{
 "tm": "1RB0RD_1LC1RE_1LA0LD_0RE0RF_1LC1RA_0LE---",
 "rank": 700,
 "class_sheet": "Shift-overflow bouncer counter",
 "class_guess": "Shift-overflow bouncer counter: SOBCv6 TM3's table started in TM3's B; halts after 194 cycles",
 "confidence": "high",
 "summary": "SOBCv6 TM3's table from TM3's B: 194 shift-overflow cycles, then a far-end carry halts it (tower ~195).",
 "mechanism": [
  "Same table as SOBCv6.v TM3 (ours A,B,C,D,E,F = TM3 B,C,A,E,D,F), blank start = TM3 started in B. Left = binary counter of words 001/111 (W0 separator, W1 b = digit b, LSB at the head); right = RC n = 1 (011)^n 01. TM3's proved rules apply verbatim: Incs fills the counter while RC grows by the fill count c; Ov turns RC(2m+2) into m new zero digits and re-reads the old counter with a shift (LOv).",
  "From blank the chain starts at step 847 (x = 1000111011): counts 5, 51, 1,308,622,847, then each count is about 2 to the previous one (shift overflow). Near the head the list freezes into segments between huge (10)^k blocks; the far tail (100-200 symbols) keeps changing.",
  "In the 195th overflow (cycle 194) LOv must increment 0^11 11 11 11 at the far end: the carry runs off the tape and the machine halts (F reads 1). Direct sims of the instantiated tape halt 64 cells from the far end for every block size tried; no earlier cycle hits any failure case."
 ],
 "diagrams": [
  {
   "mode": "default",
   "png": "img/default_site.png",
   "html": "img/viewer.html",
   "steps": 10000,
   "caption": "bbchallenge Default view, first 10k steps",
   "title": "bbchallenge Default view, 10,000 steps",
   "why": "Orientation: the counter sits at the left edge; the right part grows with each bounce.",
   "primary": false,
   "start": "blank",
   "rows": "every step 0..9,999, squeezed into 500 px by the site's bilinear filter (about one sample per 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": "every:40",
   "png": "img/cycle_every40.png",
   "html": null,
   "steps": 40000,
   "caption": "Two counter cycles: fill, then overflow at the bottom",
   "title": "Two counter cycles from blank",
   "why": "Rows every 40 steps. The fan is RC growing per bounce; watch the bottom rows near step 37,000.",
   "primary": true,
   "start": "blank",
   "rows": "every 40 steps, 0..40,000 (1,001 rows); P-configs at steps 847, 2,136 and 38,573",
   "window": "cells -13..246, 1 px = 1 cell",
   "legend": "white = 1, black = 0 or unvisited, red/green = head (last move right/left)",
   "style": "bbchallenge-default"
  },
  {
   "mode": "every:1",
   "png": "img/overflow_37120.png",
   "html": null,
   "steps": 1460,
   "caption": "One overflow, step by step: walk in, back, convert RC",
   "title": "One overflow, every step",
   "why": "Top: walk into the full counter and back. Middle: RC is cleared. Bottom zigzag: RC becomes new digits.",
   "primary": true,
   "start": "mid-run: step 37120 (tape rebuilt from the run: full counter <D 010 1 (011)^52 01)",
   "rows": "every step, 37,120..38,580 (1,461 rows, 2 px each); the next P-config is the row for step 38,573",
   "window": "cells -46..183, 1 px = 0.5 cells (each cell 2x2 px); cell 0 = the head's start cell (run cell 40)",
   "legend": "white = 1, black = 0 or unvisited, red/green = head (last move right/left)",
   "style": "bbchallenge-default"
  },
  {
   "mode": "every:1",
   "png": "img/halt_far_end.png",
   "html": null,
   "steps": 1541,
   "caption": "Cycle-194 far end alone: last carry runs off, halt",
   "title": "The halting overflow at the far end",
   "why": "Only the far 25 tokens of the cycle-194 counter, filled. Follow the last leftward run to the left edge.",
   "primary": true,
   "start": "constructed: 0^inf 1^18 0^2 1^7 0^2 1^7 0^2 1^7 0^2 1^7 0^2 1^25 0^2 1^7 0^2 1^25 0^2 1 0^2 1 <D 0 1 0 1 0 1^2 0 1^2 0 1 0^inf",
   "rows": "every step 0..1,541 (1,542 rows, 2 px each); the last row is the halt",
   "window": "cells -126..12, 1 px = 0.5 cells (each cell 2x2 px); cell 0 = the head's start cell, the counter's far end is cell -122",
   "legend": "white = 1, black = 0 or unvisited, red/green = head (last move right/left)",
   "style": "bbchallenge-default"
  }
 ],
 "history": {
