{
 "tm": "1RB0RF_1RC1RF_1LD0RE_---0LE_1LA0LE_0RA0LC",
 "rank": 30,
 "class_sheet": "Bell eats counter",
 "class_guess": "Bell eats counter, partial eat with residue (sibling of #10)",
 "confidence": "med",
 "summary": "Marker bounce feeds a 3-cell binary counter; the bell eats only its low digit, so residue seeds the next epoch",
 "mechanism": [
  "Hub K(a,b,X) = (110)^a 0 (011)^b C> X: each bounce moves the 0-marker one block and adds 1 to counter X.",
  "Clean counter D(k,s,v): digits ?01 split by 001000, 4-stage tail; when b runs out only the low digit is eaten (E1/E2).",
  "The rest seeds the next epoch as residue; forcing an eat during its digestion can halt, so we need eat time > digestion time."
 ],
 "diagrams": [
  {
   "mode": "default",
   "png": "img/default_site.png",
   "html": null,
   "steps": 10000,
   "caption": "bbchallenge default view, first 10,000 steps",
   "title": "bbchallenge default view, 10,000 steps",
   "why": "The site's view. Find the diagonal stripes inside the bell and the busy right edge.",
   "primary": true,
   "start": "blank",
   "rows": "every step: steps 0..9,999 squeezed into 500 px rows as on bbchallenge.org (each px row blends steps 20y+9 and 20y+10, so 1 px row = 20 steps)",
   "window": "cells -200..199, 1 px = 1 cell",
   "legend": "white = 1, black = 0 or never visited, red/green = head after a right/left move (pale = blended by the squeeze)",
   "style": "bbchallenge-default"
  },
  {
   "mode": "state:C:R",
   "png": "img/stateCR_right_3e5.png",
   "html": null,
   "steps": 300000,
   "caption": "One row per C-right entry: epochs as diagonals, counter at right",
   "title": "Epochs: marker diagonals and the counter",
   "why": "Each down-right diagonal is one epoch (the 0-marker crossing the blocks); the counter is the right-edge strip.",
   "primary": true,
   "start": "blank",
   "rows": "one row per entry into state C moving right, steps 0..300,000 (1,485 rows); not evenly spaced in time",
   "window": "cells -425..0, 1 px = 1 cell, counted from the right edge (each row's rightmost visited cell is pinned to the last column)",
   "legend": "white = 1, black = 0 or never visited, red = head (just moved right)",
   "style": "bbchallenge-default"
  },
  {
   "mode": "custom",
   "png": "img/epoch48_counter.png",
   "html": null,
   "steps": 43094,
   "caption": "Counter word at every hub of the N=48 epoch, with safety strip",
   "title": "One residue epoch: counter word per bounce",
   "why": "Rows are bounces. Compare the left strip's colour with the grey wedge: where does red appear?",
   "primary": true,
   "start": "blank",
   "rows": "one row per hub K(a,b,X) (C moving right after (011)^b) from the left record at step 28,266 (k=47) to the next one at step 43,094: 61 rows, steps not evenly spaced",
   "window": "cells 0..63, 1 px = 1/4 cell (4x4 px per cell), counted from the epoch's first counter cell (absolute, so the word stays put as b shrinks), plus a 6 px strip on the left",
   "legend": "white = 1, black = 0, grey = cells already absorbed into the (011) block region; left strip: red = forcing the eat at this hub (b:=1) halts, green = safe",
   "style": "custom"
  },
  {
   "mode": "records",
   "png": "img/records_1e8.png",
   "html": null,
   "steps": 100000000,
   "caption": "Left/right records to 1e8: right edge 3,283, left only 146",
   "title": "Left and right records to 100M steps",
   "why": "One row per new edge cell. The left edge moves 2 cells per epoch; the right edge carries the counter.",
   "start": "blank",
   "rows": "the start, then one per new leftmost or rightmost cell (3,430 rows, steps 0..100,000,000); not evenly spaced in time",
   "window": "cells -146..3,283, 1 px = 1 cell",
   "legend": "white = 1, black = 0 or never visited, red/green = head on a new right/left cell",
   "style": "bbchallenge-default"
  },
  {
   "mode": "blaze",
   "png": "img/blaze_1e7.png",
   "html": null,
   "steps": 10000000,
   "caption": "Blaze 1e7: one smooth bell",
   "title": "Blaze, 10 million steps",
   "why": "The bell's outline over 1e7 steps; no chaotic edge.",
   "start": "blank",
   "rows": "time blocks of 16,384 steps, averaged (611 blocks, steps 0..10,000,000), stretched to 500 px: 1 px ~ 20,000 steps",
   "window": "cells -80..1,468, 1 px = ~1.9 cells (Blaze's 1,549 columns stretched to 800 px)",
   "legend": "white = 0, orange = 1, in-between shades = both mixed inside the pixel's block (averaged); no head",
   "style": "bbchallenge-blaze"
  },
  {
   "mode": "custom",
   "png": "img/fs_digest_share.png",
   "html": null,
   "steps": 100000000000,
   "caption": "Share of each epoch's bounce budget used by residue digestion",
   "title": "Digestion share per epoch to 1e11 steps",
   "why": "Dots sit far below the yellow budget line and under the blue curve, and they sink as N grows.",
   "primary": false,
   "start": "blank",
   "rows": "no rows: one dot per epoch of the blank-tape orbit to step 100,000,000,000 (517 epochs with N >= 10 and a digestion phase)",
   "window": "x = log10 N from 1 to 4.2 (N = b at the epoch's first hub), y = beta/N from 0 (bottom) to 1 (top), 420 x 220 px",
   "legend": "white = one epoch (beta = b used until the hub word is back in grammar G), yellow = budget beta = N (the eat), blue = 8/sqrt(N) (exhaustive clean-residue maximum), grey ticks = N = 10, 100, 1000, 10000",
   "style": "custom"
  },
  {
   "mode": "custom",
   "png": "img/nest_beta_sqrtN.png",
   "html": null,
   "steps": 0,
   "caption": "Digestion cost per epoch of the rules-only model, scaled by sqrt N",
   "title": "Digestion cost / sqrt(N), model orbit",
   "why": "Green dots are TM-checked epochs, white ones the model beyond. Compare their height with the blue line.",
   "primary": false,
   "start": "blank",
   "rows": "no rows: one dot per epoch of the rules-only model orbit (epochs 3..80585; epochs 3..519 equal the TM log to 1e11 steps)",
   "window": "x = log10 N from 1 to 6.2 (N = b at the epoch start), y = beta/sqrt(N) from 0 (bottom) to 10 (top), 420 x 220 px",
   "legend": "green = epoch checked against the TM log, white = later model epoch, yellow = budget beta = N (y = sqrt N), blue = 7.95 (clean-residue maximum with 2^(k+3) <= N), grey ticks = N = 10 .. 10^6",
   "style": "custom"
  }
 ],
 "history": {
  "skeleton": "0^inf <A 101 001^k T 0^inf",
  "vars": [
   "k (left record; next epoch has b = k+1)",
   "len T (residue cells)",
   "T (residue word)"
  ],
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 },
 "rules": [
  {
   "name": "R restart",
   "from": "0^inf <A 101 001^k 000 Y 0^inf",
   "to": "0^inf 110 0 011^(k+1) C> Y 0^inf",
   "steps": "3k+7",
   "cases": "any Y not starting with 011; k <= 3000",
   "validated": {
    "instances": 600,
    "passed": 600
   },
   "source": "manual"
  },
  {
   "name": "B bounce",
   "from": "0^inf 110^a 0 011^(b+1) C> X 0^inf",
   "to": "0^inf 110^(a+1) 0 011^(b+d) C> inc(X) 0^inf",
   "steps": "8b+c(X)",
   "cases": "b >= 1; X = any of 9,730 orbit hub words; inc(X), d in {0,1} and c(X) do not depend on a, b",
   "validated": {
    "instances": 600,
    "passed": 600
   },
   "source": "manual"
  },
  {
   "name": "W clean counter",
