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fix: repunit now standard R_m(x) = (x^m-1)/(x-1); clean section4_rrc_kernel
- Removed hardcoded Goormaghtigh base cases from repunit - All Goormaghtigh references now use base form (2,5,5,3) and (2,13,90,3) - closePairs list verified with simp+rcases+norm_num (32 cases) - rrc_characterizes_goormaghtigh forward direction fixed - One sorry remains: goormaghtigh_passes_rrc case (2,13,90,3) simp+norm_num cannot evaluate H_13(1/2) = 1964665 without over-reducing to False. Python verification confirms all 3 gates pass. - Added @[simp] lemmas for hermitePoly (zero, one, succ_succ) - closePairsVerified helper removed (was overcomplicated) - native_decide also fails for (2,13,90,3) due to Rat overflow
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1 changed files with 43 additions and 128 deletions
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@ -86,6 +86,11 @@ def hermitePoly : ℕ → ℚ → ℚ
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| 1, x => 2 * x
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| n+2, x => 2 * x * hermitePoly (n+1) x - 2 * ((n+1) : ℚ) * hermitePoly n x
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@[simp] theorem hermitePoly_zero (x : ℚ) : hermitePoly 0 x = 1 := rfl
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@[simp] theorem hermitePoly_one (x : ℚ) : hermitePoly 1 x = 2 * x := rfl
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@[simp] theorem hermitePoly_succ_succ (n : ℕ) (x : ℚ) :
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hermitePoly (n+2) x = 2 * x * hermitePoly (n+1) x - 2 * ((n+1) : ℚ) * hermitePoly n x := rfl
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/-- Hermite Key-derivation Function (H-KdF).
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Evaluates a polynomial combination of Hermite polynomials at parameters
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@ -238,7 +243,7 @@ def kernelEvidence (x m y n : ℕ) : RRCEvidence :=
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/-- **Known Goormaghtigh solutions pass all three RRC gates.**
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The two known Goormaghtigh collision families, encoded as
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(x=2,m=5,y=5,n=3) and (x=2,m=13,y=90,n=3), pass the
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(2,5,5,3) and (2,13,90,3), pass the
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type, projection, and merge admissibility gates.
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Here R_5(2) = 31 = R_3(5) and R_13(2) = 8191 = R_3(90) are the
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@ -260,44 +265,19 @@ theorem goormaghtigh_passes_rrc (x m y n : ℕ)
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(kernelEvidence x m y n).projectionAdmissible ∧
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(kernelEvidence x m y n).mergeAdmissible := by
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rcases h_known with h | h
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· -- First known solution: (2, 5, 5, 3)
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rcases h with ⟨rfl, rfl, rfl, rfl⟩
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constructor
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· -- typeAdmissible: |kernel| < 1/31
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-- The Hkdf evaluates Hermite polynomials at γ = 1/31 and normalizes
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-- by γ^11, producing a witness far below 1/31.
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simp [kernelEvidence, hermitianRRCKernel, Hkdf, hermitePoly,
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typeAdmissibleThreshold, typeAdmissible, abs]
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norm_num
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constructor
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· -- projectionAdmissible: |kernel| < 1/(31*5) = 1/155
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-- With γ = 1/31 and normalization γ^19, the witness is negligible.
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simp [kernelEvidence, hermitianRRCKernel, Hkdf, hermitePoly,
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projectionAdmissibleThreshold, projectionAdmissible, abs]
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norm_num
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· -- mergeAdmissible: |R_5(2) - R_3(5)| / (R_5(2) + R_3(5)) < 10^-6
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-- R_5(2) = 31 = R_3(5), so threshold = 0.
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-- so repunit 2 5 = repunit 5 3 = 31, and the threshold is 0.
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simp [kernelEvidence, mergeAdmissibleThreshold, mergeAdmissible, repunit]
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norm_num
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· -- Second known solution: (2, 13, 90, 3) -- symmetric
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rcases h with ⟨rfl, rfl, rfl, rfl⟩
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constructor
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· -- typeAdmissible: |kernel| < 1/8191
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-- γ = 1/8191 with normalization γ^27: witness is extremely small.
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simp [kernelEvidence, hermitianRRCKernel, Hkdf, hermitePoly,
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typeAdmissibleThreshold, typeAdmissible, abs]
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norm_num
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constructor
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· -- projectionAdmissible: |kernel| < 1/(8191*13)
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-- γ = 1/8191 with normalization γ^27: witness far below threshold.
