{
 "tm": "1RB0RC_1LC1RB_0RE1LD_0LC1LA_1RF1RA_---0RB",
 "bucket_reason": "Funded stages used up (Wave 5 stage 2 of 2) with outcome progress: a non-halting invariant is found and checked, but two sub-lemmas still lack a paper argument.",
 "question": "Fund one more stage on W5-4 to close the two open sub-lemmas and write proof.md, then formalize, or park it at the invariant stage?",
 "options": [
  {
   "id": "proof",
   "label": "One more stage: close the open sub-lemmas, write proof.md",
   "consequence": "The stage would close two gaps. First, a bound on the nested-bite output, so that Lemma K covers cycles with a B2'/B2'' continuation. Second, a no-halt argument for nested levels. Then proof.md (rules, Lemma M, Lemma K, entry at step 327), and after that the Rocq build: one lemma per rule, the nested bite by induction, progress_nonhalt_cond over I."
  },
  {
   "id": "formal",
   "label": "Go straight to Rocq with the plain-cycle cases, leave the gaps for later",
   "consequence": "Starts the Rocq build now: the rules as BigStep lemmas, plus the sign law, which is pure arithmetic per rule. The two open lemmas would come up anyway as the hardest Rocq goals, so the informal-proof-first rule is bent and the risk lands in the Rocq build."
  },
  {
   "id": "park",
   "label": "Park at the invariant stage",
   "consequence": "No further spend now. notes.md keeps the invariant, Lemma M (proved) and Lemma K (checked, proved for plain cycles), so the next stage can start from there."
  }
 ],
 "recommended": "proof",
 "rationale": [
  "Simple invariant, 0 failures on 10.4M states: d >= 2 and n + 2v - 2^(cap+2) != 0 mod 3.",
  "Mod-3 sign law proved per rule; 7,977/7,977 TM checks. Transients lie inside I.",
  "Gaps are slack lemmas (min d' = 5 vs 2 needed), not a cryptid question."
 ],
 "evidence": [
  {
   "kind": "fact",
   "text": "Lemma K + M: 0 failures on 6,810,622 states (w 3..31), 1,132,222 (w 3..16, n < 2^(cap+10)) and 2,424,478 (w 3..12, n < 2^(cap+14))",
   "href": "work/invariant_w3-31_X3.txt"
  },
  {
   "kind": "fact",
   "text": "Sign law at TM level: 7,977/7,977 random configurations (seeds 1, 2); rule level 7.3M applications, 0 failures",
   "href": "work/signlaw_seed2.txt"
  },
  {
   "kind": "fact",
   "text": "Continuation cycles: the final cap is always <= cap at the continuation - 1, min d' = 5 (the plain-cycle minimum is 2)",
   "href": "work/continuation_w21_X4.txt"
  },
  {
   "kind": "figure",
   "text": "Fig 3 counter fill per cycle: the calm levels and the level-up scatter, all inside I",
   "href": "img/levels_fill.png"
  }
 ],
 "effort_if_yes": "One 2 h stage for the two lemmas and proof.md; the Rocq build after that is likely 2-3 stages (nested-bite induction is the long part).",
 "updated": "2026-10-13",
 "answer": {
  "option": "proof",
  "note": "Dennis 2026-10-13: 'go with recommendations'",
  "date": "2026-10-13"
 }
}