Foundation Physics Logic Realization Physics State

The physics state skeleton

In the Recognition Science framework, recognition (a discrete record of events) is meant to ground physical structure. The declaration PhysicsState is the smallest such grounding: a structure holding one natural number, called a tick, which represents a step in an identity sequence. It is a skeleton, not a full physics theory. The structure is indexed by the framework's own arithmetic type, LogicNat, and it supports decidable equality, meaning any two states can be checked for sameness.

The cost function on these states is deliberately trivial. physicsCost x y returns 0 if the two states are identical and 1 otherwise. This is the minimal possible cost: it only distinguishes sameness from difference, with no graded distance. Two theorems in the machine-checked library prove the basic properties: the cost of a state with itself is zero, and the cost is symmetric, so physicsCost x y = physicsCost y x. A successor function tickStep advances a state by one, and an interpretation maps each natural number to the corresponding state.

The library also proves a faithfulness theorem: the mapping from the arithmetic numbers to physics states is injective, and no zero step collapses to a successor. This means the arithmetic structure is preserved exactly under the interpretation. A further definition shows that the arithmetic extracted from this realization is equivalent to that of any other realization, establishing a kind of invariance.

In Recognition Science, this declaration establishes the stable interface for a physics realization: identity ticks as the step action, recognition states as the carrier, and equality cost as the minimal realization of physical tick arithmetic. It does not claim to derive any specific physical law, constant, or dimensional structure. It is a lightweight hook, explicitly described in the source as a skeleton, not the full forcing chain. The framework's larger claims about dimensions or constants rely on other modules, not on this one.

@[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 declaration does not derive any specific physical law, constant, or dimensional structure. The equality cost here is not the forced J(x) cost function from the main forcing chain. The faithfulness theorem does not establish that physics reduces to arithmetic in any empirical sense.

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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