BB(6) hunt
Machines / #19

1RB1RF_1RC---_1RD0RD_1LE1RC_1RA1LE_1RE0LF

#19ParkedShift-overflow structural counterbbchallenge ↗Explorer

Where it stands

Parked at the deeper analysis

Parked by Dennis 2026-10-10 (decision queue: park); deepen-2 findings are the re-entry point. Deepen pass 2: a closed-form fast-forward (validated against round 1 and against full-size runs) reaches trip 13.5e6 (about 1.27e15 steps).

Read the full status

There the base-4 counter overflows: a left record, then a full-tape recoding trip that is quadratic in zone size (about 1e16 steps at real sizes) and depends on zone parity. After it, a new counter epoch starts from a new wall. The machine is a multi-epoch shift-overflow counter. The rebuild rule and the epoch recursion are open. Confidence low; not a cryptid so far. Priority call for Dennis: continue (rebuild rule) or park.

Observe
Deepen
Informal proof
Rocq
space-time diagram, first 10,000 steps from blank
The first 10,000 steps from a blank tape. Time runs down; white is 1.
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

Bouncer whose turn-back point is a base-4 counter; digits = separators between growing (110101) zones.

Status
Parkeddeepen2 · updated 2026-10-10Not blocking · Stuck at the deepen2 stage. Answer Q1–Q3 whenever; a “no” redirects the work.
idle since Oct 10 15:46 · status.json 2026-10-10
Class
Shift-overflow structural counterours: sheet Shift-overflow structural counter ≠ ours Base-4 bouncer counter: separators between growing 6-cell zones are the digits · med confidence
On the list
not on BB6_holdouts_663.txt (line 542 of the previous list)
#19eqv ×1table ×2sim 1e15UBRRBA
Your calls on file 2
parked
Parked by Dennis 2026-10-10 (decision queue: park); deepen-2 findings are the re-entry point.
decision.json · answered 2026-10-10: park
Park #19, rather than fund a stage on its overflow/rebuild rule? Decisions →

Mechanism

How it works: 3 notes
  • Tape = left end ... Z2 S1 Z1 S0 Z0 W: zones Z repeat a 6-cell block (110101, rotated); W = fixed 99-cell right wall.
  • Separators S have 4 states: 3 bounce the head back, the 4th lets it through, so turn-back depth is a base-4 ruler.
  • Z0 gains 1 block per 2 sweeps, Z_k ~ Z_(k-1)/4. The left end, where new digits appear, is not characterised yet.

Figures for Checkpoint A

Fig 1 · Raw view, every 50 stepsQ1

Every 50 steps. Flat edges are right passes, steep edges left passes; note where each sweep turns back.

Raw view, every 50 steps
bbchallenge-defaultStart mid-run: step 1,901,496Rows every 50 stepsWindow cells -68..2,245, 1 px = 1 cellColors 1 0 head after a right/left move
Start: tape from the real run, head at cell 2,134 · Rows: steps 1,901,496 + 0, 50, ..., 200,000 (4,001 rows) · Colors: The 6-cell zones look grey; A right pass moves 50 cells per row (nearly flat edges), a left pass ~15 (steeper edges); a sweep's turn-back is where its edge stops on the left
Fig 2 · One row per round trip, 452 tripsQ2

One row per round trip; red marks each turn-back. Compare how deep each turn-back goes.

One row per round trip, 452 trips
customStart blankRows one per round trip of the headWindow cells -68..900, 1 px = 1 cellColors 1/0 on cells this trip reached 1/0 on cells it did not reach the turn-back cell not yet visited
Rows: (it leaves the right end, turns back, returns to within 120 cells of the right end), drawn at the return: 452 trips, steps 24,389..1,994,475 (1 row = 4 px tall); not evenly spaced in time · Window: the right part of the tape, up to cell 2,185 at the end, is cut off: every trip reaches it · Colors: red: deepest point; The red diagonal is S0 (3 trips in 4 turn there); every 4th trip goes on to S1, every 16th to S2, every 64th deeper still
Fig 3 · Right edge pinned, 10 million stepsQ3

Right edge pinned to 1e7 steps. Thin slanted lines are zone borders; watch the ones at the far left.

Right edge pinned, 10 million steps
bbchallenge-defaultStart blankRows the start, then one per new rightmost cellWindow cells -4,921..0, 1 px = 1 cellColors 1 0 or never visited head on the new right cell
Rows: (4,854 rows, steps 0..10,000,000); not evenly spaced in time · Window: counted from the right edge (each row's rightmost visited cell is 0, pinned to the last column; it is absolute cell 4,853 at step 1e7) · Colors: The grey texture is the 6-cell zone pattern; the thin slanted lines are the zone borders (separators)
More figures (2)Fig 4–5: Blaze view, 1 billion steps · bbchallenge Default view, first 10,000 steps
Fig 4 · Blaze view, 1 billion steps

Faint inner curves at about 1/4 and 1/16 of the edge are zone borders (the counter's digits).

Blaze view, 1 billion steps
bbchallenge-blazeStart blankRows time blocks of 1,048,576 stepsWindow cells -96..48,095, 1 px = ~60 cellsColors 0 1
Rows: averaged (954 blocks, steps 0..1,000,000,000), stretched to 500 px: 1 px ~ 2,000,000 steps · Window: Blaze averages 32 cells per column; its 1,506 columns are stretched to 800 px · Colors: in-between shades = both mixed inside the pixel's block (averaged); no head
Fig 5 · bbchallenge Default view, first 10,000 stepsviewer ↗

For orientation: the start of the run as bbchallenge.org shows it.

bbchallenge Default view, first 10,000 steps
bbchallenge-defaultStart blankRows every step: steps 0..9,999 squeezed into 500 px rows as on bbchallenge.orgWindow cells -200..199, 1 px = 1 cellColors 1 0 or never visited head after a right/left move
Rows: each px row blends steps 20y+9 and 20y+10, so 1 px row = 20 steps · Colors: red/green: pale colours = blended by the squeeze
Rules (8/8 validated)exact tape rewrites the proof will formalize, each re-checked by simulation
rulefrom ⟶ tocasesstepschecked
R1 C-pass
C> (011111)n
⟶ (110101)n C>
any context; n=1..27706n✓24/24
R2 F-sweep (0^6 packet)
(111010)n <F 06
⟶ <F 06 (111110)n
any context; n=1..189420n✓24/24
Ra: S0 bounce a->b
X 19 010 <F 04 113 (011111)n 18 W
⟶ X 19 0 17 (01)3 0 <F 04 (111110)n 113 W
W = 0^3 1^11 0^7 (111110)^2 1^13 0 1^5 0^7 1^5 0 1^11 0^7 1^7 0^inf (fixed wall); X = real left context (4255 cells) or its last 300; n=1..134226n+624✓16/16
Rc: S0 bounce c->a
X 19 (01)6 0 <F 04 (111110)n 113 W
⟶ X 19 010 <F 04 113 (011111)n+1 18 W
as Ra; n=1..129026n+688✓16/16
RbH: S0 opens, S1 bounces (H->P)
Y H(m) P0 19 0 17 (01)3 0 <F 04 (111110)n 113 W
⟶ Y P(m) P0 19 (01)6 0 <F 04 (111110)n+1 113 W
P0 = 1^7 0 1^9 (01111111)^2 (110101)^3 (01)^2 1^8 0 1^7 0 1^9 0 1^7 0 (1^8 (10)^4)^2; H(m)=1^8 (10111111)^2 (10)^6 (101110)^m; P(m)=1^8 (10111111)^2 (10)^4 1^7 0 (101110)^m; Y real; n,m=1..106052n+26m+2130✓14/14
RbM: S0 opens, S1 bounces (M->L)
Y M(m) P0 19 0 17 (01)3 0 <F 04 (111110)n 113 W
⟶ Y L(m) P0 19 (01)6 0 <F 04 (111110)n+1 113 W
M(m)=1^9 0 1^7 (01)^3 1 (10)^5 (101110)^m, L(m)=1^9 0 1^4 (11101010)^2 1^7 0 (101110)^m52n+26m+2130✓14/14
RbL: S0 opens, S1 bounces (L->H, Z1+1)
Y L(m) P0 19 0 17 (01)3 0 <F 04 (111110)n 113 W
⟶ Y H(m+1) P0 19 (01)6 0 <F 04 (111110)n+1 113 W
n,m=1..97452n+26m+2220✓14/14
RbP: carry S0->S1->S2 (P->M, Z1+1)
Y_b P(m) P0 19 0 17 (01)3 0 <F 04 (111110)n 113 W
⟶ Y_c M(m+1) P0 19 (01)6 0 <F 04 (111110)n+1 113 W
Y_b/Y_c = real S2-and-left context before/after the carry (fixed); n,m=1..50552n+26m+6340✓12/12
Compressed historythe tape as 0∞ [left end] Z3d S2 Z2c S1 Z1b S0 Z0a W C> 0∞ (Z = a rotation of 110101; first right record of each cycle) at 20 moments: where the rules are read off
0∞ [left end] Z3d S2 Z2c S1 Z1b S0 Z0a W C> 0∞ (Z = a rotation of 110101; first right record of each cycle)
stepa (Z0 blocks)b (Z1)c (Z2)deeper zonesright edge pos
24,63214---188
64,6224---326
107,80622---466
188,65646---658
329,154747--886
577,37811116--1,182
1,005,84815828--1,558
1,765,60622143--2,058
3,082,882302649-2,710
5,394,0024109115-3,574
9,433,1385531262444,714
16,512,6687411743676,222
28,876,85299023652118,214
50,564,5601,320318721610,854
88,434,5541,756427992314,338
154,800,5942,33357113532 518,954
270,879,1003,0957621836 1325,054
473,958,1644,1041,01424621 4 1333,126
829,471,8485,4391,34833042 10 1143,806
1,451,620,6947,2051,79044070 17 1157,934
a (Z0 blocks)last 7,205
7,205424,6321.5e9log xstep 24632: a (Z0 blocks) = 14step 64622: a (Z0 blocks) = 4step 107806: a (Z0 blocks) = 22step 188656: a (Z0 blocks) = 46step 329154: a (Z0 blocks) = 74step 577378: a (Z0 blocks) = 111step 1005848: a (Z0 blocks) = 158step 1765606: a (Z0 blocks) = 221step 3082882: a (Z0 blocks) = 302step 5394002: a (Z0 blocks) = 410step 9433138: a (Z0 blocks) = 553step 16512668: a (Z0 blocks) = 741step 28876852: a (Z0 blocks) = 990step 50564560: a (Z0 blocks) = 1320step 88434554: a (Z0 blocks) = 1756step 154800594: a (Z0 blocks) = 2333step 270879100: a (Z0 blocks) = 3095step 473958164: a (Z0 blocks) = 4104step 829471848: a (Z0 blocks) = 5439step 1451620694: a (Z0 blocks) = 7205
b (Z1)last 1,790
1,79073.3e51.5e9log xstep 329154: b (Z1) = 7step 577378: b (Z1) = 16step 1005848: b (Z1) = 28step 1765606: b (Z1) = 43step 3082882: b (Z1) = 64step 5394002: b (Z1) = 91step 9433138: b (Z1) = 126step 16512668: b (Z1) = 174step 28876852: b (Z1) = 236step 50564560: b (Z1) = 318step 88434554: b (Z1) = 427step 154800594: b (Z1) = 571step 270879100: b (Z1) = 762step 473958164: b (Z1) = 1014step 829471848: b (Z1) = 1348step 1451620694: b (Z1) = 1790
c (Z2)last 440
44093.1e61.5e9step 3082882: c (Z2) = 9step 5394002: c (Z2) = 15step 9433138: c (Z2) = 24step 16512668: c (Z2) = 36step 28876852: c (Z2) = 52step 50564560: c (Z2) = 72step 88434554: c (Z2) = 99step 154800594: c (Z2) = 135step 270879100: c (Z2) = 183step 473958164: c (Z2) = 246step 829471848: c (Z2) = 330step 1451620694: c (Z2) = 440
deeper zoneslast 70 17 11
2349.4e68.8e7step 9433138: deeper zones = 4step 16512668: deeper zones = 7step 28876852: deeper zones = 11step 50564560: deeper zones = 16step 88434554: deeper zones = 23
right edge poslast 57,934
57,93418824,6321.5e9log xstep 24632: right edge pos = 188step 64622: right edge pos = 326step 107806: right edge pos = 466step 188656: right edge pos = 658step 329154: right edge pos = 886step 577378: right edge pos = 1182step 1005848: right edge pos = 1558step 1765606: right edge pos = 2058step 3082882: right edge pos = 2710step 5394002: right edge pos = 3574step 9433138: right edge pos = 4714step 16512668: right edge pos = 6222step 28876852: right edge pos = 8214step 50564560: right edge pos = 10854step 88434554: right edge pos = 14338step 154800594: right edge pos = 18954step 270879100: right edge pos = 25054step 473958164: right edge pos = 33126step 829471848: right edge pos = 43806step 1451620694: right edge pos = 57934
Template comparisonSBCv6a.v TM1: partly applies · the nearest solved proof, what carries over, and the transitions that differ
01
A1RB1RF
B1RC---
C1RD0RD
D1LE1RC
E1RA1LE
F1RE0LF
file
SBCv6a.v:15
module
TM1
machine
1LB1RE_0LC0RB_1RD0LA_0RA0RD_0RF0RA_1RD--- (11 transitions apart)
diff
9 transitions after best relabel/mirror (SBCv10 TM4: 8, SOBLv2 TM3: 8 and a HALT proof). Nearest solved overall: 3 transitions, decider certificates only (NG_v1, CPS_LRU_v1, RWL_mod_v3).
note
Rules/tapes: no (template-shaped tapes collapse or halt at step 1 under this TM). Architecture: yes. SBCv6a's RIncs digit list and SBCv10's signal net are the same kind of digit counter.
Proof strategy & riskshow the formal proof will go · 3 risks to watch

Longitudinal proof in SBCv6a TM1 style: ind lemmas for R1/R2 blocks; one segRLs lemma per separator state (3 bounces + 1 pass) and one for the wall W. The digit list is an Inductive with a WF invariant, closed by multistep_nonhalt + progress_nonhalt_cond. A left-end lemma that creates new digits is still needed.

  • ▲Left end not characterised: deep zones reset at the 1.8e8 left record (Z3 32 -> new 6/13). It might be messy.
  • ▲Separators differ: two phase types (bounce offsets -10 vs -12) and different inert debris (P0 vs P1). Needs a closed family of separator words.
  • ▲Size: several separator/phase cases, so expect 20-40 KB (SBCv6a/SBCv10 scale). qsim and rulefit found nothing, so the proof is by hand.
Stats & provenance2·1010 steps simulated · 214,969 cells · the status in full · files
simulated
2·1010 steps
tape
214,969 cells · 34,836 nonzeros
growth
right edge ~ sqrt(t): step ≈ 0.432·hi² (hi=214890 at 2e10); left edge stuck at -78 since 1.8e8; nonzeros at 1e9
status now
Parked by Dennis 2026-10-10 (decision queue: park); deepen-2 findings are the re-entry point. Deepen pass 2: a closed-form fast-forward (validated against round 1 and against full-size runs) reaches trip 13.5e6 (about 1.27e15 steps). There the base-4 counter overflows: a left record, then a full-tape recoding trip that is quadratic in zone size (about 1e16 steps at real sizes) and depends on zone parity. After it, a new counter epoch starts from a new wall. The machine is a multi-epoch shift-overflow counter. The rebuild rule and the epoch recursion are open. Confidence low; not a cryptid so far. Priority call for Dennis: continue (rebuild rule) or park. (status.json, 2026-10-10)
observer's note
Mechanism found (base-4 separator counter, no Collatz-style branching). 8 rules validated, 134/134 instances. Left end still open.
files
observe.json · notes.md · status.json

Proof & checks

No proof work yet.

Files

machines/<tm>/: 8 entries · raw notes, status, scripts, images

namesizechanged
notes.md ↓26 KB2026-10-10 15:46
status.json ↓1.1 KB2026-10-10 15:46
observe.json ↓12 KB2026-10-10 04:41
deepen.json ↓11 KB2026-10-10 07:23
decision.json ↓4.0 KB2026-10-10 12:49
img/8 items2026-10-10 01:54
observe_scripts/10 items2026-10-10 01:54
scripts/34 items2026-10-10 07:22
status.json in full
{
 "tm": "1RB1RF_1RC---_1RD0RD_1LE1RC_1RA1LE_1RE0LF",
 "status": "STUCK",
 "stage": "deepen2",
 "updated": "2026-10-10",
 "reason": "Parked by Dennis 2026-10-10 (decision queue: park); deepen-2 findings are the re-entry point. Deepen pass 2: a closed-form fast-forward (validated against round 1 and against full-size runs) reaches trip 13.5e6 (about 1.27e15 steps). There the base-4 counter overflows: a left record, then a full-tape recoding trip that is quadratic in zone size (about 1e16 steps at real sizes) and depends on zone parity. After it, a new counter epoch starts from a new wall. The machine is a multi-epoch shift-overflow counter. The rebuild rule and the epoch recursion are open. Confidence low; not a cryptid so far. Priority call for Dennis: continue (rebuild rule) or park.",
 "parked": "Parked by Dennis 2026-10-10 (decision queue: park); deepen-2 findings are the re-entry point.",
 "holdouts_740": "absent: removed by equivalence (upstream Eqv_WordPairs.v TM83, commit d3c52f5) to 1LB---_1RC0RC_1LD1RB_1RE1LD_0RA1RF_1RD0LF, which stays on the 740 list; a proof of either settles both"
}