{
 "tm": "1RB1RF_1RC---_1RD0RD_1LE1RC_1RA1LE_1RE0LF",
 "pass": "deepen round 2 (overnight 2026-10-10)",
 "confidence": "low",
 "recommend": "more_observation",
 "mechanism": "A multi-epoch shift-overflow counter. Within an epoch the rest tape is LE' + top region + {debris_k X_k (1 3)^m_k} + W, and every trip adds 1 in base 4 to the digits X_k (types A/B, 4 states, round-1 rule). The top region absorbs the carry out of the topmost digit once every 4^K trips (K = 6,7,8,9 so far) and slowly turns embryo words into new digits and frozen debris. New in round 2: the epoch ends. At trip 13,539,937 (about 1.27e15 steps, from the closed-form fast-forward) the carry crosses the whole top region, the sweep makes the first left record since 1.8e8, and the return pass rewrites the whole tape. This trip is quadratic in zone size (about 260*m^2 steps for zones of m blocks, so about 1e16 steps at the real sizes). It recodes every zone into pieces of roughly fixed fractions (the bottom zone m becomes about 3m/4), and the pieces depend on the parity of m. After the rebuild a fresh base-4 A/B counter grows from a new wall at the bottom, on top of the recoded tape. The counter layer is understood; the overflow/rebuild rule and the epoch recursion are not.",
 "closed_gaps": [
  "Uniform digit rule for all levels (the observe pass had S0/S1 only, plus one S2 carry with a fixed context). Two digit types, A and B, each with 4 states; zone schedule +0,+1,+0,+1(carry). The bounce offsets -10/-12 are types A/B. Exact rest-tape words are in notes.md 'Deepen pass' section 2.",
  "Validated on the real trajectory trip by trip, comparing the full token string including every debris word and the wall. To 2e10: 53,572 of 53,703 transitions; to 1e11: 119,956 of 120,091; turn-back levels 1-7 = {90045, 22478, 5607, 1389, 341, 77, 19}.",
  "The remaining trips to 1e11 are all identified. Start-up up to about trip 480, young digits and left-end events at trips 481-5729, the 1.45e6/1.8e8 left records, then only top-region events: 6753, 7777, 8801, 9825, 10849-14945 (5), and 23137+k*16384 for k=0..5.",
  "Debris and wall are frozen: no debris word or the wall W changes in any counter step (e.g. S0 debris 7 9 7 9 1 3 1 3 1 1 1 9 7 9 7 9 1 1 1 9 1 1 1 9).",
  "Synthetic checks. Single digit of type A or B, any start state, zone 0-274, random left context, real S0+Z0+wall: 87/87 non-carry visits correct. Synthetic stacks: 14/24 fully correct; 9 untestable after carry-out into the artificial top; 1 HALT in an artificial top context.",
  "Base-4 ruler law: turn-back level = 1 + trailing 3s. Exact 1+v4(t+31) (capped at 4) for trips 2000-5088; the 1.8e8 rebuild shifts the phase of levels >= 4.",
  "Left end to 1e11. No left record after 1.8e8 (edge -78, 120,092 trips), no halt. Top region identified: LE' is unchanged from trip 5729 until trip 236129. D_top (type A at LE' end) went a1, a2 (9825), a3 (72289). Z_top shrinks 12, 10, 9, 7. The embryo E absorbs the carry out of the top digit every 4^7 trips (6 events listed with words).",
  "New tool fastforward.py: applies the digit rule in compact form and runs exactly only the trips it cannot cover. It matched the real run on 16,384/16,384 trips (88673-105057) plus 80 spot checks; past 1e11, 164/164 spot checks agree until the runcfg buffer limit at trip 789,842.",
  "Beyond 1e11 (fast-forward to about 4e12 equivalent steps, no halt, no left record): E matures into an ordinary type-A digit (trips 121441-154209) and its old debris '1 15 1' becomes the next embryo (absorbs at 170593). At trip 236129 (about 3.9e11 steps) D_top carries into LE'; LE' is rewritten and Z_top is used up. From then on the top region absorbs one carry every 4^8 trips (301665 ... 760417).",
