{ "tm": "1RB0RF_1RC1RF_1LD1RA_---0LE_1LA0LE_0RA0LC", "rank": 10, "class_sheet": "Bell eats counter", "class_guess": "Bell eats counter, partial eat (leftover digits ride in next counter)", "confidence": "med", "summary": "Mirrored FractalBEC bounce+counter; C1 makes the eat partial: N grows +1/+3/+4+r per epoch, leftovers ride along", "mechanism": [ "Mirror of FractalBEC tm10: blocks (110)^a 0 (011)^b on the left, a binary counter on the right; each bounce adds 1.", "C1 differs: when b runs out the bell eats the counter only partly, so N grows by 1, 3 or 4+r per epoch, never doubles.", "Uneaten digits stay as inert 1s inside the next counter (residue); carries pass straight through them." ], "diagrams": [ { "mode": "state:C:R", "png": "img/compare_ours_vs_template_stateCR.png", "html": null, "steps": 300000, "caption": "Ours (left) vs template: each diagonal is one epoch, additive vs doubling", "title": "Ours (left) vs the FractalBEC template (right)", "why": "Both panels: one row per entry into C moving right. Compare the down-right diagonals: how many, how long.", "primary": true, "start": "blank", "rows": "both machines from the blank tape, steps 0..300,000; one row per entry into state C moving right: ours 2,161 rows, template 1,497 (dark grey below it = no more rows). Not evenly spaced in time, and row k is a different step in each panel", "window": "cells -337..0, 1 px = 1 cell, counted from the right edge (each panel pins its own rightmost visited cell to its last column): ours = absolute cells -50..287 at step 300,000; the template panel is 779 cells wide (absolute -16..762); 24 px dark-grey gap between the panels", "legend": "left = ours, right = FractalBEC tm10 mirrored (`1RB0RF_1RC1RF_1LD1RB_---0LE_1LA0LE_0RA0LC`, only C1 differs). white = 1, black = 0 or never visited, red = head (just moved right), dark grey = padding. Each down-right diagonal = one epoch (the 0-marker crossing the blocks): ours has 25 epochs by step 300,000, each a little longer than the last; the template has 8, each ~4x longer", "style": "bbchallenge-default" }, { "mode": "state:C:R", "png": "img/counter_zoom.png", "html": "img/viewer.html", "steps": 23000, "caption": "Right-edge zoom: counter digits, residue 1s, epoch resets", "title": "Right-edge zoom: counter digits and residue", "why": "The counter at the right edge. Watch the isolated 1s between its digits while carries pass.", "primary": true, "start": "blank", "rows": "one per entry into state C moving right, steps 0..23,000 (422 rows; 1 row = 3 px tall); not evenly spaced in time", "window": "cells -89..0, 1 px = 1/3 cell (1 cell = 3x3 px), counted from the right edge (each row's rightmost visited cell is 0, pinned to the last column; it is absolute cell 112 at step 23,000)", "legend": "white = 1, black = 0 or never visited, red = head (just moved right)", "style": "bbchallenge-default" }, { "mode": "records", "png": "img/records_1e8.png", "html": "img/viewer.html", "steps": 100000000, "caption": "Records to 1e8: right edge gains 2,083 cells, left only 276", "title": "Left/right records to 100M steps", "why": "The right edge gains 2,083 cells, the left only 276.", "start": "blank", "rows": "the start, then one per new leftmost or rightmost cell (2,360 rows, steps 0..100,000,000); not evenly spaced in time", "window": "cells -276..2,083, 1 px = 1 cell", "legend": "white = 1, black = 0 or never visited, red/green = head on a new right/left cell", "style": "bbchallenge-default" }, { "mode": "default", "png": "img/default_site.png", "html": "img/viewer.html", "steps": 10000, "caption": "bbchallenge default view, first 10k steps", "title": "bbchallenge default view, first 10,000 steps", "why": "The site's view: a bouncer growing mostly to the right.", "start": "blank", "rows": "every step: steps 0..9,999 squeezed into 500 px rows as on bbchallenge.org (each px row blends steps 20y+9 and 20y+10, so 1 px row = 20 steps)", "window": "cells -200..199, 1 px = 1 cell", "legend": "white = 1, black = 0 or never visited, red/green = head after a right/left move (pale colours = blended by the squeeze)", "style": "bbchallenge-default" }, { "mode": "blaze", "png": "img/blaze_1e7.png", "html": "img/viewer.html", "steps": 10000000, "caption": "Blaze 1e7: one smooth bell, no