Cost Ndim Projector Aapply Smul

The scaling law

In linear algebra, a linear map is a function between vector spaces that respects two operations: adding inputs and multiplying inputs by a number. The theorem AApply_smul concerns the second operation, scalar multiplication. It states that for any real number c and any vector v, the operator A applied to the scaled vector c·v equals c times the operator applied to v. In symbols, A(c·v) = c·A(v). This property, called homogeneity, is one of the two defining features of a linear map; the other, additivity, is established by a separate theorem AApply_add.

The operator A itself is not arbitrary. In the Recognition Science framework, it is constructed from a covector β and an inverse metric kernel hInv. The covector is raised to a vector using the kernel, and A is then defined as the rank-one operator that sends a vector v to the scalar λ times the dot product of β with v, multiplied by the sharpened vector. This construction appears in the machine-checked library of formal theorems as the definition AApply. The theorem AApply_smul is a proved consequence of this definition, verified in the library's Projector module.

The practical content of the theorem is that the operator A behaves predictably under scaling. If you double the input, the output doubles. If you halve the input, the output halves. This is a necessary condition for A to be a linear operator, and it underpins the later results in the module: the projector P, the almost-product operator F, and the golden operator G all inherit this scaling property, as shown by the theorems PApply_smul, FApply_smul, and the quadratic laws that follow. The theorem is a building block, not a standalone discovery; its role is to certify that the algebra of these operators is consistent.

What AApply_smul does not claim is any physical content. It is a purely algebraic statement about a defined operator. It does not say that this operator represents a physical process, nor that the scaling property holds in any empirical sense. The theorem is true by construction, given the definitions in the library. Whether the operator A corresponds to something measurable in the physical world is a separate question, not addressed by this declaration. The theorem also does not establish the additivity property A(v+w) = A(v)+A(w); that is a different theorem, AApply_add, proved separately in the same module.

theorem AApply_smul {n : ℕ}
    (lam : ℝ) (hInv : Fin n → Fin n → ℝ) (β : Vec n)
    (c : ℝ) (v : Vec n) :
    AApply lam hInv β (c • v) = c • AApply lam hInv β v := by
  funext i
  unfold AApply dot
  calc
    lam * sharp hInv β i * ∑ j : Fin n, β j * (c * v j)
        = lam * sharp hInv β i * (c * ∑ j : Fin n, β j * v j) := by
            congr 1
            rw [Finset.mul_sum]
            apply Finset.sum_congr rfl
            intro j hj
            ring
    _ = lam * sharp hInv β i * (c * dot β v) := by
          simp [dot]
    _ = c * (lam * sharp hInv β i * dot β v) := by
          ring

/-- The rank-one operator `A = h^{-1} \tilde g` in coordinates, where
`\tilde g = λ β ⊗ β`. -/
noncomputable def AApply {n : ℕ}
    (lam : ℝ) (hInv : Fin n → Fin n → ℝ) (β : Vec n) : Vec n → Vec n :=
  fun v => fun i => lam * sharp hInv β i * dot β v

theorem PApply_smul {n : ℕ}
    (lam : ℝ) (hInv : Fin n → Fin n → ℝ) (β : Vec n)
    (c : ℝ) (v : Vec n) :
    PApply lam hInv β (c • v) = c • PApply lam hInv β v := by
  ext i
  simp [PApply, AApply_smul, mul_assoc, mul_comm]

theorem FApply_smul {n : ℕ}
    (lam : ℝ) (hInv : Fin n → Fin n → ℝ) (β : Vec n)
    (c : ℝ) (v : Vec n) :
    FApply lam hInv β (c • v) = c • FApply lam hInv β v := by
  ext i
  simp [FApply, PApply_smul, mul_comm]
  ring

What this page does not claim

The theorem does not claim any physical content or empirical validity for the operator A. The theorem does not establish the additivity property A(v+w) = A(v)+A(w).

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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Derived articles

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