IndisputableMonolith.Cost.Ndim.Metric
This module supplies the metric tensor entries obtained by taking the Hessian of the n-dimensional reciprocal cost function JlogN expressed in logarithmic coordinates. Researchers modeling the geometry of cost landscapes in Recognition Science would cite these definitions when assembling the full metric from second derivatives. The construction reduces the multidimensional Hessian to a form depending on a single weighted dot product, as prepared in the imported Hessian module.
claimThe metric entry derived from the Hessian of the cost $JlogN$ in log coordinates is $m_{ij} = Hess(JlogN)_{ij}$, where the n-dimensional cost depends only on the aggregate $dot α t$.
background
The module sits inside the Cost.Ndim hierarchy and imports the Hessian formulas for the n-dimensional reciprocal cost. In log-coordinates the n-dimensional cost depends only on the single weighted aggregate dot α t, so its Hessian reduces to a rank-1 update controlled by that scalar. Sibling definitions metricEntry, metricEntry_zero and metric_at_equilibrium_eq_hessian then package the resulting second-derivative entries for direct use in the metric tensor.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module completes the concrete metric construction for JlogN inside the n-dimensional cost framework, supplying the Hessian-derived entries that higher-level geometry and equilibrium statements rely on. It directly implements the reduction to the aggregate dot α t that the upstream Hessian module isolates, thereby closing the interface between the abstract cost function and its Riemannian structure.
scope and limits
- Does not compute explicit numerical values for the metric entries.
- Does not extend the construction beyond log coordinates.
- Does not address time-dependent or relativistic cost variants.
- Does not include curvature scalars or geodesic equations.