chaotic edge", "title": "Blaze, 10 million steps", "why": "One smooth bell, no chaotic edge.", "start": "blank", "rows": "time blocks of 16,384 steps, averaged (611 blocks, steps 0..10,000,000), stretched to 500 px: 1 px ~ 20,000 steps", "window": "cells -138..946, 1 px = ~1.4 cells (Blaze averages 1 cell per column; its 1,085 columns are stretched to 800 px)", "legend": "white = 0, orange = 1, in-between shades = both mixed inside the pixel's block (averaged); no head", "style": "bbchallenge-blaze" } ], "history": { "skeleton": "0^inf enc(a) 0^inf", "to": "0^inf (110)^(a+1) 0 (011)^b C> enc(a+1) 0^inf", "steps": "8b+8, plus 6t+8 if t>0 (t = trailing 1-bits of a)", "cases": "a>=1, b>=1. enc(v) = v0 010 (00 v1)(00 v2)...(00 vk), LSB first, trailing 0s trimmed; covers top-digit overflow at a=2^k-1", "validated": { "instances": 127, "passed": 127 }, "source": "manual" }, { "name": "R1g bounce, any counter word", "from": "0^inf (110)^a 0 (011)^(b+1) C> X 0^inf", "to": "0^inf (110)^(a+1) 0 (011)^b C> X' 0^inf", "steps": null, "cases": "b>=1; X = counter words with residue sampled from epochs N=37 and N=331; X' depends on X only (a,b inert; tested a<=3000, b<=1500)", "validated": { "instances": 160, "passed": 160 }, "source": "manual" }, { "name": "E0 epoch end, N even (clean start)", "from": "0^inf 110 0 (011)^(N-1) C> 101 0^inf", "to": "0^inf 110 0 (011)^N C> 110 enc3(N/2) 0^inf", "steps": null, "cases": "N even, N>=2. enc3(m) = (00 m0)(00 m1)... LSB first. N' = N+1. ~4N^2 steps", "validated": { "instances": 60, "passed": 60 }, "source": "manual" }, { "name": "E1 epoch end, N = 3 mod 4 (clean start)", "from": "0^inf 110 0 (011)^(N-1) C> 101 0^inf", "to": "0^inf 110 0 (011)^(N+2) C> F((N+1)/4) 0^inf", "steps": null, "cases": "N = 3 mod 4. F(v) = 1 0 v0 (00 v1)(00 v2)..., F(0) = 1. N' = N+3", "validated": { "instances": 40, "passed": 40 }, "source": "manual" }, { "name": "E2 epoch end, N = 1 mod 4 (clean start)", "from": "0^inf 110 0 (011)^(N-1) C> 101 0^inf", "to": "0^inf 110 0 (011)^(N+3+r) C> F(N >> (3+r)) 0^inf", "steps": null, "cases": "N = 1 mod 4; r = run of 1-bits of N starting at bit 2 (r=0..7 tested). N' = N+4+r. N up to 14077", "validated": { "instances": 50, "passed": 50 }, "source": "manual" } ], "template": { "file": "vendor/busycoq/verify/FractalBEC.v", "module": "Section FractalBEC, instance nonhalt10 (line 269)", "machine": "1LB0LF_1LC1LF_1RD1LB_---0RE_1RA0RE_0LA0RC", "diff": "mirror of tm10 = 1RB0RF_1RC1RF_1LD1RB_---0LE_1LA0LE_0RA0LC; only C1 differs: 1RB -> 1RA", "applies": "partly", "note": "On ours RInc 4/4 and Ov 8/8 hold; LInc fails for every odd counter (8/16, C reads LSB 1 -> 1RA) and config i -> i+1 fails 5/6. Bounce shape carries over; counter encoding and doubling epoch map do not." }, "strategy": "BECv7/FractalBEC shape: prove R1 (shift rule + counter increment over a residue-tolerant digit word, Inductive like BECv7) with ind; then an epoch lemma K(1,N-1,X) -->+ K(1,N'-1,X') with N'>N; nonhalt via multistep_nonhalt + progress_nonhalt_cond, invariant 'X in residue language'. E0-E2 alone do not close: real orbit almost never has a clean counter.", "risks": [ "General epoch map with residue not found yet: dN depends on residue (N=21: clean +5, with residue +10).", "Residue language may be big: 6-factors stable (23) but 8-factors still new at epoch 178; zero runs up to 20.", "Probably 15-30 KB of Rocq (counter-with-separators encoding); twin holdout 1RB0RE_1RC1RE_1LD1RA_1LA0LD_0RA0LF_1LD--- shares the trajectory." ], "checkpoint_questions": [ { "text": "Fig 1: ours has many epochs that grow a little each time; the template has few that grow ~4x. Agree this is a new epoch map, not a template tweak?", "figs": [ 0 ] }, { "text": "Fig 2: do the isolated residue 1s between counter digits stay put while carries pass through them (inert separators)?", "figs": [ 1 ] } ], "discord_check": null, "stats": { "simulated_steps": 1000000000, "cells": 5243, "nonzeros": 2610, "growth": "span ~t^0.34 (1085 / 2360 / 5243 cells at 1e7 / 1e8 / 1e9); 278 epochs to 1e9, N +~6/epoch, epoch ~4N^2 steps" }, "status": "CHECKPOINT_A", "status_reason": "Mechanism found and bounce + clean-epoch rules validated; not Collatz-like (branches on low bits of a linearly growing N). Open: residue dynamics." }