IndisputableMonolith.Cost.Ndim.XCoordinates
Defines the active x-coordinate chart for the N-dimensional reciprocal cost: direction fields αᵢ/xᵢ, diagonal corrections, and the Hessian of the multi-component cost in those coordinates. Supplies 2D specializations and an explicit determinant formula for the 2×2 Hessian. Cited by anyone computing second-variation structure or curvature of the N-dim cost away from the scalar kernel. Pure definition and algebraic identity layer on top of Cost.Ndim.Core.
claimIn the $x$-chart of the multi-component reciprocal cost, the active direction is $\alpha_i/x_i$. The module assembles the Hessian entries $H_{ij}$ (diagonal and off-diagonal), the full Hessian matrix, its $2\times 2$ specialization, and the closed-form determinant $\det H^{(2)}(r)$ of that specialization.
background
The parent module Cost.Ndim.Core lifts the scalar reciprocal cost kernel to $N$ components by a weighted log aggregate. That construction yields a multi-variable cost whose first and second derivatives are most naturally written in reciprocal (or log-ratio) coordinates.
This module fixes the active $x$-chart: each coordinate direction is the ratio $\alpha_i/x_i$ rather than a raw Cartesian increment. From that choice one obtains diagonal correction terms, individual Hessian entries $H_{ij}$, and the assembled Hessian matrix. A two-dimensional specialization (vec2, $H^{(2)}$) is singled out because many Recognition calculations reduce to a pair of conjugate channels.
Notation follows the Core conventions: the scalar kernel is the usual $J$-cost (T5 uniqueness), and the multi-component lift inherits the Recognition Composition Law structure componentwise through the weighted log sum.
proof idea
Definition module. It introduces xDirection, the diagonal correction, Hessian entry formulas (on- and off-diagonal, including the zero-cost locus), matrix assemblies, and the 2D special case. Algebraic identities such as xHessianMatrix2_eq_general and det_xHessianMatrix2OfR_formula are direct expansions of those definitions; no deep lemmas beyond Core are required.
why it matters in Recognition Science
Second-variation and curvature statements for the N-dimensional reciprocal cost need an explicit Hessian in a concrete chart. The $x$-coordinate Hessian and its 2×2 determinant feed any later positivity, signature, or stability argument for the multi-component cost surface. In the broader Recognition chain this sits under the Cost domain that ultimately supports mass-ladder and coupling calculations built on the unique $J$-kernel (T5) and the self-similar fixed point $\phi$ (T6). No downstream consumers are wired yet in the graph; the module is infrastructure for those curvature lemmas.
scope and limits
- Does not prove positive-definiteness or signature of the Hessian.
- Does not derive the multi-component cost itself (that lives in Cost.Ndim.Core).
- Does not treat charts other than the active x-direction αᵢ/xᵢ.
- Does not connect the Hessian determinant to physical observables or mass formulas.
- 2D determinant formula is algebraic only; no spectral or eigenvalue analysis.
depends on (1)
declarations in this module (15)
-
def
xDirection -
def
xDiagonalCorrection -
def
xHessianEntry -
def
xHessianMatrix -
theorem
xHessianEntry_offDiag -
theorem
xHessianEntry_diag -
theorem
xHessianEntry_zero_cost -
abbrev
vec2 -
def
xHessianMatrix2OfR -
def
xHessianMatrix2 -
theorem
xHessianMatrix2_eq_general -
theorem
det_xHessianMatrix2OfR_formula -
theorem
det_xHessianMatrix2_formula -
theorem
det_xHessianMatrix2_zero_cost -
theorem
det_xHessianMatrix2_ne_zero_of_generic