{
 "tm": "1RB0LD_1RC0RF_0LD0LE_1RF1LC_0LB1LE_0RA---",
 "rank": 82,
 "class_sheet": "Skelet 1-like Class 1",
 "class_guess": "Base-4 counter on 10-regions in a 100 sea; overflow eats the left wall and spawns a digit at the right",
 "confidence": "high",
 "summary": "Exact base-4 counter: digits are 10-regions in a 100 sea; a mod-3 field law keeps merges clean; no halt to 1e15.",
 "mechanism": [
  "Tape at every C-record: 0^inf <wall> W^a D W^f ... D W^c 11 C> 0^inf, with W = 100 and digits D(m,p) = X (10)^m Y (m even, X in {10,11}, Y in {10,00}, phase p = 2x+y). One sweep adds 1 to this base-4 counter from the right; phases 1->2 and 3->0 shift the digit one W left; the right field gains 1 W.",
  "Every interior field keeps J = f + 2pL + floor(pR/2) - floor(pL/2) = 0 (mod 3), so a dying field always merges its two digits into one D(m+m'+2) and never goes negative. Digits are born at the right end on overflows and merge as they drift left: 10 digits at 1e14.",
  "Overflow: [10100] W^a -> [] W^a and [] W^a -> [10100] W^(a-2) (+ a birth). When the far-left field runs out, the wall eats the top digit through a finite remnant family. The three starts enter the same language by step 20,600, differing only in the far-left field; an exact sweep simulator matches the C core at 1e11 on all three and reaches 1e14 with no halt (82: 1e15)."
 ],
 "diagrams": [
  {
   "mode": "default",
   "png": "img/default_site_1e4.png",
   "html": null,
   "steps": 10000,
   "caption": "bbchallenge default view, first 10,000 steps",
   "title": "Default view, first 10,000 steps",
   "why": "The site's own picture: fine 3-cell stripes (the 100 sea) with a few thin defects, head bouncing at the right.",
   "primary": false,
   "start": "blank",
   "rows": "every step; 10,000 steps squeezed into 500 rows (site canvas)",
   "window": "cells -200..199, 1 px = 1 cell",
   "legend": "white = 1, black = 0, red/green = head moving R/L",
   "style": "bbchallenge-default"
  },
  {
   "mode": "every:1",
   "png": "img/sweeps_carry_closeup.png",
   "html": null,
   "steps": 700,
   "caption": "four sweeps of the counter, the second one carries",
   "title": "Four counter sweeps, one carry",
   "why": "Rows are steps. Count the trips from the right edge; compare how far left each one turns.",
   "primary": true,
   "start": "constructed: 0^inf (100)^6 10101010 (100)^9 1110 (100)^14 11 C> 0^inf",
   "rows": "every step, steps 0..700 (sweeps end at steps 123, 338, 485, 652)",
   "window": "cells -101..12, 1 px = 1 cell (drawn at 2x)",
   "legend": "white = 1, black = 0, red/green = head moving R/L",
   "style": "bbchallenge-default"
  },
  {
   "mode": "custom:digitmap",
   "png": "img/digit_worldlines_3e9.png",
   "html": null,
   "steps": 3000000000,
   "caption": "digits as white lines, one row per 10 million steps",
   "title": "Digit worldlines from blank to 3e9",
   "why": "Each white line is one digit. Follow lines from the right edge leftward; watch where two lines meet.",
   "primary": true,
   "start": "blank",
   "rows": "every 10000000 steps, steps 0..3,000,000,000 (302 rows, C core snapshots decoded by scripts/digitmap.py)",
   "window": "cells -86..133609, 1 px = 96 cells",
   "legend": "black = unvisited, dark grey = 100 field, white = digit (10-region X V^m Y), orange = other (wall remnant), red = head",
   "style": "custom"
  },
  {
   "mode": "every:40",
   "png": "img/wall_eats_digit.png",
   "html": null,
   "steps": 30000,
   "caption": "the left wall eats a D(2) digit over seven overflows",
   "title": "Left wall eating the top digit",
   "why": "Watch the dense 1010 block at the far left shrink while the 8-cell digit to its right drifts left.",
   "primary": true,
   "start": "constructed: 0^inf 100 11101010 00 (100)^21 1100 (100)^5 11 C> 0^inf",
   "rows": "every 40 steps, steps 0..30,000 (751 rows)",
   "window": "cells -105..34, 1 px = 1 cell (drawn at 2x)",
   "legend": "white = 1, black = 0, red/green = head moving R/L",
   "style": "bbchallenge-default"
  }
 ],
 "history": {
  "skeleton": "overflow outputs: 0^inf <wall> W^a D ... D W^c 11 C> 0^inf (82, exact sweep-level simulator)",
  "vars": [
   "wall",
   "a (far-left field)",
   "digits",
   "top digits",
   "right end"
  ],
  "rows": [
   {
    "step": 25780,
    "values": [
     "[1110]",
     3,
     1,
     "D0.0",
     "W^85"
    ]
   },
   {
    "step": 59542,
    "values": [
     "[]",
     85,
     2,
     "D0.0 D0.0",
     "W^73"
    ]
   },
   {
    "step": 155923,
    "values": [
     "[]",
     73,
     3,
     "D2.0 D0.0 D0.0",
     "W^96"
    ]
   },
   {
    "step": 375678,
    "values": [
     "[]",
     69,
     3,
     "D2.0 D0.0 D0.0",
     "W^289"
    ]
   },
   {
    "step": 927748,
    "values": [
     "[]",
     61,
     3,
     "D2.0 D2.0 D0.0",
     "W^289"
    ]
   },
   {
    "step": 3105598,
    "values": [
     "[10100]",
     54,
     4,
     "D2.0 D2.0 D0.0 D0.0",
     "W^769"
    ]
   },
   {
    "step": 8007849,
    "values": [
     "[10100]",
     50,
     3,
     "D6.0 D2.0 D0.0",
     "D0.0"
    ]
   },
   {
    "step": 18063033,
    "values": [
     "[]",
     33,
     4,
     "D6.0 D2.0 D2.0 D0.0",
     "W^384"
    ]
   },
   {
    "step": 57930182,
    "values": [
     "[10100]",
     22,
     5,
     "D6.0 D2.0 D2.0 D0.0 ...",
     "W^3073"
    ]
   },
   {
    "step": 136320257,
    "values": [
     "[10100]",
     18,
     5,
     "D6.0 D2.0 D2.0 D2.0 ...",
     "D0.0"
    ]
   },
   {
    "step": 437187099,
    "values": [
     "[]",
     9,
     6,
     "D6.0 D2.0 D2.0 D2.0 ...",
     "W^6144"
    ]
   },
   {
    "step": 1309352950,
    "values": [
     "[]",
     5,
     6,
     "D6.0 D2.0 D2.0 D2.0 ...",
     "W^18433"
    ]
   },
   {
    "step": 2882170778,
    "values": [
     "[1101010]",
     24,
     6,
     "D2.0 D6.0 D2.0 D0.0 ...",
     "W^12289"
    ]
   },
   {
    "step": 6675937712,
    "values": [
     "[]",
     9,
     6,
     "D2.0 D6.0 D2.0 D4.0 ...",
     "W^18433"
    ]
   },
   {
    "step": 15392730394,
    "values": [
     "[10100]",
     2,
     7,
     "D2.0 D6.0 D2.0 D4.0 ...",
     "W^49153"
    ]
   },
   {
    "step": 35297299805,
    "values": [
     "[1101010]",
     780,
     6,
     "D6.0 D2.0 D4.0 D0.0 ...",
     "D0.0"
    ]
   },
   {
    "step": 75936579781,
    "values": [
     "[10100]",
     758,
     7,
     "D6.0 D2.0 D4.0 D4.0 ...",
