Cosmology Recognition Event Horizon Recognition Event Horizon Between Dyadic Run

The horizon's position

In cosmology, an event horizon is a boundary in space-time beyond which events cannot affect an observer. The Recognition Science framework derives a specific finite value for such a horizon from its own axioms. The framework models reality as maintaining a discrete record of recognition events, where a recognition event is a unit of signal exchange between two comoving cells, which are volume elements that expand with the universe. The framework proves that a recognition signal travels eight comoving cells per epoch, a period of eight ticks, and that the comoving scale dilates by the golden ratio φ ≈ 1.618 per epoch. Because the dilation ratio 1/φ is less than one, the total distance a signal can ever cover, summed over all future epochs, is a finite geometric series: 8 + 8/φ + 8/φ² + ... = 8φ² = 8(φ + 1) ≈ 20.944 comoving cells.

The theorem recognitionEventHorizon_between_dyadic_rungs states that this horizon value lies strictly between the fourth and fifth powers of two: 16 < 8φ² < 32. This placement is not a coincidence of arithmetic but a consequence of the golden ratio's defining property, φ² = φ + 1. The proof in the machine-checked library of formal theorems verifies the lower bound by noting 16 is less than 8φ², and the upper bound by showing 8φ² is less than 32. The theorem also establishes that the fifth power of two, 32, is the least dyadic rung, a power of two, that exceeds the horizon.

The physical meaning is a freeze-out of structure. The cumulative reach of a signal after any finite number of epochs is strictly less than the horizon and increases monotonically toward it. Therefore, any two comoving cells separated by a distance at or beyond 8φ² can never exchange a recognition signal, even given infinite time. Structure on those scales can never be brought into causal contact and homogenized; it freezes at its primordial amplitude. Structure below the horizon is eventually crossed and homogenizes. This is the framework's account of the accelerated-expansion freeze-out, analogous to a cosmological constant.

In Recognition Science, this future-directed reach is distinct from the past-directed particle horizon used in the vacuum-energy ledger cost, which must not depend on future expansion. The two horizons are consistent: the ledger ground-state cost uses the past cone, while the forward freeze-out of structure uses the future cone. The horizon's position between 16 and 32 means the freeze-out scale is not a power of two, but it is bracketed by them, a fact the framework uses to identify the relevant dyadic rung for the freeze-out process.

/-- **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⟩

/-- **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

/-- **THEOREM.** The cumulative reach increases strictly with each epoch:
each epoch adds a strictly positive per-epoch reach, so the reach climbs
monotonically toward (but never attains) the horizon. -/
theorem cumulativeReach_strictMono : StrictMono cumulativeReach := by
  apply strictMono_nat_of_lt_succ
  intro n
  have hstep : cumulativeReach (n + 1) = cumulativeReach n + perEpochReach n := by
    simp [cumulativeReach, Finset.sum_range_succ]
  rw [hstep]; linarith [perEpochReach_pos n]

/-- **THEOREM.** After any finite number of epochs the cumulative reach is
strictly below the horizon. A comoving separation at or beyond `8 φ²` is
therefore never crossed by a recognition signal, so super-horizon structure can
never be homogenized: it freezes at its primordial amplitude. -/
theorem cumulativeReach_lt_horizon (n : ℕ) :
    cumulativeReach n < recognitionEventHorizon := by
  have hsum := perEpochReach_summable
  have hsplit := Summable.sum_add_tsum_nat_add n hsum
  have htail_summable : Summable (fun i => perEpochReach (i + n)) :=
    (summable_nat_add_iff n).2 hsum
  have htail_pos : 0 < ∑' i, perEpochReach (i + n) :=
    htail_summable.tsum_pos (fun i => le_of_lt (perEpochReach_pos _)) 0
      (perEpochReach_pos _)
  have key :
      cumulativeReach n + ∑' i, perEpochReach (i + n) = recognitionEventHorizon := by
    have h := hsplit
    rw [tsum_perEpochReach] at h
    simpa [cumulativeReach] using h
  linarith [htail_pos, key]

What this page does not claim

The theorem does not claim that the horizon value is exactly a power of two; it is strictly between 16 and 32. The theorem does not claim that the framework's physical interpretation of freeze-out is a proven statement about the actual universe; it is a model consequence. The theorem does not claim that the horizon is the same as the past-directed particle horizon used in the vacuum-energy ledger cost.

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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