Cost Ndim Projector Fapply Gapply

The golden operator's law

In linear algebra, a projector is a linear map that sends a vector space onto a subspace while leaving that subspace fixed; applying it twice does nothing new. The Recognition Science framework builds one from a cost function: a covector β and an inverse metric kernel hInv determine an operator A, and normalizing A yields the projector P. From P the framework defines F = 2P - I, a reflection-like operator that flips the complement of the projected subspace. The declaration FApply_GApply proves that this F, when followed by the golden operator G, satisfies a law that mirrors the golden ratio's own equation.

The golden ratio φ is the number satisfying r² = r + 1, approximately 1.618. The framework's golden operator G is built from the same projector: G(v) = (1/2)v + (√5/2)F(v). The theorem FApply_GApply shows that F(G(v)) = (1/2)F(v) + (√5/2)v. This is not a coincidence of notation; the operator equation mirrors the scalar identity φ² = φ + 1 when F plays the role of the square root of 5. The proof relies on F being an involution, meaning F(F(v)) = v, which the framework proves separately under the condition that the scalar μ is nonzero.

The result is a formal theorem in the framework's machine-checked library of formal theorems, meaning every step is verified by a computer. It does not claim that the golden ratio is physically observed in any experiment, nor that this operator appears in nature. It establishes an algebraic fact: given the framework's definitions, the golden operator satisfies a quadratic relation that matches the golden ratio's defining equation. The theorem also generalizes: a metallic family of operators satisfies the analogous equation x² = px + q, covering the golden case when p = q = 1.

What this changes is the internal coherence of the framework. The golden ratio does not enter as an arbitrary constant; it emerges from the same projector construction that defines the cost-induced operators. The theorem shows the framework's algebra is consistent enough to reproduce a classical constant from its own definitions, a necessary step before any claim about physical relevance. The declaration does not, by itself, connect this operator to measurements or to the framework's larger claims about dimensions or particle masses.

theorem FApply_GApply {n : ℕ}
    (lam : ℝ) (hInv : Fin n → Fin n → ℝ) (β : Vec n)
    (hμ : mu lam hInv β ≠ 0) (v : Vec n) :
    FApply lam hInv β (GApply lam hInv β v)
      = ((1 : ℝ) / 2) • FApply lam hInv β v + (Real.sqrt 5 / 2) • v := by
  unfold GApply
  rw [FApply_add, FApply_smul, FApply_smul, FApply_square _ _ _ hμ]

theorem FApply_square {n : ℕ}
    (lam : ℝ) (hInv : Fin n → Fin n → ℝ) (β : Vec n)
    (hμ : mu lam hInv β ≠ 0) (v : Vec n) :
    FApply lam hInv β (FApply lam hInv β v) = v := by
  ext i
  have hPFi : PApply lam hInv β (FApply lam hInv β v) i = PApply lam hInv β v i := by
    simpa using congrFun (PApply_FApply lam hInv β hμ v) i
  calc
    FApply lam hInv β (FApply lam hInv β v) i
        = (2 • PApply lam hInv β (FApply lam hInv β v) - FApply lam hInv β v) i := by
            simp [FApply]
    _ = (2 • PApply lam hInv β v - FApply lam hInv β v) i := by
          simp [hPFi]
    _ = v i := by
          simp [FApply]

theorem MetallicApply_square {n : ℕ}
    (p q lam : ℝ) (hInv : Fin n → Fin n → ℝ) (β : Vec n)
    (hμ : mu lam hInv β ≠ 0) (hq : 0 ≤ p ^ 2 + 4 * q) (v : Vec n) :
    MetallicApply p q lam hInv β (MetallicApply p q lam hInv β v)
      = p • MetallicApply p q lam hInv β v + q • v := by
  ext i
  have hFMi :
      FApply lam hInv β
          ((p / 2) • v + (Real.sqrt (p ^ 2 + 4 * q) / 2) • FApply lam hInv β v) i
        = ((p / 2) • FApply lam hInv β v
          + (Real.sqrt (p ^ 2 + 4 * q) / 2) • v) i := by
    simpa [MetallicApply] using congrFun (FApply_MetallicApply p q lam hInv β hμ v) i
  have hsqrt : Real.sqrt (p ^ 2 + 4 * q) * Real.sqrt (p ^ 2 + 4 * q) = p ^ 2 + 4 * q := by
    nlinarith [Real.sq_sqrt hq]
  have hsqrtq :
      (Real.sqrt (p ^ 2 + 4 * q) / 2) * (Real.sqrt (p ^ 2 + 4 * q) / 2)
        = (p ^ 2 + 4 * q) / 4 := by
    nlinarith [hsqrt]
  simp [MetallicApply, hFMi]
  have hmul :
      (Real.sqrt (p ^ 2 + 4 * q) / 2) *
          ((Real.sqrt (p ^ 2 + 4 * q) / 2) * v i)
        = ((p ^ 2 + 4 * q) / 4) * v i := by
    calc
      (Real.sqrt (p ^ 2 + 4 * q) / 2) *
          ((Real.sqrt (p ^ 2 + 4 * q) / 2) * v i)
          = ((Real.sqrt (p ^ 2 + 4 * q) / 2) *
              (Real.sqrt (p ^ 2 + 4 * q) / 2)) * v i := by
              ring
      _ = ((p ^ 2 + 4 * q) / 4) * v i := by rw [hsqrtq]
  nlinarith [hmul]

What this page does not claim

The golden ratio is not claimed to be physically observed in any experiment. The theorem does not connect the golden operator to measurements or to the framework's claims about dimensions or particle masses. The framework does not claim the golden operator appears in nature.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Cost/Ndim/Projector.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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