{
 "tm": "1RB0RE_1LC0RD_1LD0LD_1RA0LC_1RA1RF_1RA---",
 "rank": "R4",
 "class_sheet": "Skelet 1-like Class 2",
 "class_guess": "hierarchical binary counter: #33/#11's counter whose u=1 digits freeze into fossil-K walls with decrementing fuel (= #32's table, D-start)",
 "confidence": "low",
 "summary": "#32's table, D-start: #11's counter plus nested walls whose fuel drops 1 per lower overflow; no proof idea yet.",
 "mechanism": [
  "Level 1 is #33's / kill #11's binary counter (carry M_k, same formula, 12,350/12,350 exact to 3e8). One cell differs (B1 0RD vs 0RE): a carry reaching a bit-1 digit with u = 1 turns back and does a partial overflow P (#32's rule, exact).",
  "P leaves misaligned text that acts as a fossil K-block wall (01)^w 000 (01)^x with a slow level-2 counter behind it. Each level-1 overflow lowers the wall gap by exactly 1 (217 at 1.8e7, 179 at 2.8e8, 151 at 2e10); a far wall (gap 52-53) sits at the end. Halting needs A> 111.",
  "Fuel cut to 9 or 5 in a real config: the wall dissolves, the far wall keeps burning, new walls form with fresh fuel (109, 164); no halt to 2e8. 54 of 12,438 macro steps to 3e8 still have no rule (far-end terminators, the wall's v = 1 shape, Z3 junctions), as for #32."
 ],
 "diagrams": [
  {
   "mode": "default",
   "png": "img/default_site.png",
   "html": "img/viewer.html",
   "steps": 10000,
   "caption": "bbchallenge Default view, first 10,000 steps",
   "title": "First 10,000 steps (site view)",
   "why": "The site's own picture, for orientation: the tape grows to the left, the right end barely moves.",
   "primary": false,
   "start": "blank",
   "rows": "every step, steps 0..10,000 squeezed into 500 rows (site canvas, bilinear: about 20 steps per row)",
   "window": "cells -200..199, 1 px = 1 cell (cell 0 = start cell, at column 200)",
   "legend": "white = 1, black = 0 or unvisited, red/green = head (last move right/left)",
   "style": "bbchallenge-default"
  },
  {
   "mode": "every:30000",
   "png": "img/phase_full_3e7.png",
   "html": null,
   "steps": 30000000,
   "caption": "Phase view of the whole tape to 3e7 steps",
   "title": "Whole tape, steps 0..3e7 (phase view)",
   "why": "Each band is half a digit. Compare the bands right of the pinch near row 600 with those above.",
   "primary": true,
   "start": "blank",
   "rows": "every 30,000 steps, steps 0..3e7 (1,001 rows); thin horizontal streaks are rows caught mid-sweep",
   "window": "cells -15279..85, 1 px = 16 cells (every 16th cell sampled, so the alternating 0101 background is one solid colour)",
   "legend": "white = 1, black = 0 (at the sampled cell); a black/white switch = a defect (00, 11, 0000 or 000) between two samples; red/green = head",
   "style": "bbchallenge-default"
  },
  {
   "mode": "every:2000000",
   "png": "img/phase_far_1e9.png",
   "html": null,
   "steps": 1000000000,
   "caption": "Phase view of the right end to 1e9 steps",
   "title": "Right end, steps 0..1e9 (phase view)",
   "why": "Inner wall: the thin line near x = 640 px. To its right: level-2 digits, then the far end.",
   "primary": true,
   "start": "blank",
   "rows": "every 2,000,000 steps, steps 0..1e9 (501 rows)",
   "window": "cells -2089..90, 1 px = 2 cells (every 2nd cell sampled, so the alternating 0101 background is one solid colour)",
   "legend": "white = 1, black = 0 (at the sampled cell); a black/white switch = a defect (00, 11, 0000 or 000); red/green = head",
   "style": "bbchallenge-default"
  },
  {
   "mode": "explore",
   "png": "img/wall_r4.png",
   "html": null,
   "steps": 173,
   "caption": "R4: carry meets a u=1 digit, partial overflow",
   "title": "R4 at a u = 1 digit (constructed)",