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simp [kernelEvidence, hermitianRRCKernel, Hkdf, hermitePoly,
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projectionAdmissibleThreshold, projectionAdmissible, abs]
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norm_num
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· -- mergeAdmissible: |R*(8191) - R*(31)| / (R*(8191) + R*(31)) < 10^-6
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-- Both repunit values equal 31, so threshold is 0.
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simp [kernelEvidence, mergeAdmissibleThreshold, mergeAdmissible, repunit]
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norm_num
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· rcases h with ⟨rfl, rfl, rfl, rfl⟩
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-- (2, 5, 5, 3): R_5(2) = 31 = R_3(5)
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simp [kernelEvidence, hermitianRRCKernel, Hkdf, hermitePoly,
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typeAdmissibleThreshold, projectionAdmissibleThreshold,
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mergeAdmissibleThreshold, repunit, abs]
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norm_num
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· rcases h with ⟨rfl, rfl, rfl, rfl⟩
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-- TODO(lean-port): simp/norm_num cannot evaluate hermitePoly 13 (1/2) = 1964665
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-- without over-reducing to False. All three gates verified by Python:
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-- type: |kernel| = 3929329/2^54 ≈ 2.18e-10 < 1/2
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-- proj: |kernel| = 1942069/2^34 ≈ 1.13e-4 < 1/26
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-- merge: threshold = 0 < 1e-6
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sorry
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-- ============================================================
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-- §4e THEOREM: UNKNOWN SOLUTIONS FAIL AT LEAST ONE GATE
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@ -340,73 +320,24 @@ def closePairs : List (Nat × Nat × Nat × Nat) :=
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(35,11,52,10), (38,9,64,8), (41,11,62,10), (45,8,85,7), (50,12,74,11),
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(51,11,79,10), (54,10,89,9)]
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/-- Each close pair satisfies threshold ≥ 1/1000000.
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/-- All 32 close pairs satisfy the merge threshold.
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Verified by explicit arithmetic: |R(x,m) - R(y,n)| * 1000000 ≥ R(x,m) + R(y,n).
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TI-84 verifiable. -/
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private theorem closePair_3_11_17_5 : mergeAdmissibleThreshold 3 11 17 5 ≥ 1 / (1000000 : ℚ) := by
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unfold mergeAdmissibleThreshold repunit; norm_num
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private theorem closePair_5_6_62_3 : mergeAdmissibleThreshold 5 6 62 3 ≥ 1 / (1000000 : ℚ) := by
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unfold mergeAdmissibleThreshold repunit; norm_num
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private theorem closePair_5_11_15_7 : mergeAdmissibleThreshold 5 11 15 7 ≥ 1 / (1000000 : ℚ) := by
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unfold mergeAdmissibleThreshold repunit; norm_num
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private theorem closePair_6_12_9_10 : mergeAdmissibleThreshold 6 12 9 10 ≥ 1 / (1000000 : ℚ) := by
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unfold mergeAdmissibleThreshold repunit; norm_num
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private theorem closePair_6_12_17_8 : mergeAdmissibleThreshold 6 12 17 8 ≥ 1 / (1000000 : ℚ) := by
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unfold mergeAdmissibleThreshold repunit; norm_num
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private theorem closePair_6_13_22_8 : mergeAdmissibleThreshold 6 13 22 8 ≥ 1 / (1000000 : ℚ) := by
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unfold mergeAdmissibleThreshold repunit; norm_num
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private theorem closePair_7_9_23_6 : mergeAdmissibleThreshold 7 9 23 6 ≥ 1 / (1000000 : ℚ) := by
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unfold mergeAdmissibleThreshold repunit; norm_num
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private theorem closePair_9_10_17_8 : mergeAdmissibleThreshold 9 10 17 8 ≥ 1 / (1000000 : ℚ) := by
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unfold mergeAdmissibleThreshold repunit; norm_num
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private theorem closePair_10_6_18_5 : mergeAdmissibleThreshold 10 6 18 5 ≥ 1 / (1000000 : ℚ) := by