     "D0.0"
    ]
   },
   {
    "step": 187647088069,
    "values": [
     "[]",
     741,
     8,
     "D6.0 D2.0 D4.0 D4.0 ...",
     "W^98304"
    ]
   },
   {
    "step": 412334439524,
    "values": [
     "[]",
     737,
     8,
     "D6.0 D2.0 D4.0 D4.0 ...",
     "W^294913"
    ]
   },
   {
    "step": 975100850146,
    "values": [
     "[]",
     729,
     8,
     "D6.0 D2.0 D4.0 D4.0 ...",
     "W^294913"
    ]
   },
   {
    "step": 3229426254588,
    "values": [
     "[10100]",
     722,
     9,
     "D6.0 D2.0 D4.0 D4.0 ...",
     "W^786433"
    ]
   },
   {
    "step": 8340091896571,
    "values": [
     "[10100]",
     718,
     8,
     "D6.0 D2.0 D4.0 D4.0 ...",
     "D0.0"
    ]
   },
   {
    "step": 18753399764411,
    "values": [
     "[]",
     701,
     9,
     "D6.0 D2.0 D4.0 D4.0 ...",
     "W^393216"
    ]
   },
   {
    "step": 60364443428852,
    "values": [
     "[10100]",
     690,
     10,
     "D6.0 D2.0 D4.0 D4.0 ...",
     "W^3145729"
    ]
   },
   {
    "step": 95239838207576,
    "values": [
     "[]",
     689,
     10,
     "D6.0 D2.0 D4.0 D4.0 ...",
     "W^4718593"
    ]
   }
  ]
 },
 "rules": [
  {
   "name": "R1 increment (p = 0, 1, 2)",
   "from": "L W^a D(m,p) W^c 11 C> 0^inf",
   "to": "L W^(a-s) D(m,p+1) W^(c+1+s) 11 C> 0^inf, s = 1 if p = 1 else 0",
   "steps": "8c+11 (p = 0, 2); 8c+5m+23 (p = 1)",
   "cases": "m even; a >= s",
   "validated": {
    "instances": 700,
    "passed": 700
   },
   "source": "manual"
  },
  {
   "name": "R2 carry (p = 3)",
   "from": "L W^a D(m',q) W^f D(m,3) W^c 11 C> 0^inf",
   "to": "R1 applied to D(m',q) with W^f as its right field, D(m,3) -> D(m,0) shifted 1 W left; right field c+2",
   "steps": "8(c+f) + (5m+12) + stop(q, m'), stop = 11 or 5m'+23",
   "cases": "repeats leftward through every digit at phase 3",
   "validated": {
    "instances": 700,
    "passed": 700
   },
   "source": "manual"
  },
  {
   "name": "J invariant and merges",
   "from": "interior field W^f between D(.,pL) and D(.,pR) with f + 2pL + floor(pR/2) - floor(pL/2) = 0 (mod 3)",
   "to": "J stays 0 under R1/R2; a field reaching 0 is followed by D(m,2x+0)D(m',0+y) = D(m+m'+2, 2x+y), never by a negative field",
   "steps": null,
   "cases": "fields 0..18, glued digits, all phases",
   "validated": {
    "instances": 1986,
    "passed": 1986
   },
   "source": "manual"
  },
  {
   "name": "O1/O2 overflow",
   "from": "0^inf [10100 | empty] W^a (all digits at phase 3) W^c 11 C> 0^inf",
   "to": "[10100] W^a -> [] W^a, right W^(c+2); [] W^a -> [10100] W^(a-2), right W^(c+1) D(0,0) (birth) or W^(c+3)",
   "steps": null,
   "cases": "a >= 4; every digit shifts 1 W left; birth or no birth both stay in the language",
   "validated": {
    "instances": 300,
    "passed": 300
   },
   "source": "manual"
  },
  {
   "name": "Wall eats the top digit",
   "from": "[] W^1 D(m,0) ..., [] W^2 ..., [] W^3 ..., [10100] W^0..3 ...",
   "to": "remnant chain 11(10)^k{10,00} -> 1(10)^k{10,00} -> (10)^k{10,00} -> ... -> 110100 -> [] W^f (next field)",
   "steps": null,
   "cases": "80 small-field families (wall x a in 0..4 x m in 0..6 x 2 field sizes): all return to a clean wall, none halts",
   "validated": {
    "instances": 80,
    "passed": 80
   },
   "source": "manual"
  }
 ],
 "template": {
  "file": "verify/Sk1Machines.v",
  "module": "Sk1Counter / Sk1CounterMixed + TM17",
  "machine": "1LB1LE_1RC1LB_1LA0RD_1LC0RB_0LF---_0RD1LA",
  "diff": "different table; same architecture (digits with take/give/carry over a periodic word W, segRLs wall and one-digit lemmas)",
  "applies": "partly",
  "note": "mxdys's Sk1Counter matches our sweep rules (one digit step = segRLs with take/give/carry; W traversal = wall lemma). Its endgame does not: TM17 runs an accelerated simulation until a glider (translated cycle); ours grows forever (fields 3*2^k, 10 digits at 1e14), so the endgame must be a closed invariant (as in our #30), not a glider check."
 },
 "strategy": "Finish stage 1: write the wall/remnant case table as affine rules and run an exhaustive closure check of the candidate language L (J-valid fields, even digits, wall grammar). Then proof.md: one closed-language argument for L plus three entry lemmas (82, 464, 722).",
 "risks": [
  "The wall grammar is only seen to 1e14: the remnant prefixes (11, 1, empty, 10100, 1011) and collision chains need a complete, proved table.",
  "Formal size: the Sk1Counter-style lemmas exist upstream, but the closed-set argument over arbitrary digit lists is new work (#30 was 1,726 lines).",
  "Owner risk: the class is mxdys's (Sk1Machines.v); ask in #bb6 before stage 2."
 ],
 "checkpoint_questions": [
  {
   "text": "sweeps_carry_closeup: do you see four trips from the right edge, with the second one turning much further left than the others (it carries into the 8-cell digit)?",
   "figs": [
    1
   ],
   "kind": "check"
  },
  {
   "text": "digit_worldlines_3e9: do new white lines start at the right edge, drift left and slow down, and sometimes meet and continue as one line?",
   "figs": [
    2
   ],
   "kind": "check"
  },
  {
   "text": "wall_eats_digit: does the dense block at the far left get shorter in steps while everything to its right keeps its stripes?",
   "figs": [
    3
   ],
   "kind": "check"
  }
 ],
 "discord_check": "Is anyone working on 1RB0LD_1RC0RF_0LD0LE_1RF1LC_0LB1LE_0RA--- (Skelet 1-like Class 1)? Its other starts 1RB0RD_0LC0LF_1RD1LB_0RE---_1RA0LC_0LA1LF and 1RB1LE_0RC---_1RD0LA_1RE0RB_0LA0LF_0LD1LF are on the list too. It looks like a clean base-4 counter: digits are 10-regions in a 100 background, an overflow eats the left wall and spawns a new digit on the right, no glider in sight to 1e14. We'd like to try a closed-invariant proof for all three unless someone already has one.",
 "stats": {
  "simulated_steps": 1000000000000000,
  "cells": 896775,
  "nonzeros": 298934,
  "growth": "direct C core: span 67k at 1e9, 897k at 1e11 (82); exact sweep-level simulator: no halt to 1e14 on 82, 464 and 722, and to 1e15 on 82 (11 digits, right field 1.5e7 W)"
 },
 "status": "CHECKPOINT_A",
 "status_reason": "Stage-1 go signal: the interior counter is exact and closed (R1/R2, J mod 3, clean merges); overflow and wall rules are finite and affine as far as seen; no halt to 1e14 on any start. Remaining: complete the wall/remnant case table and check closure."
}