Foundation Physics Logic Realization Physics Cost Symm

The symmetry of cost

In the Recognition Science framework, a ledger is a discrete record of events, and each event is a recognition, an act of matching one state against another. The cost of that act is forced, not chosen, but the framework also builds a minimal working model: a state is just a natural number tick, and the cost between two states is 0 if they are identical and 1 otherwise. This is the equality cost, the simplest possible way to say that recognizing something costs nothing when it is already the same thing.

The declaration physicsCost_symm proves, as a theorem in the machine-checked library of formal theorems, that this cost is symmetric: the cost from state x to state y equals the cost from y to x. In plain terms, the ledger does not care which way you read the comparison. Recognizing a tick as itself costs nothing in either direction, and recognizing two different ticks costs one unit regardless of order. The proof is a short case split: either the two states are equal or they are not, and both branches give the same result.

This symmetry is a sanity property, not a deep discovery. It does not say that all recognition costs in the framework are symmetric, and it does not derive the full cost function J(x) = (x + 1/x)/2 - 1 that the framework's central theorem forces. That uniqueness result lives elsewhere, under stronger conditions. The lemma here only checks that the minimal equality-cost model behaves as a ledger should: direction does not change the price.

What the declaration establishes is a small but load-bearing consistency check. It confirms that the simplest physics realization, one where states are just ticks and cost is equality, satisfies a basic requirement that any sensible cost function must meet. Without this, the framework's physical interpretation would fail at the first step. With it, the minimal model is a coherent base on which the larger forcing chain can build.

theorem physicsCost_symm (x y : PhysicsState) : physicsCost x y = physicsCost y x := by
  by_cases h : x = y
  · subst h
    simp [physicsCost]
  · have h' : y ≠ x := by intro hyx; exact h hyx.symm
    simp [physicsCost, h, h']

/-- Equality cost on physics states. -/
def physicsCost (x y : PhysicsState) : Nat :=
  if x = y then 0 else 1

What this page does not claim

This lemma does not prove that all recognition costs in the framework are symmetric. This lemma does not derive the full forced cost function J(x). This lemma does not establish any physical law beyond the minimal equality-cost model.

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:

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

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