Foundation Modular Logic Realization Modular Interpret Step

The periodic step

In ordinary arithmetic, counting never repeats: after 0 comes 1, after 1 comes 2, and the sequence continues without end. The theorem named modularInterpret_step concerns a different kind of counting, one that runs on a finite, repeating cycle, like the hours on a clock. It establishes, in the language of the Recognition Science framework, that a single step of this cyclic counting process is consistent: applying the successor operation to a number and then interpreting it on the cycle gives the same result as first interpreting the number and then taking one cyclic step. In plainer terms, the order of operations does not matter; the step commutes with the interpretation.

The framework's machine-checked library of formal theorems proves this for every choice of cycle length. Given any natural number k, the framework defines a finite carrier of size k+2, and a cyclic successor function that moves from one element to the next, wrapping around at the end. The theorem states that for any natural number n, interpreting the successor of n on this cycle equals taking the cyclic successor of the interpretation of n. This is a structural guarantee: the periodic interpretation is a homomorphism with respect to the successor operation, preserving the essential step structure even though the carrier is finite.

In Recognition Science, this result supports a broader point about the framework's model of logic. The framework's internal arithmetic remains unbounded, but a realization of that arithmetic can be periodic on its carrier. The theorem demonstrates that Universal Forcing, the framework's central mechanism, does not require every realization to embed arithmetic faithfully into its carrier. A periodic, finite interpretation is sufficient to preserve the step structure, even though it cannot represent the full content of arithmetic. This is a theorem about the framework's own construction, not a claim about physical space or time.

What the theorem does not claim is equally important. It does not assert that the cyclic carrier itself is a model of arithmetic; the finite cycle lacks the unboundedness that arithmetic requires. It does not claim that the periodic interpretation is injective, and indeed it is not: distinct natural numbers can map to the same element on the cycle. The theorem establishes only the commutativity of the step, not any property of the cyclic carrier beyond that. It is a precise, limited result about the interaction between an unbounded internal arithmetic and a finite, periodic carrier.

theorem modularInterpret_step (k : ℕ) (n : ArithmeticFromLogic.LogicNat) :
    modularInterpret k (ArithmeticFromLogic.LogicNat.succ n)
      = cycStep k (modularInterpret k n) := by
  apply Fin.ext
  change ArithmeticFromLogic.LogicNat.toNat (ArithmeticFromLogic.LogicNat.succ n) % modulus k =
    (ArithmeticFromLogic.LogicNat.toNat n % modulus k + 1) % modulus k
  rw [ArithmeticFromLogic.LogicNat.toNat_succ, Nat.succ_eq_add_one]
  have h := (Nat.add_mod (ArithmeticFromLogic.LogicNat.toNat n) 1 (modulus k)).symm
  simpa [Nat.mod_eq_of_lt (one_lt_modulus k)] using h

theorem modularInterpret_step (k : ℕ) (n : ArithmeticFromLogic.LogicNat) :
    modularInterpret k (ArithmeticFromLogic.LogicNat.succ n)
      = cycStep k (modularInterpret k n) := by
  apply Fin.ext
  change ArithmeticFromLogic.LogicNat.toNat (ArithmeticFromLogic.LogicNat.succ n) % modulus k =
    (ArithmeticFromLogic.LogicNat.toNat n % modulus k + 1) % modulus k
  rw [ArithmeticFromLogic.LogicNat.toNat_succ, Nat.succ_eq_add_one]
  have h := (Nat.add_mod (ArithmeticFromLogic.LogicNat.toNat n) 1 (modulus k)).symm
  simpa [Nat.mod_eq_of_lt (one_lt_modulus k)] using h

/-- Modular realization has invariant extracted arithmetic. -/
noncomputable def modular_arithmetic_invariant (k : ℕ) (R : LogicRealization.{0, 0}) :
    (UniversalForcing.arithmeticOf (modularRealization k)).peano.carrier
      ≃ (UniversalForcing.arithmeticOf R).peano.carrier :=
  ArithmeticOf.equivOfInitial
    (UniversalForcing.arithmeticOf (modularRealization k))
    (UniversalForcing.arithmeticOf R)

What this page does not claim

The cyclic carrier is a model of arithmetic. The periodic interpretation is injective. The theorem applies to physical space or time.

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)

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