/machines/1RB0RE_1LC0RD_1LD0LD_1RA0LC_1RA1RF_1RA---/notes.md · raw

1RB0RE_1LC0RD_1LD0LD_1RA0LC_1RA1RF_1RA---

Status: CHECKPOINT_A (2026-10-10, observe done; confidence low, so proof work waits for Dennis) Class guess (agent / Dennis): a hierarchical (nested) binary counter. Level 1 is #33's / kill #11's counter. Each higher level sits behind a "fossil K-block" wall whose fuel drops by 1 per lower-level overflow. Sheet: Skelet 1-like Class 2. / _ Spreadsheet notes / Discord thread: - Sheet: Results row 1287 and Classifications row 16. Both give the class only (Skelet 1-like Class 2), with no status, notes, links or strikethrough (tools/holdouts.py lookup --sheet). - On all 26 snapshots from 2024-06-14 to 2026-10-07_778 (line 596). A singleton group in 2026-10-07_eqv_1590.txt. - Not in data/discord/* (mxdys UBRRBA 86, dyuan01 longitudinal 24) and not in reports/discord-drafts.md. Not claimed, so not SKIPPED. - Only reports/targeting.md mentions it (reserve R4, "open 1-transition pair #33/R4"). Viewer: https://bbchallenge.org/1RB0RE_1LC0RD_1LD0LD_1RA0LC_1RA1RF_1RA---

Identity (checked by brute force over 720 renamings x mirror)

Observation

Candidate rules

All in R4's orientation, checked with tools/sim.py on R4 itself (exact tape and exact step count). Scripts are in scripts/.

M_k (carry; = #33's M_k, same formula):                                                    [validate_M.py 300/300; real run 12,350/12,350 to 3e8]
  <C (01)^b 000 (01)^a [(10)^vj (01)^uj]_{j=m..j*+1} (10)^v 00 (01)^u  S
  --> <C (01)^(b+2) 000 (01)^(a+1) [(10)^(vj-1) 00 (01)^uj]_{trailing} (10)^(v-1) (01)^(u+1)  S
  steps 32 + 8a + 8b + 4v + sum_trailing (8uj + 4vj);  needs a >= 1, v >= 1, every trailing digit uj >= 2, vj >= 1; S arbitrary (never read)
  negative control: one trailing digit with u = 1 -> #33's formula fails 60/60
P (partial overflow at the first bit-1 digit with u = 1; #32's P mirrored):              [validate_P.py 300/300; real run 34/34 to 3e8]
  <C (01)^b 000 (01)^a  d~_m .. d~_(p+1)  (10)^vp 01  S        d~_i = (10)^vi (01)^ui bit 1, ui >= 2;  S starts with 1;  a >= 1
  --> <C 01 000 (01)^(b+3) (10)^a (01)^vm [(10)^u(i+1) 00 (01)^(vi-1)]_{i=m-1..p+1} (10)^u(p+1) 00 (01)^(vp-1) 10 S[1:]
      (p = m: <C 01 000 (01)^(b+3) (10)^a (01)^vm 10 S[1:])
  steps 77 - 4k + 8a + 20b + 4*sum_{i>=p} ui + 8*sum_{i>=p} vi,  k = m-p+1, up = 1
O (= #33's overflow; all bits 1, every u >= 2, text after the digits exactly 1001):        [validate_O.py 200/200; control with one u = 1: 0/200]
  <C (01)^b 000 (01)^a [(10)^vj (01)^uj]_{j=m..1} 1001
  --> <C 01 000 (01)^(b+3) (10)^a (01)^vm [(10)^uj 00 (01)^(v(j-1)-1)]_{j=m..2} (10)^u1 00 1001
  steps 89 - 4m + 8a + 20b + 4 sum u + 8 sum v      (never fires from blank to 3e8: the far end is never the bare 1001)

Template check

Cryptid check (ground rule 6)

Checkpoint A (Dennis)

Informal proof

None. Proof work waits on Dennis (low confidence).

