{
 "tm": "1RB0LF_0RC0RB_1LC1LD_1RE0RD_---0RA_1LA0LF",
 "rank": 17,
 "class_sheet": "Unknown",
 "class_guess": "Self-similar multi-wall bouncer (x3 epochs)",
 "confidence": "med",
 "summary": "Bouncer on a 10-background with domain walls; self-similar epochs a'=3a+6, c'=3a+c+2 (exact, 5/5)",
 "mechanism": [
  "Tape = alternating 10 background plus a few walls (00, 11, (010)^2, dense 1011 blocks).",
  "Sweeps flip 10<->01 and bounce off walls; phases (fixed sweep programs) chain into epochs, span x3 per epoch.",
  "Epoch anchors P(a,c,T): a' = 3a+6, c' = 3a+c+2 exactly. The tail rule T -> T' is the open question."
 ],
 "diagrams": [
  {
   "mode": "default (every step)",
   "png": "img/default_blank_1500.png",
   "html": "img/viewer.html",
   "steps": 1500,
   "start": "blank",
   "rows": "every step: steps 0-1,500 in 4 columns of 375, plus a zoom of steps 1,176-1,280",
   "window": "cells -42..25, 1 px = 1/2 cell (each cell is 2x2 px); zoom panel: cells -38..6, each cell 7x7 px",
   "legend": "white = 1, black = 0 or never visited, red/green = head that just moved R/L",
   "style": "bbchallenge-default",
   "caption": "From blank: 1010 stripes, walls; each head pass shifts stripes one cell",
   "title": "First 1,500 steps from blank",
   "why": "Every step from blank; thicker stripes are walls. Follow the head and the thin stripes it crosses.",
   "primary": true
  },
  {
   "mode": "epoch 2 vs epoch 3, rescaled (custom)",
   "png": "img/epochs_2v3.png",
   "html": null,
   "steps": 3824945149,
   "start": "mid-run: step 45,249,547 (epoch-2 anchor P) and step 439,293,481 (epoch-3 anchor P), both on the blank-tape run",
   "rows": "450 rows per epoch, anchor to next anchor: every 875,654 steps (epoch 2), every 7,523,671 steps (epoch 3)",
   "window": "cells -36,880..79, 1 px = 80 cells (epoch 2); epoch 3: cells -110,640..239, 1 px = 240 cells; cell 0 at the right edge of both",
   "legend": "panels 1-2: as the walls view (grey background, blue wall, orange (1011)^n block, white never visited); panel 3 (overlay, walls only): blue = epoch 2 only, red = epoch 3 only, black = both",
   "style": "custom",
   "caption": "Epochs 2 and 3 rescaled: left part repeats \u00d73; tail does not",
   "title": "Epoch 2 vs epoch 3, scaled to match",
   "why": "Right panel overlays epochs 2 and 3 (matched scale): black = both, blue = 2 only, red = 3 only.",
   "primary": true
  },
  {
   "mode": "walls over time (custom)",
   "png": "img/walls_time_4e8.png",
   "html": null,
   "steps": 439500000,
   "start": "blank",
   "rows": "every 500,000 steps (steps 0-439,500,000, 880 rows)",
   "window": "cells -36,800..63, 1 px = 64 cells",
   "legend": "grey = 1010\u2026 background (either phase), blue = wall (a 00 or 11 among the pixel's cells; the head's own pair ignored), orange = at least 5/8 ones ((1011)^n-type block), white = never visited, green tick = epoch start (anchor P); head not drawn",
   "style": "custom",
   "caption": "Walls over time: each epoch 3\u00d7 wider and about 9\u00d7 longer",
   "title": "Walls over time, 440 million steps",
   "why": "Blue lines are walls; green ticks mark epoch starts. Each epoch is 3\u00d7 wider and about 9\u00d7 longer.",
   "primary": false
  },
  {
   "mode": "records (custom colours)",
   "png": "img/records_epochs.png",
   "html": null,
   "steps": 439293481,
   "start": "blank",
   "rows": "every 48th left/right record (records 48..36,816; record 36,823 = the epoch-3 anchor)",
   "window": "cells -36,768..95, 1 px = 48 cells",
