Cost Ndim Radical Distribution

The silent directions

In differential geometry, the radical distribution of a quadratic form is the set of directions along which the form vanishes: moving in one of these directions changes nothing about the form. For the rank-one log-coordinate metric used in Recognition Science's cost analysis, this set has a simple description. The metric's Hessian, the matrix of second derivatives that measures local curvature, detects only a single active direction, denoted α. The radical distribution is then exactly the hyperplane of vectors perpendicular to α: all directions v such that the dot product α · v equals zero.

The module formalizes this distribution and its geometric consequences. Its central result is that the radical distribution is integrable: the space of all directions splits into flat, parallel sheets, each defined by a constant value of the dot product α · t = c. Moving along a radical direction keeps you on the same sheet. The theorem radical_integrable_by_affine_leaves states this precisely: starting from any point on a sheet and moving along any radical direction by any amount stays on that sheet. A companion theorem, preserves_own_leaf_iff_mem_Radical, gives the converse: a direction preserves its own sheet if and only if it lies in the radical.

In plain language, the module proves that the cost function's blind spots are not scattered or twisted; they organize themselves into clean, parallel layers. This matters because integrability is a strong structural property. It means the silent directions can be treated as a coherent coordinate system, not as a collection of unrelated degeneracies. The framework's library of machine-checked theorems establishes this as a formal result, not a heuristic observation.

/-- The radical distribution of the rank-one Hessian metric. -/
def Radical {n : ℕ} (α : Vec n) : Set (Vec n) :=
  { v | dot α v = 0 }

/-- 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

/-- 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

/-- A constant direction preserves the affine leaf through `t` exactly when it
lies in the radical distribution. -/
theorem preserves_own_leaf_iff_mem_Radical {n : ℕ} (α t v : Vec n) :
    (∀ s : ℝ, affineShift t v s ∈ LevelSet α (dot α t)) ↔ v ∈ Radical α := by
  constructor
  · intro h
    have h1 := h 1
    rw [mem_LevelSet_iff, dot_affineShift] at h1
    unfold Radical
    have : dot α v = 0 := by linarith
    exact this
  · intro hv s
    exact affineShift_mem_LevelSet α (by simp [LevelSet]) hv

What this page does not claim

No claim that the radical distribution is unique to Recognition Science; it is a standard geometric object. No claim that the module proves the cost function's uniqueness; it only analyzes a given metric. No claim that the active direction α is derived in this module; it is taken as input.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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