   "from": "0^inf 110^a 0 011^(b+1) C> D(n) 0^inf",
   "to": "0^inf 110^(a+1) 0 011^(b+[n=3]) C> D(n+1) 0^inf",
   "steps": null,
   "cases": "D(n) = (v0 01 001000)...(v_(k-1) 01) tail_s, n = 2^(k+2)-4+s*2^k+v, tails '',0001,000010001,000110001; D(1..3)=1,010001,110001; n <= 1e5",
   "validated": {
    "instances": 600,
    "passed": 600
   },
   "source": "manual"
  },
  {
   "name": "E2 odd eat",
   "from": "0^inf 110^a 0 011 C> 101 E(j,s,u) 0^inf",
   "to": "0^inf <A 101 001^a 000 11 E(j,s,u+1) 0^inf",
   "steps": null,
   "cases": "E(0,s,0)=tail_s, E(j,s,u)=001000 D(j,s,u); u+1=2^j overflows to s+1; j <= 9",
   "validated": {
    "instances": 600,
    "passed": 600
   },
   "source": "manual"
  },
  {
   "name": "E1 even eat",
   "from": "0^inf 110^a 0 011 C> 001^i 000 Y 0^inf",
   "to": "0^inf 110^(a+2) 0 011^i C> Y 0^inf",
   "steps": null,
   "cases": "i >= 1 (i <= 40 tested), Y random / orbit / clean words not starting with 011",
   "validated": {
    "instances": 600,
    "passed": 600
   },
   "source": "manual"
  },
  {
   "name": "E0 empty",
   "from": "0^inf 110^a 0 011 C> 0^inf",
   "to": "0^inf <A 101 001^(a+1) 0001 0^inf",
   "steps": null,
   "cases": "a <= 3000",
   "validated": {
    "instances": 200,
    "passed": 200
   },
   "source": "manual"
  },
  {
   "name": "digestion (open)",
   "from": "0^inf 110 0 011^N C> 11 E(j,s,u) 0^inf",
   "to": "clean counter after O(len) bounces",
   "steps": null,
   "cases": "not exact yet: inc on words with extra 001 groups; forced eats halt at 1-2 bounce counts inside this window",
   "validated": {
    "instances": 0,
    "passed": 0
   },
   "source": "manual"
  }
 ],
 "template": {
  "file": "verify/FractalBEC.v:269",
  "module": null,
  "machine": "1RB0RF_1RC1RF_1LD1RB_---0LE_1LA0LE_0RA0LC",
  "diff": "C1: template (tm10 mirrored) 1RB, ours 0RE; #10 has 1RA",
  "applies": "partly",
  "note": "RInc (marker bounce) = our B. LInc (2-cell counter) and Ov/P0 (whole counter eaten, fractal doubling k -> 2k+3) do not hold: our counter has 3-cell digits + 4-stage tail and the eat is partial (E1/E2), leaving residue. Same hub as #10 (STUCK on Lemma X)."
 },
 "strategy": "Deepen: exact rules for digestion and for counters with extra 001 groups, then an abstract epoch map (N, residue) -> (N', residue') checked against the TM, then prove 'digestion < N' as an inductive size invariant; proof shaped like FractalBEC/BECv7.",
 "risks": [
  "Digestion may not close into a short rule list (as #10's Lemma X).",
  "Safety is not regular (random eat times halt), so no CTL shortcut.",
  "Non-clean counters (75/138 eats) may hide further cases."
 ],
 "checkpoint_questions": [
  {
   "text": "Fig 2: is each down-right diagonal one epoch, with the counter living in the busy strip at the right edge?",
   "figs": [
    1
   ],
   "kind": "check"
  },
  {
   "text": "Fig 3: do the red (unsafe) rows sit only inside the early part where the grey wedge is still growing, far from the epoch's last rows?",
   "figs": [
    2
   ],
   "kind": "check"
  },
  {
   "text": "Fund a deepen stage (digestion rules + epoch map + size invariant) before any proof.md, or park like #10?",
   "kind": "decision"
  }
 ],
 "discord_check": null,
 "stats": {
  "simulated_steps": 1000000000,
  "cells": 7720,
  "nonzeros": 4642,
  "growth": "left edge +2 cells/epoch; 137 epochs to 1e9; k = 2,497 at step 983,786,092; right edge ~t^0.36"
 },
 "status": "CHECKPOINT_A",
 "status_reason": "Rules R/B/W/E0/E1/E2 exact and validated; residue digestion and the size invariant (eat after digestion) are open, so confidence med."
}