Cost Ndim Neutrality Zero Cost Iff Aggregate One

The neutrality surface

The Recognition Science framework models recognition through a ledger, a discrete record of events, and assigns each event a cost, a number measuring how expensive it is to recognize. In the n-dimensional version, each event has several components, each weighted by importance. The declaration zero_cost_iff_aggregate_one proves a clean equivalence: the total cost is zero exactly when a certain weighted aggregate equals one.

To see what that means, define the aggregate as exp of the weighted sum of logarithms of the components. The theorem states, for any weights and any positive components, that this aggregate equals one if and only if the weighted log sum is zero. Since the cost function is built so that zero cost also corresponds to that same zero weighted log sum, the two conditions collapse into one. In plain terms: a recognition event costs nothing precisely when its weighted geometric mean is one.

This is a theorem in the framework's machine-checked library of formal theorems, proved for any finite number of dimensions. It does not claim that the aggregate being one is the only way to get zero cost, nor does it say anything about what the weights should be. It also does not assert that zero cost is desirable; it only identifies the condition under which it occurs.

/-- Zero-cost iff aggregate equals one. -/
theorem zero_cost_iff_aggregate_one {n : ℕ} (α x : Vec n) :
    JcostN α x = 0 ↔ aggregate α x = 1 := by
  constructor
  · intro h
    exact (aggregate_eq_one_iff α x).2 ((zero_cost_iff_dot_zero α x).1 h)
  · intro h
    exact (zero_cost_iff_dot_zero α x).2 ((aggregate_eq_one_iff α x).1 h)

/-- Aggregate equals one exactly when the weighted log sum is zero. -/
theorem aggregate_eq_one_iff {n : ℕ} (α x : Vec n) :
    aggregate α x = 1 ↔ dot α (logVec x) = 0 := by
  unfold aggregate
  constructor
  · intro h
    have : Real.exp (dot α (logVec x)) = Real.exp 0 := by simpa using h
    exact Real.exp_injective this
  · intro h
    simp [h]

What this page does not claim

The theorem does not specify what the weights should be. It does not claim that zero cost is always preferable. It does not assert that the aggregate being one is the only route to zero cost.

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/Neutrality.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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: