{
 "tm": "1RB0RA_1LC0LD_1LE1LD_0LB0RE_1RF1LB_---0RA",
 "rank": 28,
 "class_sheet": "Bell eats counter",
 "class_guess": "Bell eats counter: BECv1 tm10 with E1 = 1LB (3-cell binary counter with fillers + unary marker; eats shift the counter)",
 "confidence": "med",
 "summary": "tm10 sibling: 3-cell binary counter eats a unary run; 9 round rules predict 100% of rounds to 1e10",
 "mechanism": [
  "Left: a binary counter in 3-cell blocks (1=100, 0=000, filler f=010 that carries skip). Right: tm10's 011 unary run with a 00 marker. Each round adds 1 and moves the marker one block left.",
  "When the marker reaches the counter, one of 3 eats fires. EAT0f drops the low digit and fillers into the unary. EAT0z and EAT1 absorb the marker and shift the counter one cell right (phi).",
  "The shifted blocks are debris (s=001). A carry that reaches an s re-encodes them (CARRY_S, recursive). On the blank run the debris is always gone before the next eat."
 ],
 "diagrams": [
  {
   "mode": "default",
   "png": "img/default_site.png",
   "html": "img/viewer.html",
   "steps": 10000,
   "caption": "bbchallenge view: first rounds, sweeps and marker",
   "title": "bbchallenge default view, 10,000 steps",
   "why": "The site's view: the counter sits at the left edge, the unary run to its right.",
   "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 about 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": "state:E:L",
   "png": "img/stateEL_left_4e4.png",
   "html": "img/viewer.html",
   "steps": 40000,
   "caption": "One row per left entry into E; diagonal = 00 marker",
   "title": "One row per left entry into E",
   "why": "Left edge pinned. Follow one dark diagonal down to the left band, then watch the band and the next diagonal.",
   "primary": true,
   "start": "blank",
   "rows": "every entry into state E moving left: 552 rows to step 40,000; every round start is one of them, and carries add a few more rows",
   "window": "cells 0..119, 1 px = 1 cell; counted from each row's leftmost visited cell (frame = left), so only the 120 leftmost cells are shown",
   "legend": "white = 1, black = 0 or unvisited, red/green = head (last move right/left)",
   "style": "bbchallenge-default"
  },
  {
   "mode": "explore",
   "png": "img/eat1_explore.png",
   "html": null,
   "steps": 140,
   "caption": "Constructed EAT1: marker absorbed, counter shifted one cell",
   "title": "Close-up: an EAT1 eat",
   "why": "Compare the counter's 1-cells (columns left of the start head) in the top row and below the first left sweep.",
   "primary": true,
   "start": "constructed: 0^inf 100 000 010 1 <E 11 00 (011)^6 0^inf",
   "rows": "every step: 141 rows, step 0 to step 140 (the eat ends at step 37, then the next round runs)",
   "window": "cells -12..24, 1 px = 0.25 cells (each cell 4x4 px); cell 0 = the start head (the LSB cell b)",
   "legend": "slate = 0, white = 1, head cell coloured by state: red = A, orange = B, blue = C, green = D, magenta = E, cyan = F",
   "style": "bbchallenge-explore"
  },
  {
   "mode": "blaze",
   "png": "img/blaze_1e8.png",
   "html": "img/viewer.html",
   "steps": 100000000,
   "caption": "Blaze, 1e8 steps: bell-shaped left edge, slow right edge",
   "title": "Blaze, 100 million steps",
   "why": "Whole run at a glance: left edge grows like a bell, the right edge slowly.",
   "primary": false,
   "start": "blank",
   "rows": "Blaze time blocks of 131,072 steps (each averaged), 763 blocks to step 1e8 (Blaze image, not stretched)",
