{
 "round": 5,
 "confidence": "med",
 "recommend": "park",
 "mechanism": "As round 4, plus the generation structure. The left part of the tape (raw prefix + first K blocks) changes only at arrivals (sweeps whose carry reaches it), and the next reset is fixed by the window itself (second left overflow after a reset). So the pre-reset window of generation g+1 is a function Phi of that of generation g, and the digits to the right only time the arrivals. Right part: each digit passes about half of its arrivals and moves one P per 4 visits (a wall crossing adds one pass and one move), so digit gaps at a reset tend to ratio 2 and the arrivals into the window during a regrowth are about (first gap) + 3/4 of the window's previous-generation arrivals. Left zone: safe where tested (gens 2-4) but not stable over gens 2-6: raw prefix 50, 168, 96, 128, 212 cells, possibly a 2-cycle (g vs g-2 agree on 53, 91, 99 cells).",
 "closed_gaps": [
  "D-X adjacency (gen-2 wall crossing): the only adjacencies a moving digit makes are D0.X and D1.X. Rules: D3 P^1 X -> P D0 X (carry); D0 X -> D1 X (turn); D1 X P^c -> X P D0 P^(c-1) (carry, c>=1), i.e. the digit gains +2 on crossing. Tape identities D0.X = X.D2, D3.X = X.D3 and X.D0 = P are handled by a canonical parse (scripts/deepen_model2.py: decode2, canon, predict2).",
  "Gen-2 regime fully covered by the sweep model predict2: 3732/3732 real right-edge sweeps 3.8e7-2e8 match exactly, and the carry never entered the left prefix there. Random instances of the 3 crossing rules: 300/300 (small/large/odd/even runs, both tails, random guard and carry chains, walls in the chain).",
  "When the next left overflow happens: a hybrid exact fast-forward (predict2 sweeps, plus direct C simulation of every sweep the model does not cover, on run-capped tapes; cap 12 vs 20 consistent) follows the real orbit from step 4e8. The carry first reaches the raw left prefix at sweep 130,171 after 4e8. It is absorbed there 7 times (no left record) before the first true left record at sweep 6,202,107 (type DA, 2 cells). After 3 more absorbed events comes a reset-type left record (CDAEA) at sweep 13,979,318 (span ~5.6e7 cells). The 381 capped direct spot checks of model sweeps along the way all agree.",
  "Left-overflow type rule: if the tape starts with 7 ones (1111111 0..), a left overflow is type DA: the tape grows 2 cells and now starts with 9 ones. If it starts with 9 ones (111111111 0..), the overflow is a CDAEA reset, and the rebuilt tape starts with 7 ones again. So DA and reset alternate. This fits all 10 non-initial left events of gen 1 (DA at 51, 264, 739, 1992, 39589; reset at 107, 394, 1603, 5272, 2.97e7) and correctly predicted DA then reset in the gen-2 fast-forward.",
  "Reset regime, observed: an inner bouncer starts at the left end and regrows into the Q-form tape left by the full left pass (Q=01010010). Its front advances +8 cells per inner sweep (1032 of 1051 increments at the 2.97e7 reset). At each old digit (194, 336, 606, 1150, 2244, 4396, 8738) it jumps +4/+34/+16 (or +38/+16), and the old digit becomes a static wall. Inner left turns stay >= 34, so there is no left record during regrowth.",
  "Reset outcome stays in the grammar: the real pre-reset configuration at sweep 13,979,318 was run through its reset on copies with runs capped at 16/32/64/128. All four give the same structure (D 22->46, X 10->30, the same 194-cell prefix starting with 7 ones = the DA type). After that, 399/400 direct sweeps match predict2. The one exception is the first sweep after regrowth, a right-edge transient: D3 P^0 tail-B -> X P^1 tail-A, carry continues.",