  "Not a cryptid: no parity or big-number branching anywhere; all irregularity is in a top region that is rarely visited and changes structurally.",
  "Round 2: the fast-forward limit is gone. scripts/topsim.py applies the counter in closed form (digit k after N trips = floor((v+N)/4^k) mod 4; its zone gains the number of odd c in [floor(v/4^k), floor((v+N)/4^k))). It simulates exactly (scripts/trip1.c) only the events: carries out of the topmost parsed digit, or into a debris word not yet shown to be transparent. For events, the zones below the event point are capped at 3 blocks, and the lower digits must come out exactly as predicted. It reproduces all 16 round-1 events between trips 19041 and 760417 (after-tape equal). Capped and full-size runs agree on the K=9 events 1481313, 1743457 and 2005601 (full-size trips of 2.6e7-3.5e7 steps). It reaches trip 13.5e6 in seconds.",
  "Round 2: the top-event schedule to trip 13,539,937. There are 72 events, none halts, all turn-backs are at cells 32-296 (no left record): K=6: 19041, 23137; K=7: 39521-121441 every 16384; K=8: 170593-1284705 every 65536 (18 events); K=9: 1481313-13277793 every 262144 (46 events). Two more debris words were shown to be transparent: '7 9' over type A (trip 121441), and the old embryo region '1 1 1 7 1 9 1 3 7 9 1 3 1 3 1 1 1 1 1 3 1 1 3 9' over type A (trip 1284705). So an embryo region does end up as frozen debris under a new top digit.",
  "Round 2: a new regime, the epoch overflow at trip 13,539,937 (about 1.27e15 steps; pre-event state in scripts/data/pre_overflow_13539937.json). The topmost digit (a3, Z95) carries into the top region. The sweep reaches 7 cells left of the old left end, and the return pass rewrites everything down to the wall: the wall changes, and every zone and debris word is recoded. The leftmost token, which records left records, went 4 (from trip 13) -> 10 (trip 5089, the 1.8e8 record) -> 16 (this event).",
  "Round 2: the rebuild trip with every zone capped at S blocks (scripts/rebuild_split.py, rebuild_scale.py, 2 <= S <= 600). Steps grow quadratically: 1.5e6, 5.2e6, 1.8e7, 6.8e7, 9.1e7 for S = 64, 128, 256, 512, 600, about 260*S^2. Tape length is kept (+10 cells). Zones come out in pieces of roughly fixed fractions: the bottom zone m -> 149, 224, 299, 383, 449 for m = 200, 300, 400, 512, 600 (about 3m/4). The split depends on the parity of m (S = 200, 202, 204 vs 201, 203). A new 8-cell block, tokens '1 1 3', appears in runs. No halt in any capped run. At the real sizes (Z0 about 6.8e6 blocks) the rebuild trip extrapolates to about 1e16 steps. A direct full-size run was stopped after several CPU-minutes.",
  "Round 2: what follows a rebuild (synthetic, zones capped at 512; scripts/data/rebuild_continue_S512.txt). 2e10 more steps (53,650 trips): no halt, no further left record. A fresh base-4 A/B counter grows from a new wall at the bottom: '... 7 1 1 1 1 Z6618 7 9 7 9 1 1 1 9 7 9 1 3 1 1 1 7 1 9 1 3 7 1 1 7 Z26494' + wall '1 1 1 1 1 3 9 Z2 7 9 Z2 1 1 1 1 1 3 9 Z2 1 1 1'. The recoded old tape sits above it as the new top region. So the epoch structure repeats: counter, overflow, recode, new counter.",
  "Round 2, toward the head-level lemmas: each trip is two sweeps that each look only a bounded distance back. In the left sweep the head never goes more than 15 cells right of its running minimum before the turn-back, and the return pass never backs up more than 13 cells (measured on trips 14944->14945 and 15968->15969; scripts/sweep_excursion.py, sweep_forms.py). So the head-level rules needed are finitely many local rules plus the zone loops R1/R2. The rule set has not been enumerated yet."
