{
 "tm": "1RB0RA_1LC0LF_1LE0LD_1LB---_1RA1LB_0LB0RC",
 "rank": 29,
 "class_sheet": "Bell eats counter",
 "class_guess": "Bell eats counter: BECv1 tm108 twin (E1) with 3-cell counter digits and a grid-shifting eat",
 "confidence": "med",
 "summary": "BECv1 tm108 twin (E1): 3-cell binary counter, unary, marker; rounds ruled (99.7%), the grid-shifting eat is open",
 "mechanism": [
  "Tape: counter W of 3-cell units (0 = 000, 1 = 100, spacer 010; MSB left), unary (010)^b, marker 00, then (101)^c. Only E1 differs from tm108: B carries, so digits are 3-cell units.",
  "Each round the marker steps one unit left (b-1, c+1) and W += 1 (B carries over 100 and spacers, turns at 000); an overflow adds a 3-cell digit at the left end.",
  "At b = 0 the marker eats W: the 3-cell grid shifts by one cell (1-digits -> spacers, spacers -> 001), c becomes the new unary, the marker is reborn at the right, then mini-eats re-align W. N ~ (4t)^(1/3); no halt to 1e10."
 ],
 "diagrams": [
  {
   "mode": "default",
   "png": "img/default_site.png",
   "html": "img/viewer.html",
   "steps": 10000,
   "caption": "bbchallenge view: sweeps between counter and marker",
   "title": "bbchallenge default view, first 10,000 steps",
   "why": "The site's own view. Each zigzag is one round between the counter (left turns) and the marker (right turns).",
   "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, so 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": "default",
   "png": "img/eat_closeup_19500.png",
   "html": null,
   "steps": 900,
   "caption": "Close-up of the last rounds before an eat and after it",
   "title": "One eat, step by step",
   "why": "One row per step. Follow the right turning point leftwards until it meets the left turns, then the run right.",
   "primary": true,
   "start": "mid-run: step 19,500 (blank run; config in img/eat_start_19500.txt)",
   "rows": "every step 19,500..20,399 (900 rows, each 2 px tall)",
   "window": "cells -110..39, 1 px = 0.5 cells (each cell 2x2 px); cell numbers are those of the blank run (start cell = 0)",
   "legend": "white = 1, black = 0 or unvisited, red/green = head (last move right/left)",
   "style": "bbchallenge-default"
  },
  {
   "mode": "blaze",
   "png": "img/blaze_1e9.png",
   "html": null,
   "steps": 1000000000,
   "caption": "Blaze, 1e9 steps: smooth t^(1/3) growth, epoch banding",
   "title": "Blaze, 1 billion steps",
   "why": "Overall shape only. Compare the moving left edge with the nearly fixed right edge; faint horizontal bands are epochs.",
   "primary": true,
   "start": "blank",
   "rows": "Blaze time blocks of 1,048,576 steps (each averaged), 954 blocks to step 1e9, stretched to 500 px (1 px = about 2 million steps)",
   "window": "cells -4288..527, 1 px = 6 cells (Blaze image: 1 column per 4 cells, 1,204 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"
  },
  {
   "mode": "records:L",
   "png": "img/recordsL_left_1e7.png",
   "html": "img/viewer.html",
   "steps": 10000000,
   "caption": "Left records, left edge pinned: counter overflows",
   "title": "Left records, left edge pinned",
   "why": "One row per new leftmost cell; an overflow makes three. The pinned edge shows the counter word between epochs.",
   "primary": false,
   "start": "blank",
   "rows": "left records only: the start, then one row per new leftmost cell (839 rows to step 10,000,000; 3 rows per counter overflow; not evenly spaced in time)",
   "window": "cells 0..149, 1 px = 0.5 cells (each cell 2x2 px); counted from each row's leftmost visited cell (frame = left), the rest of the tape is cut off",
   "legend": "white = 1, black = 0 or unvisited, red/green = head (last move right/left)",
   "style": "bbchallenge-default"
  }
 ],
 "history": {
  "skeleton": "0^inf <B W (101)^b 00 (101)^c 0^inf   (at each left record; W over d = 111, S = 101)",
  "vars": [
   "W (d = digit, S = spacer; MSB left)",
   "D (digits in W)",
   "b",
   "c"
  ],
  "rows": [
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    "values": [
     "ddd",
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    "values": [
     "dddd",
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     15,
     15
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    "values": [
     "d",
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     1
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   },
