Cosmology Dark Energy Dilution From Jcost Occ Of One

The single-channel occupancy

The declaration occOf_one is a machine-checked theorem in the Recognition Science framework's library of formal theorems. It states that when a recognition ledger, a discrete record of events and their costs, has exactly one independent channel, the occupancy of that channel equals the per-channel survival ratio. In symbols, occOf 1 = s. The occupancy is the framework's measure of how much of a quantity survives after recognition costs are paid; the survival ratio s is the fraction that remains after one step of attenuation.

The proof is a short chain of equalities. The occupancy is defined as an exponential aggregate of log-costs, and with one channel that aggregate collapses to the survival ratio itself. The theorem then uses the fact that the exponential of a logarithm returns the original positive number. This is not a deep result on its own, but it is the precise statement that connects the abstract definition of occupancy to the concrete physical premise of self-similar attenuation, where each channel reduces by the reciprocal self-similarity fixed point.

In Recognition Science, the importance of occOf_one is that it supplies the one-dimensional case for a larger construction. The framework models dark energy as the occupancy of a recognition ledger across independent channels. The per-channel survival ratio is assumed to follow the self-similar attenuation law s = 1/(1+s), which forces s to be the inverse of the golden ratio. The theorem occOf_one shows that with a single channel, the occupancy is exactly that survival ratio, so the one-dimensional behavior is not an additional assumption but a consequence of the definition.

The theorem does not claim that dark energy actually is such a ledger occupancy. That identification is a modeling choice, not a proved fact. It also does not claim that the self-similar attenuation law is derived; that law remains a physical premise. What occOf_one establishes is purely internal: within the framework's definitions, the one-channel occupancy has the stated simple form, and this form is what allows the composition law across channels to be proved rather than assumed.

/-- 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]

/-- 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]

/-- 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]

What this page does not claim

The theorem does not claim that dark energy is a recognition-ledger occupancy; that identification is a modeling choice. The theorem does not claim that the self-similar attenuation law is derived; it remains a physical premise. The theorem does not claim anything about the measured 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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