   "legend": "grey = 1010\u2026 background (either phase), blue = wall (a 00 or 11 among the pixel's cells; the head's own pair ignored), orange = at least 5/8 ones ((1011)^n-type block), white = never visited, green tick = epoch start (anchor P); head not drawn",
   "style": "custom",
   "caption": "Records view: epochs nest at 3\u00d7 scale (tail excepted)",
   "title": "Records view: epochs nest at 3\u00d7 scale",
   "why": "Rows count records, so each epoch is 3\u00d7 taller and wider; left of the tail the picture repeats.",
   "primary": false
  },
  {
   "mode": "blaze",
   "png": "img/blaze_1e9_labeled.png",
   "html": "img/viewer.html",
   "steps": 1000000000,
   "start": "blank",
   "rows": "Blaze time blocks: 1 row = 1,048,576 steps, averaged (quality); 954 rows, the last one partial",
   "window": "cells -65,088..63, 1 px = 64 cells (the engine's 1018x954 image; the site stretches it to 800x500)",
   "legend": "white = no 1s or never visited, light orange = half 1s (1010\u2026 background), darker orange = 3/4 1s ((1011)^n), full orange = all 1s; green ticks in the margin = epoch starts",
   "style": "bbchallenge-blaze",
   "caption": "bbchallenge Blaze, 10\u2079 steps: overall shape; walls too thin to see",
   "title": "Blaze view, 1 billion steps",
   "why": "Overall shape only; the walls are too thin to see at this scale.",
   "primary": false
  }
 ],
 "history": {
  "skeleton": "0^inf <A (10)^3 (010)^2 (10)^a (01)^2 1 (10)^b (01)^2 1 (10)^c T",
  "vars": [
   "a",
   "b",
   "c",
   "len(T)"
  ],
  "rows": [
   {
    "step": 544201,
    "values": [
     168,
     4,
     217,
     481
    ]
   },
   {
    "step": 701071,
    "values": [
     2,
     502,
     217,
     "?"
    ]
   },
   {
    "step": 5363499,
    "values": [
     510,
     4,
     723,
     1527
    ]
   },
   {
    "step": 6401649,
    "values": [
     240,
     814,
     723,
     "?"
    ]
   },
   {
    "step": 6601149,
    "values": [
     140,
     1114,
     723,
     "?"
    ]
   },
   {
    "step": 6712239,
    "values": [
     2,
     1528,
     723,
     "?"
    ]
   },
   {
    "step": 45249547,
    "values": [
     1536,
     4,
     2255,
     4603
    ]
   },
   {
    "step": 57001297,
    "values": [
     186,
     4054,
     2255,
     "?"
    ]
   },
   {
    "step": 439293481,
    "values": [
     4614,
     4,
     6865,
     13835
    ]
   },
   {
    "step": 545008381,
    "values": [
     474,
     12424,
     6865,
     "?"
    ]
   },
   {
    "step": 3824945149,
    "values": [
     13848,
     4,
     20709,
     41563
    ]
   },
   {
    "step": 4700080419,
    "values": [
     4114,
     29206,
     20709,
     "?"
    ]
   },
   {
    "step": 35431165697,
    "values": [
     41550,
     4,
     62255,
     124667
    ]
   }
  ]
 },
 "rules": [
  {
   "name": "Epoch map (anchor to anchor)",
   "from": "0^inf <A (10)^3 (010)^2 (10)^a (01)^2 1 (10)^4 (01)^2 1 (10)^c T",
   "to": "0^inf <A (10)^3 (010)^2 (10)^(3a+6) (01)^2 1 (10)^4 (01)^2 1 (10)^(3a+c+2) T'",
   "steps": "~150a^2, depends on T",
   "cases": "a even >=2, c odd >=3 (always on orbit). c even: c'=3a+c+3. c=1: c'=3a+5. T' unknown",
   "validated": {
    "instances": 49,
    "passed": 49
   },
   "source": "manual"
  },
  {
   "name": "Burst cycle",
   "from": "0^inf <A (10)^3 (010)^2 (10)^(a+2) (01)^2 1 (10)^b r",
   "to": "0^inf <A (10)^3 (010)^2 (10)^a (01)^2 1 (10)^(b+6) r",
   "steps": "20a+210",
   "cases": "a even >=2 (odd a: other cycle, never on orbit); any b>=0; r never read",
   "validated": {
    "instances": 420,
    "passed": 420
   },
