{
 "tm": "1RB0LA_1RC---_0LD1RD_0RA0RE_0RF1RC_1LF0LA",
 "rank": 33,
 "label": "W7-6",
 "class_sheet": "Translated Counter",
 "class_guess": "Translated base-4 counter with overflow, trail erase and nested rebuild (phase cycle)",
 "confidence": "med",
 "summary": "Exact base-4 counter while counting; halts only at overflow/collision events hinging on exact coincidences.",
 "mechanism": [
  "Counting phase: a ~4w+3-cell counter moves right over blank tape and leaves a trail 1^N behind; each tick the head sweeps the whole counter left and back.",
  "At every C right record the word is 0^inf L 1^N 0^2 s_1 0 s_2 0 ... s_2w 0 1 C> 0^inf with s_i in {1^2, 1^3}; pairs are base-4 digits MSB first: 23=0, 22=1, 33=2, 32=3; '4 3' = digit 0 with a pending carry.",
  "Rule R1 (exact, 10,416/10,416 random record transitions, w = 2..40): each C right record shows value V+1. The top digit is always 3, so a w-pair counter runs at most 4^(w-1) increments.",
  "Overflow (all digits 3 + carry): A sweeps left erasing the trail (A1 = 0LA); separators 0 1^2 0 become single-1 markers; a small counter is rebuilt at the left end and counts through the gap.",
  "Collision: when the counter front reaches a marker or the old counter's remnant, it either absorbs it (counter grows) or halts. Synthetic scans: the outcome depends on the exact (value, gap) pair, e.g. w=3, V=49, gap 28 halts at step 6,290.",
  "History: phase 1 (8 pairs) overflowed at 3.73e6; nested rebuilds 3.7e6-2.1e7 (gaps 0^(2*4^k-2)); the current phase is a 23-pair counter (V = 61,170,343,443,693 at step 100,000,903) that overflows after ~9.2e12 more increments, ~4.6e15 steps, trail ~1.8e13.",
  "Family: lines 472, 569, 690 are distinct tables (not starts of 33's table) with the same architecture by end tapes (1^5408013, 1^4591407, 1^10536 0 1^2 0 1^5410013 at 1e9); qsim 0 rules on all four."
 ],
 "diagrams": [
  {
   "mode": "every:20000",
   "png": "img/phases-2.2e7.png",
   "html": null,
   "steps": 22000000,
   "primary": true,
   "title": "Phases to 2.2e7: count, erase, rebuild",
   "caption": "Counting wedges, overflow erasures, nested rebuild, then the long phase",
   "why": "White wedges are counting phases; the near-flat lines are A erasing the trail after an overflow.",
   "start": "blank",
   "rows": "every 20000 steps (1101 rows, to step 2.2e7)",
   "window": "cells -81..89280, 1 px = 128 cells",
   "legend": "white = 1, black = 0 (each pixel samples one of 128 cells)",
   "style": "bbchallenge-default"
  },
  {
   "mode": "every:1",
   "png": "img/tick-w4.png",
   "html": null,
   "steps": 700,
   "primary": true,
   "title": "Two counter ticks, w = 4 pairs",
   "caption": "Left zigzag sweep then right sweep; trail grows on the left",
   "why": "Each tick the head crosses the whole counter; compare the digit blocks before and after each sweep.",
   "start": "constructed: 0^inf 1^3 0 1^2 0 1^5 0 1^40 0^2 1^3 0 1^2 0 1^2 0 1^3 0 1^3 0 1^2 0 1^3 0 1^2 0 1 C> 0^inf",
   "rows": "every step (701 rows)",
   "window": "cells -84..15, 1 px = 1 cell (PNG upscaled 2x)",
   "legend": "white = 1, black = 0, coloured pixel = head (bbchallenge colours)",
   "style": "bbchallenge-default"
  },
  {
   "mode": "every:8",
   "png": "img/collision-halt.png",
   "html": null,
   "steps": 6290,
   "primary": true,
   "title": "A collision that halts",
   "caption": "w=3 counter (V=49) meets a marker 28 cells ahead, halts",
   "why": "The counter absorbs the marker, overflows, rebuilds and halts: halting needs this exact value/gap coincidence.",
   "start": "constructed: 0^inf 1^3 0 1^2 0 1^5 0 1^30 0^2 1^3 0 1^2 0 1^2 0 1^3 0 1^2 0 1^2 0 1 C> 0^28 1 0^inf",
   "rows": "every 8 steps; last row = halt at step 6290",
   "window": "cells -90..47, 1 px = 1 cell",
   "legend": "white = 1, black = 0, coloured pixel = head (bbchallenge colours)",
   "style": "bbchallenge-default"
  },
  {
   "mode": "default",
   "png": "img/default-site.png",
