{
 "tm": "1RB0RD_1LC0LE_0RA0LB_1RC1RF_1LB1RD_0RC---",
 "rank": 96,
 "class_sheet": "Unknown",
 "class_guess": "Bell eats counter variant (W5-5's table, 2 cells changed): hub defect + unary counter, ternary transfer",
 "confidence": "low",
 "summary": "100-block segments with one hub marker: each round moves 3^(k+1) blocks across the hub and adds 1 to a unary counter",
 "mechanism": [
  "The tape is runs of 100 blocks joined by markers: one hub marker 0 (W5-5's 0001 defect, mirrored), 101 separators, and a unary counter (101)^c at the right end. HUB(k), exact: the head zigzags left through (101)^c, makes 3^(k+1) short trips at the hub (each moves one 100 block across it), then adds one 101 to the counter; k = number of 101 separators right of the hub.",
  "When the hub's right segment b drops below 3^(k+1), the outcome branches on b and the neighbours (99 events to 1e9): the hub becomes a 11 marker, moves, or the head sweeps to the left edge. Once per deep cycle (88 cycles to 1.55e11) the head reaches the left edge; then the counter drains one 101 per long trip while inner hubs work in older segments, which later freeze (58, 106, 1452, 393, 165, 338, 838, 12269, 41798).",
  "No halt to 1.55e11. Head windows (radius 4-10) stop changing after 1e7 steps, but plain NGramCPS fails for n = 2-20. Halting needs A or D to meet 111 moving right (A1 = 0RD, D1 = 1RF, F1 = halt); the real tape has 111 only at deep sweeps, where C passes it moving left. 1 of 2,000 random segment tapes halted, and it was malformed."
 ],
 "diagrams": [
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   "mode": "default",
   "png": "img/default_site.png",
   "html": "img/viewer.html",
   "steps": 10000,
   "caption": "Site Default view, 10,000 steps from blank",
   "title": "bbchallenge Default view",
   "why": "The standard site picture: a growing block field with a slanted right edge.",
   "primary": true,
   "start": "blank",
   "rows": "10,000 steps squeezed into 500 rows (site view, about 20 steps per row)",
   "window": "cells -200..199, 1 px = 1 cell",
   "legend": "white = 1, black = 0 or unvisited, red/green = head after a move R/L",
   "style": "bbchallenge-default"
  },
  {
   "mode": "default",
   "png": "img/hub_round_k0.png",
   "html": null,
   "steps": 1360,
   "caption": "Three HUB(0) rounds from a constructed round start",
   "title": "Three hub rounds, k = 0",
   "why": "Each round: a zigzag left through the right field, short trips at the 0 marker, a sweep back right.",
   "primary": true,
   "start": "constructed: 0^inf 010^3 1 100^6 0 100^9 101^10 B> 0^inf",
   "rows": "every step, steps 0..1,360 (rounds end at steps 535, 986, 1,353); 2 px per step",
   "window": "cells -90..29, 1 px = 0.5 cells (scale 2; cell 0 = the head's start cell)",
   "legend": "white = 1, black = 0 or unvisited, red/green = head after a move R/L",
   "style": "bbchallenge-default"
  },
  {
   "mode": "default",
   "png": "img/hub_round_k1.png",
   "html": null,
   "steps": 3470,
   "caption": "Two HUB(1) rounds: one 101 separator, nine trips per round",
   "title": "Two hub rounds, k = 1",
   "why": "Same start shape plus one 101 separator; count the short trips at the marker in each round.",
   "primary": true,
   "start": "constructed: 0^inf 010^3 1 100^4 0 100^20 101 100^5 101^8 B> 0^inf",
   "rows": "every step, steps 0..3,470 (rounds end at steps 2,203 and 3,458); 2 px per step",
   "window": "cells -120..29, 1 px = 0.5 cells (scale 2; cell 0 = the head's start cell)",
   "legend": "white = 1, black = 0 or unvisited, red/green = head after a move R/L",
   "style": "bbchallenge-default"
  },
  {
   "mode": "every:2000000",
   "png": "img/every2e6_1e9.png",
   "html": null,
   "steps": 1000000000,
   "caption": "One row every 2,000,000 steps to 1e9",
   "title": "Whole run to 1e9",
   "why": "Right edge grows in steps; the lighter band is the (101) counter field filling and draining.",
   "primary": false,
   "start": "blank",
   "rows": "every 2,000,000 steps, 501 rows to step 1e9",
   "window": "cells -69..25596, 1 px = 4 cells (first cell of each 4-cell group)",
   "legend": "white = 1, black = 0 or unvisited; (100) and (101) fields show as different grey textures because of the 4-cell sampling",
   "style": "bbchallenge-default"
  }
