Foundation Arrow Of Time Before Transitive

The before relation

In mathematics, a relation is transitive when one step leads to another: if A comes before B, and B comes before C, then A comes before C. Ordinary time is transitive. The Recognition Science framework contains a machine-checked proof, named before_transitive, that its own definition of "before" has this same property. The proof is a theorem in the framework's library of formal theorems, checked by a computer, with no gaps left for human error.

What is the framework's "before"? It is not a clock or a calendar. The framework models time as a sequence of steps, each step adding a number called Berry phase, a quantity from quantum mechanics that measures how a system changes as it goes around a loop. The framework defines a quantity Z, which is the sum of the absolute values of these Berry phases up to a given step. The definition of "before" is simple: one moment is before another if its Z value is smaller. The theorem before_transitive proves that this definition is transitive, meaning the relation forms a proper ordering of moments.

The framework goes further. It proves that Z never decreases as steps are added, so the "before" relation points in a consistent direction. It also proves that the relation is irreflexive, meaning no moment is before itself, and asymmetric, meaning if A is before B, then B is not before A. These properties together mean the framework's time has a genuine arrow: it is ordered, directed, and cannot loop back on itself. This arrow emerges from the mathematics of Berry phase accumulation, not from any imported assumption about thermodynamics.

What before_transitive does not claim is equally important. It does not claim that this formal ordering is the same as physical time as measured by clocks. It does not claim that the Berry phase model explains why humans experience time passing. The theorem is a statement about a mathematical structure: given the framework's definitions, the before relation is transitive. The bridge from this formal structure to the physical world, to actual clocks and human experience, is a separate question that the framework does not settle in this theorem.

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

/-- Z-complexity at step k: sum of absolute Berry phases up to k. -/
def zAtStep (seq : TemporalSequence) (k : Fin seq.n_steps) : ℝ :=
  (Finset.univ.filter (fun i : Fin seq.n_steps => i.val ≤ k.val)).sum
    (fun i => |seq.berry_at_step i|)

/-- Forward direction: adding a step with nonzero Berry phase increases Z. -/
theorem forward_accumulates (phases : List ℝ) (new_phase : ℝ) (hn : new_phase ≠ 0) :
    let z_before := (phases.map fun p => |p|).foldl (· + ·) 0
    let z_after := ((phases ++ [new_phase]).map fun p => |p|).foldl (· + ·) 0
    z_before < z_after := by
  simp only
  rw [List.map_append, List.foldl_append]
  simp only [List.map_cons, List.map_nil, List.foldl_cons, List.foldl_nil]
  linarith [abs_pos.mpr hn]

/-- 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 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

What this page does not claim

The theorem does not claim that the framework's formal ordering is identical to physical time as measured by clocks. The theorem does not claim that the Berry phase model explains the human experience of time passing. The theorem does not claim that the framework's arrow of time is the same as the thermodynamic arrow without additional assumptions.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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