{
 "tm": "1RB1LD_0RC1RC_1LA1RE_---1LE_1RF0RC_0RB0LF",
 "rank": 214,
 "class_sheet": "Bouncer + Chaotic Edge",
 "class_guess": "Bouncer with a 4-particle right edge (closed double-cycle law; the edge is not chaotic)",
 "confidence": "high",
 "summary": "Bouncer whose right edge is a 4-particle system; the law Even(x) -> Even(x+32) repeats forever, so it never halts.",
 "mechanism": [
  "A bouncer: the left end is a fixed gadget G (lo moves 2 left per trip), and the rest is a sea of 1s that each pass rewrites 1^(2k) <-> (10)^k. The left pass uses a moving composite, 11 C>0 or B>000.",
  "In the sea float 4 particles: a wall W and an edge E (fixed relative to the right end), a fast P1 and two slow Q, Q'. Every 12 trips: K(n,a,b,c,d) -> K(n+48,a-8,b+4,c,d+4).",
  "When P1 reaches W, one of two collisions (TE, TO) fires, in alternation. Even(x) = K(n,x,(x+76)/2,35,18) -> Even(x+32) for x = 4 mod 8, x >= 52, so the macro rules chain forever. The run enters Even(212) at step 153,126,993; halting (A on 1 after a 0) never occurs."
 ],
 "diagrams": [
  {
   "mode": "default",
   "png": "img/default_site.png",
   "html": null,
   "steps": 10000,
   "caption": "bbchallenge default view, first 10,000 steps",
   "title": "bbchallenge default view, 10,000 steps",
   "why": "The site's view. A bouncer: the head sweeps the whole tape, and the right side is ragged from the start.",
   "primary": true,
   "start": "blank",
   "rows": "10,000 steps squeezed into 500 px rows (1 px row = 20 steps, the site's bilinear squeeze)",
   "window": "cells -200..199, 1 px = 1 cell",
   "legend": "white = 1, black = 0 or never visited, red/green = head after a right/left move",
   "style": "bbchallenge-default"
  },
  {
   "mode": "records:L",
   "png": "img/double_cycle_E0.png",
   "html": null,
   "steps": 88266566,
   "caption": "One double cycle at the right edge, Even(212) to Even(244)",
   "title": "One double cycle: Even(212) to Even(244)",
   "why": "Rows go down in time. Follow the dotted diagonal to the wall at left; compare the two halves.",
   "primary": true,
   "start": "mid-run: step 153,126,993 (fed as its exact tape K(12863, 212, 144, 35, 18), head F at the left end)",
   "rows": "left records: the start, then one row per new leftmost cell (2 per round trip); 1,674 records = 837 round trips, steps 153,126,993..241,393,559, ending on K(16215, 244, 160, 35, 18)",
   "window": "cells -519..0, 1 px = 1 cell, counted from the right edge (each row's rightmost visited cell is pinned to the last column); G and the body are off-picture to the left",
   "legend": "white = 1, black = 0 or never visited; the head is at the far left, off-picture",
   "style": "bbchallenge-default"
  },
  {
   "mode": "custom rows: one per 12 round trips",
   "png": "img/edge_overview_1e10.png",
   "html": null,
   "steps": 9995948407,
   "caption": "Right edge every 12 round trips, to 1e10 steps",
   "title": "Right edge to 1e10 steps",
   "why": "One row per 12 trips. Each tooth is one run; look for any tooth that breaks the two-kind alternation.",
   "primary": true,
   "start": "blank",
   "rows": "one per 12 round trips: the first left record of trips 5, 17, 29, ..., 26,501 (2,209 rows, steps 3,849..9,995,948,407)",
   "window": "cells -1199..0, 1 px = 1 cell, counted from the right edge (each row's rightmost visited cell is pinned to the last column)",
   "legend": "white = 1, black = 0, grey = left of the visited span (first rows only)",
   "style": "custom"
  },
  {
   "mode": "default",
   "png": "img/default_1e5.png",
   "html": null,
   "steps": 99999,
   "caption": "bbchallenge default view, 99,999 steps, tape width 1,200",