Formalization log

Outcome

2026-10-10, observe: CHECKPOINT_A, confidence low. - Validated: M, P, O. - Not covered: 54 of 12,438 macro steps to 3e8 (three event types, the same ones #32 left open). - The nested-wall mechanism is observed but not formalized. - Suggested next step, if Dennis wants to go on: treat #32 and R4 together. - Model one level as "K-block + digits + fossil K-block". - Turn the fuel-decrement law and the wall-birth and dissolution events into exact rules. Two fuel-cut runs (gap 9 and 5) show dissolution is benign there. - Then look for a recursive invariant: a level's fuel is refilled whenever a new wall is born.

Decision (2026-10-10)

Question: R4 and #32 are one table with one open mechanism: a hierarchical counter whose fossil-K walls lose one unit of fuel per lower overflow. Invest in a joint #32/R4 recursive invariant, or park both? Answer: park_both (Park R4 and #32 together for now). Dennis: go with all recommendations (2026-10-10). Parked by Dennis 2026-10-10 together with #32 (same table): open hierarchical counter (fossil-K walls); needs a joint multi-level invariant.

Reopened (2026-10-12)

Dennis: close out Skelet 1-like Class 2. The class's 8 kills (#3's and #11's tables) are the templates to learn from.

Class 2 close-out, stage 1 (2026-10-12)

Joint stage with #32 (same table). Outcome: progress, not closed. The full write-up is in machines/1RB0LC_1RC0RC_1LD0RB_1LA0LE_1LD1LF_1LD---/notes.md (section of the same name). It has the regime maps, the exhaustion tables, the halting cuts and the scripts; R4's orbit is its section 5. In short: - Halting is decided by residues mod 5 of fuel values at exhaustion events. A regime-I state K(W (0,0,N) (0,0,M) D; a, 1) whose N is cut to 2 halts, from every real state tried (4 of 4), and N = 7 halts after one cycle; N = 0, 1, 3, 4, 6 pass. - R4's orbit, read in #32's orientation (#32's table started in D): II(30, 7, 29) at step 51,277, then II(33, 2, 31), then I(32, 32) at 475,465, then I(27, 33) at 14,968,044,039. So R4 is in regime I with N = 27 ≡ 2 (mod 5). - If the regime-I map (N − 5, M + 1 per cycle) holds for 5 more cycles, R4 reaches N = 2 and, going by the cuts, halts. The step count would be far beyond direct simulation. - The top barely moves for now: seven digits down sits a second level, (0,0,1083), where all kicks after 1.5e10 happen, and the four digits in between only count those kicks. How it relates to the 2026-10-10 "inner wall" gaps (217 → 151 cells) was not checked. The top advances only when the deep level passes carries up, so R4's fate also rides on the residues of the deep level's exhaustions. A run of R4 to 3e11 does not halt: the top stays I(27, 33), and the deep fuel goes 1083 → 1076, one unit per two kicks. - The answered park_both decision (2026-10-10) is archived as decisions/2026-10-10-park_both.json. No new decision: stage 2 of this track (an exact abstract simulator to project R4's orbit to its N exhaustion) needs no call from Dennis.

Class 2 close-out, stage 2 (2026-10-12)

Joint stage with #32 (same table). Outcome: progress, not closed. The full write-up is in machines/1RB0LC_1RC0RC_1LD0RB_1LA0LE_1LD1LF_1LD---/notes.md (section of the same name). R4 in short, in #32's orientation (D start): - A new exact simulator (c2s2_tsim.c in #32's scripts/: run-length 2-cell tokens, chain, zig-zag and closed-form carry jumps) reproduces R4's b = 1 dump to 3e11 line for line (119/119). It agrees with itself with jumps on and off to 1e17 (186/186). - No halt to 8.5e37 steps (the 2^126 guard of its 128-bit step counter). - The top went I(27, 33) at 1.50e10, then the carry into M at 2.5e13, then kick 1 of the regime-I cycle at 6.24e15, giving W (0,0,26) (0,0,0) 1 (31,0,·). It has not moved since: the first 60 defects are unchanged from 3.5e25 to 8.5e37. The next top kick, kick 2 of 16 in this cycle, comes later than 8.5e37. - Cutting R4's own top to N = 0, 1, 2, 3, 7, 12 changes nothing by 8.5e37: the top never moves in any of them. - On #32's states, the N ≡ 2 (mod 5) halting rule now holds through four full cycles: N = 2, 7, 12, 17, 22 halt from the step-276,440,631 state, the last after 1.383e14 steps. N = 27 from that state does not halt by 8.5e37, because a lower level runs out first. - So whether R4 halts (its top needs 5 more cycles to reach N = 2) is still open. It needs bigger counters and one more acceleration level; that is stage 3.