   "source": "manual"
  },
  {
   "name": "Slow-phase cycle (epoch 1, ~1e7)",
   "from": "0^inf <A (10)^3 (010)^2 (10)^a (01)^(b+2) 0^2 (1011)^2 (10)^c (01)^d 0^2 (10)^e (01)^(f+4) (010)^2 (1110)^g R",
   "to": "0^inf <A (10)^3 (010)^2 (10)^(a+2) (01)^b 0^2 (1011)^2 (10)^(c+2) (01)^(d+2) 0^2 (10)^e (01)^f (010)^2 (1110)^(g+2) R",
   "steps": "20(a+b)+10(c+d+e+f)+340",
   "cases": "a,b,c,e,f odd; d even >=2 (d=0 fails); g any; R = right part frozen in this phase",
   "validated": {
    "instances": 86,
    "passed": 86
   },
   "source": "manual"
  },
  {
   "name": "sligocki low-level [0] (early burst-type)",
   "from": "0^inf (01)^3 (10)^2 (01)^(2a+11) (10)^3 <F (01)^(2b+4) (010)^2 1 (10)^2 1 (110)^2 1 0 1^3 (10)^2 1^6 0^inf",
   "to": "0^inf (01)^3 (10)^2 (01)^(2a+9) (10)^3 <F (01)^(2b+10) (010)^2 1 (10)^2 1 (110)^2 1 0 1^3 (10)^2 1^6 0^inf",
   "steps": "40a+344",
   "cases": null,
   "validated": {
    "instances": 10,
    "passed": 10
   },
   "source": "qsim"
  },
  {
   "name": "sligocki low-level [1] (early 5-counter phase)",
   "from": "0^inf (01)^3 (10)^2 0 (10)^2 0^2 1 (01)^(2a+4) (10)^(2b+11) 1^2 (01)^(2c+5) (10)^2 (01)^(2d+7) <A (10)^2 (01)^(2e+4) 0 (01)^2 1^2 0^inf",
   "to": "0^inf (01)^3 (10)^2 0 (10)^2 0^2 1 (01)^(2a+8) (10)^(2b+7) 1^2 (01)^(2c+9) (10)^2 (01)^(2d+5) <A (10)^2 (01)^(2e+10) 0 (01)^2 1^2 0^inf",
   "steps": "80a+80b+40c+40d+1360",
   "cases": "b, d shrink by 2: needs a mod split",
   "validated": {
    "instances": 10,
    "passed": 10
   },
   "source": "qsim"
  }
 ],
 "template": {
  "file": "vendor/busycoq/verify/LongNv1.v",
  "module": "TM1",
  "machine": "1LB0RD_0LC0LB_0RD1LB_1RD1RE_1RF---_0RA1LA",
  "diff": "12/12 transitions differ (8/12 after best mirror+renaming); Bouncer_v3.v TM1: 11/12 (9/12)",
  "applies": "partly",
  "note": "No rule or config carries over (from LongNv1 TM1's start ours halts in 3 steps).\nThe technique does: segRLs_n history induction P n -> P (S n) fits our x3 epoch induction."
 },
 "strategy": "Per-phase cycle lemmas (es + ind), chained with follow into Epoch: P(a,c,T) -->+ P(3a+6,3a+c+2,T').\nThen induct over epochs with progress_nonhalt_cond (invariant: a even, c odd, Inv(T)).\nInv(T) = tail language closed under one epoch, via Longitudinal sideRLs/segRLs_n (Bouncer_v3, LongNv1) or LongN closed sets.",
 "risks": [
  "Tail T is rewritten every epoch (73-100% of it is read); its rule T->T' is unknown and may be chaotic.",
  "Each epoch has many 5-7-counter phases; a hand proof could exceed 50 KB.",
  "Halt pattern D> 1^k 00 exists and the tail has 0^2/0^3 walls; non-halting depends on the tail invariant."
 ],
 "checkpoint_questions": [
  {
   "text": "Fig 1: does the head shuttle between walls (thicker stripes), shifting the thin stripes one cell on each pass?",
   "figs": [
    0
   ]
  },
  {
   "text": "Fig 2, right panel: are the left edge and the walls from the top left black, i.e. epochs 2 and 3 match at 3\u00d7 scale?",
   "figs": [
    1
   ]
  },
  {
   "text": "Fig 2, right panel: near the right edge (the tail), do red and blue lines disagree, so the tail changes each epoch?",
   "figs": [
    1
   ]
  }
 ],
 "discord_check": null,
 "stats": {
  "simulated_steps": 40000000000,
  "cells": 384466,
  "nonzeros": 192262,
  "growth": "span x3 per epoch (time x8.5-9.9), ~2*sqrt(t); right edge only 56 -> 84"
 },
 "status": "CHECKPOINT_A",
 "status_reason": "Left recursion is exact and affine (parities invariant, no Collatz branching). Tail rule open. Sheet: Level-0 tab, class Unknown, no notes; not in Discord lists."
}