Cosmology Dark Energy Dilution From Jcost S Forced

The forced survival ratio

In cosmology, dark energy is often modeled as a substance whose density dilutes as the universe expands. The Recognition Science framework asks what happens when that dilution obeys a strict self-similarity rule: each step of attenuation is the reciprocal of one plus the current value. The declaration s_forced proves that this rule, together with the positivity of the survival ratio, forces the ratio to be exactly the reciprocal of the golden ratio, φ⁻¹ ≈ 0.618. This is a theorem in the machine-checked library of formal theorems, meaning the proof has been verified step by step with no gaps.

The classical context matters. The golden ratio φ satisfies φ² = φ + 1, and its reciprocal φ⁻¹ = (√5 − 1)/2 ≈ 0.618. The self-similarity condition s = 1/(1 + s) is a fixed-point equation: substituting φ⁻¹ gives 1/(1 + φ⁻¹) = φ⁻¹, which holds because φ⁻¹ is the positive solution. The theorem s_forced shows this is the only positive solution, so the self-similarity premise alone pins down the number. This is the same equation that appears in continued fractions and in the geometry of the pentagon, where the diagonal-to-side ratio is φ.

In Recognition Science, the ledger, a discrete record of events, provides the underlying structure. The framework models dark-energy occupancy as an exponential aggregate over independent recognition channels, each with the same survival ratio s. The theorem s_forced establishes that this ratio must be φ⁻¹, and from there the framework derives the present amplitude A_RS = φ⁻⁴ · J(φ), where J is the forced cost function. These results are theorems conditional on two surviving premises: the identification of dark-energy occupancy with the channel-aggregate form, which is a model choice, and the self-similar attenuation premise itself. The framework does not derive dark energy from nothing; it substitutes one premise for a logically equivalent one and proves the rest.

What s_forced does not claim is equally important. It does not prove that the physical dark-energy occupancy actually has the channel-aggregate form; that identification is a model, not a theorem. It does not derive the self-similar attenuation premise from deeper principles; that premise remains an assumption. And it does not establish any numerical agreement with cosmological observations, such as the measured dark-energy density. The theorem is a conditional result: given the model and the premise, the survival ratio is forced to φ⁻¹, and the amplitude follows. The framework's library records this honestly, noting that the premise count is unchanged, only the form of one premise is re-derived.

/-- The per-channel survival ratio is forced to `φ⁻¹` by the self-similar
premise alone (via the recipShift fixed-point theorem). -/
theorem s_forced : C.s = 1 / Constants.phi :=
  DarkEnergyPhiDilutionDerivation.self_similar_attenuation_forced C.s_pos C.self_similar

/-- The occupancy through `n` independent recognition channels: the uniform-weight
J-cost log-aggregate of the per-channel survival ratio. MODEL-level
identification: this form (exp of an additive log-ledger over independent
channels) is what replaces the abstract `composes` axiom. -/
def occOf (n : ℕ) : ℝ :=
  Cost.Ndim.aggregate (fun _ : Fin n => 1) (fun _ => C.s)

/-- The channel-derived attenuated amplitude is exactly the single predicted
present amplitude `A_RS = φ⁻⁴ · J(φ)` of `DarkEnergyPhiDilutionLaw`. -/
theorem amplitude_eq_predicted :
    DarkEnergyAmplitudeDerivation.attenuatedAmplitude
        (DarkEnergyPhiDilutionDerivation.derivedTheta C.toDimensionUniformDilution)
      = DarkEnergyPhiDilutionLaw.predictedPresentAmplitude := by
  rw [DarkEnergyPhiDilutionDerivation.derivedTheta_eq_phiDilutionLaw]
  rfl

What this page does not claim

The physical dark-energy occupancy is proven to have the channel-aggregate form. The self-similar attenuation premise is derived from deeper principles. Any numerical agreement with observed dark-energy density is established.

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

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