Cost Ndim Xcoordinates X Hessian Entry Off Diag

The off-diagonal entry

In the Recognition Science framework, a ledger, a discrete record of events, assigns a positive weight to each of several coordinates. The framework's cost function measures how far a given set of weights and coordinates sits from an ideal balance. The second derivative of that cost, its Hessian, describes how the cost curves as the coordinates move. The declaration xHessianEntry_offDiag isolates one piece of that curvature: the entry that links two distinct coordinates i and j, with i not equal to j.

The theorem proves a clean formula for that off-diagonal entry. Let R be the positive aggregate, the weighted sum of the coordinates. Let α_i and α_j be the weights, and let x_i and x_j be the coordinates. Then the off-diagonal Hessian entry equals ((R + R⁻¹) / 2) times (α_i / x_i) times (α_j / x_j). The diagonal correction term, which appears in the full Hessian, drops out entirely when i and j differ. The formula holds for any number of coordinates, not just two.

This result matters because it exposes the structure of the cost's curvature. On the zero-cost locus, where R equals 1, the factor (R + R⁻¹) / 2 becomes exactly 1. The off-diagonal entry then reduces to the product (α_i / x_i)(α_j / x_j), a rank-one outer product of the active direction with itself. That means the Hessian collapses to a single direction, and the cost is flat along every other direction. The theorem is a proved statement in the machine-checked library of formal theorems, with no unproved assumptions beyond the standard axioms of the underlying type theory.

What the declaration does not claim is just as important. It does not assert that the Hessian is nondegenerate, or that the cost has a unique minimum. Those claims require additional conditions, which the library states separately. The off-diagonal formula alone says nothing about the diagonal entries, which carry their own correction term. And the theorem does not identify which coordinates are physically meaningful; it only describes the mathematics of the cost's curvature for any positive weights and coordinates. The framework's physical interpretation of these coordinates remains a separate question, not settled by this declaration.

theorem xHessianEntry_offDiag {n : ℕ} (α x : Vec n) {i j : Fin n} (hij : i ≠ j) :
    xHessianEntry α x i j
      = ((aggregate α x + (aggregate α x)⁻¹) / 2) * xDirection α x i * xDirection α x j := by
  unfold xHessianEntry xDiagonalCorrection
  simp [hij]

/-- On the zero-cost locus `aggregate α x = 1`, the `x`-Hessian collapses to
the rank-one outer product of the active direction with itself. -/
theorem xHessianEntry_zero_cost {n : ℕ} (α x : Vec n) {i j : Fin n}
    (hR : aggregate α x = 1) :
    xHessianEntry α x i j = xDirection α x i * xDirection α x j := by
  unfold xHessianEntry xDirection xDiagonalCorrection
  rw [hR]
  by_cases hij : i = j
  · simp [hij]
  · simp [hij]

theorem xHessianEntry_offDiag {n : ℕ} (α x : Vec n) {i j : Fin n} (hij : i ≠ j) :
    xHessianEntry α x i j
      = ((aggregate α x + (aggregate α x)⁻¹) / 2) * xDirection α x i * xDirection α x j := by
  unfold xHessianEntry xDiagonalCorrection
  simp [hij]

What this page does not claim

The theorem does not prove the Hessian is nondegenerate or that the cost has a unique minimum. The declaration says nothing about the diagonal Hessian entries or their correction term. The formula does not identify which coordinates are physically meaningful in the framework.

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

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