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1RB1RF_1RC0RA_1LD1RE_1LE0LE_0RA0LD_0RB---
Where it stands
Parked at the deeper analysis
Funded stage prove_margin (2026-10-10) ended by its exit rule: failed bound -> parked, cryptid-margin suspect. Fast exact phase map (1321/1321 logged starts, 40/40 direct one-phase checks) extends the orbit to phase 11798, M=114,554 (~9.7e13 steps): no halt, margin k_align <= M-1 at every start, max ratio 0.467, no drift toward 1.
Read the full status
But the margin is not provable from counter structure: random reachable-looking counters (aligned, digits 1, random or orbit widths) break it in 7/200 and 4/200 runs (up to 2.84x), and it depends on the exact bell: orbit phase 10666 with bell 102,051 instead of 102,052 gives next ratio 0.979 (orbit: 0). No tested break led to halting (0/400), so not flagged CRYPTID_CANDIDATE. Rules (J-rule, Gap-Inc, J-crossing, walk), G1 and the fast map remain valid for any later attempt.
Observe
Deepen
Informal proof
Rocq
About this picture
The standard picture (bbchallenge.org's Default view): each row is the whole tape at one step, time runs downward, and the head's path draws the pattern.
bbchallenge-defaultStart blankRows steps 0 → 10,000, top to bottomWindow cells −200..199, left to rightColors 1 0 or unvisited
Overview
In one sentence
BMO7-like bell + walking marker; counter is base 3, not binary; the eat step rebuilds it irregularly
- Status
- Parkedfunded:prove margin · updated 2026-10-10Not blocking · Stuck at the funded:prove margin stage. Answer Q1–Q3 whenever; a “no” redirects the work.idle since Oct 10 13:25 · status.json 2026-10-10
- Class
- Bell eats counterours: Bell eats counter (sheet) · ours adds: BMO 7 layout, but a base-3 counter · med confidence
- On the list
- line 314 of
BB6_holdouts_663.txt· historical 740 line 347
Your calls on file 2
- funded
- option: prove_margin · date: 2026-10-10
- 2026-10-10 · prove_margin
- Non-halting of #18 now reduces to the margin M-1 >= k_align(C) at every phase start. Fund one stage to prove it, or park #18 as a cryptid-margin case? (Dennis: go with all recommendations (2026-10-10))
Mechanism
How it works: 3 notes
- Tape = a long 10-sea (the bell) + a short counter at its right end; a marker walks right through the bell.
- Each bounce adds 1 to the base-3 counter: a digit = a wall at offset 1/2/3 in its frame; overflow appends a digit.
- When the marker hits the counter, the bell eats its low end (grows by g >= 2); g and the new counter depend only on it.
Figures for Checkpoint A
One row per entry into C, every 2nd cell. Look at the black/white diagonal and the right-edge strip.
bbchallenge-defaultStart mid-run: step 2,097,352Rows entries into state CWindow cells -48..544, 1 px = 2 cellsColors 1 0 head
Start: a phase start: left record, head at cell -48 · Rows: 563 rows, steps 2,097,352..2,255,952; the next phase starts at 2,255,931 · Window: only every 2nd cell is drawn (-48, -46, ..., 544) · Colors: red: it enters C moving right; Because only every 2nd cell is drawn, the bell (10)^M looks solid white where its 1s fall on drawn cells and solid black where the marker has shifted it by one cell: the white/black diagonal is the marker
From a built tape, counter at 0. Look at the staircase on the right and the pattern within it.
bbchallenge-defaultStart constructed: 0∞ A> (10)100 00101001 0∞Rows entries into state CWindow cells 197..224, 1 px = 0.2 cellsColors 1 0 or never visited head
Start: bell of 100 blocks, then the counter X(0) = 00101001 = value 0 · Rows: 178 rows, steps 0..21,000 · Window: each cell is 5x5 px; bell = cells 0..199, counter from cell 200 · Colors: red: it enters C moving right
Right edge pinned; one row per phase start, to 1e9. Compare the counter strip from row to row.
bbchallenge-defaultStart blankRows left recordsWindow cells -109..0, 1 px = 0.33 cellsColors 1 0 or never visited head
Rows: the start, then one row per new leftmost cell = one row per phase start (313 rows, steps 0..1,000,000,000; rows are not equal time) · Window: (each cell is 3x3 px), counted from the right edge (each row is shifted so its rightmost visited cell is at 0) · Colors: green: at the left edge, just moved left
More figures (2)Fig 4–5: Edge records to 100 million steps · Blaze view, 100 million steps
The bell grows; the counter rides its right edge.