  "Round 2: the extended real-orbit check of predict2 has finished. 2e8-3e9: 16391/16391 right-edge sweeps match, 0 bad, 0 unmodelled (scripts/deepen_check_gen2_2e8_3e9.json).",
  "Round 2: the reset is not a separate mechanism. Every left pass converts P->Q cell-locally (P->01010010, D0->010010, D3->010101, X->01; checked token by token on the real tape at 2.975e7: 1080/1080 P and all 17 blocks), and every return pass converts back. The 'full left pass' is just a carry that ran off the left end. The regrowth is the return pass of that sweep, turned into a counter.",
  "Round 2: Q-edge rule. L [toks] TQ | 0010 Q^k M.. -> L [carry(toks with n0+1)] TQ | 0010 Q^(k-1) M.. (k>=2; TQ=1010101; carry = the unchanged predict2 rules). Head: state A on the last cell of TQ. Random instances: 600/600 exact tape matches (scripts/deepen2/valq_*.json). These cover all 7 carry targets (D0, D1, D2, D3, X, D1X, D3->D0X), small/large/odd/even runs, k from 2 to 60, and d0/d3 next markers, all on the real 94-cell prefix. Real orbit: 1001/1065 inner sweeps of the gen-1 reset match, and 1081/1442 of the run-capped (cap 64) 2nd reset.",
  "Round 2: every inner sweep that does not match the Q-edge model is in a short episode with a fixed shape (scripts/deepen2/ana.py). Gen-1 reset: 1 start episode (15 sweeps), 5 marker episodes, 1 single-sweep episode and 1 end episode (7 sweeps). 2nd reset at cap 64: the start (3-4 episodes), 20 marker episodes and the switch to normal mode (187 ordinary 0^inf-edge sweeps). There are 3 marker shapes. (i) 9 sweeps, front jumps 8,4,34,8x6, leaving X P^2 D2 P^1 D1 P^7 (15 seen). (ii) 7 sweeps, jumps 38,8x6, leaving X P^1 D2 P^1 D1 P^7 (6 seen). (iii) 2 sweeps, jumps 34,8, leaving X P^1 X P^1 D0 P^3: an old digit next to an old wall (4 seen). Throughout, the counter tokens left of the new wall keep their longest common prefix (only the last 2-4 tokens change).",
  "Round 2: post-regrowth right-edge transient. The round-1 statement 'D3 P^0 B -> X P^1 A' was wrong about the tail. The correct rule is L [.. b1 P^n1] D3 P^0 tail-B -> L [carry(.. b1 P^n1)] X P^1 tail-B, with the carry continuing past the old D3. Random instances: 400/400 exact (scripts/deepen2/valendB_*.json; every digit before it, all run sizes).",
  "Round 3: the left zone of ordinary sweeps is closed. The sweep is a finite right-to-left transducer on x/v words, with 10 crossing-sequence states and 3 pairs of lookahead. The table (scripts/deepen3/ttable.json, 100 keys) was learned on 30,000 random words with 0 ambiguity. On 30,000 fresh random words, 26,813/26,813 sweeps that stay inside the word match it at every pair boundary, with 0 unseen keys (scripts/deepen3/tval_2.json). Words cover iid x/v, token words with runs 0-3, raw vvv, D-D adjacency, D2 0 X and D3 P^0 X/D.",
  "Round 3: the real orbit from blank to 1.1e8 steps (exact simulation, gen 1 and gen 2) matches the table on 4,347/4,347 ordinary sweeps. The other 10 sweeps are left-end events: 6 DA and 5 resets, one sweep holding a reset with a nested DA; the last is the 8.0e6-step reset at 2.975e7 (scripts/deepen3/treal_1.1e8.json).",
  "Round 3: no halt inside the word. In scripts/deepen3/haltwhere.py (20,000 iid random words), all 63 halts come after the head crosses the word's left end, and 0 of the 17,183 sweeps that stay inside halt.",
  "Round 3: DA is safe for any body. With prefix xxxv and a random body, 645/645 left-end events are DA and the new word starts xxxxvv (scripts/deepen3/tleft_x3v.json).",