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    "values": [
     "dd",
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     3
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   },
   {
    "step": 15062,
    "values": [
     "ddd",
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   {
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    "values": [
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   },
   {
    "step": 19592,
    "values": [
     "ddddd",
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   {
    "step": 22402,
    "values": [
     "SddSd",
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   },
   {
    "step": 24761,
    "values": [
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     23,
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   {
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    "values": [
     "ddSddSd",
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   {
    "step": 30808,
    "values": [
     "SSdSdd",
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    "step": 33397,
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    "values": [
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    "values": [
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    "step": 103300017,
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    "values": [
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    "values": [
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    "values": [
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    "values": [
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    "values": [
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    "step": 9913826129,
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   {
    "step": 9934141962,
    "values": [
     "ddSSSdddddSSSdSdSSSdd",
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     1469,
     2047
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   },
   {
    "step": 9975773479,
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   {
    "step": 9996120762,
    "values": [
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  ]
 },
 "rules": [
  {
   "name": "R1 A over spacer (= BECv1 HR1)",
   "from": "l A> 101^n r",
   "to": "l 010^n A> r",
   "steps": "7n",
   "cases": null,
   "validated": {
    "instances": 60,
    "passed": 60
   },
   "source": "manual"
  },
  {
   "name": "R2 A over carried digit",
   "from": "l A> 111 r",
   "to": "l 000 A> r",
   "steps": "3",
   "cases": null,
   "validated": {
    "instances": 60,
    "passed": 60
   },
   "source": "manual"
  },
  {
   "name": "R3 B over spacer (= HR3)",
   "from": "l 010^n <B r",
   "to": "l <B 101^n r",
   "steps": "3n",
   "cases": null,
   "validated": {
    "instances": 60,
    "passed": 60
   },
   "source": "manual"
  },
  {
   "name": "R4 carry over digit 1",
   "from": "l 100 <B r",
   "to": "l <B 111 r",
   "steps": "3",
   "cases": "E1 = 1LB hands the carry back to B (tm108: E keeps carrying)",
   "validated": {
    "instances": 60,
    "passed": 60
   },
   "source": "manual"
  },
  {
   "name": "R5 turn at digit 0",
   "from": "l 000 <B r",
   "to": "l 100 A> r",
   "steps": "5",
   "cases": null,
   "validated": {
    "instances": 60,
    "passed": 60
   },
   "source": "manual"
  },
  {
   "name": "R6 overflow at the left end",
   "from": "0^inf <B r",
   "to": "0^inf 100 A> r",
   "steps": "5",
   "cases": "new MSB digit, 3 cells",
   "validated": {
    "instances": 60,
    "passed": 60
   },
   "source": "manual"
  },
  {
   "name": "R7 marker step (= HR2)",
   "from": "l A> 101 00 101 r",
   "to": "l <B 00 101 101 r",
   "steps": "13",
   "cases": null,
   "validated": {
    "instances": 60,
    "passed": 60
   },
   "source": "manual"
  },
  {
   "name": "R8 marker birth (= HE2)",
   "from": "l 010 A> 0^inf",
   "to": "l <B 00 101 0^inf",
   "steps": "6",
   "cases": null,
   "validated": {
    "instances": 60,
    "passed": 60
   },
   "source": "manual"
  },
  {
   "name": "ROUND: one counter increment",
   "from": "0^inf W 010^(b+1) A> 101 00 101^c 0^inf",
   "to": "0^inf inc(W) 010^b A> 101 00 101^(c+1) 0^inf",
   "steps": "10b+21+10s+6o",
   "cases": "W over {010,000,100}, ends in a digit; W = U 000 V with V in {010,100}* (s, o = counts in V): inc(W) = U 100 V' (V' = V with 100 -> 000); no 000 in W: inc(W) = 100 V'. Holds for 870,221 of 872,741 K-to-K transitions from blank to 1e10 (0 mismatches)",
   "validated": {
    "instances": 372,
    "passed": 372
   },
   "source": "manual"
  },
  {
   "name": "OVERFLOW: one counter period, left record to left record",
   "from": "0^inf <B W 101^b 00 101^c 0^inf",
   "to": "0^inf <B 111 W 101^(b-2^D) 00 101^(c+2^D) 0^inf",
   "steps": null,
   "cases": "W over {111,101} (all digits carried), D = number of 111 units, b > 2^D. From blank: 2,382 of 2,382 in-epoch left-record pairs to 1e10; c = 2^D-1 at 2,911 of 2,919 clean left records",
   "validated": {
    "instances": 150,
    "passed": 150
   },
   "source": "manual"
  },
  {
   "name": "EAT shape (b = 0), aligned W: no halt, next K reached",
   "from": "0^inf W A> 101 00 101^c 0^inf (b = 0)",
   "to": "0^inf W' 010^b' A> 101 00 101^c' (r') 0^inf, marker reborn at the right end",
   "steps": null,
   "cases": "Shape check only (no halt, a K-moment follows). W' is W re-read on a grid shifted by one cell (100 -> 010, 000 -> 000, 010 -> 001) plus local changes; exact form not ruled",
   "validated": {
    "instances": 400,
    "passed": 400
   },
   "source": "manual"
  },
  {
   "name": "EAT cascade: exact rule (open)",
   "from": "eat with 001 units (shifted segments) and pseudo-markers",
   "to": "open",
   "steps": null,
   "cases": "2,186 eat/mini-eat transitions and 334 carries into shifted segments to 1e10 are not covered by any exact rule yet",
   "validated": {
    "instances": 0,
    "passed": 0
   },
   "source": "manual"
  }
 ],
 "template": {
  "file": "verify/BECv1_6x2_solved.v:872",
  "module": null,
  "machine": "1RB0RA_1LC0LF_1LE0LD_1LB---_1RA1LE_0LB0RC",
  "diff": "E1: 1LB here vs 1LE in tm108",
  "applies": "partly",
  "note": "The right half is identical: tm108's HR1, HR2, HR3 and HE2 (w0 = 101, w0' = 010, w1 = 00) hold here as R1, R7, R3 and R8. The left counter is not: in tm108, E carries over single-cell digits and m0 = 00 separates the counter from the unary. Here B carries (R4), digits are 3-cell units mixed with 010 spacers, and the eat shifts the grid, so solve_BECv1 (and its fixed-width LeftBinaryCounter) cannot apply."
 },
 "strategy": "Not ready for a proof. Next: rule the eat as a transducer on the counter word: the one-cell grid re-read (100 -> 010, 000 -> 000, 010 -> 001), then the mini-eat cascade until W is aligned again. Then look for a regular invariant on such words (BECv7/BECv8 carry-head and two-track transducer proofs are the closest style).",
 "risks": [
  "The eat cascade is long and context-dependent: random {010,000,100,001} words with no 100-001 pair still halt in about 5% of trials, so the invariant must be finer than unit bigrams.",
  "The eat consumes the counter's bit pattern at eat time (about N mod 2^D). If safety depends on those bits, this becomes arithmetic (cryptid protocol).",
  "Proof size: likely a BECv7/BECv8-class file (tens of KB), not a one-line certificate."
 ],
 "checkpoint_questions": [
  {
   "text": "eat_closeup_19500: each zigzag is one round. Does the right turning point (the 00 marker) move left by about 3 cells per round while the left turning points follow a binary-counter pattern, until they meet (the eat) and the head runs to the right end?",
   "figs": [
    1
   ],
   "kind": "check"
  },
  {
   "text": "Proof route: the counter, unary and marker are BECv1-like and ruled (99.7% of rounds to 1e10). The eat shifts the 3-cell grid and needs a transducer argument (BECv7/BECv8 style). Spend one more ~2 h stage ruling the eat, or park #29 as STUCK and move on?",
   "figs": [],
   "kind": "decision"
  },
  {
   "text": "Sibling R2 (F0 = 0LD) passes the counter rules R1, R3, R4 and R5 but fails the marker step R7. Should we analyse #29 and R2 as one family (shared eat analysis), or keep them separate?",
   "figs": [],
   "kind": "decision"
  }
 ],
 "discord_check": null,
 "stats": {
  "simulated_steps": 10000000000,
  "cells": 10628,
  "nonzeros": 6528,
  "growth": "span ~ t^(1/3): unary N ~ (4t)^(1/3) units, span ~ 3N cells (10,628 at 1e10)"
 },
 "status": "CHECKPOINT_A",
 "status_reason": "Mechanism identified at medium confidence: rounds, overflow, marker step and birth are ruled and validated; the eat (grid shift plus mini-eat cascade, 0.29% of K-transitions to 1e10) is not ruled. Proof work waits for Dennis (CLAUDE.md: med confidence)."
}