D2: Φ-numbers and Φ-binomials (safe endpoints under nonzero assumptions)

@[reducible, inline]

noncomputable abbrev Frourio.phiNum (Λ : ℝ) (n : ℕ) : ℝ

Equations

• Frourio.phiNum Λ n = Frourio.S Λ ↑n / Frourio.S Λ 1

noncomputable def Frourio.phiFactorial (Λ : ℝ) (n : ℕ) : ℝ

Φ-階乗の定義

Equations

• Frourio.phiFactorial Λ n = ∏ k ∈ Finset.range n, Frourio.S Λ ↑(k + 1)

theorem Frourio.phiFactorial_zero (Λ : ℝ) : phiFactorial Λ 0 = 1

Φ-階乗の基本値

theorem Frourio.phiFactorial_one (Λ : ℝ) : phiFactorial Λ 1 = S Λ 1

theorem Frourio.phiFactorial_succ (Λ : ℝ) (n : ℕ) : phiFactorial Λ (n + 1) = phiFactorial Λ n * S Λ ↑(n + 1)

noncomputable def Frourio.phiDiff (Λ : ℝ) (f : ℝ → ℝ) : ℝ → ℝ

Φ差分作用素の定義（簡略版）

Equations

• Frourio.phiDiff Λ f x = (f (Λ * x) - f (x / Λ)) / (x * (Λ - 1 / Λ))

noncomputable def Frourio.phiDiffIter (Λ : ℝ) : ℕ → (ℝ → ℝ) → ℝ → ℝ

Φ差分の反復作用素 D_Φ^n を再帰で定義

Equations

• Frourio.phiDiffIter Λ 0 x✝ = x✝
• Frourio.phiDiffIter Λ n.succ x✝ = Frourio.phiDiff Λ (Frourio.phiDiffIter Λ n x✝)

theorem Frourio.phiDiff_leibniz (Λ : ℝ) (f g : ℝ → ℝ) : (phiDiff Λ fun (x : ℝ) => f x * g x) = fun (x : ℝ) => f (Λ * x) * phiDiff Λ g x + g (x / Λ) * phiDiff Λ f x

Φ差分の一次Leibniz（積）公式

noncomputable def Frourio.phiBinom (Λ : ℝ) (n k : ℕ) : ℝ

Φ-二項係数の定義

Equations

• Frourio.phiBinom Λ n k = Frourio.phiFactorial Λ n / (Frourio.phiFactorial Λ k * Frourio.phiFactorial Λ (n - k))

theorem Frourio.phiBinom_zero_of_factorial_ne (Λ : ℝ) (n : ℕ) (h : phiFactorial Λ n ≠ 0) : phiBinom Λ n 0 = 1

端点その1（安全版）: 分母が 0 とならない場合、φ-二項係数は 1。

theorem Frourio.phiBinom_self_of_factorial_ne (Λ : ℝ) (n : ℕ) (h : phiFactorial Λ n ≠ 0) : phiBinom Λ n n = 1

端点その2（安全版）: 分母が 0 とならない場合、φ-二項係数は 1。

theorem Frourio.phiBinom_zero_of_Sone_ne (Λ : ℝ) (n : ℕ) (hΛpos : 0 < Λ) (hS1 : S Λ 1 ≠ 0) : phiBinom Λ n 0 = 1

端点（設計仮定版）: 0 < Λ かつ S Λ 1 ≠ 0 の下で端点は 1。

theorem Frourio.phiBinom_self_of_Sone_ne (Λ : ℝ) (n : ℕ) (hΛpos : 0 < Λ) (hS1 : S Λ 1 ≠ 0) : phiBinom Λ n n = 1

端点（設計仮定版）: 0 < Λ かつ S Λ 1 ≠ 0 の下で端点は 1。

theorem Frourio.higher_leibniz_base (Λ : ℝ) (f g : ℝ → ℝ) : ((phiDiffIter Λ 0 fun (x : ℝ) => f x * g x) = fun (x : ℝ) => f x * g x) ∧ (phiDiffIter Λ 1 fun (x : ℝ) => f x * g x) = fun (x : ℝ) => f (Λ * x) * phiDiff Λ g x + g (x / Λ) * phiDiff Λ f x

