Cosmology Dark Energy Dilution From Jcost Occ Of Pos

The positive occupancy theorem

In cosmology, the energy density of empty space, often called dark energy, is a positive quantity. The Recognition Science framework formalizes a model of how that density might dilute over time. Within that model, the declaration occOf_pos is a machine-checked theorem: for any number of independent channels, the computed occupancy is strictly greater than zero. It is a basic sanity property, the kind of fact that must hold before any physical prediction is taken seriously.

The theorem is stated in the language of the framework's library, a machine-checked collection of formal theorems. The occupancy is defined as an exponential of a sum of logarithms of per-channel survival ratios. Since the exponential of any real number is positive, the proof is a direct consequence of that definition. The declaration also confirms that a single channel carries exactly its survival ratio, and that occupancy across a union of channels is the product of the individual occupancies. These are structural facts about how the model combines independent contributions.

In Recognition Science, the framework models dark energy density as the result of a ledger, a discrete record of recognition events, where the cost of each event is forced by a proved functional equation. The occupancy is the exponential of an additive log-ledger over independent channels. This channel-aggregate form is a MODEL-level identification: the framework chooses it as the native form, but nothing in the theorem forces the physical dark energy density to have this shape. The theorem does not derive the dilution law from nothing; it replaces one abstract premise with a logically equivalent concrete form.

The theorem also does not establish the value of the dark energy density. The self-similar attenuation premise, which states that a single channel attenuates by the reciprocal self-similarity fixed point, remains a physical assumption. It is this premise, not occOf_pos, that forces the per-channel survival ratio to be the inverse of the golden ratio. The positive occupancy theorem is a necessary condition for the later amplitude claims, but it is not the source of their physical content.

What occOf_pos changes is the status of the model's internal consistency. Before this theorem, the positivity of the occupancy could have been an additional assumption. Now it is a proved consequence of the definition. This is a small but real step: it removes one potential hidden axiom from the derivation chain, making the framework's claim to a minimal premise set stronger.

/-- The channel occupancy is strictly positive. -/
theorem occOf_pos (n : ℕ) : 0 < C.occOf n := by
  rw [occOf_eq_exp]
  exact Real.exp_pos _

/-- 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)

/-- One channel carries exactly the survival ratio: `occOf 1 = s`. -/
theorem occOf_one : C.occOf 1 = C.s := by
  rw [occOf_eq_exp]
  push_cast
  rw [one_mul, Real.exp_log C.s_pos]

/-- **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

What this page does not claim

The theorem does not prove that the physical dark energy density has the channel-aggregate form. The theorem does not derive the dilution law from no premises; the self-similar attenuation premise survives. The theorem does not establish the numerical value of the dark energy density.

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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