{
 "tm": "1RB1LC_1RC1LB_1LD0RB_0LE1RD_---0LF_0LA1LF",
 "stage": "deepen2",
 "date": "2026-10-10",
 "confidence": "high",
 "recommend": "proof",
 "mechanism": "Recursive counter with marker absorption. The edge epoch Ep acts on the word W right of the edge's 1^3 block: S-words [2,2,a,2,b]+M and U-words [1,g]+M have closed forms, and a T-word [a,2,b]+M replays the machine's own history from [1]+M at frame 2 for a-1 epochs. Every epoch costs 2 or 3 decrements. The orbit O_n(M)=Ep^n([1]+M) of a marker word M (lone 1s, gaps g_i) follows an exact absorption law: step n costs 3 exactly when n=I_j=sum_{i<=j}(g_i-1), when the j-th marker comes within reach of the replay, and costs 2 otherwise. The top-level edge words are the marker-free orbit O_n([]) (edge word [1] at TM step 50), so every top-level epoch costs 2 and the edge frame never runs out. The machine cannot halt. The irregular P->P' map only enters through a 'within reach' inequality, so this is not a cryptid.",
 "closed_gaps": [
  "[round 1] Run model = TM: identical at every macro step for 3,000 steps from step 18,826; summed step counts reproduce the TM history exactly (steps, P, Q) for Fs(5)..Fs(21), to step 2,642,404,258",
  "[round 1] Short epoch, general: Ep([2,2,a,2,b]+M) = (3,[a+1,2,b+2,2,1]+M), 18 macro steps, 12a+8b+280 TM steps, for all a,b>=1, M empty or first gap>=2 (model 2,800/2,800; TM 60/60, a,b up to 400). Covers Fs->Fl and D->E",
  "[round 1] Translation lemma: if Ep(W)=(r,W'), c=5-r, then for any context L0, y>=2, z>=c+1: L0+[y] | [z,2]+W -> L0+[y-1] | [z-c+1,2]+W' in the same macro steps (3,000 random words in the model; 24 inner-context TM checks)",
  "[round 1] Long epoch recursion: for a T-word [a,2,b]+M, a 7-step transient, then a-1 inner epochs Ep^i([1]+M) at frame 2 with z_1=b+2, z_{i+1}=z_i-c_i+1; if z_i>=c_i+1 throughout, Ep(W)=(3,[z_a,2]+W_a) (405/405 random T-words, 0 disagreements)",
  "[round 1] Transitions between regimes: Fl(k) gives Fs(k+1) or D(k+1) exactly when three inner epochs cost 3 (z_a=5-#cost3=2). Observed exactly 3 for all 135 (k,P) with k=3..17 and P even; never 2 or 4; never a halt",
  "[round 1] D(k)->E(k)->Fs(k+1,2,2k) for every k=1..16",
  "[round 1] Cost table for E(j)+[g,1]: cost 3 iff g<=2j, cost 2 iff g>=2j+2 (j=1..8, g=2..20). A clean marker-absorption threshold",
  "[round 1] P->P' map tabulated for all even P, k<=17: irregular but injective; halting at the outer level does not depend on it",
  "[round 2] Top level = marker-free orbit: edge word at TM step 50 is [1] (R=[3,2,1]), and Ep^n([1]) is the whole history D(0),E(0),D(1),E(1),Fs(2,2,2),Fl(2),D(3),E(3),Fs(4,2,6),... (n<=39 checked; matches observe history)",
  "[round 2] Absorption law: for any lone-marker word M with gaps g_i>=2, the orbit O_n(M)=Ep^n([1]+M) never halts and Ep(O_n) costs 3 exactly at n in {I_j = sum_{i<=j}(g_i-1)}, else 2. 27,908 random orbits (m=1..6, to I_m+4..8 epochs), 0 failures, odd gaps included. Explains round 1's 'exactly three': Fl(k,P,Q) replays [2,1,P,1,Q,1], so cost-3 at replays 2, P+1, 2k",
  "[round 2] Shape + e-law (even gaps): every orbit word is U (n=0), S or T with lone markers, and e = #markers-(b-a+1) [S] or #markers-(b+1-a) [T] equals the number of unabsorbed original markers (307,830 words, 0 failures)",