  "skeleton": "P-config: 0^inf Lmp(x) <D 010 1 011 01 0^inf  (x = counter, D digit pairs)",
  "vars": [
   "D",
   "c",
   "n0"
  ],
  "rows": [
   {
    "step": 847,
    "values": [
     4,
     5,
     2
    ]
   },
   {
    "step": 2136,
    "values": [
     7,
     51,
     25
    ]
   },
   {
    "step": 38573,
    "values": [
     32,
     1308622847,
     654311423
    ]
   }
  ]
 },
 "rules": [
  {
   "name": "Incs (TM3, proved)",
   "from": "Lmp x <D 010 1 011 01 0^inf",
   "to": "Lmp ones(x) <D 010 1 011^(c+1) 01 0^inf",
   "steps": null,
   "cases": "c = 2^D - 1 - value(x), must be odd (lowest digit 0)",
   "validated": {
    "instances": 278,
    "passed": 278
   },
   "source": "manual"
  },
  {
   "name": "Ov (TM3, proved)",
   "from": "Lmp ones(x) <D 010 1 011^(2m+2) 01 0^inf",
   "to": "Lmp ((W1 W0)^m W1 W1 v[1:]) <D 010 1 011 01 0^inf",
   "steps": null,
   "cases": "v = LOv(ones(x)) must start with W0; LOv's increments must not carry off the far end",
   "validated": {
    "instances": 63,
    "passed": 63
   },
   "source": "manual"
  },
  {
   "name": "chain from blank",
   "from": "P x at step 847, x = 1000111011",
   "to": "P-configs at steps 2,136 and 38,573 (exact tape match)",
   "steps": null,
   "cases": "then counts 1,308,622,847 and towers; model cycles 2..193 checked by sim on instantiated tapes",
   "validated": {
    "instances": 2,
    "passed": 2
   },
   "source": "manual"
  },
  {
   "name": "far-end halt (cycle 194)",
   "from": "Lmp ones(x_193) <D 010 1 011^(2m+2) 01 0^inf",
   "to": "halt",
   "steps": null,
   "cases": "LOv at far token 10 must increment 0^11 11 11 11: carry off the far end, F reads 1",
   "validated": {
    "instances": 12,
    "passed": 12
   },
   "source": "manual"
  }
 ],
 "template": {
  "file": "SOBCv6.v:1164",
  "module": "TM3",
  "machine": "1LB0LE_1RC0RE_1LA1RD_1LA1RB_0RD0RF_0LD---",
  "diff": "same table after renaming (ours A,B,C,D,E,F = TM3 B,C,A,E,D,F); only the start state differs",
  "applies": "partly",
  "note": "All of TM3's rule lemmas carry over (rename E->D, B->A). New: init (blank -> P x at step 847), 194 step_recs instead of 2, LInc_rec through (10) blocks with a lower bound, explicit pairs on the far side of abstract blocks (TM3's layout breaks at our cycle 4), and a far-end 'overflow halts' predicate instead of TM3's near-head direct run."
 },
 "strategy": "HALT proof from TM3's module: new init by solve_init to step 847, a lower-bounded block representation, 194 step_recs by vm_compute, then an inductive 'overflow halts' predicate (deep part halts => whole halts) with a far-end base case.",
 "risks": [
  "194 vm_compute step_recs on lists up to ~38.5k items may be slow; fallback: compress the frozen near-head segments.",
  "The block representation must carry lower bounds (k >= 2) for LInc through (10) blocks; get the layout wrong and LOv_rec fails as TM3's did at our cycle 4.",
  "The final halt needs a new predicate over the whole counter (~190 abstract blocks); TM3's near-head step1 tail does not reach the far end."
 ],
 "checkpoint_questions": [
  {
   "text": "One overflow, every step: does the head walk into the full counter and back, then clear the right part and turn it into new digits (bottom zigzag), ending in the next P-config?",
   "figs": [
    2
   ],
   "kind": "check"
  },
  {
   "text": "Far-end halt: on the far 25 tokens alone, does the last leftward run reach the left edge and the machine halt in the final row?",
   "figs": [
    3
   ],
   "kind": "check"
  },
  {
   "text": "Priority: start the formal HALT proof now (TM3's module, new init, 194 step_recs, far-end halting predicate), given the class is mined and others run UBRRBA on this list?",
   "figs": [],
   "kind": "decision"
  }
 ],
 "discord_check": null,
 "stats": {
  "simulated_steps": 1000000000,
  "cells": 28806,
  "nonzeros": 18608,
  "growth": "cycle k+1 has about 2^(cycle k count) bounces: power tower; halting time a tower of height about 195"
 },
 "status": "CHECKPOINT_A",
 "status_reason": "Mechanism = SOBCv6 TM3's proved rules; model predicts a halt in cycle 194, reproduced by direct simulation of instantiated tapes. Next: formal HALT proof."
}