IndisputableMonolith.Cost.Ndim.Hessian
The module defines log-coordinate gradient entries and Hessian structures for the N-dimensional reciprocal cost JlogN. Researchers extending scalar J-cost to vector settings for metrics and projectors cite these definitions. Content consists of direct lifts from the scalar kernel via weighted log aggregates in the imported Core module.
claimGradient entry $\partial_i J_{\log N}$ and Hessian matrix $H_{ij}$ of the multi-component reciprocal cost in log coordinates.
background
The Core module defines N-dimensional reciprocal cost by lifting the scalar J-kernel through a weighted log aggregate. This Hessian module specializes to log-coordinate derivatives of that aggregate. The underlying J satisfies the Recognition Composition Law $J(xy)+J(x/y)=2J(x)J(y)+2J(x)+2J(y)$ and appears in the forcing chain as the uniqueness map T5.
proof idea
this is a definition module, no proofs
why it matters in Recognition Science
Feeds the log-coordinate cost metric, the cost-induced projector algebra with its quadratic law $A^2=\mu A$, and the radical distribution on the hyperplane orthogonal to the active direction. Supports the rank-one tensor picture in the Recognition framework and the transition from scalar to N-dimensional cost structures.
scope and limits
- Does not compute eigenvalues or signature of the Hessian.
- Does not prove positive semi-definiteness or convexity.
- Does not connect entries to the phi-ladder or mass formula.
- Does not address integrability of the radical distribution.
- Does not incorporate the eight-tick octave or D=3 forcing.
used by (3)
depends on (1)
declarations in this module (14)
-
def
gradientEntry -
def
hessianEntry -
def
hessianMatrix -
def
hessianAt -
def
applyTensor -
def
applyHessian -
def
quadraticHessian -
theorem
hessianEntry_zero -
theorem
hessianAt_zero -
theorem
hessianAt_factor -
theorem
applyHessian_eq_direction -
theorem
applyHessian_of_dot_zero -
theorem
quadraticHessian_eq -
theorem
quadraticHessian_nonneg