Foundation Primitive Recognition Calculus Grow Ratio Orbit Order Add Mono To Int

The integer bridge

A signed orbit, a discrete record of events tagged positive or negative, carries two natural-number counts: how many positive events and how many negative events. The lemma toInt_eq says that converting such an orbit to an integer is exactly the difference of those two counts. In symbols, for any signed orbit a, a.toInt = (a.pos.toNat : ℤ) - (a.neg.toNat : ℤ). This is a definitional unfold: it spells out what the conversion already meant, with no hidden choice of representatives.

The point of the lemma is not the arithmetic, which is trivial, but the route it opens. A companion lemma, le_iff_toInt_le_cf, uses it to compare two signed orbits by their integer images. That comparison deliberately avoids an older, choice-tainted bridge and instead goes through a natural-number comparison. The payoff is that the order on signed orbits becomes compatible with integer order in a way that is fully constructive: no appeal to the axiom of choice anywhere in the chain.

In Recognition Science, this is one small rivet in a larger structure. The framework models recognition as a forced cost, and from that cost derives a scaling law, an eight-tick cycle, and ultimately three spatial dimensions. Within that program, the integer bridge matters because it keeps the order-theoretic plumbing free of choice principles, so that later theorems can be audited to a minimal axiom base. The lemma itself, however, claims nothing about physics. It does not say that any particular orbit occurs, that the counts are finite in practice, or that the integer difference has physical meaning. It only says how the conversion is computed.

What the lemma does not claim is equally precise. It does not assert that every signed orbit corresponds to a real process, nor that the order comparison is meaningful for infinite counts. It does not establish that the integer difference is a recognition cost or any other framework quantity. It is a purely syntactic fact about a data type, a bridge that later theorems may walk across but that by itself carries no load.

/-- Choice-free unfold of `toInt` to its pos/neg Nat-cast difference. -/
private lemma toInt_eq (a : SignedOrbit) :
    a.toInt = (a.pos.toNat : ℤ) - (a.neg.toNat : ℤ) := by
  cases a with
  | mk pos neg => exact SignedOrbit.toInt_mk pos neg

/-- Choice-free Int order bridge, routed through the CF Nat bridge `le_iff_toNat_cf`
    (never through the choice-tainted `SignedOrbit.le_iff_toInt_le`). -/
private lemma le_iff_toInt_le_cf (a b : SignedOrbit) :
    SignedOrbit.le a b ↔ a.toInt ≤ b.toInt := by
  rw [le_iff_toNat_cf, toInt_eq a, toInt_eq b]
  constructor
  · intro hh; omega
  · intro hh; omega

What this page does not claim

The lemma does not claim that any particular signed orbit occurs in a physical process. The lemma does not claim that the integer difference is a recognition cost or any other framework quantity. The lemma does not claim that the order comparison works for infinite counts.

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

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