Foundation Physics Logic Realization Physics Realization

The minimal physics hook

In Recognition Science, a framework that derives structure from the cost of recognizing events, the declaration physicsRealization is a deliberately small bridge. It connects the framework's abstract arithmetic to a minimal notion of physical state. The construction builds a ledger, a discrete record of events, whose entries are natural numbers, and defines a state as one such entry. The step from one state to the next is simply counting upward: state n moves to state n+1. The cost of recognizing two states as the same is zero, and the cost of recognizing them as different is one.

This cost function obeys two basic properties that any reasonable measure of sameness should have. Recognizing a state against itself costs nothing, and the cost is symmetric: comparing x to y costs the same as comparing y to x. The construction also proves a faithfulness property. The mapping from arithmetic to states is injective, meaning distinct numbers land on distinct states, and no state is its own successor. These are not physical claims about force, mass, or geometry. They are structural guarantees that the arithmetic behaves as a clean, non-collapsing state space.

The larger forcing chain of Recognition Science, which derives constants and dimensions, is not rebuilt here. This construction exists to give a stable interface: identity ticks as the step action, recognition states as the carrier, and equality cost as the minimal realization of physical tick arithmetic. It is a hook, not a theory of physics. The declaration shows that the framework's arithmetic can be interpreted as a physics state space in a faithful way, and it proves the minimal properties that interpretation needs. What it does not do is assert that this particular state space is the one the real world uses. That identification, if it is ever made, would be a separate and much larger step.

@[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 claim that the natural-number state space is the actual state space of the physical world. It does not derive any physical constant, dimension, or force law. It does not assert that the full forcing chain is built on this particular realization.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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