Cosmology Dark Energy Dilution From Jcost Amplitude Eq Predicted

The amplitude identity

Dark energy is the name cosmologists give to the unknown influence that appears to accelerate the expansion of the universe. In the Recognition Science framework, dark energy is modeled as a kind of ledger, a discrete record of events, where the cost of each recognition event is forced by a proved mathematical law. The declaration amplitude_eq_predicted is a theorem in the framework's machine-checked library of formal theorems. It states that a specific model of how dark energy dilutes over time produces exactly the amplitude that the framework had earlier predicted, with no numerical fitting.

The theorem works through a chain of definitions. A channel dilution is a model where each independent recognition channel attenuates by a single survival ratio, and that ratio is self-similar, meaning it satisfies the equation s = 1/(1+s). The only positive solution to this equation is the reciprocal of the golden ratio, φ⁻¹ ≈ 0.618. The framework proves that when this survival ratio is aggregated over independent channels using its native cost function, the resulting occupancy follows a simple exponential law: the log of the occupancy is the sum of the per-channel log-costs. Because exponentials turn sums into products, the occupancy across disjoint channel blocks composes multiplicatively. This multiplicativity was previously an assumption in the framework's dilution model; the new theorem derives it from the channel-aggregate form instead.

The headline result follows from this strengthened dilution law. At the framework's forced spacetime dimension, the channel-derived occupancy θ equals φ⁻⁴, and the attenuated amplitude equals the framework's predicted present amplitude A_RS = φ⁻⁴ · J(φ), where J is the forced cost function. The theorem amplitude_eq_predicted is the final glue: it proves that the attenuated amplitude computed from the channel dilution model is exactly the predicted present amplitude, by unfolding definitions and applying the earlier derived identity. The proof is axiom-clean, with zero admitted assumptions or sorry placeholders.

What the theorem does not claim is equally important. The identification of the physical dark-energy occupancy with a channel aggregate, meaning that occupancy equals the exponential of an additive log-ledger over independent channels, is a MODEL-level structural choice. Nothing in the module forces the actual dark energy in the universe to have that form. The per-channel self-similar attenuation, the premise that one channel attenuates by the reciprocal self-similarity fixed point, also remains a physical assumption. The theorem replaces one assumption (multiplicative composition) with a logically equivalent one (the channel-aggregate form), so it is a premise substitution, not a reduction to zero premises. The physical amplitude claim is therefore a theorem only conditional on these two surviving premises.

In plain terms, the declaration proves an internal consistency result within the framework: if dark energy's dilution follows the framework's native channel-aggregate structure with self-similar attenuation, then its present amplitude is forced to be exactly the predicted value. It does not prove that dark energy actually has that structure. The framework's contribution is to show that the predicted amplitude is not an arbitrary number but the unique outcome of a specific, precisely stated model. A reader can now see that the framework's dark-energy prediction is internally coherent, and that the remaining question is empirical: whether the physical universe matches the model's structural assumptions.

/-- 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

/-- **THE `composes` PREMISE, DISCHARGED.** Occupancy across `m + n` independent
channels is the product of the block occupancies. Proof: the log-ledger is
additive over disjoint channel blocks, and `exp` turns sums into products
(`Real.exp_add`) — the multiplicative shadow of J-cost log-additivity, now a
theorem instead of an axiom. -/
theorem occOf_composes (m n : ℕ) :
    C.occOf (m + n) = C.occOf m * C.occOf n := by
  rw [occOf_eq_exp, occOf_eq_exp, occOf_eq_exp, ← Real.exp_add]
  congr 1
  push_cast
  ring

/-- 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)

What this page does not claim

The theorem does not prove that the physical dark energy in the universe is a channel aggregate; that identification is a model choice. The theorem does not derive the dilution law from nothing; it substitutes one premise (multiplicative composition) with a logically equivalent one (the channel-aggregate form). The theorem does not make any numerical prediction about the measured value of the dark energy density in SI units.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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