Foundation Physics Logic Realization Physics Cost

The minimal cost

The declaration physicsCost is a definition, not a discovery. It takes two recognition states, which are discrete records of where a process has been, and returns a natural number: 0 if the two states are identical, 1 if they differ. That is the whole function. It is the cheapest possible way to tell two things apart, a binary ledger with no gradations.

The machine-checked library of formal theorems proves two properties of this definition. First, the cost of comparing a state with itself is zero. Second, the cost is symmetric: comparing x to y costs the same as comparing y to x. These are not assumptions; they are consequences of the definition, verified by the library's checker. The definition also sits inside a larger structure that interprets arithmetic as a sequence of states, each one a successor of the last, and the library proves this interpretation is faithful: distinct numbers map to distinct states, and no state is its own successor.

In Recognition Science, this function models the minimal realization of physical tick arithmetic. The framework's larger forcing chain derives constants and dimensions from a cost function with five conditions; this definition is a lightweight hook that gives a stable interface to that chain. It is a choice of how to start, not a theorem about what must be. The equality cost is the simplest possible starting point, and the framework uses it as a placeholder for the full physics forcing chain, which is large and currently imports modules with unrelated build fragility.

The definition does not claim that physics actually works this way. It does not derive the fine-structure constant, the golden ratio, or three spatial dimensions; those results, where they exist, live elsewhere in the library and depend on the full cost function, not on this equality cost. It does not say that recognition states are physical objects. It says only: if you want a minimal, symmetric, machine-checked notion of cost between discrete states, here is one, and here is what it proves about itself.

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

@[simp] theorem physicsCost_self (x : PhysicsState) : physicsCost x x = 0 := by
  simp [physicsCost]

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']

/-- Physics tick interpretation is faithful. -/
theorem physics_faithful :
    LogicRealization.FaithfulArithmeticInterpretation physicsRealization where
  injective := by
    intro a b h
    cases h
    rfl
  zero_step_noncollapse := by
    intro n h
    have htick := congrArg PhysicsState.tick h
    exact ArithmeticFromLogic.LogicNat.zero_ne_succ n htick

What this page does not claim

This definition does not derive the fine-structure constant, the golden ratio, or three spatial dimensions. It does not claim that recognition states are physical objects. It does not say that physics actually works this way.

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

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