   "html": "img/viewer.html",
   "steps": 10000,
   "primary": false,
   "title": "bbchallenge view, 10,000 steps",
   "caption": "Start-up and the first translated counter",
   "why": "The site's own picture, for orientation.",
   "start": "blank",
   "rows": "every step, 10000 steps squeezed into 500 rows (site canvas)",
   "window": "cells -200..199, 1 px = 1 cell (site canvas 400x500)",
   "legend": "white = 1, black = 0, coloured pixel = head (bbchallenge colours)",
   "style": "bbchallenge-default"
  }
 ],
 "history": {
  "skeleton": "0^inf 1^3 0 1^2 0 1^5 0 1^1322 0 1^2 0 1^N 0^2 enc23(V) 1 C> 0^inf",
  "vars": [
   "N",
   "V"
  ],
  "rows": [
   {
    "step": 100000903,
    "values": [
     399782,
     61170343443693
    ]
   },
   {
    "step": 119999033,
    "values": [
     479471,
     61170343483540
    ]
   }
  ]
 },
 "rules": [
  {
   "name": "R1 record increment",
   "from": "0^inf L 1^N 0^2 enc_w(V) 1 C> 0^inf",
   "to": "next C right record: enc_w(V+1) (carry shown as '4 3')",
   "steps": "~20 (low pair) or one full sweep, ~502 per increment at w=23",
   "cases": "V < 4^w - 1; top digit 3",
   "validated": {
    "instances": 10416,
    "passed": 10416
   },
   "source": "manual"
  },
  {
   "name": "R2 digit code",
   "from": "pair (s1,s2) of 1-runs",
   "to": "23->0, 22->1, 33->2, 32->3",
   "steps": null,
   "cases": "w = 2..40",
   "validated": {
    "instances": 10416,
    "passed": 10416
   },
   "source": "manual"
  },
  {
   "name": "R3 trail growth",
   "from": "N",
   "to": "N + ~2 per increment (79,689 over 39,847 increments)",
   "steps": null,
   "cases": "w = 23, measured, not exact",
   "validated": {
    "instances": 1,
    "passed": 1
   },
   "source": "manual"
  },
  {
   "name": "R4 collision outcome",
   "from": "enc_w(V) 1 C> 0^g 1 0^inf",
   "to": "absorb (counter grows) or HALT, depending on exact (V, g)",
   "steps": null,
   "cases": "w=3: 3 halting cells of 8 x 48 (one orbit); w=2: V=15 halts for most g",
   "validated": {
    "instances": 432,
    "passed": 432
   },
   "source": "manual"
  }
 ],
 "template": {
  "file": "SOTCv2.v",
  "module": "TM1",
  "machine": "1RB---_0RC1LF_1LC1LD_1RE0RA_0RD1RF_0LD0LF",
  "diff": "different table (nearest SOTC/BTC proof >= 5 transitions away)",
  "applies": "partly",
  "note": "M-Abozaid's token/sweep lemmas fit the counting phase (R1); nothing upstream covers overflow -> erase -> nested rebuild -> collision, where this machine can halt."
 },
 "strategy": "No nonhalting proof attempted (taken class; needs a new decision). HALT route only: an exact phase-level model (R1 closed form + run-length transitions) to decide the transition after the 23-pair counter overflows (~4.6e15 steps).",
 "risks": [
  "Collision outcomes hinge on exact coincidences of large irregular numbers (the 23-pair start value has irregular digits): cryptid-like lottery.",
  "A HALT at ~4.6e15+ needs a checked runner with a formal increment lemma (W5-5-style base-4 digit lemmas).",
  "Translated Counter is M-Abozaid's class (SOTCv2.v); nonhalting work needs a separate call."
 ],
 "checkpoint_questions": [
  {
   "text": "Fig 1: do the white wedges (counting) and flat erase lines match 'count, overflow, erase, rebuild'?",
   "figs": [
    0
   ],
   "kind": "check"
  },
  {
   "text": "Fig 3: a synthetic counter halts at a marker only for an exact value/gap pair. Park as a cryptid-like lottery, or fund the exact phase model for the ~4.6e15 transition?",
   "figs": [
    2
   ],
   "kind": "decision"
  }
 ],
 "discord_check": null,
 "stats": {
  "simulated_steps": 1000000000,
  "cells": 3954386,
  "nonzeros": 3954346,
  "growth": "trail ~2 cells per increment, ~4.0e5 cells per 1e8 steps (linear within a phase)"
 },
 "status": "CRYPTID_CANDIDATE",
 "status_reason": "Stage 1 stop rule: counting is exact (R1), but halting happens only at overflow/collision events whose outcome depends on exact coincidences of large irregular numbers."
}