Cost Ndim Metric

The cost metric in log coordinates

The cost ndim metric is the Hessian-derived metric entry for the recognition cost function in log coordinates. It is defined as the second derivative of the cost with respect to the log-coordinate variables, evaluated at a given point. This makes it a local geometric object: it describes how the cost curves in each direction of the log-coordinate space, and it is the natural metric for measuring distances and angles in that space.

In Recognition Science, the cost function J is not chosen freely; it is forced by the five plain conditions of the framework. The metric inherits that forcing. The module establishes a precise fact about the metric at equilibrium, the point where all log-coordinate variables are zero. At that point, the metric entry for any two directions i and j is exactly the product of the corresponding cost parameters α_i and α_j. This is the outer-product form: the metric is the outer product of the parameter vector with itself.

The theorem metric_at_equilibrium_eq_hessian states this coincidence directly: at equilibrium, the metric equals the Hessian matrix of the cost function. This is not an approximation and not a modeling choice; it is a machine-checked identity. The consequence is that the local geometry of the cost function at its equilibrium point is fully determined by the parameters α, with no additional structure needed. The metric and the Hessian are the same object there.

The practical meaning is that near equilibrium, the cost surface is quadratic with a fixed curvature matrix, and that matrix is the outer product of the parameter vector. This gives a clean local picture of the recognition recognition cost: a single vector α controls the entire second-order geometry at the equilibrium point. The result is a theorem in a formal proof assistant, axiom-clean, and it is a building block for later claims about the structure of the recognition space.

MODEL metricEntry · IndisputableMonolith/Cost/Ndim/Metric.lean

THEOREM metricEntry_zero · IndisputableMonolith/Cost/Ndim/Metric.lean

THEOREM metric_at_equilibrium_eq_hessian · IndisputableMonolith/Cost/Ndim/Metric.lean

What this page does not claim

This answer does not claim that the metric is defined away from equilibrium in the same outer-product form. This answer does not claim that the parameters α are derived from the forcing chain; they are inputs to the metric definition. This answer does not claim any empirical or physical interpretation of the metric beyond the mathematical identity that is machine-checked.

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/Metric.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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