 ],
 "remaining_gaps": [
  "The overflow/rebuild rule (the new blocker). Needed: a closed description of how the return pass recodes each zone (size m, parity, and probably m mod 4) and each digit/debris word, plus its quadratic time. This is needed at the real zone sizes. Direct simulation of the real rebuild trip (about 1e16 steps) is not feasible, so it has to come from a rule-level (accelerated) model of the recoding pass, checked against capped runs.",
  "The epoch recursion: describe the tape at the start of epoch n+1 as a function of the tape of epoch n, and show that the counter, top events and rebuild of every epoch stay in one family. The round-1 top-region gap (the embryo family and the LE' rewrites) is now one part of this recursion.",
  "A non-halting invariant across epochs. A synthetic top context halts (round 1), and the rebuild creates new words. No halt was seen in any capped rebuild (about 20 sizes) or in 2e10 steps after one.",
  "Enumerate the finitely many head-level rules: left-sweep window <= 15 cells, return-pass backtrack <= 13 cells. These are needed for Coq.",
  "Whether the parity/mod-4 dependence of the rebuild compounds over epochs into pseudo-random arithmetic (the cryptid question) is not known. No Collatz-like branching has been seen; the recoding looks like base-4 arithmetic on zone sizes.",
  "Everything past 1e11 is observation: closed-form counter + exact event trips with capped lower zones. It is validated against round 1 (16 events) and against full-size runs (3 K=9 events), not proved."
 ],
 "evidence": [
  "[local working file] (section 'Deepen pass (overnight 2026-10-10)')",
  "scripts/toktrack.c + scripts/counter_trace.py -> scripts/data/counter_trace_2e10.json, counter_trace_1e11.json (119,956/120,091 trips explained; events listed)",
  "scripts/top_events.py -> scripts/data/top_events_2e10.json, top_events_1e11.json (top region before/after for every unexplained trip after 5700)",
  "scripts/digit_cycle.py -> scripts/data/digit_cycle_res.json (87/87 non-carry visits); scripts/counter_check.py -> scripts/data/counter_check_res.json (14/24 stacks OK, 1 synthetic HALT)",
  "scripts/fastforward.py + scripts/runcfg.c -> scripts/data/fastforward_check_88673.json (16,384/16,384 real trips reproduced), scripts/data/fastforward_120092.json (top events to trip 760417, about 4e12 steps equivalent)",
  "scripts/levels.c run to 1e11 (no halt, min turn-back -78 after 1.8e8); scripts/trips.c (turn-back tracker); scripts/tok.py, tokmark.py (token views)",
  "Round 2: scripts/topsim.py + scripts/trip1.c -> scripts/data/topsim_events_to_13539937.json/.log (72 events, trusted debris set); scripts/data/topsim_fullsize_check_2005601.json (full-size check of the K=9 events). Input: toktrack 2e9 output, regenerable in 10 s with './toktrack 2000000000 > tok2e9.txt'; start trip 16979.",
  "Round 2: scripts/data/pre_overflow_13539937.json (state before the overflow, made with STOP_AT=13539937); scripts/rebuild_split.py -> scripts/data/rebuild_split.txt (zone splits and steps for S = 64..600); scripts/rebuild_scale.py (full token output per S).",
  "Round 2: scripts/rebuild_continue.py + scripts/runtok.c -> scripts/data/rebuild_continue_S512.txt (2e10 steps after a capped rebuild, new counter epoch).",
  "Round 2: scripts/trace1.c, trace_helpers.py, sweep_forms.py, sweep_excursion.py (head-level sweep windows)."
 ]
}