Foundation Universal Forcing Canonical Iso Peano Equiv Unique

The canonical isomorphism

In mathematics, a bijection between two number systems only says the systems have the same number of elements. An isomorphism says more: it preserves the operations that give the systems their shape. For number systems built from zero and a successor function (the Peano structure behind counting), an isomorphism must send zero to zero and commute with the successor step. The declaration peanoEquiv_unique in the Recognition Science library proves that between any two forced arithmetics, there is exactly one such structure-preserving isomorphism.

The background is the framework's Universal Forcing program. Recognition Science starts from a single premise: reality keeps a ledger, a discrete record of recognition events, and the cost of recognition is forced, not chosen. From that premise the framework derives a chain of theorems, including the golden ratio, an eight-tick cycle, and three spatial dimensions. The arithmetic-extraction layer, of which peanoEquiv_unique is part, shows that the number systems arising from any two realizations of the framework's law are not merely in bijection; they are canonically equivalent as Peano algebras.

The theorem has three parts. First, the universal forcing bijection sends the forced zero to the forced zero. Second, it commutes with the forced step map. These two facts upgrade the bare carrier bijection to a homomorphism of Peano algebras. Third, and this is the heart of peanoEquiv_unique: any two structure-preserving isomorphisms between the same pair of forced arithmetics have the same underlying function. The isomorphism is not merely some isomorphism; it is the unique one. The library packages this into a certificate, quantified over all realizations, stating that the canonical isomorphism exists and is unique.

What the theorem does not claim is as important as what it proves. The Peano object carries only zero and step; it does not carry addition, multiplication, or order. Preservation of the ring operations and the order relation is the next step, not proved here. The docstring is explicit: the richer ordered-semiring isomorphism is remaining work. So peanoEquiv_unique establishes canonicality at the Peano-algebra layer, and stops there.

/-- **Canonicality.**  Any two structure-preserving isomorphisms between the
forced arithmetics of two realizations have the same underlying function.  The
isomorphism furnished by Universal Forcing is therefore the unique one. -/
theorem peanoEquiv_unique (R : LogicRealization.{u, v}) (S : LogicRealization.{u, w})
    (e₁ e₂ : PeanoEquiv (forcedArith R).peano (forcedArith S).peano) :
    (e₁.toEquiv : (forcedArith R).peano.carrier → (forcedArith S).peano.carrier)
      = (e₂.toEquiv : (forcedArith R).peano.carrier → (forcedArith S).peano.carrier) :=
  (forcedArith R).initial.uniq (forcedArith S).peano e₁.toHom e₂.toHom

/-- The universal forcing bijection sends the forced zero to the forced zero. -/
theorem equivOfInitial_map_zero (R : LogicRealization.{u, v}) (S : LogicRealization.{u, w}) :
    (ArithmeticOf.equivOfInitial (forcedArith R) (forcedArith S))
        (forcedArith R).peano.zero
      = (forcedArith S).peano.zero :=
  ((forcedArith R).initial.lift (forcedArith S).peano).map_zero

/-- The universal forcing bijection commutes with the forced step map. -/
theorem equivOfInitial_map_step (R : LogicRealization.{u, v}) (S : LogicRealization.{u, w})
    (x : (forcedArith R).peano.carrier) :
    (ArithmeticOf.equivOfInitial (forcedArith R) (forcedArith S))
        ((forcedArith R).peano.step x)
      = (forcedArith S).peano.step
          ((ArithmeticOf.equivOfInitial (forcedArith R) (forcedArith S)) x) :=
  ((forcedArith R).initial.lift (forcedArith S).peano).map_step x

/-- A structure-preserving isomorphism of Peano objects: an equivalence of
carriers that respects zero and step. -/
structure PeanoEquiv (A B : PeanoObject.{u}) where
  toEquiv : A.carrier ≃ B.carrier
  map_zero : toEquiv A.zero = B.zero
  map_step : ∀ x, toEquiv (A.step x) = B.step (toEquiv x)

What this page does not claim

The theorem does not claim that the isomorphism preserves addition, multiplication, or order; those are not part of the Peano object. The theorem does not claim that the framework derives physical constants; it operates at the level of number systems. The theorem does not claim that the Peano isomorphism is unique among all bijections; only among structure-preserving isomorphisms.

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/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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