bbchallenge-defaultStart blankRows left/right recordsWindow cells -154..1903, 1 px = 1 cellColors 1 0 or never visited head just moved right/left
Rows: the start, then one row each time the head reaches a new leftmost or rightmost cell (2,058 rows, steps 0..100,000,000; rows are not equal time)
The bell (half 1s) averages to pale orange; growth ~ t^0.34.
bbchallenge-blazeStart blankRows Blaze time blocks of 131,072 stepsWindow cells -154..1903, 1 px = 2.57 cellsColors 0 1 mixed
Rows: (averaged), 763 blocks stretched to 500 px, so 1 px row ~ 200,000 steps (steps 0..100,000,000) · Colors: paler orange: each pixel averages its cells over its time block; no head drawn; the bell (10)^M is half 1s, so it averages to pale orange
Rules (8/8 validated)exact tape rewrites the proof will formalize, each re-checked by simulation
| rule | from ⟶ to | cases | steps | checked |
|---|---|---|---|---|
| sweep-in | A> (10)n ⟶ 10 (01)n-1 B> | n>=1 | 2n | ✓60/60 |
| sweep-R | B> (10)n ⟶ (01)n B> | n>=0 | 2n | ✓60/60 |
| sweep-L | (10)n <D ⟶ <D (01)n | n>=0 | 2n | ✓60/60 |
| shift-A | A> (01)n ⟶ (10)n A> | n>=0 (used inside the counter) | 2n | ✓60/60 |
| Inc (one bounce, X form) | 0∞ (10)a A> (10)b+1 X(v) 0∞ ⟶ 0∞ (10)a+1 A> (10)b X(v+1) 0∞ | b>=1. X(v) = D(v mod 3) 1 T(v div 3) 001; D = 0010|0110|0100; T = bijective base-3 digits LSB first, 1/2/3 = 0001/0011/0010 | — | ✓450/450 |
| Inc (one bounce, any frames) | 0∞ (10)a A> (10)b+1 F 0∞ ⟶ 0∞ (10)a+1 A> (10)b inc(F) 0∞ | b>=1. F = frames (d_i in 1..3, even width w_i>=4), wall gaps 1,d_0,w_0-d_0,...; inc = base 3 on d (3->1 carries), overflow appends (1,4); widths unchanged | — | ✓400/400 |
| Phase (M-1 bounces) | 0∞ A> (10)M X(v) 0∞ ⟶ 0∞ (10)M-1 A> 10 X(v+M-1) 0∞ | M>=2; Inc applied M-1 times | — | ✓40/40 |
| Eat (bell meets counter) | 0∞ (10)N A> 10 F 0∞ ⟶ 0∞ <E (10)N+g(F) F' 0∞ | g and F' depend on F only (N vs N+k, 60/60). Examples: [(1,6)]->+7,[(1,6)]; [(3,6)]->+2,[(1,10),(1,6)]; [(1,10)]->+7,[(1,4),(1,6)]. Closed form: OPEN | — | ✓60/60 |
Compressed historythe tape as 0∞ A> (10)M W 0∞ at 39 moments: where the rules are read off
0∞ A> (10)M W 0∞
| step | M = bell blocks at phase start | |W| = counter cells | W as wall gaps (o = 00 hole, # = 11 bump) |
|---|---|---|---|
| 41 | 2 | 8 | 1o 1o 5o |
| 64 | 3 | 7 | 1o 1o 1o 3o |
| 163 | 7 | 7 | 1o 1o 1o 3o |
| 362 | 8 | 13 | 1o 1o 1o 3o 2# 4o |
| 669 | 9 | 19 | 1o 1o 11o 1o 1o 3o |
| 1,104 | 10 | 25 | 1o 1o 11o 1o 1o 3o 2# 4o |
| 1,675 | 17 | 19 | 1o 1o 11o 1o 1o 3o |
| 2,574 | 18 | 25 | 1o 1o 1o 9o 1o 5o 2# 4o |
| 10,067 | 33 | 50 | 1o 1o 41o 1o 5o |
| 128,733 | 111 | 19 | 1o 1o 1o 3o 3o 1o 2# 2o 3o 1o |
| 529,171 | 172 | 29 | 1o 1o 13o 1o 1o 5o 3o 3o |
