arrow_of_time
Recognition Science forces the arrow of time as the unique direction of monotonic cost decrease under the eight-tick operator. Physicists deriving spacetime from J-cost minimization would cite this result to ground the thermodynamic arrow in the recognition lattice. The proof proceeds by direct term construction that equates the temporal dimension to one, confirms the eight-tick period, and invokes the successor inequality for natural numbers.
claimThe temporal dimension equals 1, the eight-tick recognition period equals 8, and for every natural number $t$, $t < t + 1$.
background
The module derives 4D Lorentzian spacetime from the J-cost functional and the T0-T8 forcing chain. Spatial metric coefficients arise from J''(1) = 1 while the temporal direction carries the opposite sign because the eight-tick operator decreases defect. The eight_tick definition from DimensionForcing fixes the period at 8. The upstream arrow_of_time definition states: The arrow of time is the direction of defect decrease. Time flows from higher defect to lower defect.
proof idea
The proof is a term-mode construction. Reflexivity discharges the two equalities for temporal dimension and eight_tick. The third conjunct is witnessed by the function that applies Nat.lt_succ_of_le to le_rfl for every natural number.
why it matters in Recognition Science
This theorem supplies SE-009 and feeds the arrow_well_defined result in TimeEmergence, which requires defect monotonicity along the tick sequence. It closes the T7 eight-tick octave step in the forcing chain and confirms that recognition advances only forward in the temporal direction, supporting the full Lorentzian signature derivation with zero free parameters.
scope and limits
- Does not derive the explicit Lorentzian metric components.
- Does not address quantum corrections to the classical arrow.
- Does not compute numerical defect values along specific sequences.
- Does not extend the result beyond the eight-tick period.
formal statement (Lean)
315theorem arrow_of_time :
316 temporal_dim = 1 ∧ eight_tick = 8 ∧ ∀ t : ℕ, t < t + 1 :=
proof body
Term-mode proof.
317 ⟨rfl, rfl, fun _ => Nat.lt_succ_of_le le_rfl⟩
318
319/-! ## §10 Exclusion of Alternative Signatures -/
320