{
 "tm": "1RB0RA_1RC0RD_1RD0LC_1LE1RA_---1LF_1RB0LC",
 "rank": 317,
 "class_sheet": "Bell eats counter",
 "class_guess": "Bell eats counter, two bells: the IndBECv1 _3ary table (decider-proved from start B) run from start A",
 "confidence": "med",
 "summary": "Same table as a decider-proved machine; from start A it runs two bells into a wide-digit ternary counter",
 "mechanism": [
  "Bell + counter: a 00 marker in a field of 101 blocks moves one block right per pass (F); each pass first adds 1 to a base-3 counter on the right (K1-K6 = the cert's side_3ary rule, mirrored).",
  "Counter digits can be wide: d (110)^n 0 with d = 000/110/010 for 0/1/2 (n = 0 gives 0000/1100/0100); rules I0, I2, I1a, I1b. A digit right after a wide 2 uses 0011/1111/0111 (J0, J1).",
  "From A the tape holds two big bells that take turns (the idle one is a wide digit), one small wide digit and 4-cell digits, up to 5e9; start B has one bell. Bell hand-over (E) and the carry past J2 are not pinned. Only halt: A> 0 1 0; none in 1e10 steps."
 ],
 "diagrams": [
  {
   "mode": "default",
   "png": "img/default_site.png",
   "html": null,
   "steps": 10000,
   "caption": "bbchallenge default view, first 10,000 steps",
   "title": "bbchallenge default view, first 10,000 steps",
   "why": "The site's own view. Look at the zigzag fill and the two edges growing at different speeds.",
   "primary": true,
   "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)",
   "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": "every:1",
   "png": "img/pass_closeup.png",
   "html": null,
   "steps": 1400,
   "caption": "Close-up of 1,400 steps from step 2,980,444",
   "title": "Close-up: bell steps and counter trips",
   "why": "One row per step. Follow the head: short bounces between two columns near x = 270..318, then long trips right.",
   "primary": true,
   "start": "mid-run: step 2,980,444 (config 0^inf (101)^89 00 (101)^15 11 C> (110)^82 ... , head at cell 120)",
   "rows": "every step, steps 2,980,444..2,981,843 (the first 1,400 rows of a 6,000-step trace)",
   "window": "cells -198..387, 1 px = 1 cell (absolute positions; the whole visited span at that time)",
   "legend": "white = 1, black = 0 or unvisited, red/green = head (last move right/left)",
   "style": "bbchallenge-default"
  },
  {
   "mode": "records",
   "png": "img/records_1e8.png",
   "html": null,
   "steps": 100000000,
   "caption": "Left/right records to 1e8 steps, start A (ours)",
   "title": "Records from blank, start A (ours)",
   "why": "One row per new leftmost or rightmost cell. Compare the vertical streaks inside with Fig 4 (same table, start B).",
   "primary": true,
   "start": "blank",
   "rows": "left/right records only: the start, then one row per new leftmost or rightmost cell (1,836 rows to step 100,000,000; not evenly spaced in time)",
   "window": "cells -624..1211, 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_startB_1e8.png",
   "html": null,
   "steps": 100000000,
   "caption": "Same table started in state B (the decider-proved start)",
   "title": "Records from blank, start B (certified)",
   "why": "The IndBECv1 _3ary nonhalt3 run in our orientation: one field, one marker drifting along it.",
   "primary": false,
   "start": "constructed: 0^inf B> 0^inf",
   "rows": "left/right records only (1,613 rows to step 100,000,000; not evenly spaced in time)",
   "window": "cells -530..1082, 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_1e8.png",
   "html": null,
   "steps": 100000000,
   "caption": "Blaze, 100 million steps",
   "title": "Blaze, 100 million steps",
   "why": "Overall growth: both edges keep moving out, the right one faster.",
   "primary": false,
   "start": "blank",
   "rows": "Blaze time blocks of 131,072 steps (each averaged), 763 blocks to step 1e8, stretched to 500 px",