  "[round 2] Positional invariants at T-words [a,2,b]+M_n, aff=b+1-a: INT I_aff(M_n)<=a-2 (21,034/21,034); EXT next marker at (a-2)+(I_j-n-1) for n<I_j, so slack drops 1 per epoch (14,039/14,039); TAIL later original gaps unchanged (14,039/14,039); ABS-STEP at n=I_j the marker is within reach or merged into the head (6,215/6,215)",
  "[round 2] z-accounting: absorbing exactly aff markers inside replays 1..a-1 means no z-exhaustion and z_a=2 (cost 2, S-word result); absorbing aff+1 makes z fail at the last replay with z=2",
  "[round 2] Missing regime closed, exhaustion rule X: if the last replay child is pushed with z=2 (ALPHA cost-2 / GAMMA cost-3 word), Ep([a,2,b]+M) = (2,[a''+2,2,b''+2,g''+2]+rest), where W_{a-1}=[a'',2,b'',g'']+rest. Derived by hand; 88/88 random T-words and 1,345/1,345 absorption steps in orbits. BETA (z=3) never occurs in orbits",
  "[round 2] U rule Ep([1,g]+rest)=(3,[2,2,1,g-2]+rest) for g>=4, and Ep([1])=(3,[2,2,1]) (600/600); T rule at a=1 Ep([1,2,b]+M)=(3,[b+2,2,1]+M) (600/600). With S (= T at a=2), T and X, every edge-word type is covered",
  "[round 2] allP extended with the C model to k=18..25 for every even P (164 cases): exactly three cost-3 frame-2 epochs each, every one closing to Fs(k+1) or D(k+1). 299 (k,P) cases total, k=3..25",
  "[round 2] Earlier halts in family_robust explained: extra top-level markers make e>0, so the top level must pay cost 3 and later halts. AL with m=0 is exactly why the real run never does"
 ],
 "remaining_gaps": [
  "The absorption law is validated, not proven. proof.md must write out its strong induction on n (all M at once) and prove that INT/EXT/TAIL/e-law propagate through S->T, T->S (via the inner orbit at step a-1) and rule X (cost-3 T->T). It must also show that the final replay word W_{a-1} is a T-word",
  "Every rule rests on the run model, which matched the TM directly (round 1: 3,000 macro steps, and the full history to step 2.64e9). Rule X and the U rule were checked in the model only, not re-run on the raw TM",
  "Formalization will be large: TM-level R1/R2/R3, the translation lemma, the S/T/U/X epoch rules and AL as a nested induction over marker words. Likely bigger than the CounterClass examples"
 ],
 "evidence": [
  "scripts/epc.c: C port of runmodel.py with ep/orbit/c3/abs/allP modes; abs mode = absorption law + shape + e-law test. Outputs in scripts/absorption_out.txt",
  "scripts/position_check.py (+ position_check_out.txt): INT/EXT/TAIL/ABS-STEP, e-law and cost law along 2,400 random even-gap orbits",
  "scripts/exhaust_check.py: round-1 no-exhaustion rule 979/979 and rule X (ALPHA) 88/88 on random T-words; scripts/exhaust_kinds.py: rule X 1,345/1,345 at orbit absorption steps (ALPHA 1,162, GAMMA 183), no BETA",
  "scripts/u_rule_check.py: U rule 600/600, a=1 T rule 600/600",
  "scripts/allP_k18-25.txt: all even P for k=18..25, exactly 3 cost-3 epochs each, all close to Fs/D",
  "scripts/census.py: exploratory per-push census at every depth (superseded by the orbit view; kept for reference)",
  "notes.md sections 'Deepen pass (overnight 2026-10-10)' and 'Deepen pass 2 (overnight 2026-10-10)'"
 ]
}