   "window": "cells -2076..283, 1 px = 2 cells (Blaze image, 1,180 columns)",
   "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_1e9.png",
   "html": "img/viewer.html",
   "steps": 1000000000,
   "caption": "Left records to 1e9: counter digits at the pinned left edge",
   "title": "Left records, left edge pinned, 1e9",
   "why": "Left edge pinned. Each row is a new leftmost cell; the counter digits sit in the first columns.",
   "primary": false,
   "start": "blank",
   "rows": "left records only: the start, then one row per new leftmost cell (4,634 rows to step 1,000,000,000; not evenly spaced in time)",
   "window": "cells 0..149, 1 px = 1 cell; counted from each row's leftmost visited cell (frame = left), so only the 150 leftmost cells are shown",
   "legend": "white = 1, black = 0 or unvisited, red/green = head (last move right/left)",
   "style": "bbchallenge-default"
  }
 ],
 "history": {
  "skeleton": "0^inf W 0 <E 11 (011)^n 0^inf",
  "vars": [
   "n (unary blocks; the next epoch lasts about n rounds)",
   "|W| (blocks)",
   "#s (shifted debris blocks)",
   "W (1=100 0=000 f=010 s=001)"
  ],
  "rows": [
   {
    "step": 120,
    "values": [
     5,
     1,
     0,
     "f"
    ]
   },
   {
    "step": 337,
    "values": [
     6,
     3,
     1,
     "f f s"
    ]
   },
   {
    "step": 664,
    "values": [
     9,
     3,
     1,
     "f s 0"
    ]
   },
   {
    "step": 1191,
    "values": [
     10,
     5,
     2,
     "f s 0 s f"
    ]
   },
   {
    "step": 2118,
    "values": [
     19,
     1,
     0,
     "f"
    ]
   },
   {
    "step": 6335,
    "values": [
     30,
     0,
     0,
     ""
    ]
   },
   {
    "step": 10475,
    "values": [
     36,
     0,
     0,
     ""
    ]
   },
   {
    "step": 22976,
    "values": [
     41,
     5,
     1,
     "f 0 f s 0"
    ]
   },
   {
    "step": 52983,
    "values": [
     54,
     8,
     3,
     "f f 0 f f s s s"
    ]
   },
   {
    "step": 100967,
    "values": [
     68,
     8,
     3,
     "f 0 s 0 s f s 0"
    ]
   },
   {
    "step": 191998,
    "values": [
     79,
     13,
     8,
     "f 0 f s s s s 0 0 s s s s"
    ]
   },
   {
    "step": 415534,
    "values": [
     111,
     7,
     3,
     "f f s f 0 s s"
    ]
   },
   {
    "step": 880399,
    "values": [
     138,
     17,
     10,
     "f 0 0 0 f s 0 s f s s s s s s s s"
    ]
   },
   {
    "step": 1884030,
    "values": [
     189,
     10,
     4,
     "f f s 0 0 s f s f s"
    ]
   },
   {
    "step": 3995447,
    "values": [
     248,
     10,
     3,
     "f f f s f f s 0 s 0"
    ]
   },
   {
    "step": 8577850,
    "values": [
     313,
     20,
     13,
     "f 0 0 f f f s s f s s s s s s s s s s s"
    ]
   },
   {
    "step": 18895844,
    "values": [
     429,
     15,
     8,
     "f f s 0 f 0 f s s s s f s s s"
    ]
   },
   {
    "step": 39652245,
    "values": [
     560,
     10,
     1,
     "f 0 0 0 f f 0 0 s 0"
    ]
   },
   {
    "step": 84498575,
    "values": [
     720,
     20,
     12,
     "f 0 f f 0 s s f 0 s s s s 0 s s s s s s"
    ]
   },
   {
    "step": 178862950,
    "values": [
     973,
     5,
     0,
     "f f f f f"
    ]
   },
   {
    "step": 385954048,
    "values": [
     1225,
     14,
     5,
     "f 0 0 f f s 0 0 s s s f f s"
    ]
   },
   {
    "step": 820471330,
    "values": [
     1589,
     29,
     21,
     "f f 0 0 f 0 0 s s s 0 s s s s s s s s s s s s s ..."
