Cost Ndim Xcoordinates

The x-coordinate Hessian

In Recognition Science, the cost (a penalty assigned to a recognition event) generalizes to many components. The module cost-ndim-xcoordinates studies the curvature of that multi-component cost in the positive x-coordinates. The central object is the Hessian matrix, the square table of second derivatives that describes how steeply the cost bends as each coordinate moves. The module derives a closed formula for the determinant of this matrix in the two-component case, and proves exactly when that determinant vanishes.

The Hessian entry for coordinates i and j is built from a positive aggregate R = α·x, the weighted sum of the coordinates. Off-diagonal entries are a product of the active directions αᵢ/xᵢ and αⱼ/xⱼ scaled by (R + R⁻¹)/2. Diagonal entries add a correction term. The determinant formula for the 2×2 case factors cleanly: it is proportional to (R − 1)(R + 1)(R²a + R²b − R² + a + b + 1), divided by 4R²x²y².

This factorization proves three structural facts. First, when the aggregate equals 1, the determinant is zero: the Hessian is degenerate on the neutral locus. Second, away from that locus and from the secondary discriminant factor, the Hessian is nondegenerate. Third, on the zero-cost locus the Hessian collapses to a rank-one outer product of the active direction with itself, so its curvature is concentrated along a single line.

The practical payoff: the neutral locus where cost is zero is also where the curvature degenerates, a fact that constrains how the multi-component cost can be minimized. This is a technical but load-bearing step in the framework's account of how recognition events organize into higher-dimensional structure.

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]

/-- 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

/-- 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]

What this page does not claim

The module does not claim that the x-coordinate Hessian is the only meaningful curvature object for the multi-component cost. The theorems do not establish that the neutral locus is the only degeneracy locus; the secondary discriminant factor also produces degeneracy. The module does not derive the multi-component cost function itself; it assumes the cost and analyzes its curvature.

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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