{
 "pass": "funded stage inv_search (2026-10-11)",
 "confidence": "med",
 "recommend": "park (STUCK); resume only with a coupled left/right invariant method or a non-regular invariant",
 "mechanism": "Unchanged and exact: the macro model (Ee/Eo + C-pass C1-C4) covers every E-event from blank for starts A (469) and D (599); halting iff a C-pass meets 0^2. This stage searched for an inductive invariant over that model. It found none, and it found a precise obstruction. Splice two reachable E-events at a common window of the closed-block stack, keeping one's head and window: the TM halts from the splice for windows of 1-4 exact blocks, 1-5 when both stacks are spliced, and 1-7 when blocks are read as exact below 8, else mod 4. 45/45 halting splices were confirmed by direct TM simulation. Right-side-only splices never halt (0/560). So no product n-gram (CPS-style) invariant with such windows exists, under any value abstraction. The defects surface 47-6,628 macro steps after the head reaches the splice, so the constraint is not local. A global parity GP (found by GF(2) search; 0 at all 6e6 sampled E-events) is part of it: 24/45 halting splices break GP. But 21 respect GP, so more global information is needed.",
 "closed_gaps": [
  "Macro-level n-gram CPS closures built from the validated rules (absint2 tokens, absint3 groups, absint4 groups + hidden-part GP bits): 115 + 58 settings + spot runs, all FAIL (HALT or unruled patterns never seen on the real run) or LIMIT (3e6 states)",
  "Splice obstruction (splice.py + verify_splice.py): exact left windows W=1-4 halt (22 halts in ~2,400 splices, seeds 2/4/7); joint left+right W=1-5 halt (12 in ~1,240, seed 12); coarse left blocks (exact <8, else mod 4) W=1-7 halt (17 in ~1,740, seed 11); 45/45 saved halting splices halt on the TM at steps 2,595-279,060",
  "Right side is robust: exact R-group splices 0 halts in 560 (W=1-6), with the head reaching the cut within 4,000 macro steps",
  "Global parity GP = [r=1 mod 4] + [floor(a/2) odd] + #odd-j groups + #Z(=3 mod 4) + #blocks(=0 mod 4) + #blocks = 0 mod 2: constant on the real run (3e6 E-events per start), kept by every rule except C4 on a final D P1 with x = 0 mod 4 (44/1,663 Eo|D random cases)",
  "TM-level CTL widened: 191 RWL_mod + 24 NG-with-history settings, all false (with sweep1: 300 settings)"
 ],
 "remaining_gaps": [
  "The missing invariant couples the whole block stack with the right side; GP explains 24/45 halting splices, the other 21 are unexplained",
  "Product n-gram invariants with near-exact windows of 6 or more entries on both sides are not excluded (0 halts in 1,264 exact-left W5-9 and 424 joint W6-7 splices); testing them needs a compiled closure with ~1e7-1e8 states",
  "No proof.md, no formalization: nothing to formalize without an invariant"
 ],
 "evidence": [
  "scripts/absint2.py, absint3.py, absint4.py, absint_sweep.py -> absint_runs/sweep_tokens_absint2.jsonl (115 FAIL), sweep_groups4.jsonl (56 FAIL, 2 LIMIT)",
  "scripts/splice.py -> absint_runs/splice_R_s1.txt, splice_L_s2.txt, splice_L_s4.txt (+_halts.json), splice_L_s7.txt (+_halts.json), splice_Labs84_s11.txt (+_halts.json), splice_LR_s12.txt (+_halts.json)",
  "scripts/verify_splice.py -> absint_runs/verify_splice_L_s4.txt (9/9), verify_splice_L_s7.txt (7/7), verify_splice_LR_s12.txt (12/12), verify_splice_Labs84_s11.txt (17/17)",
  "scripts/gpar.py -> absint_runs/gpar_real_3e6.txt (GP = 0 at 3,000,000/3,000,000 E-events per start); rules mode seed 21",
  "scripts/ctl/sweep2.jsonl (215 settings, all false or timeout)",
  "img/splice_halt_W1.png (card Fig 6): a W=1 splice that halts at TM step 2,595"
 ],
 "rounds": [
  {
   "pass": "funded stage inv_search (2026-10-11)",