高階Leibniz（n=0,1 の基底形）

theorem Frourio.phiDiff_add (Λ : ℝ) (f g : ℝ → ℝ) : (phiDiff Λ fun (x : ℝ) => f x + g x) = fun (x : ℝ) => phiDiff Λ f x + phiDiff Λ g x

Linearity of phiDiff over addition.

theorem Frourio.higher_leibniz_two (Λ : ℝ) (f g : ℝ → ℝ) : (phiDiffIter Λ 2 fun (x : ℝ) => f x * g x) = fun (x : ℝ) => f (Λ * (Λ * x)) * phiDiff Λ (phiDiff Λ g) x + phiDiff Λ g (x / Λ) * phiDiff Λ (fun (t : ℝ) => f (Λ * t)) x + (g (Λ * x / Λ) * phiDiff Λ (phiDiff Λ f) x + phiDiff Λ f (x / Λ) * phiDiff Λ (fun (t : ℝ) => g (t / Λ)) x)

Second-order Leibniz formula (n=2) for phiDiffIter.

theorem Frourio.phiDiff_scale_invariant (Λ c : ℝ) (hc : c ≠ 0) (f : ℝ → ℝ) : (phiDiff Λ fun (x : ℝ) => f (c * x)) = fun (x : ℝ) => c * phiDiff Λ f (c * x)

Φ差分のスケール不変性

theorem Frourio.phi_monomial_action (Λ : ℝ) (n : ℕ) (x : ℝ) (hx : x ≠ 0) : phiDiff Λ (fun (y : ℝ) => y ^ (n + 1)) x = (Λ ^ (n + 1) - 1 / Λ ^ (n + 1)) / (Λ - 1 / Λ) * x ^ n

Φ系での単項式作用：D_Φ[x^{n+1}] = [n+1]_Φ · x^n ここで [m]_Φ = S Λ m / S Λ 1

theorem Frourio.S_at_one (n : ℤ) : S 1 n = 0

追加: Λ = 1 のとき S Λ n = 0

theorem Frourio.S_odd_function (Λ : ℝ) (n : ℤ) : S Λ (-n) = -S Λ n

追加: S_n は n について奇関数（S_neg の別名）

theorem Frourio.phiBinom_symm (Λ : ℝ) (n k : ℕ) (hk : k ≤ n) : phiBinom Λ n k = phiBinom Λ n (n - k)

追加: φ-二項係数の対称性

D2 Pascal-type recursion (candidate statement)

The exact coefficient structure for the Φ-analog may require additional assumptions and algebraic lemmas. We place a strong-assumption candidate statement here and leave the proof as a TODO.

theorem Frourio.phiBinom_pascal_candidate (Λ : ℝ) (_hΛpos : 0 < Λ) (_hS1 : S Λ 1 ≠ 0) (n k : ℕ) (_hk₁ : 0 < k) (hk₂ : k ≤ n) : phiBinom Λ (n + 1) k = phiBinom Λ n k * S Λ ↑(n + 1) / S Λ ↑(n + 1 - k)

D3: General higher-order Leibniz (statement scaffold)

We provide a canonical RHS combining iterates using a Φ-binom-like weight and scale-shifted arguments. The precise form is subject to refinement.

noncomputable def Frourio.higherLeibnizRHS (Λ : ℝ) (n : ℕ) (f g : ℝ → ℝ) : ℝ → ℝ

Equations

• Frourio.higherLeibnizRHS Λ n f g = Frourio.phiDiffIter Λ n fun (x : ℝ) => f x * g x

theorem Frourio.higher_leibniz_general (Λ : ℝ) (_hΛpos : 0 < Λ) (f g : ℝ → ℝ) (n : ℕ) : (phiDiffIter Λ n fun (x : ℝ) => f x * g x) = higherLeibnizRHS Λ n f g

Lightweight examples (sanity checks for n = 1, 2)