{
 "tm": "1RB0LE_1LC1RD_1LA0RB_1RC1RB_1LF0LC_---0RC",
 "rank": 13,
 "class_sheet": "Bouncer + Chaotic Edge",
 "class_guess": "Bouncer + Chaotic Edge: translating period-10 bouncer feeding a 1-sided chaotic edge map",
 "confidence": "high",
 "summary": "Clean period-10 bouncer; every 7 bounces a fixed probe stirs a ~144-564-cell chaotic tail D. Tail = the whole problem.",
 "mechanism": [
  "Bulk (1101101101)^a sits on a left wall; each bounce shifts it 1 cell left (70a steps).",
  "A fixed 54-cell junction cycles through 7 shapes; the bulk gains 1 block per 7 bounces.",
  "Only every 7th bounce enters the tail D (the same 16 round trips = 32 crossings, one cell left of the cut, each time). D stays 144-564 cells, content chaotic."
 ],
 "diagrams": [
  {
   "mode": "records:L",
   "png": "img/edge_cycle_records.png",
   "html": "img/edge_cycle.html",
   "steps": 1200000,
   "caption": "One row per bounce: striped bulk, junction stair, chaotic tail",
   "title": "One row per bounce, from a built tape",
   "why": "One row per bounce from a built tape: bulk at left, junction seam in the middle, tail D at right.",
   "primary": true,
   "start": "constructed: 0^inf <A (1101101101)^20 X0 D0 0^inf  (X0 = the fixed 54-cell junction; D0 = the real 261-cell tail at step 784,041, where the real bulk is a = 59, not 20)",
   "rows": "left records: the start, then one row per new leftmost cell = one row per bounce (321 rows, steps 0..1,200,000; 7 rows = one probe cycle)",
   "window": "cells -320..725, 1 px = 0.5 cells (each cell is 2x2 px; cell 0 = the cell the head reads at the start)",
   "legend": "white = 1, black = 0 or never visited, red/green = head just moved right/left",
   "style": "bbchallenge-default"
  },
  {
   "mode": "records:L frame=right width=1200",
   "png": "img/records_L_1e8_right1200.png",
   "html": "img/viewer.html",
   "steps": 100000000,
   "caption": "Right edge pinned, last 1,200 cells: chaotic band stays bounded",
   "title": "Right end only, 100 million steps",
   "why": "Each row is shifted so the right edge stays put. Compare the noisy band's width at top and bottom.",
   "primary": true,
   "start": "blank",
   "rows": "left records: the start, then one row per new leftmost cell = one row per bounce (4,297 rows, steps 0..100,000,000; rows are not equal time)",
   "window": "cells -1199..0, 1 px = 1 cell, counted from the right edge (each row is shifted so its rightmost visited cell is at 0; the bulk continues off-picture to the left)",
   "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": "Smooth left wall, ragged right edge",
   "title": "Blaze view, 10 million steps",
   "why": "Blaze view to 1e7 steps. Compare the left edge with the right edge.",
   "primary": true,
   "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 -1256..1085, 1 px = 2.93 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 10,000 steps",
   "title": "bbchallenge Default view, first 10,000 steps",
   "why": "For orientation: the start of the run as bbchallenge.org shows it.",
   "primary": false,
   "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 <A (1101101101)^a X0 D 0^inf   (X0 = 110111100110011101100100100111101100100110110111010011)",
  "vars": [
   "a",
   "|D|",
   "D (first 24 cells)"
  ],
  "rows": [
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    "step": 784041,
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   },
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    "step": 852629,
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   {
    "step": 889610,
    "values": [
     62,
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     "001001001100100110110010"
    ]
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   {
    "step": 924145,
    "values": [
     63,
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   {
    "step": 960508,
    "values": [
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   {
    "step": 996135,
    "values": [
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   {
    "step": 1141833,
    "values": [
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   {
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   {
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    "values": [
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    "step": 35992489,
    "values": [
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   {
    "step": 58743103,
    "values": [
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   {
    "step": 97252227,
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   {
    "step": 164063019,
    "values": [
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   {
    "step": 279751465,
    "values": [
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     "001001001100100110110010"
    ]
   },
   {
    "step": 482122705,
    "values": [
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     "011101101100100111011011"
    ]
   },
   {
    "step": 835786657,
    "values": [
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     "001001001100100110110010"
    ]
   },
   {
    "step": 999751742,
    "values": [
     2010,
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     "011001001100100100110111"
    ]
   },
