Foundation Modular Logic Realization Modulus Pos

The modulus and its proof

The declaration modulus_pos establishes a simple fact about a counting number used to build a finite, repeating carrier for logic. In the framework's library, a modulus is the size of a finite set that cycles, like the hours on a clock. The theorem states that for any natural number k, the modulus defined as k + 2 is greater than zero. This is a formal, machine-checked proof that the carrier is never empty, so it always has at least one element to hold a value.

The proof is a direct calculation. The definition sets modulus k = k + 2, and since k is a natural number (zero or more), adding 2 always yields a number at least 2. The theorem then follows by the standard arithmetic fact that any number greater than or equal to 2 is positive. The library proves this with a routine arithmetic solver, not with any special framework axiom. A companion theorem, one_lt_modulus, shows the modulus is actually greater than 1, meaning the carrier has at least two distinct elements, enough for a genuine cycle.

The point of this construction is to show that Universal Forcing, the framework's core mechanism, does not require every realization to embed arithmetic faithfully into its carrier. The modulus creates a finite, periodic interpretation: the free orbit of natural numbers is mapped onto a clock face of size k + 2, wrapping around after each full cycle. This demonstrates that the framework can operate with a periodic carrier, not just an unbounded one, while the internal logic remains free.

What modulus_pos does not claim is just as important. It does not assert that the periodic carrier itself contains arithmetic truth, only that the carrier is nonempty. The theorem says nothing about the values held in the carrier, only that the carrier exists. It also does not claim that the modular realization is the only possible one, or that it is preferred over others. It is one example among many, showing a design choice, not a necessity. The theorem is a small but load-bearing part of a larger demonstration about the flexibility of the framework's realizations.

theorem modulus_pos (k : ℕ) : 0 < modulus k := by
  unfold modulus
  omega

theorem one_lt_modulus (k : ℕ) : 1 < modulus k := by
  unfold modulus
  omega

/-- Finite cyclic Law-of-Logic realization with periodic interpretation. -/
def modularRealization (k : ℕ) : LogicRealization where
  Carrier := Fin (modulus k)
  Cost := Nat
  zeroCost := inferInstance
  compare := finCost
  zero := ⟨0, modulus_pos k⟩
  step := cycStep k
  Orbit := ArithmeticFromLogic.LogicNat
  orbitZero := ArithmeticFromLogic.LogicNat.zero
  orbitStep := ArithmeticFromLogic.LogicNat.succ
  interpret := modularInterpret k
  interpret_zero := modularInterpret_zero k
  interpret_step := modularInterpret_step k
  orbit_no_confusion := by
    intro n h
    exact ArithmeticFromLogic.LogicNat.zero_ne_succ n h
  orbit_step_injective := ArithmeticFromLogic.LogicNat.succ_injective
  orbit_induction := by
    intro P h0 hs n
    exact ArithmeticFromLogic.LogicNat.induction (motive := P) h0 hs n
  orbitEquivLogicNat := Equiv.refl ArithmeticFromLogic.LogicNat
  orbitEquiv_zero := rfl
  orbitEquiv_step := by intro n; rfl
  identity := finCost_self
  nonContradiction := finCost_symm
  excludedMiddle := True
  composition := True
  actionInvariant := True
  nontrivial := by
    refine ⟨⟨1, one_lt_modulus k⟩, ?_⟩
    simp [finCost]

What this page does not claim

The theorem does not claim the periodic carrier contains arithmetic truth. The theorem does not claim the modular realization is the only possible one. The theorem does not claim the modulus is a prime number or has any special multiplicative property.

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/ModularLogicRealization.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.

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