Cost Ndim Xcoordinates Det X Hessian Matrix2 Zero Cost

The zero-cost degeneracy

The second derivative of a function, its Hessian matrix, records how the slope changes in every direction. For the multi-component reciprocal cost, a function built from ratios of positive coordinates, the framework's library of formal theorems proves a precise statement about this matrix at a special locus. The declaration det_xHessianMatrix2_zero_cost establishes that when the aggregate R = α₁/x₁ + α₂/x₂ equals 1, the determinant of the 2×2 Hessian matrix is exactly zero.

This zero-cost degeneracy means the Hessian matrix is singular: it has a zero eigenvalue, so its rank drops from two to one. The theorem derives this from the general determinant formula for the 2×2 case, which factors as a product of terms involving (R − 1) and (R + 1). Setting R = 1 kills the first factor, leaving the determinant identically zero regardless of the other parameters. The proof requires only that both coordinates x and y be nonzero, a natural positivity condition for the cost function's domain.

The theorem does not claim that the Hessian is zero everywhere on the neutral locus, only that its determinant vanishes. The matrix itself remains nonzero: on the zero-cost locus, the Hessian collapses to the rank-one outer product of the active direction vector αᵢ/xᵢ with itself, as a companion theorem xHessianEntry_zero_cost shows. This distinction matters: rank one means the cost function is flat in one direction but still curved in another at the neutral point.

In the framework's account, this degeneracy marks the neutral locus as special: the point where recognition costs nothing is also where the cost function's second-order behavior becomes degenerate. Away from this locus, a separate theorem det_xHessianMatrix2_ne_zero_of_generic guarantees the Hessian is nondegenerate, provided a secondary discriminant factor also stays nonzero. Together these results chart where the cost function's geometry is regular and where it collapses.

What this declaration does not claim is any statement about the physical meaning of the degeneracy, nor does it assert that the neutral locus is unique or that the rank-one collapse extends to higher dimensions. The theorem is purely algebraic: a determinant computation for a 2×2 matrix under a specific condition. Its significance within the framework is that the zero-cost point carries a structural signature in the Hessian, a fact that later developments may build upon.

/-- The neutral locus `aggregate = 1` is a degeneracy locus in the `2 × 2`
model. -/
theorem det_xHessianMatrix2_zero_cost (a b x y : ℝ)
    (hx : x ≠ 0) (hy : y ≠ 0)
    (hR : aggregate (vec2 a b) (vec2 x y) = 1) :
    Matrix.det (xHessianMatrix2 a b x y) = 0 := by
  rw [det_xHessianMatrix2_formula a b x y hx hy]
  simp [hR]

theorem det_xHessianMatrix2_formula (a b x y : ℝ)
    (hx : x ≠ 0) (hy : y ≠ 0) :
    let R := aggregate (vec2 a b) (vec2 x y)
    Matrix.det (xHessianMatrix2 a b x y)
      = -(a * b * (R - 1) * (R + 1) * (R ^ 2 * a + R ^ 2 * b - R ^ 2 + a + b + 1))
          / (4 * R ^ 2 * x ^ 2 * y ^ 2) := by
  dsimp [xHessianMatrix2]
  simpa using det_xHessianMatrix2OfR_formula a b x y (aggregate (vec2 a b) (vec2 x y))
    hx hy (aggregate_pos (vec2 a b) (vec2 x y)).ne'

/-- The neutral locus `aggregate = 1` is a degeneracy locus in the `2 × 2`
model. -/
theorem det_xHessianMatrix2_zero_cost (a b x y : ℝ)
    (hx : x ≠ 0) (hy : y ≠ 0)
    (hR : aggregate (vec2 a b) (vec2 x y) = 1) :
    Matrix.det (xHessianMatrix2 a b x y) = 0 := by
  rw [det_xHessianMatrix2_formula a b x y hx hy]
  simp [hR]

/-- 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]

/-- Away from the neutral locus and the secondary discriminant factor, the
`2 × 2` `x`-coordinate Hessian is nondegenerate. -/
theorem det_xHessianMatrix2_ne_zero_of_generic (a b x y : ℝ)
    (hx : x ≠ 0) (hy : y ≠ 0)
    (ha : a ≠ 0) (hb : b ≠ 0)
    (hR1 : aggregate (vec2 a b) (vec2 x y) ≠ 1)
    (hdisc :
      (aggregate (vec2 a b) (vec2 x y)) ^ 2 * a
        + (aggregate (vec2 a b) (vec2 x y)) ^ 2 * b
        - (aggregate (vec2 a b) (vec2 x y)) ^ 2
        + a + b + 1 ≠ 0) :
    Matrix.det (xHessianMatrix2 a b x y) ≠ 0 := by
  let R := aggregate (vec2 a b) (vec2 x y)
  have hR : R ≠ 0 := (aggregate_pos (vec2 a b) (vec2 x y)).ne'
  have hRp1 : R + 1 ≠ 0 := by
    have hpos : 0 < R := by simp [R]
    linarith
  have hden : 4 * R ^ 2 * x ^ 2 * y ^ 2 ≠ 0 := by
    have hR2 : R ^ 2 ≠ 0 := pow_ne_zero 2 hR
    have hx2 : x ^ 2 ≠ 0 := pow_ne_zero 2 hx
    have hy2 : y ^ 2 ≠ 0 := pow_ne_zero 2 hy
    have h4R : 4 * R ^ 2 ≠ 0 := mul_ne_zero (by norm_num) hR2
    have h4Rx : 4 * R ^ 2 * x ^ 2 ≠ 0 := mul_ne_zero h4R hx2
    exact mul_ne_zero h4Rx hy2
  rw [det_xHessianMatrix2_formula a b x y hx hy]
  refine div_ne_zero ?_ hden
  refine neg_ne_zero.mpr ?_
  refine mul_ne_zero ?_ hdisc
  refine mul_ne_zero ?_ hRp1
  refine mul_ne_zero ?_ (sub_ne_zero.mpr hR1)
  exact mul_ne_zero ha hb

What this page does not claim

The theorem does not claim any physical interpretation of the degeneracy, only the algebraic fact of a vanishing determinant. It does not claim that the Hessian matrix itself is zero on the neutral locus, only that its determinant vanishes. It does not claim the degeneracy is unique to the 2×2 case or that it extends to higher dimensions.

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)

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: