/machines/1RB0LA_1RC---_0LD1RD_0RA0RE_0RF1RC_1LF0LA/notes.md · raw

W7-6: line 33, 1RB0LA_1RC---_0LD1RD_0RA0RE_0RF1RC_1LF0LA (Translated Counter)

Stage 1 only (2 h), 2026-10-14. Claimed in coord/claims.tsv ("Wave 7 W7-6"). Byte-exact: line 33 of data/holdouts/BB6_holdouts_740.txt.

Verdict: CRYPTID_CANDIDATE (med). The counting phase is an exact base-4 counter and cannot halt. Halting is possible only at phase events (overflow, erase, rebuild, collision), and whether an event halts depends on an exact coincidence between a counter value and a gap length. The next event is about 4.6e15 steps away. Stop rule applied; decision.json asks park vs one 2 h modelling stage.

Family (the 4 W7-6 lines)

They are four different tables (canon over all 6 starts, renaming + mirror), not starts of one table. No start-parametric argument applies. Line 33's table has no other start on the 740 list.

line machine 1e9 end tape (big runs) qsim (200k loops)
33 1RB0LA_1RC---_0LD1RD_0RA0RE_0RF1RC_1LF0LA [7] 1^1322 [3] 1^3952883 [71] 0 rules
472 1RB0LA_1RC---_0LD1RD_1RA0RE_0RF1RC_1LF0LA (D0 differs) [1] 1^5408013 [30] 0 rules
569 1RB0RD_1RC---_0LA1RA_0RE1RC_1LE0LF_0RC0LF [1] 1^4591407 [59] 0 rules
690 1RB0LA_1RC---_0LD1RD_0RB0RE_0RF0LD_1LF0LA (D0, E1 differ) [1] 1^10536 [3] 1^5410013 [48] 0 rules

472/569/690 show the same architecture (trail + counter of 30-60 cells) by end tapes only. They were not modelled. Files: work/sim1e9_.jsonl, work/qsim_.jsonl, work/compress_*.txt.

Halting transition

The only halt is B reading 1. B is entered only from A0 (1RB), and A0 follows D0 (0RA) when it matters. So the halt pattern is a right-moving sweep where D reads 0, A reads the next 0 and the cell after that is 1:

C> 1 0 0 1      or      0 C> 0 1

Counting phase (exact)

At every C right record (head moving right into fresh tape in state C):

0^inf L 1^N 0^2  s_1 0 s_2 0 ... s_2w 0  1 C> 0^inf        s_i in {1^2, 1^3}

Read the s_i in pairs, MSB first, as base-4 digits:

pair digit
23 0
22 1
33 2
32 3
4 3 0, with a carry pending into the pair on its left

R1. From one C right record to the next, the value goes V -> V+1. - Checked on 10,416 of 10,416 record transitions: random values, widths w = 2..40, top digit 3, 24 records each (scripts/validate_increment.py). - No halt occurred in any of them. - Leading pair 23 (digit 0) is not stable; every real counter has top digit 3, so a w-pair counter makes at most 4^(w-1) increments before overflow.

Per tick the head crosses the whole counter: a left zigzag sweep (A/C/D/F turns, net -3 or -4 cells per zigzag) turns the pairs into blocks 1^(3k), then a right sweep (A0 1RB, B0 1RC, C/D/E over 1s) turns them back, and the trail gains cells.

Measured at w = 23 (steps 1.0e8 to 1.2e8, 39,847 increments): - about 502 steps per increment; - trail growth 79,689 cells, so about 2 per increment (not exact: depends on carries; R3 is a measurement only).

Phase 1 (from blank) is an 8-pair counter that overflowed at step 3.73e6 (trail about 40,124).

Phase events (where halting can happen)

Synthetic scans (counter in record form, then 0^g 1 0^inf; scripts/collide_scan.py):

w=1: V=3 halts at g=1
w=2: V=13 never; V=15 (overflow value) halts for most g
w=3: V=49..63 odd, g=0..47: 3 halting cells (V=49,g=28 / V=55,g=16 / V=59,g=8), one orbit

The w=3 halting orbit (scripts/halt_orbit.py 3 49 28): 1. The counter absorbs the marker and becomes a 7-symbol word. 2. It overflows. 3. It halts at step 6,290 while rebuilding. Fig 3 shows this run.

Overflow with a short trail also halts only at specific trail lengths (scripts/overflow_scan.py): - w=1: trail 5, 40, 47; - w=2: 26, 30, 37, 41, 44; - w=3: 33, 40; - w=4: 31, 38 (of 0..60).

The sheet's "Has to halt in the counting phase" fits this picture. A halt comes from the counter, during counting, running into a marker or remnant at the wrong value.

Real orbit

step 100,000,903: N = 399,782, word 3222323333233322222223332222322232322332333222 1
V = 61,170,343,443,693  (digits 3 1 3 2 2 0 2 1 1 1 0 2 1 1 3 1 3 3 0 3 2 3 1)
4^23 - 1 - V = 9,198,400,733,970 increments to overflow
=> about 4.6e15 steps, trail about 1.8e13 cells

Its high digits look irregular. The next collision's outcome depends on these digits through exact gap and value arithmetic.

Cryptid check (stop rule)

The counting rule itself is clean arithmetic: R1 is exact. But the halting question reduces to exact coincidences at phase events, between counter values and gap lengths built from irregular big numbers. No invariant rules them out, and there is no margin to watch: the outcome is all-or-nothing per event. That is a cryptid-like lottery, so the stop rule applies.

What would settle the next event is an exact phase-level model: - R1 in closed form, plus exact trail growth and step cost per increment; - the overflow, erase, nested rebuild and collision steps simulated on a run-length tape.

It decides the event at about 4.6e15 steps: a HALT (checked-runner route as for C2-552/W5-5, which needs a formal base-4 increment lemma) or the next phase's parameters. That is next_goal, funded only by a new decision.

Template

SOTCv2.v TM1 (M-Abozaid, translated-counter level cycle) applies partly: - its tokens and sweep lemmas match our counting phase; - nothing upstream covers overflow, erase, nested rebuild and collision.

The class is M-Abozaid's, so nonhalting work needs a separate call.

Attempts