Cosmology Dark Energy Dilution From Jcost Occ Of Eq Exp

The exponential occupancy

In cosmology, a central question is how the energy density of a component changes as the universe expands. For ordinary matter, density falls as volume grows. For dark energy, the observed behavior is nearly constant. The Recognition Science framework approaches this through a discrete record of recognition events, its ledger, and asks what form of dilution is forced by its core cost function. The declaration occOf_eq_exp is a small but load-bearing piece of that derivation.

The declaration states a precise mathematical identity. If s is a per-channel survival ratio, and n is the number of independent channels, then the occupancy occOf n equals exp(n · log s). In plainer terms: the log of the occupancy is the sum of the per-channel log-costs, so the occupancy itself is an exponential of that sum. This is the multiplicative shadow of the additive log-ledger. The proof is a direct computation, unwinding definitions and applying the fact that exponentials turn sums into products.

This identity matters because it converts an abstract premise into a theorem. The framework's dilution model previously assumed that occupancy across independent dimensions composes multiplicatively. The declaration occOf_composes, which follows from occOf_eq_exp, proves that composition law for any occupancy of this channel-aggregate form. A premise that was once assumed is now derived. The surviving physical premise is per-channel self-similar attenuation, the condition that the survival ratio satisfies s = 1/(1+s), which forces s = 1/φ where φ is the golden ratio.

What the declaration does not claim is equally important. It does not prove that the physical dark energy occupancy actually has this channel-aggregate form. That identification is a modeling choice, not a theorem. The framework chooses to represent dark energy occupancy as the exponential of an additive log-ledger over independent channels; nothing forces nature to adopt that form. The declaration also does not derive the self-similar attenuation premise. That premise remains an input, not an output. The net result is a theorem conditional on two premises: the channel-aggregate identification and the self-similar attenuation. The theorem replaces one assumption with a logically equivalent one, grounding it in the framework's cost API rather than leaving it abstract.

For the reader, the practical takeaway is this: the exponential form is not an arbitrary guess. Within the framework, it is the unique consequence of combining the J-cost log-additivity with the channel-aggregate identification. The framework's library of machine-checked theorems records this as a theorem, with zero unproved axioms in the derivation itself. The physical amplitude that results, θ = φ⁻⁴, follows from these premises and the forced spacetime dimension. The chain is explicit, and the limits of the chain are explicit too.

/-- Closed exponential form of the channel occupancy: `occOf n = exp(n · log s)`.
The log of the occupancy is the sum of the `n` per-channel log-costs. -/
theorem occOf_eq_exp (n : ℕ) :
    C.occOf n = Real.exp (n * Real.log C.s) := by
  unfold occOf Cost.Ndim.aggregate Cost.Ndim.dot Cost.Ndim.logVec
  simp only [one_mul, Finset.sum_const, Finset.card_univ, Fintype.card_fin,
    nsmul_eq_mul]

/-- **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 physical dark energy occupancy is proven to have the channel-aggregate exponential form. The per-channel self-similar attenuation premise is derived from the J-cost function. The declaration reduces the number of premises in the dark energy derivation.

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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