Cost Ndim Radical Distribution Preserves Own Leaf Iff Mem Radical

The radical distribution

In the Recognition Science framework, a cost function's geometry is studied through its Hessian, the matrix of second derivatives. For the rank-one log-coordinate metric, this Hessian is degenerate: it only detects movement along a single active direction, denoted α. The radical, a term for the set of directions along which the Hessian's quadratic form vanishes, is therefore the constant hyperplane of vectors v satisfying the dot product α·v = 0. This is the formal definition given by the declaration Radical.

The framework then considers affine leaves, which are hyperplanes of the form { t | α·t = c } for a constant c. These leaves are the level sets of the active direction. The key theorem, preserves_own_leaf_iff_mem_Radical, states a precise equivalence: a constant direction v preserves the affine leaf through a point t, meaning that shifting t along v by any amount s keeps the result in the same leaf, if and only if v lies in the radical. In symbols, (∀ s, affineShift t v s ∈ LevelSet α (dot α t)) ↔ v ∈ Radical α. This is a proved theorem in the framework's machine-checked library of formal theorems.

The theorem's content is geometric and integrable. The radical distribution is integrable, meaning its integral leaves are exactly these affine hyperplanes. The theorem radical_integrable_by_affine_leaves formalizes this: starting from a point in a leaf and moving along a radical direction keeps you in the leaf. This establishes that the degenerate directions of the Hessian are precisely the directions that foliate the space into these flat, parallel hyperplanes. The result is a structural fact about the geometry, not a claim about any specific physical system.

What the theorem does not claim is broader significance. It does not assert that the radical distribution is unique or that it arises from any physical principle. It is a statement about a specific metric, the rank-one log-coordinate Hessian, and its algebraic consequences. The theorem does not say that all cost functions have this radical structure, nor does it imply anything about the values of physical constants. It is a local, formal result about the geometry of a particular metric within the framework.

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

/-- 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

/-- 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 claim that the radical distribution is unique or arises from a physical principle. It does not assert that all cost functions have this radical structure. It does not imply any specific values for physical constants.

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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: