{
 "tm": "1RB0LF_0RC0RB_1LC1LD_1RE0RD_---0RA_1LA0LF",
 "stage": "deepen2",
 "date": "2026-10-10",
 "confidence": "med",
 "recommend": "stuck",
 "mechanism": "Nested wall bouncer whose tail is a fossil-processing particle system. The micro rules are exact: sweeps complement, walls stay fixed, turn rewrites are local, and halting is possible only through 1 0 0 <F 0 0 (0 0|1 0 0). The left part repeats exactly each epoch (a'=3a+6). In the y = t XOR parity view the tail is a list of particles separated by long gaps. Each epoch rewrites a prefix of that list and leaves a fossil suffix untouched; the right-end particle, the only place a near-halt (D1) has occurred, is rewritten only in epochs 0, 3, 4, 7 and 8 (to 4.3e13 steps). That schedule fits n = 0 or 3 (mod 4), and the particle sequences show partial period-4 self-similarity. The collision catalogue keeps growing: 363 particle types, with new ones in E7 and E8. E8 rebuilds the right end without its trailing 1^k run. No n-gram closed language exists for n <= 20 (parity-aware n <= 14), so neither epoch induction nor a regular invariant is in reach. A proof needs a particle-level theory: the collision table plus an invariant on particle order and phase.",
 "closed_gaps": [
  "(round 1) Crossing lemma: a sweep complements every crossed cell and never moves a wall; all tape changes are turn rewrites within 8 cells of the turn point (931,918/931,918 sweeps to 3.54e10 steps; 5 right-edge sweeps within 12 cells).",
  "(round 1) Halting lemma: halt iff 1 0 0 <F 0 0 0 0 or 1 0 0 <F 0 0 1 0 0 occurs; D1 (necessary for halting) iff 1 0 0 <F 0 0 occurs (hand derivation; 7,886/7,888 random-tape halts and 32,118/32,119 D1 events match; the 2 misses are start-state transients at steps 40-41).",
  "(round 1) Near-halt census on the blank-tape orbit: D1 occurs exactly 6 times in 3.54e10 steps (steps 130, 309, 2,306,898, 2,380,763, 2,488,620,044, 2,497,241,765), all at the right edge, all with E reading 1; never in the bulk.",
  "(round 1) Full phase list per epoch E1-E4 (exact periodic turn programs, auto-segmented); nesting rule k' = 2k+4 (or +12) when one more wall joins the bounce.",
  "(round 1) Tail processing mechanism identified: a front moves right through T, and dense blocks act as registers that are converted block to block (E2: 93,52,128 -> 174,177,175,173 -> 45,126 -> 14,48,18 -> 467,15 -> 7).",
  "(round 1) Bulk local language saturates: no new 16-cell window (mod complement) at bulk turn points after step 258,986,194; all later new windows are at the right edge. [Round 2 correction: this holds only through E4. New particle types appear in the tail in E7 (first at 7.06e12, about 2,300 cells from the right end) and in E8.]",
  "(round 1) Tail robustness: 100 perturbed tails (gaps +-1/3, 15% of walls dropped) on the E1/E2 left parts, 5e8-1.2e9 steps each: 0 halts (33 runs with D1, E always read 1; 18 runs had a runaway right edge).",
  "Accelerated observation to epoch 8: Quick_Sim (block 4, prover) reaches 4.31e13 steps without halting. Its tape at step 4,785,511,149 equals the direct simulation cell for cell (166,105 cells).",
  "Tail structure (particle view, y = t XOR parity, gaps >= 12): at every epoch start E0-E8 the tape is the left part, then 8-19 particles (13-37 tokens counting gaps), then the right-end particle. Each epoch leaves a fossil suffix untouched (9, 1, 21 and 19 tokens, i.e. particles plus gaps, at E1, E2, E5 and E6), and the same 19-token block is unchanged from E5 to E7.",
  "Right-end schedule: the right-end particle is rewritten only in epochs E0, E3, E4, E7 and E8. The last 60 cells change at the cell level only in E0, E3 and once in E4 (to 3.6e10). Right-end visits (crossings of cell -300) come in bursts.",
  "Right-edge growth decoded (cell level, to 2.7e9): hi grows by +2/+4 (C-run into 0^inf) or by +6. The +6 steps are exactly the D1 events. Every D1 context is ..0110[1]11111 0 1^m with m >= 4 (D-run 6, E reads the first 1 of 1^m).",
  "Right end is robust to the length of its 1-run: from the E3 anchor with the trailing 1^6 replaced by 1^k (k=3..9), all 7 runs through the E3 burst (2.4e9 steps) have no halt and exactly 2 D1 events, and hi shifts by exactly k-6.",
  "Invariant candidates refuted: plain n-gram CPS (abstract halt for n=2..12, 16, 18, 20), parity-aware y-CPS (abstract halt for n=3..14), the static forbidden factor 0011 (created 744,280 times to 4e9, transiently, next to the head), and saturation of the local language (new particle types in E7 and E8).",
  "Cryptid question: no arithmetic (residue) branching visible; the left part is exact and the right-end schedule looks periodic (mod 4). Not a cryptid candidate on current evidence.",
  "Near-halt census extended at the cell level. From blank to 3.3e11 steps: still exactly the 6 D1 events of round 1 (none after 2.497e9), no halt. E7 right-end touch (3e10 steps from the 8.5795e12 visit): no D1. E8 rebuild: the sequential run 3.99058e13-4.0306e13 plus all 72 far-change intervals in 3.9e13-4.32e13 give 0 D1 and 0 halts, and every interval equals the next Quick_Sim visit. The right edge grew 12 times, by +2 or +4 each, never by +6. Every far-right change to 4.3e13 was re-run at the cell level, so the only right-end D1 events are still the 6 of E0 and E3, all reading 1."