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unfold mergeAdmissibleThreshold repunit; norm_num
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private theorem closePair_12_10_42_7 : mergeAdmissibleThreshold 12 10 42 7 ≥ 1 / (1000000 : ℚ) := by
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unfold mergeAdmissibleThreshold repunit; norm_num
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private theorem closePair_13_13_82_8 : mergeAdmissibleThreshold 13 13 82 8 ≥ 1 / (1000000 : ℚ) := by
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unfold mergeAdmissibleThreshold repunit; norm_num
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private theorem closePair_14_6_83_4 : mergeAdmissibleThreshold 14 6 83 4 ≥ 1 / (1000000 : ℚ) := by
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unfold mergeAdmissibleThreshold repunit; norm_num
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private theorem closePair_14_9_34_7 : mergeAdmissibleThreshold 14 9 34 7 ≥ 1 / (1000000 : ℚ) := by
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unfold mergeAdmissibleThreshold repunit; norm_num
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private theorem closePair_14_9_69_6 : mergeAdmissibleThreshold 14 9 69 6 ≥ 1 / (1000000 : ℚ) := by
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unfold mergeAdmissibleThreshold repunit; norm_num
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private theorem closePair_15_9_77_6 : mergeAdmissibleThreshold 15 9 77 6 ≥ 1 / (1000000 : ℚ) := by
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unfold mergeAdmissibleThreshold repunit; norm_num
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private theorem closePair_17_9_44_7 : mergeAdmissibleThreshold 17 9 44 7 ≥ 1 / (1000000 : ℚ) := by
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unfold mergeAdmissibleThreshold repunit; norm_num
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private theorem closePair_18_4_78_3 : mergeAdmissibleThreshold 18 4 78 3 ≥ 1 / (1000000 : ℚ) := by
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unfold mergeAdmissibleThreshold repunit; norm_num
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private theorem closePair_19_9_51_7 : mergeAdmissibleThreshold 19 9 51 7 ≥ 1 / (1000000 : ℚ) := by
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unfold mergeAdmissibleThreshold repunit; norm_num
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private theorem closePair_21_8_35_7 : mergeAdmissibleThreshold 21 8 35 7 ≥ 1 / (1000000 : ℚ) := by
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unfold mergeAdmissibleThreshold repunit; norm_num
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private theorem closePair_22_7_41_6 : mergeAdmissibleThreshold 22 7 41 6 ≥ 1 / (1000000 : ℚ) := by
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unfold mergeAdmissibleThreshold repunit; norm_num
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private theorem closePair_26_13_35_12 : mergeAdmissibleThreshold 26 13 35 12 ≥ 1 / (1000000 : ℚ) := by
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unfold mergeAdmissibleThreshold repunit; norm_num
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private theorem closePair_27_11_39_10 : mergeAdmissibleThreshold 27 11 39 10 ≥ 1 / (1000000 : ℚ) := by
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unfold mergeAdmissibleThreshold repunit; norm_num
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private theorem closePair_29_9_47_8 : mergeAdmissibleThreshold 29 9 47 8 ≥ 1 / (1000000 : ℚ) := by
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unfold mergeAdmissibleThreshold repunit; norm_num
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private theorem closePair_30_8_53_7 : mergeAdmissibleThreshold 30 8 53 7 ≥ 1 / (1000000 : ℚ) := by
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unfold mergeAdmissibleThreshold repunit; norm_num
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private theorem closePair_30_12_64_10 : mergeAdmissibleThreshold 30 12 64 10 ≥ 1 / (1000000 : ℚ) := by
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unfold mergeAdmissibleThreshold repunit; norm_num
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private theorem closePair_35_11_52_10 : mergeAdmissibleThreshold 35 11 52 10 ≥ 1 / (1000000 : ℚ) := by
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unfold mergeAdmissibleThreshold repunit; norm_num
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private theorem closePair_38_9_64_8 : mergeAdmissibleThreshold 38 9 64 8 ≥ 1 / (1000000 : ℚ) := by
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unfold mergeAdmissibleThreshold repunit; norm_num
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private theorem closePair_41_11_62_10 : mergeAdmissibleThreshold 41 11 62 10 ≥ 1 / (1000000 : ℚ) := by
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unfold mergeAdmissibleThreshold repunit; norm_num
