{
 "tm": "1RB0RC_1LC1RB_0RE1LD_0LC1LA_1RF1RA_---0RB",
 "rounds": [
  {
   "round": 1,
   "stage": "proof (funded option 'proof', decided 2026-10-13: 'go with recommendations'; earlier stages: notes.md Wave 5 stages 1 and 2)",
   "date": "2026-10-13",
   "confidence": "high",
   "recommend": "proof_ready: write proof.md (next stage), then the Rocq build",
   "mechanism": "Bell eats a decrementing binary counter on a spacing-3 lattice. Outer cycle starts s = (n, w, v) = R(-1, n, Z). Invariant I = {w >= 3, d = n - 2^(cap+1) >= 2, M != 0} minus (6,4,0), M = n + 2v - 2^(cap+2) mod 3 (n + v - ... when w = 1 mod 3). Lemma M (sign law) keeps M = 0 / != 0. Lemma K: from K a cycle halts only at T_c or (6,4,0) and otherwise lands in K. This stage closes Lemma K for continuation cycles and the nested levels. Shift law: a level started at R(p0, q, Z) is the cycle from (q, Z) moved by p0 + 1, so G(t) = F(t) + 1 is an inner level. The nested bite with k low set bits is k - 1 G-steps from (3, Z >> (k+1)), and a cycle whose first continuation is at R(p_c, 2, Z_c) has F(s) = (p_c + 2) + G^k(3, Z_c[3k+3:]).",
   "closed_gaps": [
    "Gap (b), no halt inside nested levels: phase lemma (H only from O2 at q = 4, which needs a phase start in H* = {w = 2, n = 2v + 4} u {w = 1, v = 0, n = 2^(L+1) + 4}), pop parity (n' = p + 1 mod 2 when w' = 1, 2; B0 f = 2 never occurs inside a cycle), Lemma J (J = {w >= 3, n >= 3, n odd if w = 1, 2} is closed under G, no halt, no rule gap). Every phase start except the outer cycle start is in J, so a cycle halts iff its start is in H*; in K that is T_c and (6,4,0).",
    "Gap (a), Lemma K for continuation cycles: A1 p_c >= 2^(L_c) - 3 (chain arithmetic, tight at w = 1 mod 3, d = 2); Lemma Z' (t in J with d <= 1: d(G t) >= 2 or cap(G t) <= cap(t) + 1, by cases with at most one overflow after the first); K-likeness persists in J. Then d(F s) = p_c + 2 + d(t_(k+2)) >= 2 in both cases (K-like inner end: >= 4; zone-dominated: cap <= L_c - 1, n >= 3). Induction on the length of the cycle computation.",
    "(6,4,0) has no predecessor in K: n' >= n - log2(n) roughly (plain) or n' >= 2^(L_c) + 2 (continuation), so a predecessor would sit inside the exhaustive window, which has none.",
    "Checks (scripts/contlemmas.py, closed rules): G-form 20,000/20,000; H* = halting set on 610,121 + 606,121 states; J1/Z'/Kp on 1,920,333 + 479,621 J states; A1/CF/A2 on 533,148 + 151,880 continuation cycles, min d' = 5; 200,000 random large states (w <= 75) 0 failures; TM level G and F agree on random J and K states."
   ],
   "remaining_gaps": [
    "proof.md not written yet: the plain-cycle part of Lemma K (stage 2 sketch: f = 1, f = 2 with B1/B2/B0, f >= 3) must be written out in full, with the new Lemmas P, J, A1, Z', the shift law and the composite form",
    "The nested-bite stack (inner walls 1 00, pop = next level's R(0, q', Z')) is justified by TM brute force (252/252) and the G-form check, not yet by a written tape argument; it is the long part of the Rocq build"
   ],
   "evidence": [
    "work/contlemmas_gform_seed7.txt: nested bite = k - 1 G-steps, 20,000/20,000",
    "work/contlemmas_hstar_w24_X4.txt, work/contlemmas_hstar_w12_X12.txt (and _w21_X4): halting set = H*",
    "work/contlemmas_jwin_w27_X4.txt, work/contlemmas_jwin_w15_X10.txt (and _w21_X4): Lemma J, Z', K-persistence",
    "work/contlemmas_kwin_w27_X4.txt, work/contlemmas_kwin_w15_X10.txt (and _w21_X4): A1, composite form, A2, min d' = 5",
    "work/contlemmas_random_seed2_w75.txt: 200,000 random large states, 0 failures",
    "work/contlemmas_tm_seed1.txt, work/contlemmas_tm_seed2.txt: TM level (scripts/macro.py), G and F agree"
   ],
   "scripts": [
    "scripts/contlemmas.py"
   ]
  }
 ],
 "updated": "2026-10-13"
}