Foundation Arrow Of Time Reverse Subtracts

Reversal and the ledger

The ledger in Recognition Science is a discrete record of events. Each step in the record carries a number called Berry phase, a geometric quantity that accumulates as the system evolves. The declaration reverse_subtracts states a simple algebraic fact: if a forward step carries phase p, then the reverse step carries phase -p, and their sum is zero. This is not a statement about physics directly; it is a statement about how the framework's bookkeeping behaves when a step is undone.

The consequence matters for the arrow of time. The framework defines Z-complexity as the sum of absolute values of the Berry phases at each step. Because absolute values ignore sign, reversing a step does not decrease Z; the magnitude of -p equals the magnitude of p. The theorem z_absolute_immune_to_reversal records this equality. The forward direction, in contrast, strictly increases Z when the new phase is nonzero, as forward_accumulates proves. The asymmetry is topological: the loop traversed backward subtracts phase, but the complexity measure only sees the size, not the direction.

This gives the framework an intrinsic before and after without importing thermodynamics. The relation isBefore is defined by comparing Z values: if Z at time t₁ is less than Z at time t₂, then t₁ is before t₂. The theorems before_transitive, before_irrefl, and before_asymm show that this relation orders time like a strict total order. Thermodynamic entropy is then defined as a monotone function of Z, and entropy_monotone derives the second law from the Berry phase accumulation. The framework models the arrow of time as a consequence of complexity growth, not as a primitive assumption.

/-- Reversing a loop subtracts phase (opposite sign). -/
theorem reverse_subtracts (phase : ℝ) :
    let forward_phase := phase
    let reverse_phase := -phase
    forward_phase + reverse_phase = 0 := by
  simp only
  ring

/-- Z-complexity uses absolute values, so reversal adds to Z, not subtracts. -/
theorem z_absolute_immune_to_reversal (phase : ℝ) (hp : phase ≠ 0) :
    |phase| = |-phase| := by
  rw [abs_neg]

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

/-- Entropy is monotone in Z (second law from Berry phase). -/
theorem entropy_monotone (z₁ z₂ d : ℝ) (hd : 0 < d) (hz : 0 ≤ z₁) (h : z₁ < z₂) :
    entropyFromZ z₁ d < entropyFromZ z₂ d := by
  unfold entropyFromZ
  apply Real.log_lt_log (by nlinarith)
  nlinarith

What this page does not claim

The theorem does not claim that reversing a step in any physical process decreases a measurable quantity. The framework does not claim that the Berry phase is the only source of temporal asymmetry. The definition of entropy as log(1 + z * density) is a modeling choice, not a derivation from first principles.

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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