Cosmology Recognition Event Horizon Recognition Event Horizon Eq

The finite horizon

In cosmology, an event horizon is the boundary between events a signal sent now can eventually reach and events it never will. Recognition Science derives a particular value for this boundary from its own first principles. The framework's model of recognition has signals traveling one comoving cell per tick, where a tick is one step in a discrete eight-step recognition cycle. That cadence is forced by the framework's theorems, not chosen by hand.

Because the framework's self-similar dilation expands the comoving scale by the golden ratio φ per epoch, a signal covers less comoving ground in each successive epoch. The reach in epoch m is 8 / φ^m. Summing this geometric series over all future epochs gives the finite limit 8φ², which equals 8(φ + 1), approximately 20.944 comoving cells. The theorem recognitionEventHorizon_eq in the framework's machine-checked library of formal theorems proves this equality: the horizon is exactly 8φ², with no fitted constant. The ingredients are the cadence 8 and the dilation sum φ², both derived rather than assumed.

The physical consequence is a freeze-out. The partial reach after any finite number of epochs stays strictly below 8φ² and approaches it monotonically. So a comoving separation at or beyond 8φ² is never crossed by a recognition signal. Structure on those scales can never be brought into causal contact and therefore never homogenizes; 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 associated with the cosmological constant Ω_Λ.

The declaration does not claim that this horizon is the only horizon in the framework. A separate construction, VacuumHorizonForcing, selects the past-directed particle horizon for the vacuum-energy ledger cost, and there the de Sitter event horizon is excluded because that calculation must not depend on future expansion. The two are consistent: the ledger ground-state cost uses the past cone, while the forward freeze-out of structure uses the future cone. The declaration also does not claim that the horizon value 20.944 is a measured cosmological quantity; it is a derived number within the framework, and the link between this discrete model and observed cosmology remains a target.

/-- 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 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]

What this page does not claim

The declaration does not claim that 8φ² is a measured cosmological quantity. The declaration does not claim that this is the only horizon in the framework; VacuumHorizonForcing defines a different, past-directed horizon. The declaration does not claim that the framework's discrete model has been observationally confirmed as the correct description of cosmic expansion.

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.

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