Explanation of metricEntry_zero

(1) Plain English

The theorem metricEntry_zero states that the metric entry at the zero vector equals the product of the corresponding components of α. In other words, when the second argument to metricEntry is the constant-zero function, the result simplifies directly to α i * α j.

(2) Importance in Recognition Science

This lemma shows that the Hessian-derived cost metric in log coordinates reduces to an outer-product form at equilibrium. It supports the structural claim that the minimal-cost configuration is governed by the α weights, consistent with the J-cost symmetry and recognition ledger in the broader framework.

(3) Reading the formal statement

The declaration is:

@[simp] theorem metricEntry_zero {n : ℕ} (α : Vec n) (i j : Fin n) : metricEntry α (fun _ => 0) i j = α i * α j

• Implicit dimension n and vector α.
• Indices i, j of type Fin n.
• The zero vector is written as the anonymous function (fun _ => 0).
• Proof: unfold dot to obtain 0, then simp on metricEntry and hessianEntry.

(4) Visible dependencies

The theorem is defined in module IndisputableMonolith.Cost.Ndim.Metric and depends on the local definition metricEntry. It is accompanied by the related result metric_at_equilibrium_eq_hessian asserting that the equilibrium metric equals the Hessian matrix. The module imports IndisputableMonolith.Cost.Ndim.Hessian but supplies no further declarations from it.

(5) What it does not prove

The theorem supplies only the zero-vector case; it does not establish symmetry, positive-definiteness, or other metric axioms, does not derive physical constants or the forcing chain, and does not address applications outside the cost-metric slice.