{
 "tm": "1RB0LE_1RC0RC_1LD0RB_1LA0LE_1LD1LF_1LD---",
 "rank": 11,
 "class_sheet": "Skelet 1-like Class 2",
 "class_guess": "Binary counter over a block list with re-pairing overflow (TM21-like Skelet 1-like Class 2)",
 "confidence": "high",
 "summary": "Binary counter on (10)^u[00](01)^v blocks; carries + re-pairing overflow explain every step to 1e9",
 "mechanism": [
  "Tape = digits (10)^u J (01)^v (J = 00: bit 0, empty: bit 1), then a growing (10)^a 000 (10)^b block at the right end.",
  "Every 4 right records = +1: the head runs left to the first 0-bit (ruler-depth carry); each touched v drops by 1.",
  "All bits 1 -> overflow: a run to the far left re-pairs the blocks. A digit whose v runs out fuses with its neighbour."
 ],
 "diagrams": [
  {
   "mode": "every:10",
   "png": "img/counter_from_K.png",
   "html": null,
   "steps": 5600,
   "caption": "From a constructed K-config: ruler-depth carries, then one overflow to left",
   "title": "Carry sweeps from a built counter tape",
   "why": "Every 10 steps from a built counter tape. Dotted diagonals are head sweeps; compare how deep each one goes.",
   "primary": true,
   "start": "constructed: 0^inf 10 01 (10)^2 00 (01)^3 (10)^3 00 (01)^5 (10)^3 00 (01)^9 (10)^3 00 (01)^17 (10)^4 0^3 B> 0^inf",
   "rows": "every 10 steps, steps 0..5,600 after the start (561 rows, 3 px each; no head drawn on row 0). The head moves up to 10 cells between rows, so each sweep shows as a dotted diagonal",
   "window": "cells -116..66, 1 px = 1/3 cell (each cell 3x3 px); cell 0 = the head's start cell",
   "legend": "white = 1, black = 0 or unvisited, red/green = head (last move right/left)",
   "style": "bbchallenge-default"
  },
  {
   "mode": "records",
   "png": "img/records_3e5.png",
   "html": "img/viewer.html",
   "steps": 300000,
   "caption": "Dashed walls = digits; each band break = one overflow",
   "title": "Edge records to 300,000 steps",
   "why": "Edge records to 300,000 steps. Look at the dashed vertical walls and each band break (left edge steps out).",
   "primary": true,
   "start": "blank",
   "rows": "left/right records: the start, then one row per new leftmost or rightmost cell (1,153 rows to step 300,000; not evenly spaced in time)",
   "window": "cells -57..1095, 1 px = 1 cell",
   "legend": "white = 1, black = 0 or unvisited, red/green = head (last move right/left); the 1010 background is 1-px stripes that look gray when shrunk",
   "style": "bbchallenge-default"
  },
  {
   "mode": "records",
   "png": "img/records_1e7.png",
   "html": null,
   "steps": 5122482,
   "caption": "5000 records (to step 5.1e6): digit walls flip and drift",
   "title": "Edge records to 5.1 million steps",
   "why": "The same view to 5.1M steps. Compare its walls and band breaks with Fig 2.",
   "primary": true,
   "start": "blank",
   "rows": "left/right records: the start, then one row per new record; stopped at the 5,000-row cap at step 5,122,482 (not evenly spaced in time)",
   "window": "cells -87..4912, 1 px = 1 cell",
   "legend": "white = 1, black = 0 or unvisited, red/green = head (last move right/left); the 1010 background is 1-px stripes that look gray when shrunk",
   "style": "bbchallenge-default"
  },
  {
   "mode": "default",
   "png": "img/default_site.png",
   "html": "img/viewer.html",
   "steps": 10000,
   "caption": "bbchallenge Default view, 10k 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"
  },
  {
   "mode": "blaze",
   "png": "img/blaze_1e7.png",
   "html": null,
   "steps": 10000000,
   "caption": "Blaze 1e7: span grows ~t^0.5, faint wall lines",
   "title": "Blaze view, 10 million steps",
   "why": "The right edge grows like the square root of time; the faint vertical lines are digit walls.",
   "primary": false,
   "start": "blank",
   "rows": "Blaze time blocks of 16,384 steps (each averaged), 611 blocks to step 1e7, stretched to 500 px (1 px = about 20,000 steps)",
   "window": "cells -104..6903, 1 px = 8.8 cells (Blaze image: 1 column per 8 cells, averaged; 876 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; the 1010 background averages to pale orange",
   "style": "bbchallenge-blaze"
  }
 ],
 "history": {
