Cost Ndim Radical Distribution Radical Integrable By Affine Leaves

What the theorem establishes

In the Recognition Science framework, the cost of recognition is measured by a function whose behavior near any point is summarized by a Hessian, a kind of second-derivative table. For the framework's rank-one log-coordinate metric, this Hessian is degenerate: it only detects change along a single active direction, which the framework writes as the vector α. The radical distribution is the set of all directions v along which this Hessian sees no change at all, defined by the equation dot α v = 0, meaning the direction v is perpendicular to α.

The theorem radical_integrable_by_affine_leaves, proved in the machine-checked library of formal theorems, establishes that this radical distribution is integrable. In plain terms, the directions of zero curvature can be organized into flat, parallel slices called affine leaves, each one a hyperplane of the form dot α t = c for some constant c. The theorem proves that if you start at any point t on one of these slices and move in any direction v that lies in the radical (so dot α v = 0), you stay on the same slice, no matter how far you go. The slices are invariant under all motion along the degenerate directions.

This is a structural fact about the framework's geometry, not a statement about the physical world. The theorem's content is that the degenerate directions form a foliation: the space is cleanly divided into these flat leaves, and the radical distribution is exactly the tangent bundle to that foliation. The companion theorem preserves_own_leaf_iff_mem_Radical sharpens this: a constant direction preserves the slice through a point if and only if it lies in the radical. The geometry is completely characterized by this perpendicularity condition.

In Recognition Science, this result matters because it clarifies what the rank-one metric can and cannot distinguish. The metric only sees the active direction α; everything orthogonal to it is invisible to the Hessian. The theorem proves that this invisibility is not a local accident but a global structure: the invisible directions organize into entire flat slices that the metric cannot tell apart. This is a precise statement about the framework's cost geometry, and it is the foundation for understanding what information the metric actually carries.

/-- 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

/-- 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

/-- 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

This theorem does not claim anything about the physical world; it is a statement about the framework's cost geometry. This theorem does not claim that the radical distribution is the only integrable structure in the framework. This theorem does not claim that the affine leaves are the same as the level sets of the cost function itself.

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.

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