Cost Ndim Xcoordinates Det X Hessian Matrix2 Of R Formula

The two-component cost curvature

The cost function in Recognition Science measures the price of a recognition event. In the multi-component setting, the cost depends on positive coordinates recognition (a discrete record of events) through an aggregate value R. The second derivative of this cost, its Hessian matrix, describes how the cost curves as the coordinates move. For two components, this matrix is a 2 by 2 table of numbers, and its determinant summarizes the curvature in one number.

The declaration det_xHessianMatrix2OfR_formula is a proved theorem in the framework's machine-checked library of formal theorems. It gives a closed formula for that determinant in terms of the two coordinates x and y, their weights a and b, and the aggregate R. The formula factors into three pieces: a factor (R - 1)(R + 1), a secondary factor involving the weights, and a denominator of 4 R² x² y². This factorization is the key structural fact: the determinant vanishes exactly when R equals 1, or when the secondary factor vanishes.

The case R = 1 is the zero-cost locus, where the aggregate equals one. Here the theorem shows the Hessian collapses to a rank-one outer product, meaning the cost is flat in one direction. The secondary factor, R²(a + b - 1) + a + b + 1, is a separate degeneracy condition. Away from both of these loci, and with nonzero coordinates and weights, the determinant is nonzero, so the Hessian is nondegenerate and the cost curves in all directions.

In Recognition Science, this result is a step in understanding the geometry of the cost function. It does not claim that the secondary factor has a special physical meaning, nor that the determinant formula extends to more than two components. The formula is a precise, proved statement about the two-component case, and its value is in showing exactly where the cost flattens and where it does not.

theorem det_xHessianMatrix2OfR_formula (a b x y R : ℝ)
    (hx : x ≠ 0) (hy : y ≠ 0) (hR : R ≠ 0) :
    Matrix.det (xHessianMatrix2OfR a b x y R)
      = -(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
  simp [xHessianMatrix2OfR, Matrix.det_fin_two]
  field_simp [hx, hy, hR]
  ring

theorem det_xHessianMatrix2OfR_formula (a b x y R : ℝ)
    (hx : x ≠ 0) (hy : y ≠ 0) (hR : R ≠ 0) :
    Matrix.det (xHessianMatrix2OfR a b x y R)
      = -(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
  simp [xHessianMatrix2OfR, Matrix.det_fin_two]
  field_simp [hx, hy, hR]
  ring

/-- 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

What this page does not claim

The secondary factor has a physical interpretation. The determinant formula applies to more than two components. The theorem proves the cost function itself is unique; that is a separate result.

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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