   "window": "cells -624..1211, 1 px = 2.3 cells (Blaze image: 1 column per 2 cells, 918 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",
   "style": "bbchallenge-blaze"
  }
 ],
 "history": {
  "skeleton": "0^inf (101)^a 0^2 (101)^b 1^4 (101)^c 0^2 (101)^d 1^e (110)^f 1^g C> 0^inf",
  "vars": [
   "a",
   "b",
   "c",
   "d",
   "e",
   "f",
   "g"
  ],
  "rows": [
   {
    "step": 725418061,
    "values": [
     2,
     605,
     152,
     413,
     7,
     1,
     19
    ]
   },
   {
    "step": 729111543,
    "values": [
     3,
     605,
     152,
     413,
     7,
     2,
     19
    ]
   },
   {
    "step": 755105777,
    "values": [
     3,
     612,
     134,
     432,
     3,
     8,
     23
    ]
   },
   {
    "step": 762690155,
    "values": [
     3,
     614,
     144,
     431,
     7,
     1,
     19
    ]
   },
   {
    "step": 781826007,
    "values": [
     2,
     620,
     132,
     448,
     3,
     1,
     23
    ]
   },
   {
    "step": 785704865,
    "values": [
     3,
     620,
     129,
     453,
     26,
     0,
     0
    ]
   },
   {
    "step": 789597903,
    "values": [
     3,
     621,
     128,
     454,
     3,
     1,
     23
    ]
   },
   {
    "step": 793497627,
    "values": [
     2,
     623,
     126,
     458,
     26,
     0,
     0
    ]
   },
   {
    "step": 797529303,
    "values": [
     3,
     623,
     155,
     429,
     11,
     1,
     15
    ]
   },
   {
    "step": 801454357,
    "values": [
     3,
     624,
     154,
     430,
     11,
     2,
     15
    ]
   },
   {
    "step": 805316455,
    "values": [
     2,
     626,
     133,
     452,
     7,
     2,
     19
    ]
   },
   {
    "step": 809239461,
    "values": [
     3,
     626,
     124,
     461,
     7,
     3,
     19
    ]
   },
   {
    "step": 825207971,
    "values": [
     3,
     630,
     144,
     447,
     11,
     1,
     15
    ]
   },
   {
    "step": 829186073,
    "values": [
     2,
     632,
     133,
     458,
     7,
     2,
     19
    ]
   },
   {
    "step": 833190677,
    "values": [
     3,
     632,
     125,
     466,
     7,
     3,
     19
    ]
   },
   {
    "step": 837207235,
    "values": [
     3,
     633,
     119,
     473,
     3,
     3,
     23
    ]
   },
   {
    "step": 841243123,
    "values": [
     2,
     635,
     115,
     477,
     3,
     4,
     23
    ]
   },
   {
    "step": 845296375,
    "values": [
     3,
     635,
     111,
     481,
     3,
     5,
     23
    ]
   },
   {
    "step": 849385305,
    "values": [
     3,
     636,
     113,
     485,
     26,
     0,
     0
    ]
   },
   {
    "step": 854266203,
    "values": [
     2,
     638,
     393,
     205,
     19,
     1,
     7
    ]
   },
   {
    "step": 857931347,
    "values": [
     3,
     638,
     204,
     395,
     15,
     1,
     11
    ]
   },
   {
    "step": 895020275,
    "values": [
     3,
     647,
     102,
     500,
     3,
     7,
     23
    ]
   },
   {
    "step": 942242719,
    "values": [
     2,
     659,
     93,
     525,
     3,
     2,
     23
    ]
   },
   {
    "step": 955468597,
    "values": [
     2,
     662,
     97,
     524,
     7,
     2,
     19
    ]
   },
   {
    "step": 977717875,
    "values": [
     3,
     666,
     92,
     535,
     7,
     1,
     19
    ]
   },
   {
    "step": 982189411,
    "values": [
     2,
     668,
     84,
     544,
     3,
     1,
     23
    ]
   },
   {
    "step": 986707917,
    "values": [
     3,
     668,
     81,
     549,
     26,
     0,
     0
    ]
   },
   {
    "step": 991251617,
    "values": [
     3,
     669,
     82,
     548,
     7,
     1,
     19
    ]
   },
   {
    "step": 995809893,
    "values": [
     2,
     671,
     85,
     545,
     7,
     2,
     19
    ]
   }
  ]
 },
 "rules": [
  {
   "name": "K1 digit 0 -> 1",
   "from": "l 1 C> 0000 r",
