Cost Ndim Metric Metric Entry

The metric entry

In mathematics, a metric is a way of measuring distances and angles on a space. The declaration metricEntry defines one such metric for the Recognition Science cost function, but only in a special coordinate system. The cost function itself, written J(x) = (x + 1/x)/2 - 1, measures the price of a recognition event: a discrete record of something happening. The metric entry takes the second derivative of the cost in logarithmic coordinates, which is a standard way to turn a function into a geometric shape. The declaration is a definition, not a theorem; it chooses what the metric will be.

What the definition establishes is a precise formula. For any number of dimensions n, with parameters α and a point t, the metric entry at coordinates i and j is exactly the Hessian entry, the second derivative of the cost. At the special point where all coordinates are zero, the equilibrium point, the metric simplifies to the product α_i times α_j. This means the metric at equilibrium coincides with the outer-product Hessian model, a fact proved in the library as a theorem. In plain terms, at the resting point of the system, the geometry is completely determined by the parameters α alone, independent of the position t.

In Recognition Science, this metric entry is part of a larger effort to understand the cost function geometrically. The framework models recognition as a forced cost, and this definition gives a way to measure how that cost changes in different directions. But the declaration does not claim that this metric is unique, nor that it has any special properties like being flat or having constant curvature. It does not claim that the metric applies outside logarithmic coordinates, nor that it describes physical spacetime. Those would be separate claims requiring separate proofs.

The practical consequence is that the framework now has a well-defined geometric object attached to its cost function, at least in the logarithmic coordinate system. This allows future work to ask geometric questions: Is the metric curved? What are its geodesics? Does it have symmetries? The definition is a tool, not an answer. It opens a door for further investigation rather than closing one.

/-- Hessian-derived metric entry for `JlogN` in log coordinates. -/
noncomputable def metricEntry {n : ℕ} (α t : Vec n) (i j : Fin n) : ℝ :=
  hessianEntry α t i j

@[simp] theorem metricEntry_zero {n : ℕ} (α : Vec n) (i j : Fin n) :
    metricEntry α (fun _ => 0) i j = α i * α j := by
  have hdot : dot α (fun _ => 0) = 0 := by
    unfold dot
    simp
  simp [metricEntry, hessianEntry, hdot]

/-- The metric at equilibrium coincides with the outer-product Hessian model. -/
theorem metric_at_equilibrium_eq_hessian {n : ℕ} (α : Vec n) :
    metricEntry α (fun _ => 0) = hessianMatrix α := by
  funext i j
  simp [hessianMatrix]

What this page does not claim

The metric entry is not claimed to be unique or to have special geometric properties like flatness. The metric entry does not describe physical spacetime; it is defined only in logarithmic coordinates for the cost function. The metric entry does not determine the fine-structure constant or any other physical constant by itself.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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