Cost Ndim Neutrality Zero Cost Iff Dot Zero

The zero-cost condition

In the Recognition Science framework, a ledger (a discrete record of recognition events) assigns a cost to each possible state. The central question is when that cost is zero. The framework's machine-checked library of formal theorems proves a clean answer: the cost is zero exactly when a particular weighted sum of logarithms is zero. The statement is theorem zero_cost_iff_dot_zero in the file IndisputableMonolith/Cost/Ndim/Neutrality.lean.

To see what this means, suppose the ledger has n entries, each with a weight α_i and a value x_i. The theorem says JcostN α x = 0 if and only if the dot product of α with the vector of logarithms of x equals 0. In symbols: JcostN α x = 0 ↔ ∑ α_i log(x_i) = 0. This is a direct consequence of the definition of the cost function and the injectivity of the exponential function. The proof is short: if the cost is zero, then the exponential of the dot product equals the exponential of zero, so the dot product itself must be zero; conversely, if the dot product is zero, the cost collapses to zero by definition.

The theorem also connects to a second formulation. A companion result, zero_cost_iff_aggregate_one, shows that zero cost is equivalent to the aggregate (a weighted product of the x_i) equaling one. Both statements are proved in the same file and are logically equivalent to each other through the first theorem. The practical upshot is that the framework gives a precise, checkable condition for when a recognition event costs nothing: the weighted log sum must vanish, or equivalently the aggregate must be exactly one.

What the theorem does not claim is equally important. It does not say that a zero cost is achievable for any particular ledger or that the weights α are derived from anything. It is a purely algebraic equivalence, a definitional consequence of how the framework defines cost. It does not assert that the cost function J is the unique one satisfying the five axioms of the framework; that uniqueness is a separate theorem. The theorem also does not say anything about what the entries x_i represent physically, only that the algebraic condition holds.

/-- Zero-cost iff weighted log sum vanishes. -/
theorem zero_cost_iff_dot_zero {n : ℕ} (α x : Vec n) :
    JcostN α x = 0 ↔ dot α (logVec x) = 0 :=
  JcostN_eq_zero_iff α x

/-- 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)

What this page does not claim

The theorem does not prove that zero cost is achievable for any particular ledger. It does not derive the weights α from any physical principle. It does not establish the uniqueness of the cost function J.

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)

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Derived articles

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