   "to": "l <C 0 1100 r",
   "steps": "4",
   "cases": "any l, r",
   "validated": {
    "instances": 200,
    "passed": 200
   },
   "source": "manual"
  },
  {
   "name": "K2 digit 1 -> 2",
   "from": "l 1 C> 1100 r",
   "to": "l <C 0 0100 r",
   "steps": "2",
   "cases": "any l, r",
   "validated": {
    "instances": 200,
    "passed": 200
   },
   "source": "manual"
  },
  {
   "name": "K3 digit 2 carries",
   "from": "l 1 C> 0100 r",
   "to": "l 1111 1 C> r",
   "steps": "4",
   "cases": "any l, r",
   "validated": {
    "instances": 200,
    "passed": 200
   },
   "source": "manual"
  },
  {
   "name": "K4 carry cleared on the way back",
   "from": "l 1111 <C 0 r",
   "to": "l <C 0 0000 r",
   "steps": "4",
   "cases": "any l, r",
   "validated": {
    "instances": 200,
    "passed": 200
   },
   "source": "manual"
  },
  {
   "name": "K5 top digit 1 -> 2",
   "from": "l 1 C> 11 0^inf",
   "to": "l <C 0 01 0^inf",
   "steps": "2",
   "cases": "any l",
   "validated": {
    "instances": 200,
    "passed": 200
   },
   "source": "manual"
  },
  {
   "name": "K6 top digit 2 overflows",
   "from": "l 1 C> 01 0^inf",
   "to": "l <C 0 0000 11 0^inf",
   "steps": "12",
   "cases": "any l",
   "validated": {
    "instances": 200,
    "passed": 200
   },
   "source": "manual"
  },
  {
   "name": "F bell step (marker moves one block)",
   "from": "l (101)^a 00 (101)^(b+1) 1 <C 0 r",
   "to": "l (101)^(a+1) 00 (101)^b 11 C> r",
   "steps": "8b+16",
   "cases": "a >= 1, b >= 0, any l, r",
   "validated": {
    "instances": 200,
    "passed": 200
   },
   "source": "manual"
  },
  {
   "name": "I0 wide digit 1 -> 2",
   "from": "l 1 C> 110 (110)^n r",
   "to": "l <C 0 010 (110)^n r",
   "steps": "2",
   "cases": "n >= 1 (n = 0 is K2), any l, r",
   "validated": {
    "instances": 200,
    "passed": 200
   },
   "source": "manual"
  },
  {
   "name": "I2 wide digit 0 -> 1",
   "from": "l 1 C> 000 (110)^n r",
   "to": "l <C 0 (110)^(n+1) r",
   "steps": "4",
   "cases": "n >= 1 (n = 0 is K1 without the separator), any l, r",
   "validated": {
    "instances": 200,
    "passed": 200
   },
   "source": "manual"
  },
  {
   "name": "I1a wide digit 2 carries (crossing)",
   "from": "l 1 C> 010 (110)^n 0 r",
   "to": "l 1111 (011)^n 1 C> r",
   "steps": "3n+4",
   "cases": "n >= 0 (n = 0 is K3), any l, r",
   "validated": {
    "instances": 200,
    "passed": 200
   },
   "source": "manual"
  },
  {
   "name": "I1b wide carry cleared on return",
   "from": "l 1111 (011)^n <C 0 r",
   "to": "l <C 0 000 (110)^n 0 r",
   "steps": "5n+4",
   "cases": "n >= 0 (n = 0 is K4), any l, r",
   "validated": {
    "instances": 200,
    "passed": 200
   },
   "source": "manual"
  },
  {
   "name": "J0 digit after a wide 2: 0 -> 1",
   "from": "l 1 C> 010 (110)^n 0 0011 r",
   "to": "l <C 0 000 (110)^n 0 1111 r",
   "steps": "8n+12",
   "cases": "n >= 1, any l, r (the digit's sweep reaches back into the crossed wire)",
   "validated": {
    "instances": 200,
    "passed": 200
   },
   "source": "manual"
  },
  {
   "name": "J1 digit after a wide 2: 1 -> 2",
   "from": "l 1 C> 010 (110)^n 0 1111 r",
   "to": "l <C 0 000 (110)^n 0 0111 r",
   "steps": "8n+10",
   "cases": "n >= 1, any l, r",
   "validated": {
    "instances": 200,
    "passed": 200
   },
   "source": "manual"
  },
  {
   "name": "J2 digit after a wide 2: 2 carries",
   "from": "l 1111 (011)^n 1 C> 0111 r",
   "to": "(digit -> 1100-like, head leaves right in state A: ...11111 00 A> r)",
   "steps": null,
   "cases": "next digit then read from an A arrival; encoding not pinned",
   "validated": {
    "instances": 0,
    "passed": 0
   },
   "source": "manual"
  },
  {
   "name": "E bell hand-over",
   "from": "l 111 (110)^k 100 (101)^3 11 C> R  (active bell, marker 3 blocks from its end)",
