Cosmology Recognition Event Horizon Dyadic Freeze Rung Is Least

The freeze rung

A recognition signal in this framework moves one cell per tick, with eight ticks per epoch. Each epoch also stretches the scale by the golden ratio φ, about 1.618, so the distance a signal covers in later epochs shrinks by that factor. Adding up the reach of all future epochs gives a finite total, about 20.944 cells. This finite total is the recognition event horizon: a boundary that a signal launched now can never cross.

The horizon's value is 8φ², which equals 8(φ + 1). A geometric series with ratio 1/φ converges to this limit because 1/φ is less than one. The number sits between 16 and 32, the fourth and fifth powers of two. The theorem dyadicFreezeRung_is_least states that 5 is the smallest whole number of doublings that exceeds the horizon. Any doubling below five, meaning 16 or less, falls short; the horizon itself is strictly below 32.

In Recognition Science, this boundary drives a freeze-out. Structure separated by more than the horizon can never exchange a signal, so it can never homogenize. Its primordial amplitude stays frozen. Structure closer than the horizon eventually gets crossed and smooths out. The framework presents this as the source of accelerated-expansion freeze-out, with no fitted constant in the calculation.

The theorem is a statement about the reach of signals, not about the past. It uses the future-directed reach of a signal launched now, which forms an event horizon. A separate calculation for the vacuum energy cost deliberately uses the past-directed particle horizon instead, so that the ground-state cost does not depend on future expansion. The two horizons are consistent, not contradictory.

What the theorem does not claim is broader. It does not say that the horizon is exactly at a power of two; it only places it between 16 and 32. It does not claim that structure beyond the horizon cannot exist, only that it cannot be reached. And it does not assert that the framework's cosmology matches observations; the freeze-out mechanism is a derived consequence, not a measured fact.

/-- **THEOREM.** `dyadicFreezeRung = 5` is the least power-of-two rung strictly above the
recognition horizon: `2^5 > 8 φ²`, while every smaller rung `2^k` (`k < 5`) is strictly below
it. So the freeze-out selects exactly the dyadic scales at or above `2^5 = 32`. -/
theorem dyadicFreezeRung_is_least :
    recognitionEventHorizon < (2 : ℝ) ^ dyadicFreezeRung ∧
    ∀ k : ℕ, k < dyadicFreezeRung → (2 : ℝ) ^ k < recognitionEventHorizon := by
  refine ⟨horizon_lt_two_pow_five, ?_⟩
  intro k hk
  simp only [dyadicFreezeRung] at hk
  have hk4 : k ≤ 4 := by omega
  have hnat : (2 : ℕ) ^ k ≤ 2 ^ 4 := Nat.pow_le_pow_right (by norm_num) hk4
  have hmono : (2 : ℝ) ^ k ≤ (2 : ℝ) ^ 4 := by exact_mod_cast hnat
  exact lt_of_le_of_lt hmono two_pow_four_lt_horizon

/-- The horizon in closed form, purely from `φ² = φ + 1`. Numerically
`8 (φ + 1) ≈ 20.944` comoving cells, matching the numeric simulation. -/
theorem recognitionEventHorizon_eq : recognitionEventHorizon = 8 * (φ + 1) := by
  unfold recognitionEventHorizon; rw [phi_sq_eq]

/-- **THEOREM.** The total cumulative reach over all epochs equals the forced
event horizon `8 φ²`. A recognition signal can only ever traverse a finite
comoving distance, even given infinitely many epochs. -/
theorem tsum_perEpochReach :
    ∑' m : ℕ, perEpochReach m = recognitionEventHorizon := by
  unfold perEpochReach recognitionEventHorizon
  rw [tsum_mul_left, tsum_phi_inv_pow]

/-- **THEOREM.** The recognition event horizon `8 φ²` sits strictly between the dyadic rungs
`2^4 = 16` and `2^5 = 32`. So a self-similar (dyadic) structure freezes exactly at and above
the scale `2^5 = 32` and homogenizes at and below `2^4 = 16`: the freeze break is forced to
the fifth dyadic rung, with no fitted scale. This is the arithmetic anchor of the Phase-16
freeze-out scale selection (`scripts/cosmogenesis/foam_freeze_out.py`). -/
theorem recognitionEventHorizon_between_dyadic_rungs :
    (2 : ℝ) ^ 4 < recognitionEventHorizon ∧ recognitionEventHorizon < (2 : ℝ) ^ 5 :=
  ⟨two_pow_four_lt_horizon, horizon_lt_two_pow_five⟩

/-- **RECOGNITION EVENT HORIZON, ONE STATEMENT.** The forced φ-dilation (one
φ-rung per eight-tick epoch, T-6 and T-7) gives a finite de Sitter recognition
horizon equal to `8 φ² = 8 (φ + 1)`; the cumulative reach converges to it from
strictly below and increases monotonically, so any comoving separation at or
beyond `8 φ²` is never crossed. This is the law-derived freeze-out mechanism:
no tuned Hubble rate and no fitted coupling enter. -/
theorem recognition_event_horizon_one_statement :
    (∑' m : ℕ, perEpochReach m = recognitionEventHorizon) ∧
    recognitionEventHorizon = 8 * (φ + 1) ∧
    (∀ n : ℕ, cumulativeReach n < recognitionEventHorizon) ∧
    StrictMono cumulativeReach :=
  ⟨tsum_perEpochReach, recognitionEventHorizon_eq, cumulativeReach_lt_horizon,
    cumulativeReach_strictMono⟩

What this page does not claim

The horizon is not exactly at a power of two; it lies strictly between 16 and 32. The theorem does not say that structure beyond the horizon is absent, only that it is unreachable. The freeze-out mechanism is a derived consequence within the framework, not a measured observation.

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

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