Foundation Primitive Recognition Calculus Orbit Succ Injective

The successor step

A successor function is the operation that moves from one counting number to the next: from 2 to 3, from 3 to 4. The statement that this function is injective means that it never maps two different starting points to the same ending point. If succ(a) equals succ(b), then a must equal b. In plain terms, each step lands on a fresh position; no two distinct places in the sequence collapse into one. This is a foundational property of the natural numbers, and it is what makes counting a reliable way to tell things apart.

The Recognition Science framework builds its own counting structure, called the orbit, a discrete record of repeated acts of distinction, from a starting point called zero and a successor step. Its machine-checked library of formal theorems proves that this successor is injective. The proof is short: if two successor results are equal, the definition of the successor constructor forces the inputs to be equal. This single fact, together with the companion result that zero is not a successor, guarantees that the orbit never loops back on itself and never identifies two different positions. The framework then shows that this orbit is equivalent to the familiar natural numbers: there is a reversible translation between the two, and it preserves the successor step.

In Recognition Science, this injectivity is the first rung of a longer ladder. The framework uses the orbit to model how a system can keep a ledger of distinct events, and from that ledger it derives, step by step, the golden ratio as a self-similar scaling, an eight-tick recognition cycle, the number 2^3, and three spatial dimensions. The injectivity theorem is what guarantees the ledger has room for all those distinctions: no two events are ever recorded as the same event.

The theorem does not claim that the successor function is the only way to build a counting structure, nor does it say anything about what the orbit is made of. It is a statement about the formal structure, not about any physical process. The framework's later results, such as the derivation of three spatial dimensions, depend on additional theorems and on the physical interpretation of the orbit as a ledger of recognition events; injectivity alone does not force any of those conclusions.

/-- R8. Successor is injective. -/
theorem succ_injective :
    Function.Injective succ := by
  intro a b h
  cases h
  rfl

/-- K4.5. The δ-orbit is equivalent to Lean Nat as a verifier display. -/
def equivNat : DistinctionNat ≃ Nat where
  toFun := toNat
  invFun := ofNat
  left_inv := ofNat_toNat
  right_inv := toNat_ofNat

/-- R8. Successor is injective. -/
theorem succ_injective :
    Function.Injective succ := by
  intro a b h
  cases h
  rfl

What this page does not claim

The successor function is the only possible way to define a counting structure. Injectivity alone forces the golden ratio, the eight-tick cycle, or three spatial dimensions. The orbit is a physical object rather than a formal construction.

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

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