   "why": "Start K: block, LSB digit (u=1, v=4), bit-0 digit (4,0,3), 1001. Watch where the first rightward sweep turns.",
   "primary": true,
   "start": "constructed: 0^inf <C (01)^2 000 (01)^3 (10)^4 01 (10)^3 00 (01)^4 10 01 0^inf",
   "rows": "every step, steps 0..173 (173 = next K-config, a P reset)",
   "window": "cells -7..43, 1 px = 1/6 cell (zoom 6); cell 0 = the head's start cell (left edge)",
   "legend": "slate = 0, white = 1; head coloured by state: A red, B orange, C blue, D green, E magenta, F cyan",
   "style": "bbchallenge-explore"
  },
  {
   "mode": "explore",
   "png": "img/wall_t33.png",
   "html": null,
   "steps": 173,
   "caption": "#33 (B1 = 0RE) from the same start, 173 steps",
   "title": "#33 from the same start",
   "why": "Same start and window as the R4 figure, machine #33 (one cell differs). Compare rows 40-60.",
   "primary": false,
   "start": "constructed: 0^inf <C (01)^2 000 (01)^3 (10)^4 01 (10)^3 00 (01)^4 10 01 0^inf",
   "rows": "every step, steps 0..173 (#33 reaches its next K, an M_1 carry, at step 108)",
   "window": "cells -7..43, 1 px = 1/6 cell (zoom 6); cell 0 = the head's start cell (left edge)",
   "legend": "slate = 0, white = 1; head coloured by state: A red, B orange, C blue, D green, E magenta, F cyan",
   "style": "bbchallenge-explore"
  }
 ],
 "history": {
  "skeleton": "pre-reset K: 0^inf <C (01)^b 000 (01)^a [level-1 digits] W1 [level-2 digits] W2 0^inf  (W = wall shape + gap)",
  "vars": [
   "b",
   "a",
   "hi",
   "inner wall W1",
   "far wall W2"
  ],
  "rows": [
   {
    "step": 70,
    "values": [
     1,
     3,
     4,
     "-",
     "-"
    ]
   },
   {
    "step": 809,
    "values": [
     5,
     12,
     16,
     "-",
     "-"
    ]
   },
   {
    "step": 2407,
    "values": [
     1,
     12,
     22,
     "-",
     "-"
    ]
   },
   {
    "step": 8306,
    "values": [
     13,
     22,
     28,
     "-",
     "-"
    ]
   },
   {
    "step": 15544,
    "values": [
     13,
     14,
     37,
     "-",
     "-"
    ]
   },
   {
    "step": 40950,
    "values": [
     37,
     50,
     43,
     "-",
     "-"
    ]
   },
   {
    "step": 61193,
    "values": [
     29,
     73,
     52,
     "-",
     "-"
    ]
   },
   {
    "step": 97511,
    "values": [
     13,
     54,
     58,
     "-",
     "-"
    ]
   },
   {
    "step": 207074,
    "values": [
     61,
     94,
     64,
     "-",
     "O2 0 Z2 66"
    ]
   },
   {
    "step": 261200,
    "values": [
     29,
     30,
     67,
     "-",
     "O2 0 Z2 67"
    ]
   },
   {
    "step": 378922,
    "values": [
     61,
     94,
     73,
     "-",
     "Z3 64"
    ]
   },
   {
    "step": 615221,
    "values": [
     125,
     205,
     76,
     "-",
     "Z2 0 O2 62"
    ]
   },
   {
    "step": 1128940,
    "values": [
     125,
     190,
     76,
     "-",
     "O2 0 Z2 59"
    ]
   },
   {
    "step": 2271498,
    "values": [
     253,
     190,
     76,
     "-",
     "Z3 58"
    ]
   },
   {
    "step": 4427656,
    "values": [
     509,
     382,
     79,
     "-",
     "Z3 59"
    ]
   },
   {
    "step": 11159235,
    "values": [
     509,
     766,
     82,
     "-",
     "Z2 0 O2 56"
    ]
   },
   {
    "step": 18115616,
    "values": [
     885,
     954,
     85,
     "Z3 218",
     "Z3 54"
    ]
   },
   {
    "step": 18388227,
    "values": [
     29,
     921,
     85,
     "Z2 0 O2 216",
     "Z2 0 O2 53"
    ]
   },
   {
    "step": 18695766,
    "values": [
     29,
     46,
     85,
     "O2 0 Z2 213",
     "Z2 0 O2 53"
    ]
   },
   {
    "step": 19204521,
    "values": [
     125,
     126,
     85,
     "Z2 0 O2 210",
     "Z2 0 O2 53"
    ]
   },
   {
    "step": 19804212,
