Cost Ndim Radical Distribution Sub Mem Radical

The radical distribution

In the Recognition Science framework, a cost function measures the price of a recognition event, a discrete record of something being recognized. The framework's central theorem forces a specific one-dimensional cost function, but its multi-dimensional generalization studies what happens when many variables move at once. The key object is the radical distribution: the set of all directions in which the cost's curvature, the second-order sensitivity, vanishes completely.

Formally, for a vector α of weights, the radical distribution is the set of all vectors v whose weighted dot product with α is zero. The declaration sub_mem_Radical, a theorem in the framework's machine-checked library of formal theorems, proves that this set is closed under subtraction: if v and w are both in the radical, then v - w is also in it. This closure property, together with the companion theorems for addition and scalar multiplication, establishes that the radical is a linear subspace, a flat plane through the origin.

The framework then shows this subspace is integrable: the radical directions can be organized into affine leaves, parallel hyperplanes of the form dot α t = c, where c is a constant. A direction preserves such a leaf exactly when it lies in the radical. This means the cost's degenerate directions, where curvature vanishes, foliate the space into flat slices. The theorem sub_mem_Radical is the algebraic backbone of this foliation; without closure under subtraction, the leaves would not be well-defined.

What this declaration does not claim is any physical content. It is a purely geometric statement about a vector space with a weighted dot product. It does not assert that these directions are physically realizable, that the radical has any particular dimension, or that the cost function itself is the unique one forced by the framework's axioms. The theorem holds for any choice of α and any finite dimension n; it is a structural fact about linear algebra, not a statement about the physical world.

@[simp] theorem mem_Radical_iff {n : ℕ} (α : Vec n) (v : Vec n) :
    v ∈ Radical α ↔ dot α v = 0 := Iff.rfl

theorem sub_mem_Radical {n : ℕ} (α : Vec n) {v w : Vec n}
    (hv : v ∈ Radical α) (hw : w ∈ Radical α) :
    v - w ∈ Radical α := by
  simpa [sub_eq_add_neg] using
    add_mem_Radical α hv (smul_mem_Radical α (-1) hw)

/-- The radical distribution is integrable: its integral leaves are the affine
hyperplanes `dot α = c`. -/
theorem radical_integrable_by_affine_leaves {n : ℕ} (α : Vec n) (c : ℝ) :
    ∀ ⦃t v : Vec n⦄, t ∈ LevelSet α c → v ∈ Radical α →
      ∀ s : ℝ, affineShift t v s ∈ LevelSet α c := by
  intro t v ht hv s
  exact affineShift_mem_LevelSet α ht hv

What this page does not claim

The theorem does not assert that radical directions are physically realizable. The theorem does not specify the dimension of the radical distribution. The theorem does not depend on the specific form of the cost function forced by the framework's axioms.

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/RadicalDistribution.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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