Foundation Arrow Of Time Before Irrefl

The before relation

The statement before_irrefl is a formal theorem in the Recognition Science framework's machine-checked library of formal theorems. It proves that the framework's "before" relation is irreflexive: no moment in time is before itself. In plain terms, if a moment has a certain amount of accumulated complexity, that moment cannot also be earlier than itself. The proof is a direct consequence of the definition of "before" as strict inequality between two real numbers, and the well-known fact that no number is strictly less than itself.

The framework builds its arrow of time from a quantity it calls Z-complexity, a non-negative real number that accumulates as steps are taken. The framework models time's direction by this accumulation: if Z at one moment is less than Z at another, the first is before the second. The theorem before_irrefl then guarantees this ordering has no loops, a basic requirement for any sensible notion of time. It is one of three ordering properties proved together: transitivity (if A is before B and B is before C, then A is before C), asymmetry (if A is before B, then B is not before A), and irreflexivity.

The theorem does not claim that Z-complexity itself is a fundamental physical quantity, nor that the framework's model of time is the only one. It does not assert that the arrow of time is objective in all physical theories, nor that thermodynamics is reducible to this complexity measure. The theorem only establishes a logical property of a relation defined within the framework. The framework's broader claim, that thermodynamic entropy emerges as a coarse-grained version of Z, is a separate derivation, not a theorem in this file.

/-- The before relation is irreflexive (a moment is not before itself). -/
theorem before_irrefl (z : ℝ) : ¬isBefore z z := by
  unfold isBefore; exact lt_irrefl z

/-- The arrow of time: if Z(t₁) < Z(t₂), then t₁ is before t₂. -/
def isBefore (z1 z2 : ℝ) : Prop := z1 < z2

/-- The before relation is transitive (time is ordered). -/
theorem before_transitive (z1 z2 z3 : ℝ) (h12 : isBefore z1 z2) (h23 : isBefore z2 z3) :
    isBefore z1 z3 := by
  unfold isBefore at *; linarith

/-- The before relation is asymmetric (if t1 < t2, then not t2 < t1). -/
theorem before_asymm (z1 z2 : ℝ) (h : isBefore z1 z2) : ¬isBefore z2 z1 := by
  unfold isBefore at *; linarith

/-- The before relation is irreflexive (a moment is not before itself). -/
theorem before_irrefl (z : ℝ) : ¬isBefore z z := by
  unfold isBefore; exact lt_irrefl z

What this page does not claim

The theorem does not claim Z-complexity is a fundamental physical quantity. The theorem does not assert the arrow of time is objective in all physical theories. The theorem does not claim thermodynamics is fully reducible to this complexity measure.

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

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