Cosmology Omega Lambda Derivation Tick Addressing Is Power2

The tick addressing

The declaration tick_addressing_is_power2 establishes a single arithmetic fact: the number called tick_addressing, which the framework uses to count the addressing slots of an eight-step recognition cycle, equals 16, and 16 is 2 raised to the fourth power. In plain terms, the framework models a discrete record of events, its ledger, as an eight-tick cycle, and it assigns each tick a four-bit address, so the total number of addresses is 2 × 2 × 2 × 2 = 16. This is a proved theorem in the machine-checked library of formal theorems, meaning the equality is verified by the kernel's rules of inference.

The fact matters because this 16 appears in the denominator of a proposed formula for the dark energy fraction of the universe, written as Ω_Λ = 11/16 − α/π. The numerator 11 counts a subset of frequency modes that the framework identifies as symmetric under a particular group structure, and the denominator 16 is the total addressing capacity. The theorem tick_addressing_is_power2 does not by itself derive the formula, prove the mode count, or establish any physical law; it only certifies that the denominator is exactly the fourth power of two, a small but necessary piece of the arithmetic.

In Recognition Science, the framework proves that the eight-tick cycle and its 2^3 structure are forced by the cost function, and this declaration extends that pattern to the addressing scheme. The theorem is a definitional consequence: given the definitions of tick_addressing and the power operation, the equality holds by computation. It does not claim that the universe literally uses four-bit addresses, nor that the number 16 has independent physical meaning outside the framework's model. It is a formal statement about the framework's own construction, not a measurement or a prediction.

/-- 16 = 2^4 (addressing bits from 4 half-epochs in the 8-tick cycle). -/
theorem tick_addressing_is_power2 : tick_addressing = 2 ^ 4 := by decide

/-- Canonical form: Ω_Λ = 11/16 − α/π with the measured CODATA α. -/
theorem omega_lambda_canonical_form :
    omega_lambda = 11/16 - Constants.ExternalAnchors.alpha_CODATA / Real.pi := by
  rw [omega_lambda_one_measured_input]
  unfold N_modes_saturated tick_addressing
  norm_num

/-- 16 = 2^4 (addressing bits from 4 half-epochs in the 8-tick cycle). -/
theorem tick_addressing_is_power2 : tick_addressing = 2 ^ 4 := by decide

What this page does not claim

The theorem does not prove the dark energy formula itself. It does not claim the number 16 has independent physical meaning outside the framework's model. It does not derive the value of the fine-structure constant α.

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

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