Cost Ndim Radical Distribution Quadratic Hessian Eq Zero Iff

The flat directions

In the Recognition Science framework, a cost function measures how expensive it is for reality to recognize one state from another. The framework builds these costs in coordinates where the cost of comparing two states depends on their ratio. In many dimensions, the local shape of this cost is described by a Hessian, a matrix of second derivatives. The theorem quadraticHessian_eq_zero_iff pins down the directions in which this Hessian is flat: it vanishes exactly along a hyperplane perpendicular to a single active direction vector α.

Concretely, the theorem states that for any point t and any direction v in n-dimensional space, the Hessian quadratic form evaluated at v equals zero if and only if the dot product of α and v is zero. The set of such v is called the radical distribution, a linear subspace of directions along which the cost does not curve at all. The proof is a direct calculation: it unfolds the definitions of the Hessian and the dot product, then uses the fact that a sum of products equals zero exactly when the vector is orthogonal to α.

The framework's library also proves that this distribution is integrable, meaning the flat directions can be assembled into smooth surfaces. These surfaces are the affine hyperplanes defined by the equation dot α t = c, for any constant c. The theorem affineShift_mem_LevelSet shows that moving along a radical direction keeps you on the same hyperplane, and preserves_own_leaf_iff_mem_Radical proves the converse: a direction preserves its own hyperplane exactly when it lies in the radical.

What this does not claim is broader. The theorem is purely local and algebraic: it describes the null directions of one Hessian at a point. It does not assert that the cost function is globally flat, nor that these hyperplanes are geodesics, nor that the active direction α is unique across the whole space. It also does not claim that the radical distribution has any physical meaning beyond being the set of directions where the local quadratic approximation stops changing. The framework's larger claims about three spatial dimensions or particle masses do not follow from this single theorem; they depend on the full forcing chain.

/-- The Hessian quadratic form vanishes exactly on the radical distribution. -/
theorem quadraticHessian_eq_zero_iff {n : ℕ} (α t v : Vec n) :
    quadraticHessian α t v = 0 ↔ v ∈ Radical α := by
  rw [quadraticHessian_eq]
  constructor
  · intro hq
    unfold Radical
    have hcosh : 0 < Real.cosh (dot α t) := by positivity
    have hsq : (dot α v) ^ 2 = 0 := by
      exact (mul_eq_zero.mp (by simpa using hq)).resolve_left hcosh.ne'
    have hdot : dot α v = 0 := sq_eq_zero_iff.mp hsq
    exact hdot
  · intro hv
    rw [mem_Radical_iff] at hv
    simp [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

/-- 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

The theorem does not claim the radical distribution is geodesically complete or globally flat. It does not claim the active direction α is unique or constant across the entire space. It does not claim that the flat directions have direct physical significance beyond the local quadratic approximation.

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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