Cost Ndim Hessian Apply Hessian Eq Direction

The single active direction

In the Recognition Science framework, the cost of a recognition event is not a free choice but a forced function. The framework's library, a machine-checked collection of formal theorems, works with this cost in log-coordinates, where the n-dimensional version depends on a single weighted sum of its inputs. The theorem applyHessian_eq_direction states that the Hessian, the matrix of second derivatives that describes local curvature, always maps any vector to a multiple of the special weight vector α. In plain terms, the curvature of the cost function acts only along one active direction, and it is silent in every perpendicular direction.

This is a strong structural result. The Hessian at any point is not a general symmetric matrix; it is a rank-one object, built from the outer product of α with itself. The theorem proves that applying this Hessian to a vector v yields cosh(dot α t) times α times the dot product of α and v. The consequence is that the quadratic form, which measures how the cost bends in a given direction, depends only on the single number dot α v. Vectors orthogonal to α lie in the kernel of the Hessian, meaning the cost is flat along those directions, and the quadratic form is always nonnegative, so the cost is locally convex.

The theorem is proved in the machine-checked library, so its status is THEOREM. It derives this rank-one structure from the definitions of the log-coordinate cost, its gradient, and its Hessian. The proof is a direct computation using the hyperbolic cosine and the properties of the dot product. The declaration does not claim that the weight vector α is derived from anything else, nor does it claim that this rank-one structure is unique to this cost function. It also does not claim anything about the physical meaning of α or the vectors it acts on; those are modeling choices, not theorems.

What this fact changes is the picture of the cost surface. Instead of a surface that bends in many directions, the framework's cost surface has exactly one direction of curvature at any point. All other directions are flat, contributing nothing to the local bending. This is a precise, proved statement about the geometry of the cost function, and it is the kind of structural fact that the framework uses to build further results, such as the forcing of three spatial dimensions.

/-- The Hessian action is always parallel to `α`. -/
theorem applyHessian_eq_direction {n : ℕ} (α t v : Vec n) :
    applyHessian α t v = fun i => Real.cosh (dot α t) * α i * dot α v := by
  funext i
  unfold applyHessian applyTensor hessianAt hessianEntry dot
  calc
    ∑ j : Fin n, (α i * α j * Real.cosh (dot α t)) * v j
        = ∑ j : Fin n, (α i * Real.cosh (dot α t)) * (α j * v j) := by
            apply Finset.sum_congr rfl
            intro j hj
            ring
    _ = (α i * Real.cosh (dot α t)) * ∑ j : Fin n, α j * v j := by
          rw [Finset.mul_sum]
    _ = Real.cosh (dot α t) * α i * dot α v := by
          simp [dot, mul_comm, mul_assoc]

/-- The full Hessian is a scalar multiple of the equilibrium outer-product model. -/
theorem hessianAt_factor {n : ℕ} (α t : Vec n) :
    hessianAt α t = fun i j => Real.cosh (dot α t) * hessianMatrix α i j := by
  funext i j
  unfold hessianAt hessianEntry hessianMatrix
  ring

/-- Vectors orthogonal to `α` lie in the kernel of the Hessian. -/
theorem applyHessian_of_dot_zero {n : ℕ} (α t v : Vec n)
    (hv : dot α v = 0) :
    applyHessian α t v = 0 := by
  funext i
  simp [applyHessian_eq_direction, hv]

theorem quadraticHessian_nonneg {n : ℕ} (α t v : Vec n) :
    0 ≤ quadraticHessian α t v := by
  rw [quadraticHessian_eq]
  positivity

What this page does not claim

The theorem does not derive the weight vector α from any more basic principle. The theorem does not claim the rank-one structure is unique to this particular cost function. The theorem does not assign any physical meaning to the vectors it acts on.

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/Hessian.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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