{
 "tm": "1RB0LC_1LA0RE_0RD1LF_1LC1LB_0LD0RB_0LA---",
 "pass": "deepen (overnight 2026-10-10)",
 "confidence": "med",
 "recommend": "proof",
 "mechanism": "The tape is a word over four tokens 0=00, 1=10, O=100, I=110: a left counter word w, then the background (10110)^g = (1 I)^g and a little-endian digit counter D (digit 1=110, 0=100) with gaps G, then 11. Each sweep (one departure of E into the background to the next) applies inc to D and inc_L ('+2': skip the last token of w, carry 1->0, stop at 0->1 or O->I, and an I-crossing rule) to w; when the lowest digit is 1 at distance g=1 it is absorbed into w by one of three rules chosen by the end of w. All 1,096,578 sweeps from step 2,000 to 1e10 match this 4-rule model exactly; non-halting reduces to an inductive invariant (isolated O/I tokens in w, G non-increasing with G_1>=1 at absorption) that is not proven yet.",
 "model": {
  "config": "(w, g, D, G): 0^inf  w  E> (10110)^g  d_1 (10110)^G_1 ... d_k (10110)^G_k  11 0^inf, E at the first cell of the first background block (a 'departure'). w = tokens 0=00, 1=10, O=100, I=110 (unique parse); first cell of w is 1.",
  "phase_X_Y": "old hub phase X/Y = last token of w (1/0); old separator regime = a token O/I inside w (old s=0 / s=110 are O / I with the bit before them).",
  "inc": "right-counter increment (old R3/R4/R5): first 0-digit j becomes 1; digits 0..j-1 become 0 and move one block left (g-1, G_j-1 +1); all ones: g-1, two new 0-digits with gaps 0.",
  "inc_L": "w = u t (t = last token). Let u = v 1^k (k maximal). v = '' -> 1 0^k t (left edge moves 2 cells). v = x0 -> x1 0^k t. v = xO -> xI 0^k t. v = xI -> inc_L(x) O (0^(k-1) t 1 if k>=1 else 1). inc_L('') = 10.",
  "strip": "before every step: while w ends in I1 (= 11010, a C block that the F return converts to 10110): drop it from w, g += 1.",
  "rules": [
   {"name": "GEN", "when": "g>=2 or d_1=0", "to": "(inc_L(w), inc(g, D, G))", "trajectory_1e10": 1094493},
   {"name": "ABS1", "when": "g=1, d_1=1, w ends in 1 (not O1)", "to": "(drop_last(inc_L(inc_L(w))), inc(G_1+2, D[2..], G[2..]))", "trajectory_1e10": 1561},
   {"name": "ABS0", "when": "g=1, d_1=1, w = u 0", "to": "(w', inc(G_1+1, D[2..], G[2..])) with u = v0 0 1^k (k>=1) -> v0 1 0^(k-1) O 1 1; u = v0 0 1^k 0 (k>=0) -> v0 O 0^(k+1) 1 1; with no 0 left of the 1-run (u = 1^k or 1^k 0): 1 0^(k-1) O11 or O 0^(k+1) 11", "trajectory_1e10": 524},
   {"name": "ABSO1", "when": "g=1, d_1=1, w = x O 1", "to": "(inc_L(x), 1, [1] + D'[2..], G') where (0, D', G') = inc(1, D, G)", "trajectory_1e10": 0}
  ]
 },
 "closed_gaps": [
  "Phase-Y absorption: no separate rule is needed. Absorption is decided by the last token of w (1 -> ABS1, 0 -> ABS0, O1 -> ABSO1), and the right part follows inc(G_1+2 or G_1+1, D[2..], G[2..]).",
  "Separator regime entry: ABS0 appends O11, so the separator is an O/I token. A carry crossing the left edge also creates a leading O/I (1^k 0 -> O 0^(k+1) 11; I-crossing at the left edge -> 10 O).",
  "Separator dynamics: carry turns O into I, or crosses I (I 1^k t -> O 0^(k-1) t 1, prefix +2). This is exactly the old 'W1 +2 and W2 shifted' carry-out and the period 2^m - 1 cycle.",
  "Separator exit: ABS1 leaves ...I1 or ...O1 -> I1. A trailing I1 is a C block (11010) that the returning F (R2) turns into 10110, so the token leaves w and g grows by 1.",
  "Configurations with two O/I tokens or a leading O/I (e.g. 100 10 10 10 100 ... at step 86,806 and 110 00 00 ... at 88,920) were unparsed by the hub parser. They are covered by the token model.",
  "One exact rule table (GEN, ABS1, ABS0 and the side case ABSO1) replaces R6, R7 and R8 and their open cases. Each rule has exact formulas, no fitted constants.",
  "The rules cover the whole run from the first departure after step 2,000 to 1e10 (1,096,578 consecutive sweeps, 0 mismatches)."