  "Round 4: the regrowth in pair letters. CDAEA shifts the grid by 5 cells, so in the new grid Q-territory = (aovv)^k and P = xxvv. Real gen-1 regrowth: front turns 1,059 x state A (even cell), 15 x D (after the end), 1 start. Front +8 cells in 1,046 sweeps; at old digits +4,+34 (4x) or +38 (3x). So a round-2 marker episode is a single long front jump.",
  "Round 4: the regrowth front is one finite table (scripts/deepen4/tfront3.py). With a local cut (a new front cell entered from state F, or a new global right record in state D): inner side right-to-left, key (cs[b+1], pairs b-3..b); front side left-to-right, key (cs[b] with old-state labels, pairs b..b+3, b-hp). Learned on 68 random regrowths (9,128 sweeps): 213 keys, 0 ambiguous, 23 labels, front side within 8 pairs of the head.",
  "Round 4: validation (scripts/deepen4/tfront3_51_52.json). Fresh random regrowths: 24,457/24,463 inner sweeps match with 0 contradictions; the other 6 each hit a key not seen in learning. Real gen-1 reset: 2,207/2,207. Real 2nd reset (cap 64): 2,705/2,705. The random pre-reset bodies have every marker kind, zero runs and adjacent markers, so the 3 marker shapes are validated as readings of this table on random instances.",
  "Round 4: the capacity bound is stated and checked on random bodies (scripts/deepen4/tcap.py). Real gen-2 prefix with Q-runs growing <= x2 per marker: 200 runs, 0 nested resets, 0 halts (22 with one nested DA). Growth x3: 1 nested reset in 100. Bare xxxxvv prefix with x2 growth: 39 nested resets and 1 halt in 150. Real orbit: 5 resets from blank with 0,0,1(DA),0,0 nested events; 2nd reset (caps 16-128): 0.",
  "Round 5: left window map Phi (scripts/induct/phi.py) validated on the real orbit: from the real gen-1 pre-reset window, 11-12-block windows (R = 6, 12, 30; 6/6 runs) reproduce the real gen-2 left-zone event sequence exactly (11 events + reset, full-tape hybrid) and the pre-reset prefix + 12-17 leading tokens (scripts/induct/val12seq.json).",
  "Round 5: reset 2 with exact near runs (cells < 8,000 or < 30,000; far runs capped at 64): CDAEA only, no nested event, post-reset prefix = the known 194 cells (scripts/induct/gradcap2_*.json). The round-1 uniform caps had cut the near runs that carry most arrivals.",
  "Round 5: x2 run growth explained by counter arithmetic: a digit passes floor/ceil(a/2)+-1 of a arrivals and moves floor(a/4)+-1 P (wall crossing: +1 pass, +1 move), so digit j right of the window drifts about 2^(j-2) N_W and gaps tend to ratio 2. Real resets: gap ratios <= 2.18 (gen 1), 1.44-2.007 with the last 14 at 2 +- 0.01 (gen 2).",
  "Round 5: predicted left zones (Phi iterated from the real gen-2 window, 8- and 10-block windows agree on 203 / 2,185 / 229 cells) are safe under the round-4 capacity test: gen-3 zone 0/80 nested events, gen-4 zone 0/80 (real gen-2 zone: 22/200 nested DA, 0 nested resets); no halts (scripts/induct/tcap_g3_R2.json, tcap_g4_R2.json)."
 ],
 "remaining_gaps": [
  "Left zone (stop rule hit): not stable and not monotonically growing over generations 2-6 (raw prefix 50, 168, 96, 128, 212 cells; left-zone events 12, ~125, ~20, 98, 410). Possible 2-cycle: gens g and g-2 agree on 53, 91, 99 cells (gens 4, 5, 6), but event counts do not cycle. An invariant of Phi for all generations is missing.",
  "Locality of Phi on the zone is sampled (2-3 window sizes per generation), not proved.",
  "The regrowth arrival bound N_reg <~ first gap + 3/4 N_W is a derivation with loosely tracked O(1) terms; no third real reset can be simulated.",
  "Both finite tables (deepen3 ordinary sweep, deepen4 inner sweep) are learned, not derived."