    ]
   },
   {
    "step": 1738588788,
    "values": [
     2071,
     18,
     8,
     "f 0 0 0 0 0 s f 0 0 0 s s s s s s s"
    ]
   },
   {
    "step": 3720429457,
    "values": [
     2668,
     18,
     8,
     "f 0 f 0 0 s f s s f f s f s s s 0 s"
    ]
   },
   {
    "step": 7958424339,
    "values": [
     3518,
     20,
     10,
     "f f 0 s s s f f 0 f f s s s f s f s s s"
    ]
   },
   {
    "step": 9996109695,
    "values": [
     3804,
     22,
     12,
     "f f f 0 f f 0 f f s s s f s s s s s s s s s"
    ]
   }
  ]
 },
 "rules": [
  {
   "name": "R1 A-sweep (= BECv1 HR1)",
   "from": "l A> (011)^n r",
   "to": "l (010)^n A> r",
   "steps": "5n",
   "cases": null,
   "validated": {
    "instances": 25,
    "passed": 25
   },
   "source": "manual"
  },
  {
   "name": "R2 marker step (= HR2)",
   "from": "l A> 011 00 011 r",
   "to": "l <B 00 011 011 r",
   "steps": null,
   "cases": null,
   "validated": {
    "instances": 25,
    "passed": 25
   },
   "source": "manual"
  },
  {
   "name": "R3 B-sweep (= HR3)",
   "from": "l (010)^n <B r",
   "to": "l <B (011)^n r",
   "steps": "3n",
   "cases": null,
   "validated": {
    "instances": 25,
    "passed": 25
   },
   "source": "manual"
  },
  {
   "name": "INC0",
   "from": "RS(W,0,n+1,m)",
   "to": "RS(W,1,n,m+1)",
   "steps": "8n+16",
   "cases": "RS(W,b,n,m) = 0^inf W b <E 11 (011)^n 00 (011)^m 0^inf; W over blocks 1=100 0=000 f=010 s=001",
   "validated": {
    "instances": 100,
    "passed": 100
   },
   "source": "manual"
  },
  {
   "name": "CARRY",
   "from": "RS(W,1,n+1,m)",
   "to": "RS(inc W,0,n,m+1)",
   "steps": "8n+16+carry",
   "cases": "inc = binary +1 from the right: f skip, 1->0, 0->1 stop, overflow prepends 1",
   "validated": {
    "instances": 100,
    "passed": 100
   },
   "source": "manual"
  },
  {
   "name": "CARRY_S (carry meets debris)",
   "from": "RS(X s Y,1,n+1,m)",
   "to": "RS(inc(incC X) f Y0,0,n,m+1)",
   "steps": null,
   "cases": "Y in {1,f}*, Y0 = Y with 1->0; incC = +1 with f=1, 0=0, s skipped (overflow prepends f); recursive when inc meets another s",
   "validated": {
    "instances": 200,
    "passed": 200
   },
   "source": "manual"
  },
  {
   "name": "BIRTH (marker reborn at the right end)",
   "from": "NM(W,0,n+1)",
   "to": "RS(W,1,n,1)",
   "steps": "8n+16",
   "cases": "NM(W,b,n) = 0^inf W b <E 11 (011)^n 0^inf; BIRTHC: NM(W,1,n+1) -> RS(inc W,0,n,1) (validated, never fires from blank)",
   "validated": {
    "instances": 200,
    "passed": 200
   },
   "source": "manual"
  },
  {
   "name": "EAT0f",
   "from": "RS(W' d f^k 1^j,0,0,m)",
   "to": "RS(W',d,k-1,m+2+j)",
   "steps": null,
   "cases": "k >= 1, d in {0,1}, j >= 0; marker stays at the counter",
   "validated": {
    "instances": 100,
    "passed": 100
   },
   "source": "manual"
  },
  {
   "name": "EAT0z",
   "from": "RS(W' 0 1^j,0,0,m)",
   "to": "NM(phi W',0,m+2+j)",
   "steps": null,
   "cases": "phi: 1->f, 0->0, f->s (each block shifted one cell right); W = 1^j uses W' empty",
   "validated": {
    "instances": 200,
    "passed": 200
