Cosmology Recognition Event Horizon Cumulative Reach Strict Mono

The finite reach

In cosmology, an event horizon is the boundary between what a signal launched now can eventually influence and what it never will. Recognition Science, a framework that derives physical structure from a discrete record of recognition events, builds such a horizon from two forced numbers: the eight ticks of its recognition cycle and the golden ratio φ ≈ 1.618, which its self-similar scaling law fixes. A recognition signal travels one comoving cell per tick, so per epoch it covers eight cells. But the framework's dilation expands the comoving scale by φ each epoch, shrinking the signal's effective reach per epoch by the factor 1/φ. The cumulative reach after n epochs is the finite sum of 8(1/φ)^m from m = 0 to n−1.

The theorem cumulativeReach_strictMono states that this cumulative reach is strictly increasing: each additional epoch adds a positive amount, so the total never stagnates or decreases. The proof is direct: the per-epoch reach is always positive, and adding it to the previous total strictly increases it. This monotonicity is one half of a dichotomy. The other half, proved separately, says the infinite sum converges to a finite limit, the recognition event horizon, equal to 8φ² ≈ 20.944 comoving cells. So the cumulative reach grows without pause but is forever bounded above by that horizon. A signal launched now can cross any comoving separation smaller than 8φ², given enough epochs, but can never cross one at or beyond it.

The physical consequence is a freeze-out of structure. Comoving separations at or beyond the horizon are never brought into causal contact by recognition signals, so they never homogenize; their primordial amplitude freezes. Separations below the horizon are eventually crossed and homogenize. This is the framework's account of accelerated-expansion freeze-out, with no fitted Hubble rate or coupling constant. The horizon value itself, 8φ², sits between 2⁴ = 16 and 2⁵ = 32 comoving cells, and the framework identifies the fifth dyadic rung as the least power of two above it.

What the theorem does not claim is equally precise. It establishes monotonic growth of a finite partial sum, not the value of the infinite limit; that convergence is a separate theorem. It does not assert that any particular physical structure lies beyond the horizon, only that if a separation is at or beyond 8φ², no recognition signal crosses it. The framework's bridge from this topological statement to physical cosmology, the claim that recognition signals correspond to causal contact in the actual universe, is itself an open target, not a proved theorem. The monotonicity result stands on its own as a statement about a geometric series with ratio 1/φ.

/-- **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]

/-- **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

/-- **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

The theorem does not prove the value of the infinite limit, only the monotonic growth of the partial sums. It does not claim that any particular observed structure lies beyond the horizon. It does not assert that recognition signals correspond to physical causal contact; that bridge remains open.

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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