{ "tm": "1RB0LD_0RC1RA_1RD1RF_1LA0LE_0RA0LC_0RA---", "rank": 9, "class_sheet": "Shift-overflow structural counter", "class_guess": "Shift-overflow counter: base-4 digit blocks in a bouncer, plus a one-time reset that leaves walls", "confidence": "med", "summary": "Bouncer on 11111010 with 4-state digit blocks (base-4 carry); 1 reset at 3e7 left static walls; no halt to 2e10", "mechanism": [ "Tape: runs of P=11111010 split by 6-cell digit blocks D0-D3; the head sweeps from the right edge to the first D0/D1.", "Each sweep steps every visited digit D0>D1>D2>D3>D0 (base-4 carry); D3>D0 moves one P across, so runs double leftward.", "At 2.97e7 the carry hit the left end: an 8e6-step rebuild left static walls X=11; digits cross them unchanged (to 2e10)." ], "diagrams": [ { "mode": "blaze", "png": "img/blaze_1e8.png", "html": "img/viewer.html", "steps": 100000000, "caption": "sqrt bouncer; flat step at 3e7 = the reset", "title": "Blaze, 100M steps", "why": "Time runs down; step 3e7 is about 30% of the way down. Look along both edges for any break.", "primary": true, "start": "blank", "rows": "Blaze time blocks of 131,072 steps (averaged), 763 blocks stretched to 500 px, so 1 px row ~ 200,000 steps (steps 0..100,000,000)", "window": "cells -48..16559, 1 px = 20.8 cells", "legend": "white = 0, orange = 1, paler orange = mixed (each pixel averages its cells over its time block); no head drawn", "style": "bbchallenge-blaze" }, { "mode": "phase", "png": "img/phase_1e8.png", "html": null, "steps": 100000000, "caption": "Phase colour per 8-cell block, row = every 2nd sweep", "title": "Phase colours, one row per 2nd sweep", "why": "One row per 2nd sweep; colour shows how the 8-cell word is shifted. Watch the top half's colour edges.", "primary": true, "start": "blank", "rows": "one row per 2nd sweep end (head at the right edge in state D): 2,069 rows, steps 17..99,998,776. Not equal time: rows ~200 steps apart at the top, ~100,000 at the bottom, and between rows 1,093 and 1,094 (from the top) 8.0M steps are skipped (29,754,243 -> 37,788,160: the reset never reaches the right edge)", "window": "cells -40..16551, 1 px = 8 cells", "legend": "away from digits the tape repeats the 8-cell word P = 11111010, and colour = how P is shifted in that 8-cell block (red, orange, yellow, green, cyan, blue, purple, magenta = shift 0..7); colour edge = a digit block (6 cells: colour steps back 2, e.g. green -> orange) or a wall X = 11 (2 cells: colour steps forward 2); white = block that is not a copy of P; dark slate = not visited yet", "style": "custom" }, { "mode": "phase", "png": "img/phase_1e9_left.png", "html": null, "steps": 1000000000, "caption": "After reset: vertical walls stay put; digit lines cross them", "title": "After the reset: left quarter to 1e9", "why": "The left quarter after the reset, to 1e9. Look where the slanted lines meet the vertical ones.", "primary": true, "start": "blank", "rows": "one row per 6th sweep end (head at the right edge in state D): 2,177 rows, steps 527..999,316,836. Not equal time: rows ~1,000 steps apart at the top, ~950,000 at the bottom, and between rows 363 and 364 (from the top) 8.1M steps are skipped (29,646,275 -> 37,788,160: the reset)", "window": "cells -40..13559, 1 px = 8 cells: the left quarter of the tape, an exact crop of img/phase_1e9.png; all 7 walls left by the reset (cells ~190..8,750) are in it, and cells 13,560..52,247 (not shown) hold only the newest digit runs", "legend": "away from digits the tape repeats the 8-cell word P = 11111010, and colour = how P is shifted in that 8-cell block (red, orange, yellow, green, cyan, blue, purple, magenta = shift 0..7); colour edge = a digit block (6 cells: colour steps back 2, e.g. green -> orange) or a wall X = 11 (2 cells: colour steps forward 2); white = block that is not a copy of P; dark slate = not visited yet", "style": "custom" }, { "mode": "blaze:from-config@29754243", "png": "img/blaze_reset_2.98e7.png", "html": null, "steps": 8100000, "caption": "The reset: inner bouncer regrows from the left", "title": "The reset, Blaze