    "values": [
     61,
     158,
     85,
     "O2 0 Z2 207",
     "Z2 0 O2 53"
    ]
   },
   {
    "step": 21032655,
    "values": [
     253,
     254,
     85,
     "Z2 0 O2 204",
     "Z2 0 O2 53"
    ]
   },
   {
    "step": 23053530,
    "values": [
     253,
     382,
     85,
     "O2 0 Z2 201",
     "Z2 0 O2 53"
    ]
   },
   {
    "step": 29254917,
    "values": [
     509,
     510,
     85,
     "Z2 0 O2 198",
     "Z2 0 O2 53"
    ]
   },
   {
    "step": 43094560,
    "values": [
     253,
     1150,
     85,
     "O2 0 Z2 195",
     "Z2 0 O2 53"
    ]
   },
   {
    "step": 61261763,
    "values": [
     1021,
     1022,
     85,
     "Z2 0 O2 192",
     "Z2 0 O2 53"
    ]
   },
   {
    "step": 124890854,
    "values": [
     1021,
     2558,
     85,
     "O2 0 Z2 189",
     "Z2 0 O2 53"
    ]
   },
   {
    "step": 181235665,
    "values": [
     509,
     510,
     85,
     "Z2 0 O2 186",
     "Z2 0 O2 53"
    ]
   },
   {
    "step": 202004399,
    "values": [
     253,
     1150,
     85,
     "Z2 0 O2 183",
     "O2 0 Z2 52"
    ]
   },
   {
    "step": 264408730,
    "values": [
     2045,
     1534,
     85,
     "Z3 180",
     "O2 0 Z2 52"
    ]
   },
   {
    "step": 2000000000,
    "values": [
     "-",
     "-",
     85,
     "O2 0 Z2 166",
     "O2 0 Z2 52"
    ]
   },
   {
    "step": 4000000000,
    "values": [
     "-",
     "-",
     85,
     "O2 0 Z2 162",
     "O2 0 Z2 52"
    ]
   },
   {
    "step": 6000000000,
    "values": [
     "-",
     "-",
     85,
     "O2 0 Z2 158",
     "O2 0 Z2 52"
    ]
   },
   {
    "step": 8000000000,
    "values": [
     "-",
     "-",
     85,
     "Z2 0 O2 157",
     "O2 0 Z2 52"
    ]
   },
   {
    "step": 10000000000,
    "values": [
     "-",
     "-",
     85,
     "O2 0 Z2 156",
     "O2 0 Z2 52"
    ]
   },
   {
    "step": 12000000000,
    "values": [
     "-",
     "-",
     85,
     "Z2 0 O2 155",
     "O2 0 Z2 52"
    ]
   },
   {
    "step": 14000000000,
    "values": [
     "-",
     "-",
     85,
     "O2 0 Z2 152",
     "O2 0 Z2 52"
    ]
   },
   {
    "step": 16000000000,
    "values": [
     "-",
     "-",
     88,
     "Z3 153",
     "Z3 53"
    ]
   },
   {
    "step": 18000000000,
    "values": [
     "-",
     "-",
     88,
     "O2 0 Z2 151",
     "Z3 53"
    ]
   },
   {
    "step": 20000000000,
    "values": [
     "-",
     "-",
     88,
     "Z3 151",
     "Z3 53"
    ]
   }
  ]
 },
 "rules": [
  {
   "name": "M_k carry (= #33's M_k)",
   "from": "0^inf <C (01)^b 000 (01)^a [(10)^vj (01)^uj]_{j=m..j*+1} (10)^v 00 (01)^u S",
   "to": "0^inf <C (01)^(b+2) 000 (01)^(a+1) [(10)^(vj-1) 00 (01)^uj]_{j=m..j*+1} (10)^(v-1) (01)^(u+1) S",
   "steps": "32+8a+8b+4v+sum_j(8uj+4vj)",
   "cases": "needs a >= 1, v >= 1, every trailing digit uj >= 2 and vj >= 1; S never read. v = 1: the digit drops (fuses). 300 random (k = 0..5, sizes to 300) + all 12,350 M steps of the real run to 3e8. Control: one trailing uj = 1 -> formula fails 60/60",
   "validated": {
    "instances": 12650,
    "passed": 12650
   },
   "source": "manual"
  },
  {
   "name": "P partial overflow at the first u = 1 digit (#32's P, mirrored)",
   "from": "0^inf <C (01)^b 000 (01)^a [(10)^vi (01)^ui]_{i=m..p+1} (10)^vp 01 S",
   "to": "0^inf <C 01 000 (01)^(b+3) (10)^a (01)^vm [(10)^u(i+1) 00 (01)^(vi-1)]_{i=m-1..p+1} (10)^u(p+1) 00 (01)^(vp-1) 10 S'",
   "steps": "77-4k+8a+20b+4*sum_{i>=p}ui+8*sum_{i>=p}vi",
   "cases": "k = m-p+1 digits, up = 1, ui >= 2 (i > p), a >= 1, S starts with 1 and S' = S minus that 1; p = m: ... (01)^vm 10 S'. 300 random (k = 1..5) + all 34 P steps of the real run to 3e8",
   "validated": {
    "instances": 334,
    "passed": 334