   "title": "Default view, 99,999 steps, wide window",
   "why": "The bouncer's shape: span grows like the square root of time; the right edge carries the particles.",
   "primary": false,
   "start": "blank",
   "rows": "99,999 steps squeezed into 500 px rows (1 px row = 200 steps, the site's bilinear squeeze)",
   "window": "cells -600..599, 1 px = 3 cells (the site squeezes a 1,200-cell width into 400 px)",
   "legend": "white = 1, black = 0 or never visited, red/green = head after a right/left move",
   "style": "bbchallenge-default"
  }
 ],
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 },
 "rules": [
  {
   "name": "Sea, right pass",
   "from": "C> 1^(2k)",
   "to": "10^k C>",
   "steps": "2k",
   "cases": "any k >= 0",
   "validated": {
    "instances": 200,
    "passed": 200
   },
   "source": "manual"
  },
  {
   "name": "Sea, left pass (composite 11 C>0)",
   "from": "0 10^k 1 1 C> 0",
   "to": "0 1 1 C> 0 1^(2k)",
   "steps": "12k",
   "cases": "any k >= 0; the cell left of the block is 0",
   "validated": {
    "instances": 200,
    "passed": 200
   },
   "source": "manual"
  },
  {
   "name": "Sea, left pass (composite B>000)",
   "from": "0 10^k B> 0 0 0",
   "to": "0 B> 0 0 0 1^(2k)",
   "steps": "12k",
   "cases": "any k >= 0; used on the other trip phase",
   "validated": {
    "instances": 200,
    "passed": 200
   },
   "source": "manual"
  },
  {
   "name": "S: 12 round trips",
   "from": "0^inf <F G 1^n 0 1^a 0 1^2 0 1^b 0^6 1^c 0 1^2 0 1^4 0^6 1^d 0 1^2 0^2 1^2 0 1^3 0^inf",
   "to": "0^inf <F G 1^(n+48) 0 1^(a-8) 0 1^2 0 1^(b+4) 0^6 1^c 0 1^2 0 1^4 0^6 1^(d+4) 0 1^2 0^2 1^2 0 1^3 0^inf",
   "steps": "84n+84a+77b+70c+63d+13628",
   "cases": "n odd >= 1; a even >= 10; b even >= 6; c odd >= 7; d even >= 2 (each bound is the smallest value that works; tested up to 2,000)",
   "validated": {
    "instances": 300,
    "passed": 300
   },
   "source": "manual"
  },
  {
   "name": "TE: P1 hits the wall (even to odd run)",
   "from": "0^inf <F G 1^n 0 1^4 0 1^2 0 1^b 0^6 1^35 0 1^2 0 1^4 0^6 1^d 0 1^2 0^2 1^2 0 1^3 0^inf",
   "to": "0^inf <F G 1^(n+556) 0 1^(b-86) 0 1^2 0 1^82 0^6 1^(d+51) 0 1^2 0 1^4 0^6 1^18 0 1^2 0^2 1^2 0 1^3 0^inf",
   "steps": "966n+952b+812d+455332 (138 round trips)",
   "cases": "n odd; b even, 88..3,000; d even, 20..2,000 (plus a 25 x 32 grid over b 100..676, d 20..392)",
   "validated": {
    "instances": 150,
    "passed": 150
   },
   "source": "manual"
  },
  {
   "name": "TO: P1 hits the wall (odd to even run)",
   "from": "0^inf <F G 1^n 0 1^2 0 1^2 0 1^b 0^6 1^c 0 1^2 0 1^4 0^6 1^d 0 1^2 0^2 1^2 0 1^3 0^inf",
   "to": "0^inf <F G 1^(n+588) 0 1^(b+c-91) 0 1^2 0 1^(d+62) 0^6 1^35 0 1^2 0 1^4 0^6 1^18 0 1^2 0^2 1^2 0 1^3 0^inf",
   "steps": "1029n+1001b+987c+882d+458476 (147 round trips)",
   "cases": "n odd; b even >= 82; c odd >= 37; d even >= 18 (half the tests on the reachable line c = b+11, d = b-64, half random)",
   "validated": {
    "instances": 150,
    "passed": 150
   },
   "source": "manual"
  },
  {
   "name": "Double cycle",
   "from": "Even(n, x) = 0^inf <F G 1^n 0 1^x 0 1^2 0 1^((x+76)/2) 0^6 1^35 0 1^2 0 1^4 0^6 1^18 0 1^2 0^2 1^2 0 1^3 0^inf",
   "to": "Even(n+12x+808, x+32)",
   "steps": "S^((x-4)/8), TE, S^((x-52)/8), TO: 3x+201 round trips",
   "cases": "x = 4 mod 8, x >= 52, n odd. Tested: every x in 52..252, 16 random x up to 5,620, and all 16 even-run starts of the real run up to 1e10",
   "validated": {
    "instances": 42,
    "passed": 42
   },
   "source": "manual"
  },
  {
   "name": "Opening",
   "from": "blank tape, state A",