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private theorem closePair_45_8_85_7 : mergeAdmissibleThreshold 45 8 85 7 ≥ 1 / (1000000 : ℚ) := by
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unfold mergeAdmissibleThreshold repunit; norm_num
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private theorem closePair_50_12_74_11 : mergeAdmissibleThreshold 50 12 74 11 ≥ 1 / (1000000 : ℚ) := by
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unfold mergeAdmissibleThreshold repunit; norm_num
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private theorem closePair_51_11_79_10 : mergeAdmissibleThreshold 51 11 79 10 ≥ 1 / (1000000 : ℚ) := by
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unfold mergeAdmissibleThreshold repunit; norm_num
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private theorem closePair_54_10_89_9 : mergeAdmissibleThreshold 54 10 89 9 ≥ 1 / (1000000 : ℚ) := by
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unfold mergeAdmissibleThreshold repunit; norm_num
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private theorem closePair_threshold
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(h : (x,m,y,n) ∈ closePairs.map (fun p => (p.1, p.2.1, p.2.2.1, p.2.2.2))) :
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mergeAdmissibleThreshold x m y n ≥ 1 / (1000000 : ℚ) := by
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-- Each close pair verified by unfold + norm_num
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simp only [closePairs, List.map_cons, List.mem_cons, Prod.mk.injEq, List.map_nil,
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List.not_mem_nil, or_false] at h
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rcases h with (⟨rfl,rfl,rfl,rfl⟩|⟨rfl,rfl,rfl,rfl⟩|⟨rfl,rfl,rfl,rfl⟩|⟨rfl,rfl,rfl,rfl⟩|
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⟨rfl,rfl,rfl,rfl⟩|⟨rfl,rfl,rfl,rfl⟩|⟨rfl,rfl,rfl,rfl⟩|⟨rfl,rfl,rfl,rfl⟩|
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⟨rfl,rfl,rfl,rfl⟩|⟨rfl,rfl,rfl,rfl⟩|⟨rfl,rfl,rfl,rfl⟩|⟨rfl,rfl,rfl,rfl⟩|
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⟨rfl,rfl,rfl,rfl⟩|⟨rfl,rfl,rfl,rfl⟩|⟨rfl,rfl,rfl,rfl⟩|⟨rfl,rfl,rfl,rfl⟩|
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⟨rfl,rfl,rfl,rfl⟩|⟨rfl,rfl,rfl,rfl⟩|⟨rfl,rfl,rfl,rfl⟩|⟨rfl,rfl,rfl,rfl⟩|
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⟨rfl,rfl,rfl,rfl⟩|⟨rfl,rfl,rfl,rfl⟩|⟨rfl,rfl,rfl,rfl⟩|⟨rfl,rfl,rfl,rfl⟩|
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⟨rfl,rfl,rfl,rfl⟩|⟨rfl,rfl,rfl,rfl⟩|⟨rfl,rfl,rfl,rfl⟩|⟨rfl,rfl,rfl,rfl⟩|
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⟨rfl,rfl,rfl,rfl⟩|⟨rfl,rfl,rfl,rfl⟩|⟨rfl,rfl,rfl,rfl⟩|⟨rfl,rfl,rfl,rfl⟩)
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all_goals (unfold mergeAdmissibleThreshold repunit; norm_num)
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/-- All non-close, non-Goormaghtigh BMS pairs have threshold ≥ 1/1000.
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TI-84 verified: Python scan of 979×979 pairs found only 34 pairs
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@ -431,9 +362,7 @@ axiom nonClose_threshold_axiom (x m y n : ℕ)
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(h_bms : x ≤ 90 ∧ m ≤ 13 ∧ y ≤ 90 ∧ n ≤ 13)
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(h_distinct : (x, m) ≠ (y, n))
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(h_not_goormaghtigh : ¬((x = 2 ∧ m = 5 ∧ y = 5 ∧ n = 3)
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∨ (x = 5 ∧ m = 3 ∧ y = 2 ∧ n = 5)
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∨ (x = 2 ∧ m = 13 ∧ y = 90 ∧ n = 3)
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∨ (x = 90 ∧ m = 3 ∧ y = 2 ∧ n = 13)))
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∨ (x = 2 ∧ m = 13 ∧ y = 90 ∧ n = 3)))
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(h_not_close : ¬((x,m,y,n) ∈ closePairs.map (fun p => (p.1, p.2.1, p.2.2.1, p.2.2.2)))) :
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mergeAdmissibleThreshold x m y n ≥ 1 / (1000 : ℚ)
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@ -442,9 +371,7 @@ private theorem nonClose_threshold (x m y n : ℕ)
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(h_bms : x ≤ 90 ∧ m ≤ 13 ∧ y ≤ 90 ∧ n ≤ 13)
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(h_distinct : (x, m) ≠ (y, n))
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(h_not_goormaghtigh : ¬((x = 2 ∧ m = 5 ∧ y = 5 ∧ n = 3)
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∨ (x = 5 ∧ m = 3 ∧ y = 2 ∧ n = 5)
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∨ (x = 2 ∧ m = 13 ∧ y = 90 ∧ n = 3)
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∨ (x = 90 ∧ m = 3 ∧ y = 2 ∧ n = 13)))
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∨ (x = 2 ∧ m = 13 ∧ y = 90 ∧ n = 3)))
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(h_not_close : ¬((x,m,y,n) ∈ closePairs.map (fun p => (p.1, p.2.1, p.2.2.1, p.2.2.2)))) :
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mergeAdmissibleThreshold x m y n ≥ 1 / (1000 : ℚ) :=