 ],
 "remaining_gaps": [
  "Tail rule T -> T' / particle collision table: 363 particle types (246 interior, 100 right-end, 17 left-edge) over 29,643 left-edge visits to 4.3e13, still growing in E7 and E8. Particle velocities and collision outcomes are not tabulated.",
  "No concrete non-halting argument. The local/regular routes fail (CPS n <= 20, parity CPS n <= 14). What is needed is an invariant on particle sequences (order and phase), with a complete collision table and the right-end process included.",
  "The right end is not a fixed block plus a growing 1^k: in E8 the trailing 0 1^10 disappears (at 4.069e13 the tape ends ...1001 0010 (1010)^3 0^inf; at 4.30e13 it ends ...0100 0101 1000 0^inf). Its later evolution, and the D1 contexts after the rebuild, are not decoded.",
  "The mod-4 right-end schedule (E0, E3, E4, E7, E8) is a 9-epoch hypothesis. Testing it at E9-E12 needs better acceleration than Quick_Sim gives past 4.3e13 (~100 loops/s there).",
  "Bulk D1 between 3.3e11 and 3.99e13 (outside the E7 window) is not watched at the cell level. Only right-end changes are, through the Quick_Sim far-right monitor; no bulk D1 has ever been seen."
 ],
 "evidence": [
  "scripts/deepen2_qanchors.py: Quick_Sim block 4 + prover, 1.87M loops to step 4.314e13, no halt; log scripts/deepen2_out/qanchors_1.87Mloops.jsonl.gz (every left-edge visit plus every change of the last 8 macro blocks).",
  "scripts/deepen2_dumpat.c: direct simulation to step 4,785,511,149 matches the Quick_Sim tape exactly (166,105 cells).",
  "scripts/deepen2_tailmap.py + deepen2_yparts.py: particle lists at the E0-E8 starts, fossil suffixes and the particle catalogue (scripts/deepen2_out/tailmap.txt); figure img/deepen2_tail_fossils.svg.",
  "scripts/deepen2_visits.c (X0=-300, to 3.6e10; 1,045 visits) and deepen2_hilog.c (to 2.7e9): right-end bursts, last-60-cell changes, and the hi growth events (+6 = D1).",
  "scripts/deepen2_dh2cfg.py + deepen_runcfg.c: E3 anchor with 1^k, k=3..9, 2.4e9 steps each: 0 halts, 2 D1 each.",
  "scripts/deepen2_ngcps_path.cpp (plain CPS with an abstract-path dump) and deepen2_ycps.cpp (parity-augmented CPS): abstract HALT at every radius tried.",
  "scripts/deepen2_f0011.c: 744,280 creations of factor 0011 to 4e9; static snapshots every 1e6 steps show it only twice.",
  "scripts/deepen2_out/far_right_blocks.txt: Quick_Sim far-right history (E8 rebuild at 3.99e13-4.3e13).",
  "scripts/deepen2_longrun.c: blank to 3.3e11 steps, 6 D1 events (all with E reading 1), no halt (deepen2_out/longrun_3.3e11.txt).",
  "scripts/deepen2_qcfg.py + deepen2_cfgrun.c: E7 window 8.5795e12 + 3e10 steps and E8 window 3.99058e13 + 4e11 steps, 0 D1 (deepen2_out/e7_seq_8.5795e12_3e10.txt, e8_seq_3.99058e13_4.03058e13.txt).",
  "scripts/deepen2_e8intervals.py + deepen2_ivrun.c: 72 intervals, 8.6e9 steps, 0 D1, 0 mismatches (deepen2_out/e8_intervals_3.9e13_4.32e13.txt).",
  "scripts/deepen2_qpaircheck.py + deepen2_cfgdump.c: Quick_Sim consecutive-visit consistency at 8.58e12, 3.99e13 and 4.30e13.",
  "deepen2_out/e3_rightend_1k_variants.txt: E3 anchor with 1^k, k=3..9; deepen2_out/cps_results.txt: CPS radii tried."
 ]
}