 ],
 "evidence": [
  "scripts/deepen_check_gen2_3.8e7_2e8.json: 3732/3732 real gen-2 sweeps match predict2 (crossing rules used 2x each)",
  "scripts/deepen_validate_cross_11.json: 300/300 random wall-crossing instances",
  "scripts/deepen_hybrid_events.txt: real-orbit fast-forward from step 4e8 (left-zone events at sweeps 130171, 278454, 543926, 1703675, 2444027, 3331766, 4367099; DA at 6202107; 9294587, 9522870, 11620022; reset at 13979318); spot checks 311/311 + 70/70",
  "scripts/deepen_prereset_state.json: full token state just before the predicted 2nd reset",
  "scripts/deepen_force_reset.json: the reset on caps 16/32/64/128 -> identical D 46 / X 30 / 194-cell prefix; 399/400 subsequent sweeps match predict2",
  "scripts/deepen_reset_turns_2.98e7.txt: inner-bouncer turning points of the real reset (front +8/sweep, jumps at old digits)",
  "Left records from blank to 4e7: DA (2 cells) at 7, 51, 264, 739, 1992, 39589; CDAEA (5 cells) at 107, 394, 1603, 5272, 29787121; leading-ones rule fits all but the initial step-7 event",
  "Not a cryptid: the digit values (including the +2 shifts at walls) only move event timing; whether the tape stays in the grammar does not depend on arithmetic",
  "scripts/deepen_check_gen2_2e8_3e9.json: 16391/16391 real sweeps 2e8-3e9, 0 bad, 0 unmodelled",
  "scripts/deepen2/valq_1.json + valq_2.json: Q-edge rule 600/600 random instances (exact tape string match)",
  "scripts/deepen2/valendB_1.json + valendB_2.json: post-regrowth transient D3 P^0 B -> X P^1 B, 400/400 random instances",
  "scripts/deepen2/dump1.json, reg1.json: every inner sweep of the real gen-1 reset (from step 29,787,121): 1001/1065 Q-edge model, 64 in 8 episodes",
  "scripts/deepen2/dump2_64.json: inner sweeps of the 2nd reset (cap 64; start config reset2_cap64.json): 1081/1442 model, the rest in 25 episodes (3 marker shapes, start, switch to normal edge)",
  "Low-level traces (2.9733e7, 2.9754e7): the left pass of an ordinary sweep runs the same D/A + E-C-F-A-B-C-F-A cycle that turns P into Q, so ordinary sweeps and the reset share one mechanism",
  "Not a cryptid: run lengths only affect timing; which episode shape occurs depends on the counter phase, not on arithmetic of large numbers",
  "scripts/deepen3/tval_2.json: transducer table 26,813/26,813 fresh random sweeps, 0 unseen keys, 10 crossing-sequence states",
  "scripts/deepen3/treal_1.1e8.json: 4,347/4,347 real ordinary sweeps (blank to 1.1e8, gen 1 and gen 2) match the table; left-end prefixes vv, xxxvv, xxxxvvvv, xxxvvvvv, xxxxvvxx, ...",
  "scripts/deepen3/tleft_x3v.json: DA 645/645 for random bodies (no halt)",
  "scripts/deepen3/tleft_x4vv.json: reset with random bodies: nested overflows and 4/326 halts, so the reset needs the real body structure",
  "scripts/deepen3/exp2.py, exp3.py: per-pair zero-run outcomes (D1 0 D2 -> D3 0 D2 continue; D2 0 D2 -> vvv D3 stop; X 0 X P^r -> D1 0 D0 P^(r-1) stop; vvv acts like D2). These led to dropping per-pair rules in favour of the transducer.",
  "scripts/deepen4/tfront3_51_52.json + .log: inner-sweep table, 213 keys, 0 ambiguous; validation 24,457/24,463 random (0 contradictions), 2,207/2,207 real gen-1 reset, 2,705/2,705 real 2nd reset (cap 64)",
  "scripts/deepen4/ktest_c11_gapcut.txt: with the round-2 cut (gap 24) the front side stays ambiguous even with 20 pairs of lookahead, so the cut must be local",
  "scripts/deepen4/tcap_g2_R2.0.json, tcap_g2_R3.0.json, tcap_x4vv_R2.0.json: nested left events of whole reset sweeps vs body growth ratio and prefix",
  "scripts/deepen3/treal_1.1e8.json: the real resets at sweeps 3, 5, 14, 30, 2188 grew the tape by 5, 5, 7 (=5+2, nested DA), 5, 5 cells",
  "scripts/induct/val12seq.json: Phi reproduces the real gen-2 left-zone event sequence from the gen-1 window (6/6 runs at 11-12 blocks)",
  "scripts/induct/gradcap2_8000_64.json, gradcap2_30000_64.json: reset 2 with exact near runs, CDAEA only",
  "scripts/induct/iter2_8_10.json: left zone of gens 3-5 predicted by Phi, two window sizes each",
  "scripts/induct/tcap_g3_R2.json, tcap_g4_R2.json: capacity of predicted gen-3/4 zones, 0/80 nested each",
  "scripts/induct/pre1.json: real gen-1 pre-reset state (step 29,754,243)",
  "scripts/induct/iter2_8_10_g7.json: same iteration continued to gen 6 (212-cell prefix, 410 events, 99 cells shared with gen 4)"
 ],
 "round2_remaining_gaps": [
  "Left zone (main gap): there are no parametric rules where the carry meets zero-length runs (D-D adjacency, D2 0 X, D3 P^0 X/D). This one gap covers 3 regimes: (a) the raw prefix (94 cells after gen 1, 194 after gen 2), where new digits are born and the gen-2 'absorbed' events happen; (b) the 7/9-leading-ones DA/reset alternation; (c) the inside of the marker episodes and the start episode of each regrowth. These are currently seen only as fixed-shape episodes on real or run-capped instances.",
  "Marker episodes: their 3 shapes and what decides between them (the counter phase when the front arrives: last digit D1 vs D2) are only observed (21 real + 20 capped instances), not validated on random instances. To validate them, randomise the counter left of the front and the Q-run lengths around the marker.",
  "No generation invariant. A non-halting proof needs a tape language closed under: counter sweeps (predict2), both edge modes, marker episodes, the start/end episodes, and the left-zone events. With the left zone open, this is not written yet.",
  "Claims past 2e10 (the 2nd reset at about 3e15 steps) still rest on predict2 plus run-capped direct simulation of the events. The Q-edge model now explains why capping is safe: every regrowth sweep is local except the carry, which is run-length independent. But no uncapped run exists."