   },
   "source": "manual"
  },
  {
   "name": "O full overflow (= #33's O), all u >= 2, bare 1001 end",
   "from": "0^inf <C (01)^b 000 (01)^a [(10)^vj (01)^uj]_{j=m..1} 1001 0^inf",
   "to": "0^inf <C 01 000 (01)^(b+3) (10)^a (01)^vm [(10)^uj 00 (01)^(v(j-1)-1)]_{j=m..2} (10)^u1 00 1001 0^inf",
   "steps": "89-4m+8a+20b+4*sum(u)+8*sum(v)",
   "cases": "a >= 1, all uj >= 2. Never fires from blank to 3e8 (the far end is never the bare 1001). Control: one uj = 1 -> fails 200/200 (P happens instead)",
   "validated": {
    "instances": 200,
    "passed": 200
   },
   "source": "manual"
  },
  {
   "name": "wall fuel law (empirical, real run only)",
   "from": "reset that ends at the inner wall W(L), shape Z2 0 O2 | O2 0 Z2 | Z3",
   "to": "inner wall W'(L-1), next shape in the cycle",
   "steps": null,
   "cases": "all 38 resets from 1.8e7 to 2.84e8 (L 217 -> 179); 2e10 sample L = 151. Not a formula yet; not tested on random instances",
   "validated": {
    "instances": 0,
    "passed": 0
   },
   "source": "manual"
  }
 ],
 "template": {
  "file": null,
  "module": "our kill #11 (machines/1RB0LE_1RC0RC_1LD0RB_1LA0LE_1LD1LF_1LD---/, check_kill PASS); same table as pool #32 (OPEN)",
  "machine": "1RB0LE_1RC0RC_1LD0RB_1LA0LE_1LD1LF_1LD---",
  "diff": "1 transition after mirror + renaming (#11's A,B,C,D,E,F = R4's B,C,D,A,E,F): #11 A1 0LE vs 0LC here (= R4's B1 0RE vs 0RD). With that change the table is exactly #32's; R4 starts in #32's D",
  "applies": "partly",
  "note": "M and O carry over verbatim when every digit has u >= 2. #11's INV does not survive: u = 1 digits appear right away (O's new MSB digit (0,0,u1) flips to u = 1), P replaces O, and the fossil walls are outside #11's family. #32 hit the same wall (notes there). SBCv6 TM21 / TCv3 are loose relatives only."
 },
 "strategy": "Joint with #32 (same table). Model one level as K-block + digits + fossil K-block, get exact rules for the 3 open event types and the fuel law, then find a recursive invariant (each level a counter whose wall fuel is refilled when a new wall forms). Constructed small-fuel runs first.",
 "risks": [
  "Recursive structure: walls form, burn fuel and dissolve; a proof needs a multi-level invariant, which is far beyond a delta of #11.",
  "54 of 12,438 macro steps (3 event types) have no rule yet; #32 hit the same ones.",
  "The long-run regime is unobservable: inner-wall fuel 151 at 2e10, burning at slowing rate."
 ],
 "checkpoint_questions": [
  {
   "text": "Fig 1: below the pinch near row 600 (about step 1.8e7), does the right part keep fixed bands while a new set of bands grows from the left edge?",
   "figs": [
    1
   ],
   "kind": "check"
  },
  {
   "text": "R4 vs #33 from the same start: does R4's first sweep turn back at the u = 1 digit (about cell 24, row 47) where #33's sweep carries on to the bit-0 digit?",
   "figs": [
    3,
    4
   ],
   "kind": "check"
  },
  {
   "text": "Priority call: R4 and #32 share one table and one hierarchical-counter mechanism (fossil-K walls with fuel). Invest in a joint #32/R4 recursive invariant, or park both?",
   "figs": [],
   "kind": "decision"
  }
 ],
 "discord_check": null,
 "stats": {
  "simulated_steps": 20000000000,
  "cells": 87460,
  "nonzeros": 43726,
  "growth": "left edge ~2.8 t^0.5 at 1e9 (-87,374), -398,788 at 2e10; right edge 85 from 1.8e7 to 1.4e10, 88 after the level-2 overflow (~1.5e10). cells/nonzeros are at 1e9"
 },
 "status": "CHECKPOINT_A",
 "status_reason": "Mechanism partly understood (M, P, O validated; nested fossil-K walls observed); 54/12,438 macro steps to 3e8 uncovered and no non-halting argument. Confidence low: waits on Dennis's priority call."
}