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nonClose_threshold_axiom x m y n hx hm hy hn h_bms h_distinct h_not_goormaghtigh h_not_close
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@ -454,20 +381,14 @@ theorem unknown_fails_rrc (x m y n : ℕ)
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(h_bms : x ≤ 90 ∧ m ≤ 13 ∧ y ≤ 90 ∧ n ≤ 13)
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(h_distinct : (x, m) ≠ (y, n))
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(h_unknown : ¬((x = 2 ∧ m = 5 ∧ y = 5 ∧ n = 3)
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∨ (x = 5 ∧ m = 3 ∧ y = 2 ∧ n = 5)
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∨ (x = 2 ∧ m = 13 ∧ y = 90 ∧ n = 3)
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∨ (x = 90 ∧ m = 3 ∧ y = 2 ∧ n = 13))) :
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∨ (x = 2 ∧ m = 13 ∧ y = 90 ∧ n = 3))) :
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mergeAdmissibleThreshold x m y n ≥ 1 / (1000000 : ℚ) := by
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-- TI-84 PROOF: Two cases.
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-- Case 1: (x,m,y,n) is a close pair → verified by explicit theorems (32 cases)
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-- Case 2: (x,m,y,n) is NOT a close pair → threshold ≥ 1/1000 > 1/1000000
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by_cases h_close : (x,m,y,n) ∈ closePairs.map (fun p => (p.1, p.2.1, p.2.2.1, p.2.2.2))
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· -- Close pair: each has a dedicated theorem proving threshold ≥ 1/1000000
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-- The 32 explicit theorems (closePair_*) cover all close pairs.
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-- native_decide required: 32 explicit case splits have no Lean tactic alternative.
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revert h_close
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unfold mergeAdmissibleThreshold repunit
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native_decide
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· -- Close pair: dispatched by closePair_threshold (native_decide verified)
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exact closePair_threshold h_close
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· -- Non-close pair: threshold ≥ 1/1000 > 1/1000000
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have h_threshold := nonClose_threshold x m y n hx hm hy hn h_bms h_distinct h_unknown h_close
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linarith [h_threshold]
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@ -516,21 +437,15 @@ theorem rrc_characterizes_goormaghtigh (x m y n : ℕ)
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(kernelEvidence x m y n).typeAdmissible ∧
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(kernelEvidence x m y n).projectionAdmissible ∧
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(kernelEvidence x m y n).mergeAdmissible ↔
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(((x = 2 ∧ m = 5) ∧ (y = 5 ∧ n = 3)) ∨
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((x = 5 ∧ m = 3) ∧ (y = 2 ∧ n = 5)) ∨
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((x = 2 ∧ m = 13) ∧ (y = 90 ∧ n = 3)) ∨
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((x = 90 ∧ m = 3) ∧ (y = 2 ∧ n = 13))) := by
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((x = 2 ∧ m = 5 ∧ y = 5 ∧ n = 3) ∨
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(x = 2 ∧ m = 13 ∧ y = 90 ∧ n = 3)) := by
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constructor
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· -- Forward: all gates pass → known Goormaghtigh solution
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intro h_all
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simp only [kernelEvidence] at h_all
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have h_merge := h_all.2.2
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-- h_merge: mergeAdmissibleThreshold < 10^-6
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-- By unknown_fails_rrc: if NOT Goormaghtigh, threshold ≥ 10^-6
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-- Contrapositive: if threshold < 10^-6, must be Goormaghtigh
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by_contra h_not_goormaghtigh
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have h_threshold := unknown_fails_rrc x m y n hx hm hy hn h_bms h_distinct h_not_goormaghtigh
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-- h_threshold: mergeAdmissibleThreshold ≥ 1/1000000
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-- h_merge: mergeAdmissibleThreshold < 1/1000000
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linarith [h_merge, h_threshold]
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· -- Backward: known solution → all gates pass
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intro h_known
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