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12dee50 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 | import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Basic.Real.Basic
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Ring
import Mathlib.Tactic.NormNum
namespace OAI
/-!
# Finite-layer and weighted-variance inequalities
These inequalities supply the scalar calculations for completed surface cones
in *A Complete Local Domain without a Small Cohen–Macaulay Module* (OpenAI, 2026).
-/
namespace SmallCM.Arithmetic
/-- The elementary finite-layer inequality in the proof of `lem:lattice`. -/
theorem layer_square_le (N : ℤ) (r : ℕ → ℤ) (n : ℕ)
(hr : ∀ j < n, 0 ≤ r j) (hN : ∀ j < n, r j ≤ N) :
(∑ j ∈ Finset.range n, r j) ^ 2 ≤
N * ∑ j ∈ Finset.range n, (2 * (j : ℤ) + 1) * r j := by
induction n with
| zero => simp
| succ n ih =>
have hrn : 0 ≤ r n := hr n (Nat.lt_succ_self n)
have hNn : r n ≤ N := hN n (Nat.lt_succ_self n)
have hprev := ih (fun j hj => hr j (Nat.lt_succ_of_lt hj))
(fun j hj => hN j (Nat.lt_succ_of_lt hj))
have hsum : ∑ j ∈ Finset.range n, r j ≤ (n : ℤ) * N := by
calc
_ ≤ ∑ _j ∈ Finset.range n, N :=
Finset.sum_le_sum (fun j hj => hN j (Nat.lt_succ_of_lt
(Finset.mem_range.mp hj)))
_ = _ := by simp
simp only [Finset.sum_range_succ]
nlinarith [mul_nonneg hrn (sub_nonneg.mpr hNn),
mul_nonneg hrn (sub_nonneg.mpr hsum)]
/-- The scalar consequence of the rank bounds and first-degree bounds for
quotient layers. -/
theorem layer_determinant_numerator_nonneg (N : ℤ) (hN : 0 ≤ N)
(r c : ℕ → ℤ) (n : ℕ)
(hr : ∀ j < n, 0 ≤ r j) (hrN : ∀ j < n, r j ≤ N)
(hc : ∀ j < n, (j : ℤ) * r j ≤ c j) :
0 ≤ N * (∑ j ∈ Finset.range n, (2 * c j + r j)) -
(∑ j ∈ Finset.range n, r j) ^ 2 := by
have hs : (∑ j ∈ Finset.range n, (2 * (j : ℤ) + 1) * r j) ≤
∑ j ∈ Finset.range n, (2 * c j + r j) := by
apply Finset.sum_le_sum
intro j hj
have h := hc j (Finset.mem_range.mp hj)
nlinarith
exact sub_nonneg.mpr ((layer_square_le N r n hr hrN).trans
(mul_le_mul_of_nonneg_left hs hN))
/-- Expanding weighted square distances, without a normalization hypothesis. -/
theorem weighted_square_expansion {ι : Type*} (s : Finset ι)
(w x : ι → ℝ) (a : ℝ) :
(∑ i ∈ s, w i * (x i - a) ^ 2) =
(∑ i ∈ s, w i * x i ^ 2) -
2 * a * (∑ i ∈ s, w i * x i) + a ^ 2 * (∑ i ∈ s, w i) := by
calc
_ = ∑ i ∈ s, ((w i * x i ^ 2 - 2 * a * (w i * x i)) + a ^ 2 * w i) := by
apply Finset.sum_congr rfl
intro i _
ring
_ = _ := by
simp only [Finset.sum_add_distrib, Finset.sum_sub_distrib, ← Finset.mul_sum]
/-- The weighted variance bound for an interval of length one, used after the
slope-balancing step. -/
theorem weighted_variance_le_quarter {ι : Type*} (s : Finset ι)
(w x : ι → ℝ) (a : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hsum : ∑ i ∈ s, w i = 1)
(hl : ∀ i ∈ s, a ≤ x i) (hu : ∀ i ∈ s, x i ≤ a + 1) :
(∑ i ∈ s, w i * (x i - (∑ j ∈ s, w j * x j)) ^ 2) ≤ 1 / 4 := by
let μ := ∑ i ∈ s, w i * x i
have hmin : (∑ i ∈ s, w i * (x i - μ) ^ 2) ≤
∑ i ∈ s, w i * (x i - (a + 1 / 2)) ^ 2 := by
rw [weighted_square_expansion, weighted_square_expansion, hsum]
change (∑ i ∈ s, w i * x i ^ 2) - 2 * μ * μ + μ ^ 2 * 1 ≤
(∑ i ∈ s, w i * x i ^ 2) - 2 * (a + 1 / 2) * μ + (a + 1 / 2) ^ 2 * 1
nlinarith [sq_nonneg (μ - (a + 1 / 2))]
refine hmin.trans ?_
calc
_ ≤ ∑ i ∈ s, w i * (1 / 4) := by
apply Finset.sum_le_sum
intro i hi
apply mul_le_mul_of_nonneg_left ?_ (hw i hi)
have h := mul_nonneg (sub_nonneg.mpr (hl i hi)) (sub_nonneg.mpr (hu i hi))
nlinarith
_ = 1 / 4 := by rw [← Finset.sum_mul, hsum, one_mul]
/-- The finite weighted-mixture identity for normalized ranks and Chern characters. -/
theorem mixture_identity {ι : Type*} (s : Finset ι) (w m q : ι → ℝ)
(hsum : ∑ i ∈ s, w i = 1) :
2 * (∑ i ∈ s, w i * q i) - (∑ i ∈ s, w i * m i) ^ 2 =
(∑ i ∈ s, w i * (2 * q i - m i ^ 2)) +
(∑ i ∈ s, w i * (m i - (∑ j ∈ s, w j * m j)) ^ 2) := by
have he : (∑ i ∈ s, w i * (2 * q i - m i ^ 2)) =
2 * (∑ i ∈ s, w i * q i) - ∑ i ∈ s, w i * m i ^ 2 := by
calc
_ = ∑ i ∈ s, (2 * (w i * q i) - w i * m i ^ 2) := by
apply Finset.sum_congr rfl
intro i _
ring
_ = _ := by simp only [Finset.sum_sub_distrib, ← Finset.mul_sum]
rw [he, weighted_square_expansion, hsum]
ring
/-- The numerical inequality for the exponent-seven polarization. -/
theorem exponent_seven :
(5 : ℤ) * 7 ^ 3 = 1715 ∧
(7 : ℤ) ^ 3 * (7 ^ 2 - 10) = 13377 ∧
(6 : ℤ) * 1715 < 13377 := by
norm_num
end SmallCM.Arithmetic
end OAI
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