Cosmology Recognition Event Horizon Two Pow Four Lt Horizon

The finite horizon

In cosmology, an event horizon is the boundary separating regions of the universe that can ever be reached by a signal from regions that cannot. The Recognition Science declaration two_pow_four_lt_horizon establishes a specific numerical fact about such a horizon: the number 16 is strictly less than the horizon value 8φ², where φ is the golden ratio. Since φ² = φ + 1, this horizon equals 8(φ + 1), approximately 20.944. The theorem proves this inequality in a machine-checked library of formal theorems, with no fitted constants.

The result follows from a model in which a recognition signal, a discrete record of events, travels one comoving cell per tick. The framework's forced self-similar dilation expands the comoving scale by φ per epoch, so a signal's reach in epoch m is 8(1/φ)^m. Summing this geometric series over all epochs gives the finite horizon 8φ². The theorem two_pow_four_lt_horizon then states that this horizon exceeds 16, placing it between the powers of two: 2⁴ < 8φ² < 2⁵. This places the horizon on a dyadic rung, a binary scale, which the framework identifies as the fifth freeze-out rung.

The physical consequence is a freeze-out of structure. 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 homogenized. It freezes at its primordial amplitude. Structure below the horizon is eventually crossed and homogenizes. This is the framework's account of accelerated-expansion freeze-out, its Ω_Λ term.

The declaration does not claim that the horizon value is exactly a power of two, nor that the number 16 itself has special physical meaning. It only establishes the strict inequality. The horizon is approximately 20.944, not 16 or 32. The theorem also does not claim that the framework's model of recognition signals is the correct description of physical cosmology; it proves a mathematical statement within the framework's axioms. The physical interpretation, that structure freezes at this scale, is a derived consequence of the model, not an independent observational claim.

/-- The horizon strictly exceeds the dyadic rung `2^4 = 16`: `16 < 8 φ²`, equivalently
`2 < φ²`, equivalently `1 < φ`. So a self-similar (dyadic) structure at scale `2^4 = 16` is
sub-horizon and is homogenized. -/
theorem two_pow_four_lt_horizon : (2 : ℝ) ^ 4 < recognitionEventHorizon := by
  have h16 : (2 : ℝ) ^ 4 = 16 := by norm_num
  rw [recognitionEventHorizon_eq, h16]
  linarith [one_lt_phi]

/-- 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 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.** The real-space freeze-out dichotomy. A comoving separation `r` is eventually
crossed by a recognition signal launched at the σ = 0 seed iff it lies strictly below the
horizon: for every `r < 8 φ²` there is a finite epoch whose cumulative reach exceeds `r` (so
structure at radius `r` is eventually homogenized), while for every `r ≥ 8 φ²` no finite epoch
ever reaches `r` (so structure at radius `r` freezes at its primordial amplitude). This is the
real-space form of the Phase-9 horizon and the law-derived statement behind the inner
homogenized ball / outer frozen foam split in
`scripts/cosmogenesis/foam_real_space_freeze_out.py`: the freeze surface is the comoving sphere
of radius `8 φ²`, which by §6 sits strictly between the dyadic shells `2^4 = 16` and
`2^5 = 32`. -/
theorem reach_dichotomy :
    (∀ r : ℝ, r < recognitionEventHorizon → ∃ n : ℕ, r < cumulativeReach n) ∧
    (∀ r : ℝ, recognitionEventHorizon ≤ r → ∀ n : ℕ, cumulativeReach n < r) := by
  refine ⟨?_, ?_⟩
  · intro r hr
    have hsum : HasSum perEpochReach recognitionEventHorizon := by
      have h := perEpochReach_summable.hasSum
      rwa [tsum_perEpochReach] at h
    have hT : Filter.Tendsto cumulativeReach Filter.atTop (nhds recognitionEventHorizon) := by
      simpa [cumulativeReach] using hsum.tendsto_sum_nat
    have hev : ∀ᶠ n in Filter.atTop, r < cumulativeReach n :=
      hT.eventually (eventually_gt_nhds hr)
    exact hev.exists
  · intro r hr n
    exact lt_of_lt_of_le (cumulativeReach_lt_horizon n) hr

What this page does not claim

The horizon value is not a power of two; it is approximately 20.944, strictly between 16 and 32. The theorem does not assert that 16 has any special physical meaning beyond being a lower bound. The physical freeze-out interpretation is a derived consequence of the framework's model, not an independent observational measurement.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: