{
 "tm": "1RB1LE_0RC1RC_0RD0RB_1RE0RF_1LA0LE_---1RB",
 "rank": 20,
 "class_sheet": "Chaotic Fractal Type 1",
 "class_guess": "Fractal call-stack bouncer (FractalType7/TM20 shape)",
 "confidence": "med",
 "summary": "Nested bouncer call stack: Y(a,b,c)->Y(a+2,b-1,c+3) per round; epochs grow x2.5-3; mod-3 gap cases",
 "mechanism": [
  "Inner loop Y(a,b,c) = 1^a 001 (10)^b 11 (01)^c E>: each round a+2, b-1, c+3, eating 6 blank cells.",
  "When b hits 0, E wipes the 1^a block and walks left (Return); the level to its left resumes the same loop.",
  "When that gap runs out it calls the right level again: a call stack, so each epoch replays shrunken copies."
 ],
 "diagrams": [
  {
   "mode": "state:E:R",
   "png": "img/state_ER_3e6.png",
   "html": "img/viewer.html",
   "steps": 3000000,
   "caption": "Rows: E entered moving right; triangles = inner loop, staircases = replays",
   "title": "State-E entries, to 3 million steps",
   "why": "Rows: entries into E moving right. Compare the small staircases left of each wide white band with earlier rows.",
   "primary": true,
   "start": "blank",
   "rows": "every entry into state E moving right: 1,937 rows to step 3,000,000 (1,402 at the right edge = right turnarounds, 535 inside the tape), 9 to 5,473 steps apart",
   "window": "cells -16..4214, 1 px = 1 cell",
   "legend": "white = 1, black = 0 or unvisited, red/green = head (last move right/left); (10)^b blocks are 1-px stripes that look gray when shrunk, 1^a blocks are solid white",
   "style": "bbchallenge-default"
  },
  {
   "mode": "records",
   "png": "img/records_1e6.png",
   "html": "img/viewer.html",
   "steps": 999391,
   "caption": "White 1^a triangle grows while (10)^b shrinks; next epoch ~3x",
   "title": "Edge records to 1 million steps",
   "why": "Edge records to 1M steps. Follow each white triangle and the gray (10)^b block beside it.",
   "primary": true,
   "start": "blank",
   "rows": "left/right records: the start, then one row per new record; stopped at the 2,400-row cap at step 999,391 (not evenly spaced in time)",
   "window": "cells -16..2383, 1 px = 1 cell",
   "legend": "white = 1, black = 0 or unvisited, red/green = head (last move right/left); (10)^b blocks are 1-px stripes that look gray when shrunk",
   "style": "bbchallenge-default"
  },
  {
   "mode": "blaze",
   "png": "img/blaze_1e9.png",
   "html": null,
   "steps": 1000000000,
   "caption": "Two top epochs at 1e9 with shrunken replay between them",
   "title": "Blaze view, 1 billion steps",
   "why": "Blaze view to 1e9 steps. Look between the two big domes.",
   "primary": true,
   "start": "blank",
   "rows": "Blaze time blocks of 1,048,576 steps (each averaged), 954 blocks to step 1e9, stretched to 500 px (1 px = about 2,000,000 steps)",
   "window": "cells -64..75135, 1 px = 94 cells (Blaze image: 1 column per 64 cells, averaged; 1,175 columns stretched to 800 px)",
   "legend": "white = 0, orange = 1, paler orange = 1 only part of the time (or in part of the cells) of that pixel's block; no head; solid orange = 1^a blocks, pale orange = (10)^b blocks (half ones)",
   "style": "bbchallenge-blaze"
  },
  {
   "mode": "default",
   "png": "img/epoch_end_site.png",
   "html": null,
   "steps": 9000,
   "caption": "From step 15000: epoch end, return left, replay, new epoch",
   "title": "Epoch end, steps 15,000 to 24,000",
   "why": "Default-view zoom: the epoch ends, the head returns left, replays, and a new epoch starts.",
   "primary": false,