view", "why": "8.1M steps from step 29,754,243: the inner bouncer regrows from the left.", "start": "mid-run: step 29754243 (the last right-edge visit before the reset; head at cell 8,747)", "rows": "Blaze time blocks of 8,192 steps (averaged), 989 blocks stretched to 500 px, so 1 px row ~ 16,200 steps (steps 29,754,243..37,854,243)", "window": "cells -37..8770, 1 px = 11 cells", "legend": "white = 0, orange = 1, paler orange = mixed (each pixel averages its cells over its time block); no head drawn", "style": "bbchallenge-blaze" }, { "mode": "records", "png": "img/records_1e7.png", "html": "img/viewer.html", "steps": 9473093, "caption": "Left/right records to 9.47e6: digit boundaries are straight lines", "title": "Left/right records to 9.47M steps", "why": "Before the reset: digit boundaries are straight lines.", "start": "blank", "rows": "left/right records: the start, then one row each time the head reaches a new leftmost or rightmost cell (5,000 rows = the row cap, steps 0..9,473,093; rows are not equal time)", "window": "cells -32..4967, 1 px = 1 cell", "legend": "white = 1, black = 0 or never visited, red/green = head just moved right/left", "style": "bbchallenge-default" } ], "history": { "skeleton": "0^inf L .. b3 P^n3 b2 P^n2 b1 P^n1 b0 P^n0 T D> 0^inf ; P=11111010, D0=111010 D1=101111 D2=101011 D3=111111 X=11; T=A: last P ends under head (..1111101 D> 0^inf), T=B: P^n0 101 D> 0^inf", "vars": [ "T", "n0", "b0", "n1", "b1", "n2", "b2", "n3", "b3", "#D", "#X" ], "rows": [ { "step": 10009637, "values": [ "B", 314, "D3", 155, "D0", 79, "D3", 39, "D1", 12, 5 ] }, { "step": 10027525, "values": [ "A", 314, "D0", 156, "D1", 79, "D3", 39, "D1", 12, 5 ] }, { "step": 10039479, "values": [ "B", 314, "D1", 156, "D1", 79, "D3", 39, "D1", 12, 5 ] }, { "step": 10051439, "values": [ "A", 315, "D2", 156, "D1", 79, "D3", 39, "D1", 12, 5 ] }, { "step": 10069371, "values": [ "B", 315, "D3", 156, "D2", 79, "D3", 39, "D1", 12, 5 ] }, { "step": 10091867, "values": [ "A", 315, "D0", 157, "D3", 78, "D0", 40, "D2", 12, 5 ] }, { "step": 10103859, "values": [ "B", 315, "D1", 157, "D3", 78, "D0", 40, "D2", 12, 5 ] }, { "step": 10115857, "values": [ "A", 316, "D2", 157, "D3", 78, "D0", 40, "D2", 12, 5 ] }, { "step": 10136867, "values": [ "B", 316, "D3", 156, "D0", 79, "D1", 40, "D2", 12, 5 ] }, { "step": 10154869, "values": [ "A", 316, "D0", 157, "D1", 79, "D1", 40, "D2", 12, 5 ] }, { "step": 10166899, "values": [ "B", 316, "D1", 157, "D1", 79, "D1", 40, "D2", 12, 5 ] }, { "step": 10178935, "values": [ "A", 317, "D2", 157, "D1", 79, "D1", 40, "D2", 12, 5 ] }, { "step": 10196981, "values": [ "B", 317, "D3", 157, "D2", 79, "D1", 40, "D2", 12, 5 ] }, { "step": 10218073, "values": [ "A", 317, "D0", 158, "D3", 79, "D2", 40, "D2", 12, 5 ] }, { "step": 10230141, "values": [ "B", 317, "D1", 158, "D3", 79, "D2", 40, "D2", 12, 5 ] }, { "step": 10242215, "values": [ "A", 318, "D2", 158, "D3", 79, "D2", 40, "D2", 12, 5 ] }, { "step": 10266327, "values": [ "B", 318, "D3", 157, "D0", 80, "D3", 40, "D3", 12, 5 ] }, { "step": 10284443, "values": [ "A", 318, "D0", 158, "D1", 80, "D3", 40, "D3", 12, 5 ] }, { "step": 10296549, "values": [ "B", 318, "D1", 158, "D1", 80, "D3", 40, "D3", 12, 5 ] }, { "step": 10308661, "values": [ "A", 319, "D2", 158, "D1", 80, "D3", 40, "D3", 12, 5 ] }, { "step": 10326821, "values": [ "B", 319, "D3", 158, "D2", 80, "D3", 40, "D3", 12, 5 ] }, { "step": 10350381, "values": [ "A", 319, "D0", 159, "D3", 79, "D0", 40, "D0", 12, 5 ] }, { "step": 10362525, "values": [ "B", 319, "D1", 159, "D3", 79, "D0", 40, "D0", 12, 5 ] }, { "step": 10374675, "values": [ "A", 320, "D2", 159, "D3", 79, "D0", 40, "D0", 12, 5 ] }, { "step": 40005946, "values": [ "A", 56, "X", 378, "D3", 81, "D3", 82, "X", 20, 11 ] }, { "step": 100058178, "values": [ "B", 892, "D0", 83, "X", 229, "D3", 312, "X", 20, 11 ] }, { "step": 400090666, "values": [ "B", 1948, "D0", 839, "D1", 300, "X", 322, "D3", 20, 11 ] }, { "step": 1600152606, "values": [ "A", 3965, "D3", 1846, "D0", 1125, "D1", 183, "X", 20, 11 ] }, { "step": 6400057808, "values": [ "B", 