from them reproduces all 36 phase boundaries in history from step 1507 (its start) to 2.3e9 steps. Parity branch present but halting does not depend on it."
}