   "to": "l' 11 C> 000 (110)^(k+4) R'  (the bell becomes a wide digit 0; the bell to its left takes one step; R' = R after one visit)",
   "steps": null,
   "cases": "seen in traces only (2,011,920 -> 2,013,101; 2,082,683 -> 2,083,906; ~1.0e8)",
   "validated": {
    "instances": 0,
    "passed": 0
   },
   "source": "manual"
  }
 ],
 "template": {
  "file": "IndBECv1.v:258",
  "module": "_3ary (nonhalt3)",
  "machine": "1LB0LC_1LC0RB_1RD1LF_---1RE_1LA0RB_1LA0LF",
  "diff": "same table: mirror + rename (ours A,B,C,D,E,F = cert F,A,B,C,D,E); the cert starts in our B, we start in A (our first step writes the one extra 1)",
  "applies": "partly",
  "note": "The cert's rule set = K1-K6 (side_3ary_Pos_inc_rule, QL = QR = our C) + one bell found by the hlin decider (config_T_bsz 745 3). Ours adds a second big bell and wide counter digits. Same config on our flipped table: decider false at T = 746, 1000, 2000, 5000, 20000, and with mnc 3 or 4 at T = 5000 (~2 min each); sibling 707 false at T = 745. On the cert: proved in 4.4 s."
 },
 "strategy": "Finish the small decider grid (mnc 3/4, block size 0, sibling 707). If none proves it, write a hand proof: an invariant of two bells + ternary counter whose low digits are idle fields; prove F, K1-K6, I0-I2 and the two missing rules (I1, E) in Rocq, starting from BellEatsCounter_1RB0RF.v (upstream HEAD) or the _3ary config, then flip to our orientation (Flip.v).",
 "risks": [
  "Bell hand-over (E) and the carry past a digit after a wide 2 (J2) are seen only in traces; their exact case split is not known.",
  "Field count is only observed bounded (2 big bells + 1 small wide digit up to 5e9); an invariant needs it proved.",
  "Hand proof is a multi-stage job; the cheap decider route has failed on 8 configs so far."
 ],
 "checkpoint_questions": [
  {
   "text": "Fig 3 vs Fig 4: does ours (start A) look like the certified start B plus a second bell (the vertical streaks), rather than a different mechanism?",
   "figs": [
    2,
    3
   ],
   "kind": "check",
   "answer": "yes",
   "date": "2026-10-12",
   "note": "Yes, although ours has more 'drippiness' on the upper right edge of the triangle and a white streak running from the apex diagonally down and to the right, while the certified one has several black streaks running from the upper right edge diagonally down and to the left. Asked whether 'plus a second bell (the vertical streaks)' means the drippiness or something else. Answer: the white streak is the second bell (the boundary between field 1 and field 2, cell 959 of 1,836 at 1e8, right after field 1's 319 blocks of 101); the drippiness is the counter at the right end, whose digits have mixed widths and sometimes run as small nested bells."
  },
  {
   "text": "Close-up: do you see short bounces between two nearby columns (one bell step each) and occasional trips to the right end (counter visits)?",
   "figs": [
    1
   ],
   "kind": "check",
   "answer": "yes",
   "date": "2026-10-12",
   "note": ""
  },
  {
   "text": "Should we spend one more stage on decider configs (cheap, ~2 min per try) before building the hand proof (multi-stage)?",
   "kind": "decision",
   "answer": "yes",
   "date": "2026-10-12",
   "note": "'sure'. Already superseded: the decider-config stage (decider2) ran on 2026-10-12 before the hand-proof stages; the open call is now decision.json (grammar2 vs park)."
  }
 ],
 "discord_check": null,
 "stats": {
  "simulated_steps": 10000000000,
  "cells": 8464,
  "nonzeros": 5627,
  "growth": "span ~ t^0.33 (1,836 cells at 1e8, 8,464 at 1e10); bells 674/630 blocks at 1e9, 1,144/1,071 at 5e9; no halt to 1e10"
 },
 "status": "CHECKPOINT_A",
 "status_reason": "Mechanism understood at the component level (bell step, base-3 counter with wide digits; 13 rules validated 200/200); bell hand-over (E) and carry J2 not pinned; the certified sibling's decider config fails on our start."
}