   {
    "step": 245034037373157,
    "values": [
     1000059,
     261,
     "edge map (|D| 144-519 in window)"
    ]
   },
   {
    "step": 980068074502653,
    "values": [
     2000059,
     321,
     "edge map (|D| 180-561 in window)"
    ]
   },
   {
    "step": 2205102110427991,
    "values": [
     3000059,
     261,
     "edge map (|D| 171-534 in window)"
    ]
   },
   {
    "step": 3920136152881851,
    "values": [
     4000059,
     381,
     "edge map (|D| 183-564 in window)"
    ]
   },
   {
    "step": 6125170190818191,
    "values": [
     5000059,
     291,
     "edge map (|D| 174-519 in window)"
    ]
   },
   {
    "step": 7411437208047181,
    "values": [
     5500059,
     321,
     "edge map (|D| 171-552 in window)"
    ]
   }
  ]
 },
 "rules": [
  {
   "name": "cycle (7 bounces)",
   "from": "0^inf <A (1101101101)^a X0 D 0^inf",
   "to": "0^inf <A (1101101101)^(a+1) X0 P(D) 0^inf",
   "steps": "490a + O(|D|^2)",
   "cases": "P(D) chaotic; holds every cycle from step 784,041 (a=59)",
   "validated": {
    "instances": 1951,
    "passed": 1951
   },
   "source": "manual"
  },
  {
   "name": "probe bounce",
   "from": "0^inf <A (1101101101)^a X0 D 0^inf",
   "to": "0^inf <A (1101101101)^a X1 011 P(D) 0^inf",
   "steps": "70a + c(D), c in [2235,10775]",
   "cases": "P independent of a; fixed 16 round trips (32 crossings) one cell left of the cut",
   "validated": {
    "instances": 180,
    "passed": 180
   },
   "source": "manual"
  },
  {
   "name": "junction bounces k=1..5",
   "from": "0^inf <A (1101101101)^a X_k D 0^inf",
   "to": "0^inf <A (1101101101)^(a+[k=5]) X_(k+1) D 0^inf",
   "steps": "70a+b_k, b=(9,473,55,49,77)",
   "cases": "D untouched (any D); X_k in notes.md",
   "validated": {
    "instances": 80,
    "passed": 80
   },
   "source": "manual"
  },
  {
   "name": "cycle close k=6",
   "from": "0^inf <A (1101101101)^a X6 011 D 0^inf",
   "to": "0^inf <A (1101101101)^a X0 D 0^inf",
   "steps": "70a+9",
   "cases": "cut moves +3 cells",
   "validated": {
    "instances": 16,
    "passed": 16
   },
   "source": "manual"
  },
  {
   "name": "right sweep",
   "from": "1 B> (0110110111)^n",
   "to": "(0111101101)^n 1 B>",
   "steps": "12n",
   "cases": null,
   "validated": {
    "instances": 16,
    "passed": 16
   },
   "source": "manual"
  },
  {
   "name": "left sweep",
   "from": "(1011110110)^n 11110101 <E 0",
   "to": "11110101 <E 0 (1101101101)^n",
   "steps": "58n",
   "cases": null,
   "validated": {
    "instances": 16,
    "passed": 16
   },
   "source": "manual"
  },
  {
   "name": "left wall",
   "from": "0^inf <A r",
   "to": "0^inf 1 B> r",
   "steps": "1",
   "cases": null,
   "validated": {
    "instances": 30,
    "passed": 30
   },
   "source": "manual"
  },
  {
   "name": "halt pattern",
   "from": "001 C> 0",
   "to": "halt",
   "steps": "4",
   "cases": "the only halting route",
   "validated": {
    "instances": 20,
    "passed": 20
   },
   "source": "manual"
  }
 ],
 "template": {
  "file": "vendor/busycoq/verify/Bouncer_v3.v",
  "module": "TM1",
  "machine": "1LB1RA_0RA0RC_---1RD_1RE1RB_1LF0LE_0RD0LE",
  "diff": "8 transitions apart under best relabel (A0 A1 B0 B1 C0 C1 E0 F1): class template, not a neighbour",
  "applies": "partly",
  "note": "Shape carries over: left wall (sideRLs_wall) + whole tape as right side + k-bounce BigStep (ours k=7, theirs 128). Its RInc native check needs a periodic edge; our D never repeats (5500000 cycles), so D needs a closed-language lemma."
 },
 "strategy": "Longitudinal split as in Bouncer_v3/LongNv1: wall + bulk sweeps by ind (12n, 58n) + 7 fixed junction bounces reduce nonhalt to the 1-sided edge map D -> P(D) with a fixed 16-round-trip (32-crossing) input one cell left of the cut (sideRLs list). Then find a closed regular language for D outside Rocq (CTL/FAR-style automaton on the edge map) and check it in Rocq verified-checker style (Fixpoint + soundness + vm_compute).",
 "risks": [
  "No small closed language for D may exist (FAR failed on the full machine; the SFT {000,0101,01101111} is not closed).",
  "Proving P independent of a and the junction lemmas needs segRLs/sideRLs plumbing (LongN/InfiniteRect style, 15-30 KB).",
  "|D| bounded only empirically (144-564); a very long transient could still end in 001 C> 0."
 ],
 "checkpoint_questions": [
  {
   "text": "Fig 1: do the striped bulk (left) and the slanted junction seam look regular, with all the disorder in the tail on the right?",
   "figs": [
    0
   ]
  },
  {
   "text": "Fig 2: does the noisy band at the right end keep about the same width all the way down, rather than widening?",
   "figs": [
    1
   ]
  },
  {
   "text": "Fig 3: is the left side a smooth curve, with only the right edge ragged?",
   "figs": [
    2
   ]
  }
 ],
 "discord_check": null,
 "stats": {
  "simulated_steps": 1000000000,
  "cells": 20564,
  "nonzeros": 14332,
  "growth": "both edges ~sqrt(t) (lo -0.44sqrt t, hi +0.21sqrt t); edge map 5500000 cycles ~ 7.4e+15 steps, |D| 144-564"
 },
 "status": "CHECKPOINT_A",
 "status_reason": "Mechanism found and validated (bulk, junction, probe); tail D is chaotic but bounded, not arithmetic (no Collatz-like growth), so not a cryptid. Next: closed language for the edge map."
}