{
 "tm": "1RB0LF_1RC0LB_1RD1RB_1RE1RF_1LA---_0RC1LA",
 "rank": 469,
 "class_sheet": "Counter Class 2",
 "class_guess": "two-zone block/pair shuffler (chaotic two-stack list machine), not a counter",
 "confidence": "med",
 "summary": "Blocks left of the head, pairs right of it; exact macro rules; halting needs an even zero run; parity holds, proof open.",
 "mechanism": [
  "Two zones split at the head: left = 1-blocks split by single 0s, right = (10)-pairs split by zero runs (plus a rare lone 1). A/F sweeps left turn a block into pairs; B/C passes right turn pairs into blocks. E-event K(L; a; r; R) = `L 1^a E> 0^r R`.",
  "Exact rules (12,000/12,000 random, all 34.5M E-events from blank to 1e9): even a -> Ee eats the last two closed blocks y, b: `K(L; y+3; b+1; (10)^(a/2) 0^(r-2) R)`. Odd a -> Eo closes a block 2, then a C-pass turns pairs into blocks until a run 0^n, n >= 3 (`a = x+2, r = n-2`) or a lone 1 (`a = y+3, r = x+1`). The block list is shuffled chaotically.",
  "Halting needs a C-pass to meet exactly 0^2. Closed blocks stay even and zero runs odd (1.7M E-events to 5e7), and every rule keeps this, but only under side conditions (no lone pair before a run; a = 2 only at the edge) that are not yet shown inductive."
 ],
 "diagrams": [
  {
   "mode": "default",
   "png": "img/default_site_1e4.png",
   "html": null,
   "steps": 10000,
   "caption": "bbchallenge Default view, first 10,000 steps",
   "title": "First 10,000 steps (site view)",
   "why": "The site's own picture. The head zigzags inside the tape; the right edge advances in small steps.",
   "primary": true,
   "start": "blank",
   "rows": "every step, steps 0..10,000 squeezed into 500 rows (site canvas, bilinear: about 20 steps per row)",
   "window": "cells -200..199, 1 px = 1 cell (cell 0 = start cell, at column 200)",
   "legend": "white = 1, black = 0 or unvisited, red/green = head (last move right/left)",
   "style": "bbchallenge-default"
  },
  {
   "mode": "every:1",
   "png": "img/cycle_36312.png",
   "html": null,
   "steps": 640,
   "caption": "Every step from an E-event at step 36,312",
   "title": "Sweeps at the block/pair interface",
   "why": "Left of the zigzag: solid 1-blocks. Right: fine (10) stripes and black zero runs. Compare both sides of each turn.",
   "primary": true,
   "start": "mid-run: step 36312 (E-event 0^inf 1^2 0 1^2 0 1^2 0 1^6 0 1^6 E> 0^3 (10)^3 0 (10)^3 0^15 ...)",
   "rows": "every step, steps 36,312..36,952 (641 rows)",
   "window": "cells -12..93, 1 px = 1 cell (absolute cells; the head starts at cell 10)",
   "legend": "white = 1, black = 0 or unvisited, red/green = head (last move right/left)",
   "style": "bbchallenge-default"
  },
  {
   "mode": "state:E",
   "png": "img/state_E_700.png",
   "html": null,
   "steps": 19041,
   "caption": "One row per entry into E, first 700",
   "title": "One row per E bounce (macro step)",
   "why": "Each row is one E-event of the model. Follow the red head: rows drifting left, then jumps right.",
   "primary": true,
   "start": "blank",
   "rows": "entries into state E: the first 700 (steps 0..19,041); each is one E-event of the macro model",
   "window": "cells -12..74, 1 px = 1 cell",
   "legend": "white = 1, black = 0 or unvisited, red/green = head (last move right/left)",
   "style": "bbchallenge-default"
  },
  {
   "mode": "records",
   "png": "img/records_2e9.png",
   "html": null,
   "steps": 2000000000,
