Cosmology Recognition Event Horizon Recognition Event Horizon One Statement

The finite horizon

In cosmology, an event horizon is the boundary separating events that can ever influence an observer from events that cannot. The Recognition Science declaration recognition_event_horizon_one_statement establishes a specific finite value for such a horizon in its own model: 8 φ², where φ is the golden ratio (1 + √5)/2 ≈ 1.618. This equals 8(φ + 1) ≈ 20.944, a number with no fitted constant.

The value arises from a simple geometric series. The framework models a recognition signal as traveling one comoving cell per tick, with eight ticks per epoch. Each epoch dilates the comoving scale by the golden ratio, so the signal's reach in epoch m is 8(1/φ)^m. Summing this over all epochs gives the finite limit 8φ² because 1/φ < 1. The declaration bundles four proved facts: the infinite sum equals the horizon value, the horizon equals 8(φ + 1), every finite partial sum stays below the horizon, and the partial sums increase strictly.

The physical consequence is a freeze-out dichotomy. Any comoving separation at or beyond 8φ² is never crossed by a recognition signal, so structure on those scales can never be brought into causal contact and never homogenized; it freezes at its primordial amplitude. Structure below the horizon is eventually crossed and homogenizes. The framework identifies this as its Ω_Λ freeze-out, the accelerated-expansion phase.

In Recognition Science, this is a theorem in the machine-checked library of formal theorems, proved with no unproved assumptions. It is not a claim about any observed astronomical horizon. The number 8φ² is a derived constant of the framework's own model, not a measurement of the actual universe's event horizon. The framework does not assert this value matches any telescope data.

/-- **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]

/-- 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.** 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]

/-- **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.** 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⟩

What this page does not claim

This is not a measurement of any observed astronomical event horizon. The framework does not claim the value 8φ² matches any telescope data. The declaration does not prove the existence of an actual physical horizon in the real universe.

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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