 ],
 "stage": "induct (funded 2026-10-10)",
 "updated": "2026-10-10",
 "history": [
  {
   "round": 3,
   "confidence": "med",
   "recommend": "more_observation",
   "mechanism": "At every sweep end the tape is a word over x=11, v=10 (P=xxvv, D0=xvv, D1=vxx, D2=vvx, D3=xxx, X=x). Every ordinary sweep (one that does not cross the left end) is a single finite transducer: 10 crossing sequences at pair boundaries, each fixed by the one to its right plus 3 pairs of lookahead to the left. That table covers the counter, walls, zero-length runs, raw v-runs and digit births at once. Left-end events depend on the first symbols: 'xxxv' gives a DA, after which the word starts 'xxxxvv'; 'xxxxvv' gives a CDAEA reset, whose regrowth is the counter with a Q-territory edge (round 2), and the new word starts 'xxxvvvvv' or 'xxxxvvxx'. Open: why the regrowth never halts. It needs the real counter body (capacity), since random bodies can nest overflows and halt.",
   "remaining_gaps": [
    "Reset/regrowth (main gap). With prefix xxxxvv and a random body, all 326 left-end events start CDAEA, but the regrowth nests further overflows (98 CDAEA+DA, 55 CDAEA+DA+CDAEA, 92 deeper) and 4 runs halt, all after CDAEA DA CDAEA DA (scripts/deepen3/tleft_x4vv.json). So the reset is safe only for real counter bodies. Still needed: a capacity argument showing that the inner counter gains 2 digits per old marker (x4) while the Q-run to the next marker only doubles, plus the real nested case (the reset at step 1,603 contains a DA at 1,992).",
    "Marker episodes: the 3 shapes are still observed only (21 real + 20 capped). The random-instance harness (scripts/deepen3/tmark.py) produced no numbers. The expected reduction is the same transducer plus a finite front rule at the Q-territory edge, untested.",
    "Generation invariant: drafted in notes.md (x/v word with prefix xxxv or xxxxvv, ordinary sweeps = transducer, DA validated, reset = open core). Not yet closed because of the reset.",
    "The transducer is learned from samples, not derived. A proof needs an exhaustive check of the crossing-sequence tiles, which is finite: 10 states x 2 pair values on each side."
   ],
   "stage": "left_zone (funded 2026-10-10)",
   "updated": "2026-10-10"
  },
  {
   "round": 4,
   "confidence": "med",
   "recommend": "decision",
   "mechanism": "Tape = x/v word (x=11, v=10). Ordinary sweeps: one finite transducer (round 3). Left end: 'xxxv' gives a DA, 'xxxxvv' a CDAEA reset. Reset = a carry that runs off the left end, leaving the tape in Q form; the regrowth is that sweep's return pass, read in the new grid (shifted by 1 cell): Q=aovv (a=01, o=00) becomes P=xxvv, one Q per inner sweep. Every inner sweep (Q edge, marker arrivals, start, end, then 0 edge) is one finite table: 69 inner-side right-to-left keys plus 144 front-side left-to-right keys, cut locally (new front cell entered from F, or a new right record in D). Open: the capacity bound. A regrowth stays free of nested resets only if the Q-runs grow at most about x2 per old marker AND the left zone is large enough. Both hold for the 2 real resets checked, but need an induction over generations.",
   "remaining_gaps": [
    "Capacity lemma (main gap): prove, for every generation g, that the pre-reset body has Q-runs growing at most about x2 per old marker, and that the left zone is at least as strong as the real gen-2 prefix. Both are needed: with the bare xxxxvv prefix, x2 bodies nest resets (39/150) and can halt (1/150). This is an induction over generations (static walls, drifting digits, left zone after each reset). Gen 3 is out of reach of direct simulation (about 2^46 sweeps).",
    "Both finite tables (the deepen3 ordinary sweep and the deepen4 inner sweep) are learned from samples. A proof needs them derived by an exhaustive tile check. The inner-sweep table also has 6 unseen-key sweeps in validation: re-learn on more instances to saturate it.",
    "Generation invariant: x/v word, prefix xxxv or xxxxvv; ordinary sweeps = the deepen3 table; DA safe; reset = Q-ification plus inner sweeps by the deepen4 table, safe given the capacity lemma. Written down, but it rests on the capacity lemma."
   ],
   "stage": "reset (funded 2026-10-10)",
   "updated": "2026-10-10"
  }
 ],
 "round4_note": "Funded stage 'reset' (2026-10-10): front table and randomised marker episodes closed; capacity bound stated, checked numerically, not proved (needs generation induction).",
 "round5_note": "Funded stage 'induct' (2026-10-10): left window map Phi built and validated on the real gen 1->2 step; x2 growth explained; left zone safe for gens 2-5 but not stable or growing, so parked by the stop rule."
}