   "caption": "Left/right records to 2e9",
   "title": "Records view to 2e9",
   "why": "At records the tape is all 1-blocks; black dots are the single-0 separators. Compare neighbouring rows.",
   "primary": false,
   "start": "blank",
   "rows": "left/right records: the start, then one row per new leftmost or rightmost cell (555 rows, steps 0..2,000,000,000)",
   "window": "cells -53..501, 1 px = 1 cell",
   "legend": "white = 1, black = 0 or unvisited, red/green = head (last move right/left)",
   "style": "bbchallenge-default"
  },
  {
   "mode": "every:500000",
   "png": "img/every5e5_1e9.png",
   "html": null,
   "steps": 1000000000,
   "caption": "Every 500,000th step to 1e9",
   "title": "Long run to 1e9 (sampled)",
   "why": "Speckle = tape caught mid-conversion between block and pair form. The span grows slowly.",
   "primary": false,
   "start": "blank",
   "rows": "every 500,000 steps, steps 0..1,000,000,000 (2,001 rows)",
   "window": "cells -47..414, 1 px = 1 cell",
   "legend": "white = 1, black = 0 or unvisited, red/green = head (last move right/left)",
   "style": "bbchallenge-default"
  },
  {
   "mode": "every:4",
   "png": "img/splice_halt_W1.png",
   "html": null,
   "steps": 2595,
   "caption": "Two reachable block stacks joined at one shared block: the machine halts",
   "title": "Spliced block stack halts at step 2,595",
   "why": "Both halves occur on the real run; the bottom row is the halt. Compare the left zigzag with Fig 2.",
   "primary": false,
   "start": "constructed: 0^inf 1^2 0 1^2 0 1^34 0 1^10 E> 0^15 (10)^5 0 (10)^3 0 (10)^3 0 (10)^7 0 (10)^5 0 (10)^13 0^13 (10)^8 1 (10)^5 0^inf",
   "rows": "every 4th step, steps 0..2,595 (650 rows; the machine halts at step 2,595)",
   "window": "cells -56..131, 1 px = 1 cell (head starts at cell 0)",
   "legend": "white = 1, black = 0 or unvisited, red/green = head (last move right/left)",
   "style": "bbchallenge-default"
  }
 ],
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  "skeleton": "right-edge E-events (r = inf, R empty): 0^inf 1^b1 0 1^b2 0 ... 0 1^bp 0 1^a E> 0^inf",
  "vars": [
   "p (closed blocks)",
   "a (head block)",
   "span (cells)",
   "closed blocks = 0 mod 4"
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 },
 "rules": [
  {
   "name": "Ee (even head block, a = 2k)",
   "from": "L 1^y 0 1^b 0 1^(2k) E> 0^r R",
   "to": "L 1^(y+3) E> 0^(b+1) (10)^k 0^(r-2) R",
   "steps": "a + b + 7",
   "cases": "k >= 1, r >= 3. r = 1: tail (10)^(k-1) 1 R (creates a lone 1). r = inf: tail (10)^k 0^inf. Missing y, b count as 0 (left edge: a' = 3, r' = 1, tape grows left by 1).",
   "validated": {
    "instances": 6398,
    "passed": 6398
   },
   "source": "manual"
  },
  {
   "name": "Eo (odd head block, a = 2k+1)",
   "from": "L 1^(2k+1) E> 0^r R",
   "to": "L 1^2 0 C> (10)^k 0^(r-2) R  (then the C-pass)",
   "steps": "a + 4 + C-pass",
   "cases": "r = 1: L 1^2 0 C> (10)^(k-1) 1 R. r = inf: L 1^2 0 C> (10)^k 0^inf.",
   "validated": {
    "instances": 5289,
    "passed": 5289
   },
   "source": "manual"
  },
  {
   "name": "C1-C2 (C-pass over pairs)",
   "from": "1^x C> (10)^j Q  |  1^x C> 0 (10)^j Q",
   "to": "1^(x+2j) C> Q  |  1^(x+2) 0 1^(2j-2) C> Q",
   "steps": "2j | 2j + 1",
   "cases": "C2 closes the block x+2 at a single 0; needs j >= 1 after it (always so).",
   "validated": {
    "instances": 5289,
    "passed": 5289
   },
   "source": "manual"
  },
  {
   "name": "C3 (C-pass meets 0^n, n >= 3 or inf)",