   "start": "mid-run: step 15000",
   "rows": "every step 15,000..23,999, squeezed into 500 px by the site's bilinear filter: px row y blends steps 15,000+18y+8 and 15,000+18y+9, so it is effectively one sample every 18 steps",
   "window": "cells -144..255, 1 px = 1 cell (x-translation 0.85: the start head, cell 196, sits at x = 340)",
   "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": "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..9,999, squeezed into 500 px by the site's bilinear filter: px row y blends steps 20y+9 and 20y+10, so it is effectively 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 1^a 001 10^b 1^2 01^c E> 0^inf",
  "vars": [
   "a",
   "b",
   "c",
   "span"
  ],
  "rows": [
   {
    "step": 1323,
    "values": [
     5,
     10,
     5,
     86
    ]
   },
   {
    "step": 2485,
    "values": [
     19,
     3,
     26,
     128
    ]
   },
   {
    "step": 4853,
    "values": [
     2,
     30,
     2,
     160
    ]
   },
   {
    "step": 5015,
    "values": [
     4,
     29,
     5,
     166
    ]
   },
   {
    "step": 10829,
    "values": [
     38,
     12,
     56,
     268
    ]
   },
   {
    "step": 16949,
    "values": [
     58,
     2,
     86,
     328
    ]
   },
   {
    "step": 24233,
    "values": [
     4,
     74,
     5,
     356
    ]
   },
   {
    "step": 50947,
    "values": [
     78,
     37,
     116,
     578
    ]
   },
   {
    "step": 78289,
    "values": [
     120,
     16,
     179,
     704
    ]
   },
   {
    "step": 114965,
    "values": [
     4,
     224,
     5,
     816
    ]
   },
   {
    "step": 343847,
    "values": [
     226,
     113,
     338,
     1482
    ]
   },
   {
    "step": 574549,
    "values": [
     348,
     52,
     521,
     1848
    ]
   },
   {
    "step": 880549,
    "values": [
     4,
     676,
     5,
     2176
    ]
   },
   {
    "step": 3584549,
    "values": [
     804,
     276,
     1205,
     4576
    ]
   },
   {
    "step": 6300119,
    "values": [
     1226,
     65,
     1838,
     5842
    ]
   },
   {
    "step": 9905729,
    "values": [
     4,
     1782,
     5,
     6248
    ]
   },
   {
    "step": 24021257,
    "values": [
     1768,
     900,
     2651,
     11540
    ]
   },
   {
    "step": 38137595,
    "values": [
     2722,
     423,
     4082,
     14402
    ]
   },
   {
    "step": 56966297,
    "values": [
     5,
     5350,
     5,
     16956
    ]
   },
   {
    "step": 213376947,
    "values": [
     6055,
     2325,
     9080,
     35106
    ]
   },
   {
    "step": 369764769,
    "values": [
     9249,
     728,
     13871,
     44688
    ]
   },
   {
    "step": 578233633,
    "values": [
     4,
     13412,
     5,
     49072
    ]
   },
   {
    "step": 704796961,
    "values": [
     3548,
     11640,
     5321,
     59704
    ]
   },
   {
    "step": 831281491,
    "values": [
     6038,
     10395,
     9056,
     67174
    ]
   }
  ]
 },
 "rules": [
  {
   "name": "YY (one round, tail)",
   "from": "L 1^a 001 10^(b+1) 1^2 01^c E> 0^(d+6) R",
   "to": "L 1^(a+2) 001 10^b 1^2 01^(c+3) E> 0^d R",
   "steps": "4b+8c+30",
   "cases": "a,b,c,d >= 0; L any; R = 1... (or d=inf)",
   "validated": {
    "instances": 87,
    "passed": 87
   },
   "source": "manual"
  },
  {
   "name": "YX (half round)",
   "from": "L 1^a 001 10^(b+1) 1^2 01^c E> 0^(d+3) R",
   "to": "L 1^(a+2) 001 10^b 0 01^(c+2) E> 0^d R",
   "steps": "4b+4c+17",
   "cases": "a,b,c,d >= 0; tail form (R=0^inf) 63/63 too",
   "validated": {