Class 2 close-out, stage 3 (2026-10-12)

Joint stage with #32 (same table), the last of this track. Outcome: cryptid. R4 and #32 are both flagged CRYPTID_CANDIDATE (confidence med), and proof work stops. The full write-up is in machines/1RB0LC_1RC0RC_1LD0RB_1LA0LE_1LD1LF_1LD---/notes.md, in the section of the same name. The decision for Dennis is decision.json in this folder (recommended: stop_cryptid). R4 in short, in #32's orientation (D start): - Simulator. c2s3_tsim.c in #32's scripts/ is the stage-2 simulator with wide integers (2048-bit steps, 512-bit run counts, jump cap 2^450). - With the old cap it reproduces R4's stage-2 b = 1 dump to 1e37 byte for byte (903 lines). - With the new cap it gives the same b = 1 lines. - R4's orbit. No halt to 1.0e80 steps. - The first 40 defects of the left text are unchanged from 4.8e17 to 1e80. - The first 8 have been fixed since kick 1 of the current top cycle (6.24e15), which leaves W (0,0,26) (0,0,0) 1 .... That cycle started at N = 27 ≡ 2 (mod 5). - What decides halting (from #32's cut runs; R4 has the same table): - A regime-I top halts exactly when its fuel N runs out at N ≡ 2 (mod 5). Of N = 0..44, only 2, 7, 12, 17 and 22 halt by 1e80. - But a lower digit that runs out before N does changes the outcome irregularly. For N = 12, 14 of the 24 small values of that digit's u still halt and 10 do not. For N = 11 and N = 13, none of the 50 halt. - So R4's top having the halting residue does not settle anything. The outcome depends on residues at the deep levels, which have to resolve first, beyond 1e80 steps. - Heuristic only. A random-residue model leans to an astronomically late halt rather than to non-halting, because the top keeps cycling and each regime-I exhaustion is another draw. The recursive_model option in decision.json is the route that could show a halt. No route to a non-halting proof is in sight.

Class 2 close-out, stage 2 (2026-10-12), re-run

The workflow issued stage 2 of this joint track a second time; stages 2 and 3 had already run (sections above). With both machines flagged and the priority call in decision.json unanswered (lint ok), the re-run did no new proof work. Outcome: cryptid, unchanged; confidence med. The write-up is in #32's notes.md, in the section of the same name. R4 in short: - Reproduction. Fresh builds of the kept simulators reproduce R4's stage-3 b = 1 dump to 1e80 byte for byte (2,145 lines; no halt), and #32's stage-2 and stage-3 dumps likewise. - New probe, on #32's base state (same table). Cutting the second lower digit u2 as well as N and u1 shows that, for N = 12, a halt that is fixed for large u2 moves or disappears once u2 runs out first, and not monotonically: u1 = 11 halts for u2 = 47..98 and u2 = 5, but not for u2 = 13..40. - With N = 11 or 13, none of 42 (u1, u2) cuts halts. So far a lower exhaustion only moves or cancels a halt of the N ≡ 2 (mod 5) residue. - For R4 this means its top's halting residue (N = 27 at the start of the current cycle) is decided only after the deep levels below it run out, one residue condition per level. That is the cryptid flag, now confirmed one level deeper.