Cost Ndim Radical Distribution Smul Mem Radical

The radical distribution

A ledger, a discrete record of recognition events, assigns a cost to each possible state. In the Recognition Science framework, near a given state the cost's curvature is captured by a quadratic form, the Hessian. For the rank-one log-coordinate metric, this Hessian is degenerate: it only detects changes along a single active direction, denoted α. The radical distribution, the set of directions along which the quadratic form vanishes, is therefore the hyperplane of vectors v whose weighted dot product with α is zero: { v | dot α v = 0 }.

The declaration smul_mem_Radical proves a closure property of this set: if a vector v lies in the radical distribution, then any scalar multiple s • v also lies in it. In plain terms, scaling a direction that costs nothing produces another direction that costs nothing. This is a theorem in the framework's machine-checked library of formal theorems. It is one of a family of closure results, alongside add_mem_Radical and sub_mem_Radical, which together show that the radical distribution forms a linear subspace: closed under addition, subtraction, and scaling.

The geometric picture is concrete. The radical distribution is the plane through the origin orthogonal to α. The theorem states that this plane is a genuine vector subspace, not merely a cone or a collection of rays. This matters because the framework uses these directions to define integrable leaves: affine hyperplanes of the form { t | dot α t = c }, where c is a constant. The related theorem affineShift_mem_LevelSet shows that moving along a radical direction from a point on such a leaf stays on the same leaf, making the distribution integrable in the sense of foliation theory.

In Recognition Science, this result supports the claim that the degenerate directions of the cost metric organize into flat, parallel sheets. It does not claim that the radical distribution is the whole space, nor that the active direction α is unique, nor that the metric is non-degenerate. It also does not claim that the radical distribution is integrable in the sense of Frobenius; that is a separate theorem, radical_integrable_by_affine_leaves, which the library proves separately. The closure under scaling is a local algebraic fact about the Hessian, not a global statement about the cost function's level sets.

theorem smul_mem_Radical {n : ℕ} (α : Vec n) {v : Vec n} (s : ℝ)
    (hv : v ∈ Radical α) :
    s • v ∈ Radical α := by
  unfold Radical dot at hv ⊢
  calc
    ∑ i : Fin n, α i * (s * v i)
        = s * ∑ i : Fin n, α i * v i := by
            rw [Finset.mul_sum]
            apply Finset.sum_congr rfl
            intro i hi
            ring
    _ = 0 := by rw [hv]; ring

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

/-- Directions in the radical stay inside the affine leaves `dot α = c`. -/
theorem affineShift_mem_LevelSet {n : ℕ} (α : Vec n) {c s : ℝ} {t v : Vec n}
    (ht : t ∈ LevelSet α c) (hv : v ∈ Radical α) :
    affineShift t v s ∈ LevelSet α c := by
  rw [mem_LevelSet_iff] at ht ⊢
  have hv' : dot α v = 0 := hv
  rw [dot_affineShift, ht, hv']
  ring

What this page does not claim

The radical distribution is not the whole space, and the active direction α is not claimed to be unique. The closure under scaling does not imply the metric is non-degenerate or that the radical is integrable in the Frobenius sense. The theorem does not assert any global property of the cost function's level sets beyond the local algebraic closure.

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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