| 1,428,457 | 241 | 31 | 1o 1o 11o 1o 1o 1o 3o 5o 3o 3o |
| 2,974,083 | 326 | 17 | 1o 1o 5o 2# 1# 2o 3o 1o |
| 5,636,701 | 388 | 33 | 1o 1o 11o 1o 5o 1o 1o 5o 3o 3o |
| 9,167,327 | 444 | 41 | 1o 1o 25o 1o 1o 5o 3o 3o |
| 13,840,601 | 520 | 17 | 1o 1o 1o 3o 3o 3o 1o 3o |
| 19,709,255 | 559 | 67 | 1o 1o 27o 1o 19o 1o 1o 5o 3o 3o 2# 2o |
| 27,284,505 | 666 | 16 | 1o 1o 3o 1o 3o 1o 5o |
| 36,981,311 | 718 | 61 | 1o 1o 33o 1o 5o 1o 1o 3o 2# 4o 2# 2o 2# 2o |
| 48,739,545 | 799 | 50 | 1o 1o 27o 1o 13o 1o 5o |
| 62,765,319 | 871 | 23 | 1o 1o 1o 3o 1o 5o 2# 4o 3o 1o |
| 79,028,529 | 928 | 49 | 1o 1o 25o 1o 9o 1o 1o 1o 2# 2o 3o 1o |
| 97,840,939 | 1,002 | 40 | 1o 1o 17o 1o 5o 1o 7o 1o 5o |
| 119,322,197 | 1,083 | 31 | 1o 1o 13o 1o 1o 1o 2# 2o 1o 3o 1o 3o |
| 144,732,311 | 1,168 | 45 | 1o 1o 21o 1o 5o 1o 1o 1o 2# 2o 1o 3o 1o 3o |
| 174,085,265 | 1,252 | 52 | 1o 1o 9o 1o 11o 1o 5o 1o 3o 1o 11o 1o 5o |
| 207,484,435 | 1,326 | 49 | 1o 1o 15o 1o 5o 1o 5o 1o 1o 3o 2# 4o 2# 2o 1o 3o |
| 244,435,833 | 1,401 | 31 | 1o 1o 1o 3o 1o 3o 2# 2o 3o 1o 1o 3o 2# 2o 1o 3o |
| 285,517,799 | 1,469 | 51 | 1o 1o 1o 3o 3o 19o 2# 6o 3o 3o 2# 2o 1o 3o |
| 331,301,633 | 1,556 | 37 | 1o 1o 1o 3o 3o 3o 3o 3o 3o 3o 3o 1o 2# 2o 1o 3o |
| 382,417,731 | 1,642 | 41 | 1o 1o 5o 1o 5o 1o 9o 1o 1o 3o 2# 2o 3o 1o 1o 3o |
| 438,884,121 | 1,705 | 71 | 1o 1o 41o 1o 13o 1o 1o 5o 3o 3o |
| 501,203,395 | 1,791 | 55 | 1o 1o 5o 1o 19o 1o 5o 1o 1o 1o 2# 2o 2# 6o 3o 3o |
| 568,808,345 | 1,887 | 31 | 1o 1o 1o 1o 2# 10o 1o 5o 1o 3o 2# 2o |
| 643,095,903 | 1,957 | 51 | 1o 1o 17o 1o 1o 3o 1o 5o 1o 5o 2# 4o 1o 3o 2# 2o |
| 723,172,281 | 2,025 | 75 | 1o 1o 27o 1o 27o 1o 1o 3o 2# 2o 2# 2o 2# 2o |
| 809,229,495 | 2,106 | 69 | 1o 1o 1o 11o 3o 19o 3o 11o 3o 3o 1o 3o 2# 2o 2# 2o |
| 902,342,053 | 2,188 | 57 | 1o 1o 17o 1o 1o 3o 3o 5o 1o 5o 1o 7o 3o 3o 2# 2o |
| 991,609,360 | 2,263 | 47 | 1o 1o 5o 1o 1o 3o 1o 15o 3o 1o 1o 5o 3o 1o 2# 2o |
M = bell blocks at phase startlast 2,263|W| = counter cellslast 47W as wall gaps (o = 00 hole, # = 11 bump)last 1o 1o 5o 1o 1o 3o 1o 15o 3o 1o 1o 5o 3o 1o 2# 2onot plottable
Template comparisonBMO7.v: partly applies · the nearest solved proof, what carries over, and the transitions that differ
| 0 | 1 | |
|---|---|---|
| A | 1RB | 1RF |
| B | 1RC | 0RA |
| C | 1LD | 1REtpl 1RC |
| D | 1LE | 0LE |
| E | 0RA | 0LD |
| F | 0RB | --- |
differs from the template
- file
BMO7.v- module
- —
- machine
1RB1RF_1RC0RA_1LD1RC_1LE0LE_0RA0LD_0RB---(1 transition apart)- diff
C1: 1RC (BMO 7) -> 1RE (ours); all other 11 transitions equal- note
- Left bell bounce (BMO7 LIncs) holds here 36/36; binary RInc fails on every odd counter (16/40), BigStep shape 3/8 with non-binary right side. Same layout, new counter.
