Foundation Physics Logic Realization Tick Step

The tick step

A tick is a single, discrete forward step in a sequence. In the Recognition Science framework, physical states are arranged in such a sequence, and the tickStep function is the formal definition of that step: it takes one state and returns the next one by incrementing the state's tick counter. The definition is a model, a chosen way to represent the idea of a step, not a theorem about the world.

The framework's machine-checked library of formal theorems proves two small facts about this setup. One, a state is never equal to its own successor, so the sequence never collapses into a loop of length one. Two, the mapping from natural numbers to states is injective, meaning two different numbers always produce two different states. Together these establish that the tickStep generates an unending, non-repeating chain of states, a faithful copy of the natural numbers.

The same library also defines a cost for moving between states: the cost is zero if the states are identical and one otherwise. This is the simplest possible cost, a choice that models a minimal recognition event. The framework proves this cost is symmetric and that moving from a state to itself costs nothing.

In Recognition Science, this declaration is a lightweight hook, a stable interface that connects the abstract machinery of universal forcing to a concrete, minimal picture of physical states. It does not claim that a tick corresponds to any specific unit of physical time, nor that the real universe is discrete in this way. It establishes a consistent, minimal arithmetic skeleton on which further physics could be built.

/-- Identity-tick successor. -/
def tickStep (x : PhysicsState) : PhysicsState :=
  ⟨ArithmeticFromLogic.LogicNat.succ x.tick⟩

/-- 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

/-- 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

/-- 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 does not claim that a tick corresponds to any specific unit of physical time, such as a Planck time. This does not claim that the real universe is discrete or that time advances in these steps. This does not claim that the equality cost function is the unique or forced cost for physical interactions.

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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