Foundation Universal Forcing Canonical Forcing Forcing Map Iff

The canonical forcing map

In mathematics, a Peano structure is a set with a distinguished starting point, usually called zero, and a successor operation that steps from one element to the next. The natural numbers are the standard example: zero, then its successor one, then two, and so on. A structure-preserving map between two such systems is a function that sends the first system's zero to the second system's zero and commutes with the successor operation, so that stepping first and then mapping gives the same result as mapping first and then stepping.

The Recognition Science framework builds its own arithmetic from a primitive notion of a ledger, a discrete record of events, and derives a cost function that forces a specific structure. Within this framework, the declaration forcing_map_iff establishes a complete characterization: a function between the carriers of two forced arithmetics is the canonical forcing map if and only if it preserves zero and the successor operation. This is an equivalence, not merely a one-way implication. The theorem proves that any function satisfying these two simple conditions must be the unique structure-preserving map, and conversely, that this unique map always satisfies the conditions.

The significance lies in what the theorem removes: representational freedom. When two forced arithmetics share a carrier universe, the framework proves there is exactly one structure morphism between them, determined entirely by the zero and step data. The forcing map is not one isomorphism among many; it is the only one. This canonicity result holds for any two realizations of the framework's logic, not just a single preferred example, and it extends to a certificate that packages existence, preservation, and uniqueness into a single structure.

What the theorem does not claim is equally important. It does not assert that the carrier universes themselves are identical, only that a unique map exists between them. It does not claim that any function preserving zero and step exists; it characterizes those that do as being the forcing map. The theorem is a statement about the uniqueness of a morphism given a shared carrier universe, not about the construction of that universe or about any physical property of the framework's constants. It leaves open how the carrier universe is chosen and what the forced arithmetic represents in physical terms.

/-- Complete characterization (general realizations). -/
theorem forcing_map_iff
    (A : ArithmeticOf.{u, v, w, w, w} R) (B : ArithmeticOf.{u', v', w, w, w} S)
    (f : A.peano.carrier → B.peano.carrier) :
    (f A.peano.zero = B.peano.zero ∧ ∀ x, f (A.peano.step x) = B.peano.step (f x))
      ↔ f = (equivOfInitial A B).toFun := by
  constructor
  · rintro ⟨hz, hs⟩
    exact forcing_map_unique A B f hz hs
  · rintro rfl
    exact ⟨equivOfInitial_map_zero A B, equivOfInitial_map_step A B⟩

/-- **General canonicity.** Any function preserving zero and step between two
forced arithmetics (sharing a carrier universe) *is* the forcing map. The map is
determined by the zero/step data alone — no representational freedom. -/
theorem forcing_map_unique
    (A : ArithmeticOf.{u, v, w, w, w} R) (B : ArithmeticOf.{u', v', w, w, w} S)
    (f : A.peano.carrier → B.peano.carrier)
    (hz : f A.peano.zero = B.peano.zero)
    (hs : ∀ x, f (A.peano.step x) = B.peano.step (f x)) :
    f = (equivOfInitial A B).toFun := by
  have h := A.initial.uniq B.peano
      (⟨f, hz, hs⟩ : PeanoObject.Hom A.peano B.peano) (A.initial.lift B.peano)
  simpa [equivOfInitial] using h

/-- **The Universal-Forcing objective at full generality.** For *any* two
Law-of-Logic realizations `R`, `S` (in arbitrary universes) and any forced
arithmetics `A` over `R`, `B` over `S` sharing a carrier universe, there is a
structure-preserving equivalence between the carriers that is *the unique*
zero/step-preserving map. This is the precise content of
`universal-forcing-program.mdc`'s "canonical equivalence of `ArithmeticOf R` and
`ArithmeticOf S` across admissible realizations": not bare existence of some
iso, but a canonical, determined morphism.

Unlike the strict-realization certificate below (where the carrier is concretely
`LogicNat` and the map is the identity), here `A.peano.carrier` and
`B.peano.carrier` may be genuinely different Peano objects, so the equivalence is
a nontrivial iso — yet still the only structure morphism between them. -/
theorem universal_objective
    (A : ArithmeticOf.{u, v, w, w, w} R) (B : ArithmeticOf.{u', v', w, w, w} S) :
    ∃ e : A.peano.carrier ≃ B.peano.carrier,
      e A.peano.zero = B.peano.zero
      ∧ (∀ x, e (A.peano.step x) = B.peano.step (e x))
      ∧ (∀ f : A.peano.carrier → B.peano.carrier,
            f A.peano.zero = B.peano.zero →
            (∀ x, f (A.peano.step x) = B.peano.step (f x)) →
            f = e.toFun) :=
  ⟨equivOfInitial A B,
   equivOfInitial_map_zero A B,
   equivOfInitial_map_step A B,
   fun f hz hs => forcing_map_unique A B f hz hs⟩

What this page does not claim

The theorem does not claim that the carrier universes of two realizations are identical. The theorem does not establish the existence of any zero and step preserving function; it only characterizes those that do exist. The theorem does not assign any physical meaning to the forced arithmetic or its elements.

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/CanonicalForcing.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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