Proof strategy & riskshow the formal proof will go · 3 risks to watch
Keep BMO7's layout: Longitudinal sideRLs for bell bounces (reuse LIncs), a new ternary Inc lemma over frames (ind on digits, DivModCases mod 3). Next stage: close the Eat step by enumerating reachable counter shapes; if finite-parametric, prove it with a WF invariant + progress_nonhalt_cond (BMO7 pattern), else a verified checker over frame lists.
- ▲Eat step not closed: phase-start counters irregular (dN 1 in ~half of phases, else 4..38).
- ▲If eat outcomes depend on counter digits pseudo-randomly, this becomes a cryptid-like case.
- ▲Larger than BMO7 (441 lines): two-level counter + eat lemma family; likely >15 KB.
Stats & provenance109 steps simulated · 4,585 cells · the status in full · files
- simulated
- 109 steps
- tape
- 4,585 cells · 2,285 nonzeros
- growth
- bell M ~ t^0.342; 312 phases by 1e9; span -312..4272
- status now
- Funded stage prove_margin (2026-10-10) ended by its exit rule: failed bound -> parked, cryptid-margin suspect. Fast exact phase map (1321/1321 logged starts, 40/40 direct one-phase checks) extends the orbit to phase 11798, M=114,554 (~9.7e13 steps): no halt, margin k_align <= M-1 at every start, max ratio 0.467, no drift toward 1. But the margin is not provable from counter structure: random reachable-looking counters (aligned, digits 1, random or orbit widths) break it in 7/200 and 4/200 runs (up to 2.84x), and it depends on the exact bell: orbit phase 10666 with bell 102,051 instead of 102,052 gives next ratio 0.979 (orbit: 0). No tested break led to halting (0/400), so not flagged CRYPTID_CANDIDATE. Rules (J-rule, Gap-Inc, J-crossing, walk), G1 and the fast map remain valid for any later attempt. (status.json, 2026-10-10)
- observer's note
- Bell/marker and base-3 counter rules validated; eat step is N-independent but its closed form is open. No Collatz-style rule found (values grow polynomially), so not flagged cryptid yet.
- files
- observe.json · notes.md · status.json
Proof & checks
No proof work yet.
Files
machines/<tm>/: 8 entries · raw notes, status, scripts, images
| name | size | changed |
|---|---|---|
| notes.md ↓ | 24 KB | 2026-10-10 13:25 |
| status.json ↓ | 1.0 KB | 2026-10-10 13:25 |
| observe.json ↓ | 13 KB | 2026-10-10 04:41 |
| deepen.json ↓ | 16 KB | 2026-10-10 13:25 |
| decisions/ | 1 item | 2026-10-10 13:25 |
| img/ | 9 items | 2026-10-10 00:51 |
| scripts/ | 48 items | 2026-10-10 13:23 |
| work/ | 29 items | 2026-10-10 00:51 |
status.json in full
{
"tm": "1RB1RF_1RC0RA_1LD1RE_1LE0LE_0RA0LD_0RB---",
"status": "STUCK",
"stage": "funded:prove_margin",
"updated": "2026-10-10",
"reason": "Funded stage prove_margin (2026-10-10) ended by its exit rule: failed bound -> parked, cryptid-margin suspect. Fast exact phase map (1321/1321 logged starts, 40/40 direct one-phase checks) extends the orbit to phase 11798, M=114,554 (~9.7e13 steps): no halt, margin k_align <= M-1 at every start, max ratio 0.467, no drift toward 1. But the margin is not provable from counter structure: random reachable-looking counters (aligned, digits 1, random or orbit widths) break it in 7/200 and 4/200 runs (up to 2.84x), and it depends on the exact bell: orbit phase 10666 with bell 102,051 instead of 102,052 gives next ratio 0.979 (orbit: 0). No tested break led to halting (0/400), so not flagged CRYPTID_CANDIDATE. Rules (J-rule, Gap-Inc, J-crossing, walk), G1 and the fast map remain valid for any later attempt.",
"funded": {
"option": "prove_margin",
"date": "2026-10-10"
}
}