 ],
 "remaining_gaps": [
  "Inductive invariant, not proven: (a) word shape: first token not 0, last token 0 or 1, O/I tokens at least 4 apart, and at an ABS0 step no O/I right before the trailing 1-run. Random words outside this set break the rules: adjacent O/I or I1I gives BADRIGHT, and e.g. 1O100 at ABS0 gives a left part that is not a token word. (b) Spacing: g>=1, G non-increasing, G_1>=1 at every absorption (ABS0 with d_2=1 would otherwise give g=0). (c) The left-tail/d_1 parity coupling: the transient tails xI0, xO1, xI1 must never meet an absorption. Informal argument: after ABS1, g>=2 when G_1>=1, and d_1 alternates each sweep.",
  "ABSO1 is validated only on random instances (95/95). It never fires on the trajectory up to 1e10, and its reachability is open.",
  "ABS0 with an O/I directly left of the trailing 1-run (model returns 'ABS0?') is not covered. It never occurs to 1e10; it needs an exclusion lemma or its own rules (some instances leave the token language).",
  "Init segment: steps 0 to 2,000 are not in the model (the first departures have D empty or G_1=0, e.g. step 2,343 has G=[0,0,0]). Plan: simulate directly to the first configuration in the invariant.",
  "Step counts per rule are not derived. Validation compares configurations at the next departure, not step formulas.",
  "No proof yet that F never reads 1 (F1 is the only halt). It held in every simulated rule application, but needs the per-rule proofs."
 ],
 "evidence": [
  "scripts/deepen/dep.c ran to 1e10 steps (1,096,600 departures, lo -4214, hi 1055, no halt). lockstream.py: GEN 1,094,493, ABS1 1,561, ABS0 524, all exact, 0 failures. The only unparsed row is step 79 (init).",
  "Earlier horizons: 1e8 (50,280 sweeps) and 1e9 (235,360 sweeps), same result.",
  "valreal.py, seeds 21-23: 8,540/8,540 random instances pass by direct simulation to the next departure (GEN 5,487; ABS1 1,647; ABS0 1,311; ABSO1 95). Size bands: small, medium, large. Words of 4-40 tokens, with leading/trailing O/I in 25%/60% of draws. k = 2-16 digits, non-increasing gaps up to ~600, g up to 300. All residues of d_2 and the GEN sub-cases: plain, carry through O/I, strip, d_1 set/carry.",
  "Invariant probes on 1e10 data (rinv.py, lockstream.py): G non-increasing at all 1,096,579 departures. G_1>=1 everywhere except once in the init segment (step 2,343). Min g = 1. Word length <= 14 tokens, at most 2 O/I tokens, 39 word shapes. Digits k = 2-12 (k only 10-12 near 1e10).",
  "Tails at the 2,085 absorptions to 1e10 (parity.py): ABS0 always has an all-0/1 tail (0000, 0110, 0100, 1100, 1010, 1110, 0010, 1000). ABS1 sees tails like 0I11, 0O11, 0I01, I001. Never xI0, xO1 or xI1.",
  "Not a cryptid: every rule is a binary-counter increment or a fixed rewrite plus a shift of a list. There is no multiplicative growth or parity of large numbers."
 ],
 "scripts": "scripts/deepen/: dep.c (C departure dumper: gcc -O2 -o dep dep.c; ./dep 1e10 > dep.txt), model3.py (abstract model), linc.py (inc_L), lockstream.py (lockstep check: python3 -B lockstream.py dep.txt), valreal.py (random validation: python3 -B valreal.py 3000 SEED), nextdep.py (direct simulator to next departure), rinv.py/parity.py (invariant probes), valreal_21..23.json (validation counts)."
}