7952, "D2", 3841, "D3", 2121, "D0", 994, "D2", 20, 11 ] }, { "step": 20000291798, "values": [ "A", 14060, "D1", 6895, "D3", 3648, "D0", 1757, "D0", 20, 11 ] } ] }, "rules": [ { "name": "edge", "from": "l P^n A", "to": "l' P^n B | l P^n B --> l' P^(n+1) A", "steps": null, "cases": "every sweep; tail alternates A/B, edge +4 cells", "validated": { "instances": 120, "passed": 120 }, "source": "manual" }, { "name": "turn@D0", "from": "l D0 P^a A", "to": "l D1 P^a B (from B: l D1 P^(a+1) A)", "steps": "38a+22 (A) / 38a+30 (B)", "cases": "head turns at D0; l untouched", "validated": { "instances": 61, "passed": 61 }, "source": "manual" }, { "name": "turn@D1", "from": "l D1 P^a A", "to": "l D2 P^a B (from B: l D2 P^(a+1) A)", "steps": "38a+20 (A) / 38a+28 (B)", "cases": "head turns at D1; l untouched", "validated": { "instances": 59, "passed": 59 }, "source": "manual" }, { "name": "carry@D2", "from": "l D2 P^c R", "to": "l* D3 P^c R'", "steps": "+38c+14 on top of the turn", "cases": "carry continues into l (l* = l after its own visit)", "validated": { "instances": 69, "passed": 69 }, "source": "manual" }, { "name": "carry@D3", "from": "l D3 P^c R", "to": "l* P D0 P^(c-1) R'", "steps": "+38c+36 on top of the turn", "cases": "c>=1; one P moves left across the digit", "validated": { "instances": 71, "passed": 71 }, "source": "manual" }, { "name": "wall pass", "from": "l X P^c R", "to": "l* X P^c R'", "steps": null, "cases": "X=11 (phase +2 wall) is transparent to the carry", "validated": { "instances": 24, "passed": 24 }, "source": "manual" }, { "name": "D1 meets wall", "from": "l D1 X P^c R", "to": "l* X P D0 P^(c-1) R'", "steps": null, "cases": "empty run between D1 and X; carry continues", "validated": { "instances": 24, "passed": 24 }, "source": "manual" }, { "name": "full-sweep model (real orbit)", "from": "E(T; n_i, b_i) at a right-edge D record", "to": "next D record = rules above applied once", "steps": null, "cases": "gen 1 at 1e7 (297 sweeps) + gen 2 at 1e9 (596); D-X adjacency sweeps excluded", "validated": { "instances": 893, "passed": 893 }, "source": "manual" } ], "template": { "file": "verify/SBCv6a.v", "module": "TM1", "machine": "1LB1RE_0LC0RB_1RD0LA_0RA0RD_0RF0RA_1RD---", "diff": "9/12 transitions differ after best renaming+mirror; SBCv10 TM4/TM5, SOBLv1 TM2, Bouncer_v1 TM1: 8; look-alike Bouncer_v1 TM2: 10", "applies": "partly", "note": "No proved neighbour is within 8 transitions, and its blocks (10011001) never appear here. Only the method transfers: Longitudinal segRLs/sideRLs passes, plus RIncs/WF over a digit list, fit gen 1; nothing covers the reset." }, "strategy": "Mirror SBCv6a TM1 (Longitudinal, flip tm): pass lemmas for P^n, D0-D3, X, D1.X via es/ind; BigStep over an Inductive digit list with a WF invariant; multistep_nonhalt to a post-reset config, then progress_nonhalt_cond. First close the block alphabet (D2.X, X.D1 transients; left-prefix overflow).", "risks": [ "Reset not modelled: if the left overflow recurs (after 2e10, beyond our sim), each rebuild may nest new walls, so the proof needs a generation invariant.", "Rule table not closed: D2.X and X.D1 adjacency create extra transient blocks ((10)^5 1^3 ..).", "No template within 8 transitions: new-mechanism proof, likely SBCv10-sized (~50 KB)." ], "checkpoint_questions": [ { "text": "Fig 1: is the flat step near 3e7 the only break in otherwise smooth growth (the one-time reset)?", "figs": [ 0 ] }, { "text": "Fig 2: in the top half, do the colour edges fanning out from the left read as the digit lines of one base-4 counter?", "figs": [ 1 ] }, { "text": "Fig 3: do the slanted digit lines cross the vertical walls without bending, so the walls can be treated as inert?", "figs": [ 2 ] } ], "discord_check": null, "stats": { "simulated_steps": 20000000000, "cells": 227260, "nonzeros": 146198, "growth": "~1.61*sqrt(t) cells (+4 cells per sweep); 56,806 sweeps by 2e10" }, "status": "CHECKPOINT_A", "status_reason": "Counter rules validated (8 rules, random + 893 real sweeps); no Collatz-like arithmetic; reset and D-X transients still open" }