Cosmology Dark Energy Density4 From Jcost Dedensity4 Cert

What the certificate proves

The declaration DEDensity4Cert is a formal certificate, a machine-checked bundle of three proved statements about a cost function. The cost function, written domainCost, takes two positive numbers and returns the value Jcost(m/e), where Jcost is the framework's forced recognition cost. The three proved facts are: the cost is zero when the two numbers are equal, the cost is never negative for positive inputs, and a certain threshold, phi minus 3/2, is positive. These are general properties of the cost function, not statements about cosmology.

The certificate's name suggests dark energy, but the declaration itself proves nothing specific to dark energy. The function domainCost is defined as Jcost(m/e) without any reference to a physical mass or energy. A research note in the file records an intended application: setting rho_Lambda = Lambda/(8 pi G) and comparing it to the Planck density, one gets rho_Lambda/rho_Pl = phi^5/(45 pi) which is about 0.0784, leading to a value around 3.9e95 kg/m^3, far from the measured dark energy density. That note is explicitly marked as a research note, not a result. The three proved facts are shared verbatim with 2383 sibling modules, so they are not specific to this subject.

What the certificate does establish is a template. It shows that the cost function has the properties one would want for a ledger of recognition events: zero cost at identity and nonnegativity. The positive threshold is a small fact about the golden ratio, phi, being greater than 1.5. These facts are proved in the machine-checked library of formal theorems and are axiom-clean. But the leap from these general properties to a dark energy density is not made. To make this declaration a theorem about dark energy, one would need a definition of m and e in terms of physical quantities, and that definition is absent.

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

What this page does not claim

The certificate does not derive the dark energy density from the cost function. The certificate does not define m and e in physical terms. The research note's numerical comparison is not a proved result.

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/DarkEnergyDensity4FromJCost.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.

Derived articles

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