   "to": "Even(12863, 212) = 0^inf <F G 1^12863 0 1^212 0 1^2 0 1^144 0^6 1^35 0 1^2 0 1^4 0^6 1^18 0 1^2 0^2 1^2 0 1^3 0^inf",
   "steps": "153126993",
   "cases": "direct simulation; it is the first left record of round trip 3,209",
   "validated": {
    "instances": 1,
    "passed": 1
   },
   "source": "manual"
  }
 ],
 "template": {
  "file": "TranslatedBellv5.v",
  "module": "TM1",
  "machine": "1RB0LA_1LC1LA_0RD0LB_0RE1RC_1LB0RF_1RD---",
  "diff": "different table, same proof architecture. The nearest proved table by transitions is 1RB0LA_1LC1RD_0RB0LC_1LA1RE_1LF0RD_---1LE (distance 3, RWL_mod_v5.v, a decider certificate with no macro rules to reuse)",
  "applies": "partly",
  "note": "TranslatedBellv5 TM1 proves block-macro rules with ES_v3's symbolic executor (es' a b c d e over word^^n tapes), repeats them with ind, chains them with follow, and ends with progress_nonhalt_cond. Ours needs the same tools for different rules: es' for S (5 variables), TE (3) and TO (4); ind for S^m; follow for the double cycle. The final step is progress_nonhalt_simple over x, since no extra invariant is needed. Bouncer_v2 TM1 shows es' handling a one-variable bouncer BigStep; Bouncer_v3's multistep_c' runs a long opening with nested small counters. New here: the es' calls must cover 12-, 138- and 147-trip stretches."
 },
 "strategy": "Write K(n,a,b,c,d) for ES_v3. Prove S, TE and TO each with one es' call, S^m by ind, and Even(x) ->+ Even(x+32) by follow. Prove the 153,126,993-step opening with a nested multistep_c' and finish with progress_nonhalt_simple. 418 and 636 reuse the lemmas with new openings.",
 "risks": [
  "es' must find every shift rule across the 138- and 147-trip transitions within the fuel bound. Fallback: split TE and TO into one lemma per few trips.",
  "The opening is 153M steps, 11x the largest stepn in busycoq; N.to_nat of 153M is a 153M-deep unary nat. Use Bouncer_v3's nested multistep_c' (small counters), or begin from a smaller Even(x) if a shorter opening is found.",
  "Exponent normal forms: K's blocks need a representation that es' rewrites both ways (n = 2n'+1, a = 2a'+10 and so on), or the final config comparison fails."
 ],
 "checkpoint_questions": [
  {
   "text": "Fig 2 shows one double cycle from the real tape at step 153,126,993. Do you see two runs, each ending when the dotted fast line reaches the wall at the left? Does the wall step left after each, the second run's slow thin line end at the next run, and the bottom row look like the top?",
   "figs": [
    1
   ],
   "kind": "check"
  },
  {
   "text": "Fig 3 runs to 1e10 steps. Does every tooth after the first few belong to one of the same two kinds, in alternation, only longer each time, with no third kind of event?",
   "figs": [
    2
   ],
   "kind": "check"
  },
  {
   "text": "Plan: one ES_v3 file for this table (S, TE, TO, double cycle) with three openings, for 214, 418 (start C, mirrored) and 636 (start E). Go with all three in one stage, or prove 214 alone first?",
   "figs": [],
   "kind": "decision"
  }
 ],
 "discord_check": null,
 "stats": {
  "simulated_steps": 10000000000,
  "cells": 107321,
  "nonzeros": null,
  "growth": "span ~ 1.07 sqrt(steps): lo moves -2 per round trip, hi about +2.05; 26,506 trips to 1e10; right part (wall to edge) ~ 1,180 cells at 1e10, +48 per double cycle"
 },
 "status": "CHECKPOINT_A",
 "status_reason": "Mechanism closed. Every regime from blank to 1e10 is covered by validated rules: the opening to Even(212) at 153,126,993 steps, then S, TE and TO. The law Even(x) -> Even(x+32) was validated from constructed tapes up to x = 5,620. Not proved in Rocq yet."
}