   "date": "2026-10-11",
   "confidence": "med",
   "recommend": "park (STUCK); resume only with a coupled left/right invariant method or a non-regular invariant",
   "mechanism": "Unchanged and exact: the macro model (Ee/Eo + C-pass C1-C4) covers every E-event from blank for starts A (469) and D (599); halting iff a C-pass meets 0^2. This stage searched for an inductive invariant over that model. It found none, and it found a precise obstruction. Splice two reachable E-events at a common window of the closed-block stack, keeping one's head and window: the TM halts from the splice for windows of 1-4 exact blocks, 1-5 when both stacks are spliced, and 1-7 when blocks are read as exact below 8, else mod 4. 45/45 halting splices were confirmed by direct TM simulation. Right-side-only splices never halt (0/560). So no product n-gram (CPS-style) invariant with such windows exists, under any value abstraction. The defects surface 47-6,628 macro steps after the head reaches the splice, so the constraint is not local. A global parity GP (found by GF(2) search; 0 at all 6e6 sampled E-events) is part of it: 24/45 halting splices break GP. But 21 respect GP, so more global information is needed.",
   "closed_gaps": [
    "Macro-level n-gram CPS closures built from the validated rules (absint2 tokens, absint3 groups, absint4 groups + hidden-part GP bits): 115 + 58 settings + spot runs, all FAIL (HALT or unruled patterns never seen on the real run) or LIMIT (3e6 states)",
    "Splice obstruction (splice.py + verify_splice.py): exact left windows W=1-4 halt (22 halts in ~2,400 splices, seeds 2/4/7); joint left+right W=1-5 halt (12 in ~1,240, seed 12); coarse left blocks (exact <8, else mod 4) W=1-7 halt (17 in ~1,740, seed 11); 45/45 saved halting splices halt on the TM at steps 2,595-279,060",
    "Right side is robust: exact R-group splices 0 halts in 560 (W=1-6), with the head reaching the cut within 4,000 macro steps",
    "Global parity GP = [r=1 mod 4] + [floor(a/2) odd] + #odd-j groups + #Z(=3 mod 4) + #blocks(=0 mod 4) + #blocks = 0 mod 2: constant on the real run (3e6 E-events per start), kept by every rule except C4 on a final D P1 with x = 0 mod 4 (44/1,663 Eo|D random cases)",
    "TM-level CTL widened: 191 RWL_mod + 24 NG-with-history settings, all false (with sweep1: 300 settings)"
   ],
   "remaining_gaps": [
    "The missing invariant couples the whole block stack with the right side; GP explains 24/45 halting splices, the other 21 are unexplained",
    "Product n-gram invariants with near-exact windows of 6 or more entries on both sides are not excluded (0 halts in 1,264 exact-left W5-9 and 424 joint W6-7 splices); testing them needs a compiled closure with ~1e7-1e8 states",
    "No proof.md, no formalization: nothing to formalize without an invariant"
   ],
   "evidence": [
    "scripts/absint2.py, absint3.py, absint4.py, absint_sweep.py -> absint_runs/sweep_tokens_absint2.jsonl (115 FAIL), sweep_groups4.jsonl (56 FAIL, 2 LIMIT)",
    "scripts/splice.py -> absint_runs/splice_R_s1.txt, splice_L_s2.txt, splice_L_s4.txt (+_halts.json), splice_L_s7.txt (+_halts.json), splice_Labs84_s11.txt (+_halts.json), splice_LR_s12.txt (+_halts.json)",
    "scripts/verify_splice.py -> absint_runs/verify_splice_L_s4.txt (9/9), verify_splice_L_s7.txt (7/7), verify_splice_LR_s12.txt (12/12), verify_splice_Labs84_s11.txt (17/17)",
    "scripts/gpar.py -> absint_runs/gpar_real_3e6.txt (GP = 0 at 3,000,000/3,000,000 E-events per start); rules mode seed 21",
    "scripts/ctl/sweep2.jsonl (215 settings, all false or timeout)",
    "img/splice_halt_W1.png (card Fig 6): a W=1 splice that halts at TM step 2,595"
   ]
  }
 ]
}