   },
   "source": "manual"
  },
  {
   "name": "EAT1",
   "from": "RS(W,1,0,m)",
   "to": "NM(phi(inc W),0,m+1)",
   "steps": null,
   "cases": "W in {1,0,f}*",
   "validated": {
    "instances": 100,
    "passed": 100
   },
   "source": "manual"
  }
 ],
 "template": {
  "file": "verify/BECv1_6x2_solved.v:88 (+ verify/BECv1.v Section; structure: verify/BECv7.v)",
  "module": "tm10 / nonhalt10, cert Build_BEC_cert_v1 B A E A [0] [1] [1] [0] [1] ... 1 0 0",
  "machine": "1RB0RA_1LC0LD_1LE1LD_0LB0RE_1RF1LE_---0RA",
  "diff": "E1: 1LB (ours) vs 1LE (tm10); the other 11 transitions are equal (#7 has E1 = 0LA)",
  "applies": "partly",
  "note": "Our TM on tm10's cert: 10/11 hypotheses hold (HL1, HL3-5, HR1-3, HE0-3: 25/25 each); HL2, the 1-cell carry 1<E1 -> <E11, fails 0/25 (E1 = 1LB sends the carry through B and C, so digits are 3 cells apart).\nThe unary half carries over; LeftBinaryCounter becomes a block-list counter with fillers and a phase-shifted debris region, closest to BECv7.v's inductive RIncs carry relation over 110/000/100 blocks."
 },
 "strategy": "Proof on round starts RS/NM, with W over {1,0,f,s}. Invariant: no 1 left of an s, and no s at an eat (the flush lemma). Unary lemmas from BECv1; inc/incC/CARRY_S as an inductive carry relation in BECv7 style.\nThen progress_nonhalt_cond. The open piece is the flush lemma: before the marker returns, the counter carries through every s block.",
 "risks": [
  "The flush lemma is a size bound (2^(digits right of the leftmost s) < epoch length, about m+1); it holds with slack on the blank run (worst ratio 0.44, below 0.11 after 1e9) but is not proved.",
  "An eat with s still present leaves the block alphabet; random words that do this can halt (26 of 1,200 fuzz samples), so the invariant must carry the bound.",
  "CARRY_S is recursive (debris inside debris), so the Coq carry relation must be mutually inductive, like BECv7's RIncs."
 ],
 "checkpoint_questions": [
  {
   "text": "Fig 2: does each dark diagonal (the 00 marker) drift left until it meets the left band (the counter), and does a fresh diagonal then start further right?",
   "figs": [
    1
   ],
   "kind": "check"
  },
  {
   "text": "Fig 3: once the head is back at the left, has the 00 gap gone, and is each counter 1-cell one cell further right than in the top row (plus one new 1 from the carry)?",
   "figs": [
    2
   ],
   "kind": "check"
  },
  {
   "text": "Decision: should the next stage prove the flush lemma and write proof.md (hand route), or run a native CTL search first?",
   "figs": [],
   "kind": "decision"
  }
 ],
 "discord_check": null,
 "stats": {
  "simulated_steps": 10000000000,
  "cells": 11492,
  "nonzeros": 5201,
  "growth": "span ~t^0.34 (5,197 cells at 1e9, 11,492 at 1e10); epoch length n ~t^0.35 (1,589 at 8.2e8, 3,804 at 1e10)"
 },
 "status": "CHECKPOINT_A",
 "status_reason": "Mechanism found: 9 round rules (1,000/1,000 random-instance sims) predict all 1,008,130 rounds to 1e10, 0 mismatches. Not cryptid. Non-halting needs the unproved flush lemma, so confidence is med."
}