Cosmology Recognition Event Horizon Cumulative Reach Lt Horizon

The finite reach

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 value for such a horizon from its own principles. The framework models the universe as a discrete grid of cells, where a recognition signal, a unit of information transfer, moves one cell per tick, the framework's fundamental time step. The framework proves that the cumulative distance such a signal can cover, after any finite number of epochs, is always strictly less than a fixed limit. That limit is the recognition event horizon, defined as 8 times the square of the golden ratio, or approximately 20.944 cells.

The proof is a direct consequence of the framework's forced geometry. The signal covers 8 cells in the first epoch. The framework's self-similar dilation then expands the grid by the golden ratio each epoch, so the physical distance covered in each subsequent epoch shrinks by that same factor. The total reach is the sum of a geometric series: 8 + 8/φ + 8/φ² + ... . Because the golden ratio is greater than one, this series converges. The theorem cumulativeReach_lt_horizon in the framework's machine-checked library of formal theorems establishes that every finite partial sum of this series is below the limit, and a companion result shows the partial sums increase monotonically toward it.

The physical consequence the framework draws is a freeze-out of structure. Any two points separated by more than the horizon can never exchange a recognition signal. They are permanently outside each other's causal reach, so any initial differences in their properties can never be smoothed out or homogenized. The framework identifies this as the origin of a cosmological constant-like term, driving accelerated expansion and leaving primordial structure frozen at its original amplitude. The horizon value of about 20.9 cells is not fitted; it follows from the framework's forced cadence of eight ticks per epoch and the golden ratio dilation.

In Recognition Science, this future-directed horizon is distinct from the past-directed particle horizon used in the vacuum energy calculation. The past horizon concerns signals that could have reached us from the past, while this one concerns signals we can send to the future. The framework treats both as consistent aspects of the same discrete geometry. The theorem does not claim that the universe is literally 20.9 cells across, nor does it assign a physical size in meters to a cell. It establishes a structural limit within the framework's model, a limit that then serves as the basis for further derivations about the large-scale behavior of the model universe.

/-- **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]

/-- 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 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]

What this page does not claim

The theorem does not assign a physical size in meters to a comoving cell. The theorem does not claim that the recognition event horizon is the same as the past-directed particle horizon used in the vacuum energy calculation. The theorem does not by itself establish the existence of dark energy or a cosmological constant in the observable 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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