Foundation Growth Bounds Density Exceeds Threshold

The density bound

In mathematics, exponential growth eventually defeats polynomial growth. No matter how big a constant you multiply a cubic term by, an exponential function with a base greater than one will, from some point onward, be larger. This is a classical fact, and it is the core of what the framework's density_exceeds_threshold declaration establishes.

The statement is precise. Let phi be the golden ratio, about 1.618. The declaration proves that for any positive numbers K0 and V0, and for any positive threshold, there exists a natural number N such that K0 times phi raised to the power N, divided by V0 times (N+1) cubed, is greater than the threshold. In plainer terms: if you have a quantity that grows like phi to the N, and you divide it by a volume that grows like a cubic polynomial in N, the ratio eventually exceeds any fixed bound you choose. The proof uses Bernoulli's inequality and a known result that phi to the N exceeds any linear function of N.

This is a theorem in the framework's machine-checked library of formal theorems. The declaration is not a physical law by itself. It is a pure real-analysis result. It does not say that any particular physical density in the universe obeys this growth. It does not define what K0 or V0 are in physical terms. It only establishes the mathematical fact that such an N exists for any positive inputs.

Within the framework, this result is used to close a specific argument about growth bounds. The framework models a ledger, a discrete record of events, where the number of events grows like phi to the N. The declaration shows that this growth eventually overwhelms any cubic volume growth, which the framework associates with three-dimensional space. This is a mathematical step in a larger chain; it is not an empirical measurement.

The consequence for a reader is a clean, checkable inequality. It is a building block, not a conclusion about the world. The value of the declaration is that it is proved, not assumed, and it is proved in a way that any mathematician can verify.

/-- **LOCAL DENSITY EVENTUALLY EXCEEDS ANY THRESHOLD**

    K₀ * φ^N / (V₀ * (N+1)³) → ∞ as N → ∞. -/
theorem density_exceeds_threshold (K₀ : ℝ) (hK₀ : 0 < K₀)
    (V₀ : ℝ) (hV₀ : 0 < V₀) (threshold : ℝ) (hT : 0 < threshold) :
    ∃ N : ℕ, K₀ * phi ^ N / (V₀ * ((N : ℝ) + 1) ^ 3) > threshold := by
  -- Need phi^N > (threshold * V₀ / K₀) * (N+1)^3
  have hC : 0 < threshold * V₀ / K₀ := by positivity
  obtain ⟨N, hN⟩ := phi_exp_defeats_cubic_succ (threshold * V₀ / K₀) hC
  refine ⟨N, ?_⟩
  have hdenom_pos : 0 < V₀ * ((N : ℝ) + 1) ^ 3 := by positivity
  rw [gt_iff_lt, lt_div_iff₀ hdenom_pos]
  -- Goal: threshold * (V₀ * (N+1)^3) < K₀ * phi^N
  -- From hN: phi^N > (threshold*V₀/K₀) * (N+1)^3
  -- So K₀ * phi^N > K₀ * (threshold*V₀/K₀) * (N+1)^3 = threshold*V₀*(N+1)^3
  have hphi_pos : 0 < phi ^ N := pow_pos phi_pos N
  have hNN3 : 0 < ((N : ℝ) + 1) ^ 3 := by positivity
  have hK0phi : K₀ * phi ^ N > K₀ * (threshold * V₀ / K₀) * ((N : ℝ) + 1) ^ 3 := by
    have := mul_lt_mul_of_pos_left hN hK₀
    simp only [mul_comm, mul_assoc] at this ⊢
    linarith
  have hsimp : K₀ * (threshold * V₀ / K₀) * ((N : ℝ) + 1) ^ 3 =
               threshold * V₀ * ((N : ℝ) + 1) ^ 3 := by
    have hK0ne : K₀ ≠ 0 := ne_of_gt hK₀
    field_simp [hK0ne]
  rw [hsimp] at hK0phi
  linarith

/-- Bernoulli's inequality: for a ≥ 1, a^n ≥ 1 + n*(a-1). -/
theorem exp_ge_linear (a : ℝ) (ha : 1 ≤ a) (n : ℕ) :
    a ^ n ≥ 1 + (n : ℝ) * (a - 1) := by
  induction n with
  | zero => simp
  | succ k ih =>
    have ha_nonneg : 0 ≤ a := by linarith
    have hk_nn : (0 : ℝ) ≤ k := Nat.cast_nonneg k
    calc a ^ (k + 1) = a ^ k * a := pow_succ a k
      _ ≥ (1 + (k : ℝ) * (a - 1)) * a := by
          exact mul_le_mul_of_nonneg_right ih ha_nonneg
      _ = a + (k : ℝ) * a * (a - 1) := by ring
      _ ≥ a + (k : ℝ) * 1 * (a - 1) := by
          nlinarith [mul_nonneg hk_nn (sub_nonneg.mpr ha), sq_nonneg (a - 1)]
      _ = 1 + ((k : ℝ) + 1) * (a - 1) := by ring
      _ = 1 + (↑(k + 1) : ℝ) * (a - 1) := by push_cast; ring

/-- φ eventually exceeds any bound. -/
theorem phi_pow_exceeds (M : ℝ) : ∃ N : ℕ, phi ^ N > M :=
  exponential_exceeds_bound phi one_lt_phi M

What this page does not claim

This claim does not state that any physical density in the universe obeys this growth law. This claim does not define the physical meaning of K0 or V0. This claim does not prove that the framework's ledger model is correct.

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/Foundation/GrowthBounds.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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