Arithmetic is Categorical Across Realizations

1. Setting

Arithmetic must be the same across admissible realizations. This is the Universal Forcing program in miniature: different realizations extract canonically equivalent Peano surfaces.

2. Equations

(E1)

$$ \mathrm{ArithmeticOf}(R)\simeq \mathrm{ArithmeticOf}(S) $$

Categorical equivalence of extracted arithmetic.

3. Prediction or structural target

• extracted arithmetic: predicted canonical equivalence (dimensionless); empirical Lean theorem target. Source: Foundation.ArithmeticOf

This entry is one of the marquee derivations. The numerical or formal target is explicit, and the falsifier identifies the failure mode.

4. Formal anchor

The primary anchor is Foundation.ArithmeticOf..logicNat_initial.

/-- `LogicNat` is initial among Peano objects. -/
def logicNat_initial : PeanoObject.IsInitial logicNatPeano where
  lift := logicNatLift
  uniq := by
    intro B f g
    rw [logicNatLift_unique_fun B f, logicNatLift_unique_fun B g]

/-- The Peano object extracted from a realization's own orbit. -/
def realizationPeano (R : LogicRealization) : PeanoObject where
  carrier := R.Orbit

5. What is inside the Lean module

Key theorems:

• logicNatLift_unique_fun
• realizationLift_unique_fun
• extracted_peanoSurface
• canonical_peanoSurface

Key definitions:

• PeanoObject
• Hom
• id
• comp
• IsInitial
• ArithmeticOf
• PeanoSurface
• logicNatPeano

6. Derivation chain

• logicNat_initial - Logic-arithmetic is initial
• logicNatLift_unique_fun - Lift uniqueness
• extracted_peanoSurface - Extracted Peano surface
• equivOfInitial - Equivalence of initial objects

7. Falsifier

Two admissible recognition realizations whose extracted arithmetic is not equivalent refute categorical arithmetic forcing.

8. Where this derivation stops

Below this page the chain reduces to the RS forcing sequence: J-cost uniqueness, phi forcing, the eight-tick cycle, and the D=3 recognition substrate. If any upstream theorem changes, this page must be versioned rather than patched silently. The published URL is stable, but the version field is the contract.

9. Reading note

The minimal way to audit this page is to open the first Lean anchor and then walk the supporting declarations listed above. If the primary theorem is a module-level anchor, the key theorems section names the internal declarations that carry the mathematical load. This keeps the public derivation readable without severing it from the proof object.

10. Audit path

To audit arithmetic-is-categorical, start with the primary Lean anchor Foundation.ArithmeticOf.logicNat_initial. Then inspect the theorem names listed in the module-content section. The page is intentionally built so the public explanation is not a substitute for the proof object; it is a map into it. The mathematical dependency is the same in every case: reciprocal cost fixes J, J fixes the phi-ladder, the eight-tick cycle fixes the recognition clock, and the domain theorem listed above supplies the last step. If that last step is empirical, the falsifier section names what observation would break it. If that last step is formal, a Lean-checkable counterexample is the relevant failure mode.