    "instances": 85,
    "passed": 85
   },
   "source": "manual"
  },
  {
   "name": "XY (half round)",
   "from": "L 1^a 001 10^b 0 01^c E> 0^(d+3) R",
   "to": "L 1^a 001 10^b 1^2 01^(c+1) E> 0^d R",
   "steps": "4c+5",
   "cases": "a,b,c,d >= 0; tail form 63/63 too",
   "validated": {
    "instances": 87,
    "passed": 87
   },
   "source": "manual"
  },
  {
   "name": "YX1 (eat ones)",
   "from": "L 1^a 001 10^(b+1) 1^2 01^c E> 1^(d+1) R0",
   "to": "L 1^(a+2) 001 10^b 0 01^(c+1) E> 1^d R0",
   "steps": "4b+4c+15",
   "cases": "a,b,c,d >= 0; R0 = 0...",
   "validated": {
    "instances": 83,
    "passed": 83
   },
   "source": "manual"
  },
  {
   "name": "XY1 (eat ones)",
   "from": "L 1^a 001 10^b 0 01^(c+1) E> 1^(d+1) R0",
   "to": "L 1^a 001 10^b 1^2 01^(c+1) E> 1^d R0",
   "steps": "4c+7",
   "cases": "a,b,c,d >= 0 (fails at c=0: needs >= 1 copy of 01)",
   "validated": {
    "instances": 87,
    "passed": 87
   },
   "source": "manual"
  },
  {
   "name": "Return (b = 0)",
   "from": "L0 1^a 001 1^2 01^c E> 0^(d+1) R",
   "to": "L0 <E 0^(a+1) 1100 1^(2c+1) 0^d R",
   "steps": "a+2c+14",
   "cases": "L0 ends in 0; tail form 63/63 too. Gap leftover 0/1/2 (mod 3) = 3 different call paths, not yet ruled",
   "validated": {
    "instances": 85,
    "passed": 85
   },
   "source": "manual"
  }
 ],
 "template": {
  "file": "vendor/busycoq/verify/FractalType7v4.v",
  "module": "TM20",
  "machine": "1RB0LF_1RC0RA_0RD0RB_1LE0RF_1LB0LE_1LE---",
  "diff": "7 of 12 transitions (A1 B0 B1 D0 E0 F0 F1; halt moves F1->F0); no relabel/mirror does better; best in FractalType7v2/v3 is 8",
  "applies": "partly",
  "note": "Same architecture: S1 loop (Inc/Incs) + Empty_b return + P call + Column + Good invariant. Its S1/Inc configs do not carry over: 0/20 Inc, our machine halts in 6 steps from S1."
 },
 "strategy": "Copy TM20's skeleton, not its rules: prove XY/YX/XY1/YX1/Return with es + ind (Incs-style), define P u r r' (call returns 0^u), prove one Column lemma per gap residue mod 3, then a Good-style inductive invariant with inequality bounds on the column chain; init via chunks + native_compute; nonhalt via progress_nonhalt.",
 "risks": [
  "Gap leftovers mod 3 vary in the run (start residues 1:41, 2:23, 0:17 of 81 runs); TM20 kept all counts = 0 mod 3. Invariant must track residues.",
  "Epoch ratio alternates x2.5 / x3 (not affine), so only an inequality invariant can work; exact epoch map unknown.",
  "Resisted mxdys's FractalType7 LLM batch; template is 7 transitions away, so new Column lemmas, not copy-and-modify."
 ],
 "checkpoint_questions": [
  {
   "text": "Fig 1: do the small staircases left of each wide white band look like shrunken copies of the earlier picture?",
   "figs": [
    0
   ]
  },
  {
   "text": "Fig 2: does each white triangle end exactly where the gray block runs out, before the next, ~3× bigger one starts?",
   "figs": [
    1
   ]
  },
  {
   "text": "Fig 3: between the two big domes, is there a small shrunken copy of the earlier pattern?",
   "figs": [
    2
   ]
  }
 ],
 "discord_check": null,
 "stats": {
  "simulated_steps": 1000000000,
  "cells": 75124,
  "nonzeros": 59751,
  "growth": "span ~ sqrt(5.6 t) (t ~ 0.18 span^2); left edge frozen at -16 after step 4428; qsim: 1.99e13 steps, 8.5M nonzeros, no halt"
 },
 "status": "CHECKPOINT_A",
 "status_reason": "Mechanism found (TM20-shaped call-stack fractal); inner rules validated; epoch-level map and mod-3 call cases still open. Not Collatz-like so far."
}