 ],
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 "rules": [
  {
   "name": "HUB(0)",
   "from": "0^inf L 1 100^a 0 100^(b+3) 101^c B> 0^inf",
   "to": "0^inf L 1 100^(a+3) 0 100^b 101^(c+1) B> 0^inf",
   "steps": "24c+36b+79",
   "cases": "a>=1, c>=2; L never read",
   "validated": {
    "instances": 8081,
    "passed": 8081
   },
   "source": "manual"
  },
  {
   "name": "HUB(1)",
   "from": "0^inf L 1 100^a 0 100^(b+9) 101 100^m 101^c B> 0^inf",
   "to": "0^inf L 1 100^(a+9) 0 100^b 101 100^m 101^(c+1) B> 0^inf",
   "steps": "24c+108b+36m+643",
   "cases": "a>=1, m>=1, c>=2",
   "validated": {
    "instances": 1043,
    "passed": 1043
   },
   "source": "manual"
  },
  {
   "name": "HUB(k)",
   "from": "0^inf L 1 100^a 0 100^b [101 100^m_i] (i=1..k) 101^c B> 0^inf",
   "to": "0^inf L 1 100^(a+3^(k+1)) 0 100^(b-3^(k+1)) [101 100^m_i] 101^(c+1) B> 0^inf",
   "steps": "24c + 12*3^(k+1)*b + sum_i 12*3^(k+1-i)*m_i + d_k; d_0..d_3 = -29, -329, -3821, -37625",
   "cases": "b >= 3^(k+1), a>=1, every m_i>=1, c>=2; 1,200 random (k=0..3) + 8,363 real rounds to 1e9",
   "validated": {
    "instances": 9563,
    "passed": 9563
   },
   "source": "manual"
  },
  {
   "name": "EXHAUST",
   "from": "0^inf L 1 100^a 0 100^b [101 100^m_i] 101^c B> 0^inf, b < 3^(k+1)",
   "to": "branches on b and the neighbours (not modelled); k=0, b=2: 2^a 3 2^2 (10)^(c-1) 1 -> 2^(a+4) 0 (10)^(c-1) 1 in gap tokens",
   "steps": null,
   "cases": "99 events at the right edge to 1e9; b=0/1 usually start a deep sweep",
   "validated": {
    "instances": 0,
    "passed": 0
   },
   "source": "manual"
  }
 ],
 "template": {
  "file": null,
  "module": "W5-5's own .v (KILLED_HALT): bellpart, round = callw + bellpart, base-4 digit lemmas abs0/abs1/absm/pass2/callw_inc",
  "machine": "1RE0RD_1LC0LE_0RA0LB_1RC1RF_1LB0LD_0RC---",
  "diff": "A0: ours 1RB, W5-5 1RE; E1: ours 1RD, W5-5 0LD (W5-5's table started in its D, in our naming)",
  "applies": "partly",
  "note": "Same 3-cell words and the same bell-with-defect round, mirrored: our hub marker 0 is W5-5's 0001 defect. W5-5's defect moves 1 block per round and feeds a signed base-4 counter; here it jumps 3^(k+1) blocks and the counter is unary (101)^c. HUB(0) is W5-5's round with a unary counter; none of the digit lemmas carry over. W5-5's checked-runner route needs a halt, and none is in sight."
 },
 "strategy": "Route 1 (NONHALT, preferred): look for a closed regular language on 3-cell words (block CPS / CTL with the gap-token vocabulary); it would not need the residue arithmetic. Route 2: model the exhaustion branches and the deep cycle exactly, then look for a closed residue table. Stop if the exhaustion law is a pseudo-random residue map.",
 "risks": [
  "Exhaustion branches on b mod 3^(k+1) and re-cuts segments; frozen segment lengths and counter values look irregular (Collatz-type signal). A 111 met by A or D moving right halts, and 111 does occur at deep sweeps.",
  "Plain NGramCPS fails for n = 2..20, so any regular invariant must use phase or block structure; it may not exist.",
  "Deep cycles get long (2.9e10 steps per cycle at 1.5e11), so long-run evidence grows slowly."
 ],
 "checkpoint_questions": [
  {
   "text": "hub_round_k0: in each of the three rounds, after the zigzag through the right field, does the head make three short trips at the 0 marker, and does the black gap (the marker) move three blocks right each round?",
   "figs": [
    1
   ],
   "kind": "check"
  },
  {
   "text": "hub_round_k1: with one 101 separator to the right of the marker, do you count nine short trips per round instead of three?",
   "figs": [
    2
   ],
   "kind": "check"
  },
  {
   "text": "Next stage: (a) search for a closed regular language on 3-cell words, (b) model the exhaustion branches and deep cycle exactly, or (c) park as a Collatz-type candidate?",
   "kind": "decision"
  }
 ],
 "discord_check": null,
 "stats": {
  "simulated_steps": 155348041670,
  "cells": 25666,
  "nonzeros": 8997,
  "growth": "deep cycles: left edge -1 per cycle; right edge 25,596 at 1e9, 319,363 at 1.55e11"
 },
 "status": "CHECKPOINT_A",
 "status_reason": "HUB(k) exact and validated (1,200 random + 8,363 real rounds); exhaustion, drain and deep-cycle regimes observed but not modelled; no halting branch found"
}