{
 "tm": "1RB0LC_1LC0RE_1LA1LD_1LA1LB_0RF1RA_---0RE",
 "stage": "stage3",
 "date": "2026-10-13",
 "confidence": "med",
 "recommend": "park (stop rule of the funded stage): no closed-form invariant; reopen only with a history-carrying invariant (w, k0, n, m)",
 "mechanism": "Turn configuration R(u,e) = 0^inf u <C 0 1^e, u = w 1 (101)^k. Bounces increment the counter w (binary, digit 1 = 10, digit 0 = 00) and add a middle block; at bell residue 0/2 the eat increments twice and drops the low digit. Odd eats create a fresh S (100) at the lowest 1 above bit 0; S freezes into an inert F (110) at its first carry; F's drift down one digit per eat and are absorbed into the middle field. An odd eat whose S would land in an F gives E junk; with F at depth 1 this is an L (lone 1) at the low end with k0 = 1, whose first bounce turns 1 0 1 1 0 0 L into 1 0 1 F 0 F 0. Cycle length n' = n + k0 + [r=2] + [z ends in 00] never decreases; Phi (place values of the 1-digits minus one per live S) is exact under inc.",
 "closed_gaps": [
  "Greatest closed set at 22 cells: 179,610 states, g = 1 for 178,926, max g = 16; max g doubles every 2 cells (data/stage3/gfix22_n1.out)",
  "Forward-least closed set: 0 halts at 18/20/22 cells, junk grammar G* = (d|F)* (nothing | L (k0=1) | S 0^j 1 | L S 0^j 1), n >= Phi+1 and h >= 2^(D-1) on all states (scripts/stage3/hmin.c)",
  "Real orbit to step 17,829,276: 204/204 cycle starts in G*, n >= Phi+1 on 191/191 parsable (data/stage3/realcyc_1.78e7.txt)"
 ],
 "remaining_gaps": [
  "No inductive invariant: windows max(Phi+1, 2^(D-1)) (571 exits), + S-freeze bound (472-483 exits, 268 outside G*), digit windows 2^(D-C) (exits, halts for C=2..4) all fail at 18 cells",
  "The coupling needs history (n at the last odd eat / last L); not a function of the word",
  "Then: eat table, E/L rule, proof.md (88 and 280), Rocq"
 ],
 "evidence": [
  "data/stage3/window_tests.txt",
  "data/stage3/gfix22_n1.out",
  "data/stage3/g22_gt1.txt",
  "data/stage3/hmin18.txt",
  "data/stage3/hmin20.txt.gz",
  "data/stage3/hmin22.txt.gz",
  "data/stage3/realcyc_1.78e7.txt"
 ],
 "scripts": [
  "scripts/stage3/gfixl.c",
  "scripts/stage3/gfixp.c",
  "scripts/stage3/hmin.c",
  "scripts/stage3/swin.c",
  "scripts/stage3/README.txt"
 ],
 "rounds": [
  {
   "date": "2026-10-13",
   "option": "wave5 stage1",
   "pass": "counter-word language search",
   "confidence": "med",
   "result": "no counter-word invariant exists (free closure halts); token rules T1-T4; coupled closures 0 halts"
  },
  {
   "date": "2026-10-13",
   "option": "wave5 stage2",
   "pass": "effective value and windows",
   "confidence": "med",
   "result": "Phi closed form exact; inc on G0 exact; upper windows impossible; greatest closed set g at 16-20 cells"
  },
  {
   "date": "2026-10-13",
   "option": "stage3",
   "pass": "funded stage: closed-form invariant from the computed closed set",
   "confidence": "med",
   "recommend": "park (stop rule)",
   "result": "g at 22 cells (max 16, doubling); forward-least set has a small exact grammar G* and n >= Phi+1, 0 halts to 22 cells; no closed window (571+ exits): the coupling needs history; parked with 280; STUCK"
  }
 ],
 "updated": "2026-10-13"
}