Foundation Modular Logic Realization One Lt Modulus

A small but necessary bound

In modular arithmetic, numbers wrap around after reaching a fixed size, the modulus. The declaration one_lt_modulus proves, for every natural number k, that the modulus defined as k + 2 is greater than 1. In plain terms, it establishes that this particular cyclic structure always has at least three distinct positions: 0, 1, and 2. The proof is a direct consequence of the definition, verified by the omega decision procedure, which automates reasoning about linear arithmetic.

This bound matters because the modular construction is used as a periodic interpretation for a free logical orbit. The framework's library models a ledger, a discrete record of events, where each event is assigned a cost. Here, the cost function assigns 0 when two positions are equal and 1 otherwise, a simple equality cost on a finite carrier. The theorem one_lt_modulus ensures that the carrier is not degenerate: with at least three positions, the cyclic successor step and the periodic interpretation can meaningfully distinguish between different states. Without this bound, the structure could collapse into a trivial two-element or one-element loop, undermining the demonstration that Universal Forcing does not require every realization to embed arithmetic faithfully.

The declaration itself is a small lemma, not a grand claim. It does not assert that the modular realization is unique, that it computes any particular value, or that it models the full structure of arithmetic. It only guarantees a lower bound on the size of the carrier. The larger significance comes from how this bound supports the modular realization's role: showing that a periodic, finite interpretation can coexist with an invariant extracted arithmetic, a point about the flexibility of realizations within the framework.

What the theorem does not claim is equally precise. It does not say that the modulus is prime, that the cyclic structure is the only possible realization, or that the bound is optimal. It does not establish any property of the cost function beyond the trivial equality cost defined alongside it. The declaration is a building block, verified in the machine-checked library of formal theorems, that secures one necessary condition for the modular construction to function as intended.

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

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

What this page does not claim

The modulus is prime or has any special number-theoretic property beyond being greater than 1. The modular realization is the unique or preferred realization for Universal Forcing. The bound is optimal or that no smaller carrier could serve a similar purpose.

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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