Foundation Universal Forcing Strict Canonical Iso Strict Peano Equiv Unique

The unique bridge

In mathematics, a Peano algebra is a structure built from a starting point called zero and a single operation called successor, which takes one number to the next. The natural numbers with zero and the "plus one" operation form the standard example. A structure-preserving isomorphism between two such algebras is a translation that sends zero to zero and commutes with the successor map, meaning the translation of a successor is the successor of the translation. Such a map guarantees the two systems behave identically under their basic operations.

The Recognition Science framework begins with a ledger, a discrete record of events, and derives arithmetic as a consequence of the laws that govern recognition costs. Its machine-checked library of formal theorems contains a result called strict_peanoEquiv_unique. This theorem states that for any two strict realizations of the Law of Logic, the bijection between their forced arithmetics is a structure-preserving isomorphism of Peano algebras, and this isomorphism is unique. In plain terms: if two systems each generate their own arithmetic purely from the native law data, without any externally supplied orbits, then there is exactly one way to translate between them that respects zero and successor.

The uniqueness is the key content. The theorem does not merely say a translation exists; it says any two structure-preserving isomorphisms between the forced arithmetics have the same underlying function. There is no freedom of choice, no arbitrary convention in how the two systems correspond. The strictness condition matters: a strict realization supplies only native comparison, composition, identity, invariance, and non-triviality data, and its arithmetic is derived rather than supplied. The canonical isomorphism is therefore genuinely forced by the native law data, with no orbit handed in by the caller.

In Recognition Science, this result upgrades an earlier bijection to a unique structure-preserving isomorphism, reusing the universal forcing equivalence from the lightweight case. The certificate structure StrictUniversalForcingIsoCert packages both the existence and the uniqueness, showing the strict canonical iso inhabits the certificate. This is a theorem in the framework's library, proved with the standard axioms and no framework-specific assumptions.

What the theorem does not claim is equally precise. It does not assert that the forced arithmetic is the standard natural numbers, nor that it matches any particular empirical counting system. It only governs the relationship between two systems that already satisfy the strict realization conditions. It also does not claim that the isomorphism extends to other operations like addition or multiplication; it covers zero and successor only. The theorem is about the uniqueness of the translation, not about what the arithmetic contains beyond the Peano structure.

/-- **Strict canonicality.**

Any two structure-preserving isomorphisms between the forced arithmetics of two
strict realizations have the same underlying function.  The strict universal
forcing isomorphism is therefore the unique one. -/
theorem strict_peanoEquiv_unique (R S : StrictLogicRealization)
    (e₁ e₂ : PeanoEquiv (StrictLogicRealization.arith R).peano
      (StrictLogicRealization.arith S).peano) :
    (e₁.toEquiv : (StrictLogicRealization.arith R).peano.carrier
        → (StrictLogicRealization.arith S).peano.carrier)
      = e₂.toEquiv :=
  peanoEquiv_unique (StrictLogicRealization.toLightweight R)
    (StrictLogicRealization.toLightweight S) e₁ e₂

/-- **Strict canonicality.**

Any two structure-preserving isomorphisms between the forced arithmetics of two
strict realizations have the same underlying function.  The strict universal
forcing isomorphism is therefore the unique one. -/
theorem strict_peanoEquiv_unique (R S : StrictLogicRealization)
    (e₁ e₂ : PeanoEquiv (StrictLogicRealization.arith R).peano
      (StrictLogicRealization.arith S).peano) :
    (e₁.toEquiv : (StrictLogicRealization.arith R).peano.carrier
        → (StrictLogicRealization.arith S).peano.carrier)
      = e₂.toEquiv :=
  peanoEquiv_unique (StrictLogicRealization.toLightweight R)
    (StrictLogicRealization.toLightweight S) e₁ e₂

What this page does not claim

The theorem does not claim the forced arithmetic equals the standard natural numbers. It does not claim the isomorphism preserves addition or multiplication. It does not claim the strict realization conditions are satisfied by any particular physical system.

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/UniversalForcing/Strict/CanonicalIso.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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