Natural Numbers Emerge from Logic

1. Setting

The natural numbers are not imported. RS reconstructs successor, addition, multiplication, induction, and cancellation inside the logic-realization surface.

2. Equations

(E1)

$$ 0,\quad S(n),\quad n+m,\quad n\cdot m $$

Peano arithmetic operations reconstructed from logic.

3. Prediction or structural target

• Peano arithmetic: predicted reconstructed (dimensionless); empirical Lean theorem target. Source: Foundation.ArithmeticFromLogic

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.ArithmeticFromLogic..succ_injective.

injectivity of multiplication by the generator on the orbit. -/
theorem succ_injective : Function.Injective succ := by
  intro a b h
  cases h
  rfl

/-- **Peano P3 (induction)**: any property closed under successor and
holding at zero holds for every `LogicNat`. -/
theorem induction
    {motive : LogicNat → Prop}

5. What is inside the Lean module

Key theorems:

• zero_ne_succ
• succ_ne_zero
• succ_injective
• induction
• add_def
• zero_def
• one_def
• add_zero
• add_succ
• zero_add
• succ_add
• add_assoc

Key definitions:

• LogicNat
• zero
• succ
• add
• mul
• toNat
• fromNat
• equivNat

6. Derivation chain

• succ_injective - Successor is injective
• induction - Induction
• add_assoc - Addition is associative
• add_comm - Addition is commutative
• mul_add - Multiplication distributes over addition

7. Falsifier

A failure of successor injectivity, induction, associativity, commutativity, or distributivity in the logic-derived arithmetic refutes this derivation.

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-from-logic, start with the primary Lean anchor Foundation.ArithmeticFromLogic.succ_injective. 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.