Foundation Primitive Recognition Calculus Grow Eta Completion M0a Eta Respects C

The embedding lemma

In the Recognition Science framework, the declaration eta_respects_crossEq is a theorem about how rational numbers are embedded into a larger system of real-like objects. The framework builds real numbers not from decimal expansions but from sequences of rationals, where two sequences are considered equivalent if their terms eventually become close in a precise sense. The declaration states that if two rational numbers have the same cross-difference, a value computed from their numerators and denominators, then the constant sequences built from them are equivalent in that same sense.

The cross-difference of two rational numbers a and b is the integer a.num * b.den - b.num * a.den. When this value is zero, the two rationals are equal in the usual sense. The theorem proves that this equality is preserved when each rational is turned into a constant sequence, a sequence that repeats the same value at every step. The proof is immediate: the sequences are constant, so their cross-difference at every step is zero, and the equivalence condition holds with N equal to zero.

This lemma is not a deep result but a structural one. It guarantees that the embedding of rationals into the new system is well-defined, meaning that equal rationals do not get mapped to different objects. The declaration also supports the injectivity of the embedding, which is proved separately as etaQ_injective: distinct rationals remain distinct in the larger system. Together these facts establish that the rational numbers sit inside the new system exactly as they should, without collapsing or duplicating points.

The lemma does not claim that the new system is complete, nor that it contains all real numbers. It only concerns the constant sequences built from rationals and their equivalence. The framework's larger completion process, which adds limits of more general sequences, is a separate matter not addressed by this declaration.

/-- `eta` respects `crossEq`: if `q` and `r` are cross-equal (represent the same
delta-rational), then `eta q` and `eta r` are equivalent regular sequences.
This is the key well-definedness property of the unit map. -/
theorem eta_respects_crossEq (q r : RatioOrbit) (h : RatioOrbit.crossEq q r) :
  equiv (eta q) (eta r) := by
  intro k
  refine ⟨0, ?_⟩
  intro n _
  rw [eta_seq q n, eta_seq r n, crossDiff_of_crossEq q r h,
    Int.natAbs_zero, Nat.zero_mul]
  exact Nat.zero_le _

/-- A rational minus itself has zero cross-difference. -/
theorem crossDiff_self (a : RatioOrbit) : crossDiff a a = 0 := by
  unfold crossDiff
  omega

/-- The unit η : ℚδ → ℝδ_pre is injective: no two distinct δ-rationals collapse
in the completion. Together with well-definedness this makes η a genuine
embedding of the rational base into the M0a real line, choice-free. -/
theorem etaQ_injective : Function.Injective etaQ := by
  intro a b h
  induction a using Quot.ind with
  | mk q =>
    induction b using Quot.ind with
    | mk r =>
      have hq : RealDelta.mk (eta q) = RealDelta.mk (eta r) := h
      have hequiv : equiv (eta q) (eta r) := by
        have := Quot.eqvGen_exact hq
        -- Exactness gives `EqvGen`; collapse it with the proved equivalence.
        exact (Equivalence.eqvGen_iff equiv_equivalence).mp this
      exact Quot.sound (crossEq_of_equiv_eta hequiv)

What this page does not claim

The declaration does not prove that the new system is complete or contains all real numbers. It does not address the equivalence of non-constant sequences or the general completion process. It does not establish that the cross-difference is the only way to compare rationals in the framework.

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/EtaCompletionM0a.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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