  "skeleton": "0^inf 10 01 [(10)^u_i (01)^v_i]_{i=1..n} (10)^a 0^3 (10)^b B> 0^inf",
  "vars": [
   "n",
   "u_1..u_n",
   "v_1..v_n",
   "a",
   "b"
  ],
  "rows": [
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  ]
 },
 "rules": [
  {
   "name": "M_k carry (k = trailing 1-bits)",
   "from": "l (10)^u_k 0^2 (01)^v_k [(10)^u_i (01)^v_i]_{i=k-1..0} (10)^a 0^3 (10)^b B> 0^inf",
   "to": "l (10)^(u_k+1) (01)^(v_k-1) [(10)^u_i 0^2 (01)^(v_i-1)]_{i=k-1..0} (10)^(a+1) 0^3 (10)^(b+2) B> 0^inf",
   "steps": "32 + 8a + 8b + 8*sum_{i<k} u_i + 4*sum_{i<=k} v_i",
   "cases": "k=0..5 tested; needs v_i>=1 (i<=k), u_i>=1 (i<k), u_k>=0, a>=1, b>=0; v=1 gives a fused (merged) digit",
   "validated": {
    "instances": 84,
    "passed": 84
   },
   "source": "manual"
  },
  {
   "name": "O overflow (all bits 1)",
   "from": "0^inf 10 01 [(10)^u_i (01)^v_i]_{i=1..n} (10)^a 0^3 (10)^b B> 0^inf",
   "to": "0^inf 10 01 00 (01)^u_1 [(10)^(v_i-1) 0^2 (01)^u_(i+1)]_{i=1..n-1} (10)^v_n (01)^a (10)^(b+3) 0^3 10 B> 0^inf",
   "steps": "89 - 4n + 8a + 20b + 4*sum u_i + 8*sum v_i",
   "cases": "n=1..7 tested; needs v_i>=1 (i<n); left edge moves 3 cells; counter restarts at 1",
   "validated": {
    "instances": 118,
    "passed": 118
   },
   "source": "manual"
  },
  {
   "name": "init (blank tape to first K)",
   "from": "0^inf A> 0^inf",
   "to": "0^inf 10 01 (10)^3 01 (10)^2 0^3 10 B> 0^inf",
   "steps": "142",
   "cases": "twins: start B -> K((1,2);2,1) in 122 steps; start C -> K((1,1);2,1) in 101 steps",
   "validated": {
    "instances": 1,
    "passed": 1
   },
   "source": "manual"
  },
  {
   "name": "B sweep right",
   "from": "B> (01)^n",
   "to": "(10)^n B>",
   "steps": "2n",
   "cases": null,
   "validated": {
    "instances": 10,
    "passed": 10
   },
   "source": "manual"
  },
  {
   "name": "D sweep left",
   "from": "(01)^n <D",
   "to": "<D (10)^n",
   "steps": "2n",
   "cases": null,
   "validated": {
    "instances": 10,
    "passed": 10
   },
   "source": "manual"
  }
 ],
 "template": {
  "file": "vendor/busycoq/verify/SBCv6.v",
  "module": "TM21",
  "machine": "1RB1LB_0LC1RD_0RA1LC_0RB1RE_0RD0LF_---1LE",
  "diff": "10 of 12 transitions differ under the best renaming/mirror (Sk1like_TC_v2 TM1: 9 of 12)",
  "applies": "partly",
  "note": "Same architecture (LIncs = carries, Ov = re-pairing overflow, BigStep), but TM21's S' configs never recur here (0 real hits in 2e5 steps, all 6 states). Its LWF invariant fails: here digits run out and merge."
 },
 "strategy": "Copy TM21's shape: K(list,a,b) as a Fixpoint over (u,bit,v) blocks; prove M_k by induction on k (like LIncs_spec) and O by induction on the list (like Ov_spec) from es-level sweep lemmas. Then nonhalt via multistep_nonhalt + progress_nonhalt_cond with a merge-aware invariant that keeps every v reached by a carry >= 1. Same rules give the 2 twins (new init lemmas only).",
 "risks": [
  "Invariant is the real work: digits do run out and merge (9 -> 5 digits in one cycle); TM21's LWF does not carry over.",
  "Twin on the 'worked on extensively' tab (hipparcos, Oct 2025 thread) is this same table: duplicate-work risk until he answers.",
  "Size: list bookkeeping plus merge cases likely 20-50 KB of Rocq (TM21 is ~300 lines)."
 ],
 "checkpoint_questions": [
  {
   "text": "Fig 1: do the sweep depths follow a ruler pattern (shallow, deeper, shallow, deepest), ending in one sweep to the left edge?",
   "figs": [
    0
   ]
  },
  {
   "text": "Fig 2: does the tape look like blocks separated by dashed walls, with the whole pattern reshuffling at each band break?",
   "figs": [
    1
   ]
  },
  {
   "text": "Fig 3: further out, are the walls and band breaks the same kind as in Fig 2, with nothing new appearing?",
   "figs": [
    2,
    1
   ]
  }
 ],
 "discord_check": "see reports/discord-drafts.md message 2",
 "stats": {
  "simulated_steps": 1000000000,
  "cells": 77685,
  "nonzeros": 38836,
  "growth": "right edge ~t^0.51; left edge -3 cells per overflow (54 by 1e9; rules-only model: 122 by 1e15, 23 digits)"
 },
 "status": "CHECKPOINT_A",
 "status_reason": "M_k + O validated, exact to 1e9; rules-only model runs to 1e15 with no stuck case. Not cryptid. Ask hipparcos first (same table)."
}