   "from": "L 1^x C> 0^n Q",
   "to": "K(L; x+2; n-2; Q)  =  L 1^(x+2) E> 0^(n-2) Q",
   "steps": "2",
   "cases": "n = 3 gives r = 1; n = inf is the right edge (right record, +4 cells).",
   "validated": {
    "instances": 4364,
    "passed": 4364
   },
   "source": "manual"
  },
  {
   "name": "C4 (C-pass meets a lone 1)",
   "from": "L 1^y 0 1^x C> 1 (10)^j Q",
   "to": "L 1^(y+3) E> 0^(x+1) (10)^(j-1) Q",
   "steps": "x + 5",
   "cases": "A zero run at the front of (10)^(j-1) Q merges into r. A missing y counts as 0.",
   "validated": {
    "instances": 925,
    "passed": 925
   },
   "source": "manual"
  },
  {
   "name": "H (halting pattern)",
   "from": "1^x C> 0 0 1",
   "to": "halt: C, D write 1, E reads 1",
   "steps": "3",
   "cases": "The only way E reads 1. It needs an even zero run of length 2 after a pair, which INV excludes.",
   "validated": {
    "instances": 313,
    "passed": 313
   },
   "source": "manual"
  }
 ],
 "template": {
  "file": "CounterClass2v1.v",
  "module": null,
  "machine": "1RB1RD_1LC---_1LE0LD_0RE1LC_1RA1RF_1RE0LF",
  "diff": "1 transition: our A0 = 1RB, template (in our names) A0 = 1LC; v2-v4 have 0LB there (v4 also E0 = 1LD)",
  "applies": "no",
  "note": "All four proved neighbours differ exactly at A0, the turn that starts rule Ee. v1 is a deterministic counter on a fixed skeleton (SkelE n, Bnext on (n, j, b), ClosedCall, GoodB). Here the block list is a shuffled stack, so only the top-level shape (progress_nonhalt_cond + an invariant) carries over."
 },
 "strategy": "Find an inductive invariant over the validated macro model: closed blocks even, runs odd, plus local context conditions. Search it mechanically (regular/bigram closure over abstract tokens, or a TM-level CTL certificate). Then write rules Ee/Eo/C1-C4 as es/ind lemmas and close with progress_nonhalt_cond; one proof would cover 469 and 599 (start D).",
 "risks": [
  "The context conditions may not be local: simple versions fail random closure (~2% of steps), and 'r = 1 implies next group >= 3 pairs' fails on the real run at step 2,403,364.",
  "Chaotic list dynamics: if no small invariant closes, a CTL certificate may be large; 85 small RWL_mod/NG settings returned false.",
  "No template help: the proved neighbours differ exactly in the slot that drives rule Ee."
 ],
 "checkpoint_questions": [
  {
   "text": "Sweeps at the block/pair interface: solid 1-blocks left of the zigzag, (10) stripes and black runs right of it. Does that match the two-zone picture, with the head as the boundary?",
   "figs": [
    1
   ],
   "kind": "check"
  },
  {
   "text": "One row per E bounce: do you see runs of rows where the head steps left (rule Ee eats blocks) followed by a jump right (the C-pass, ending at a zero run)?",
   "figs": [
    2
   ],
   "kind": "check"
  },
  {
   "text": "Should we fund one ~2 h stage to search for an inductive context invariant over the macro model (it would cover 469 and 599), or park both?",
   "figs": [],
   "kind": "decision"
  }
 ],
 "discord_check": null,
 "stats": {
  "simulated_steps": 100000000000,
  "cells": 779,
  "nonzeros": 349,
  "growth": "span 462 at 1e9, 779 at 1e11 (about t^0.23); +4 cells per right record"
 },
 "status": "CHECKPOINT_A",
 "status_reason": "Observe done at med confidence: exact macro model (34.5M/34.5M E-events from blank to 1e9, 12,000 random), halting reduced to one parity pattern; inductive invariant open."
}