Cosmology Cosmology Eosdeep4 Cert

The certified core

In mathematics and physics, a certificate is a compact, machine-checkable package that vouches for a set of claims. The Recognition Science declaration EOSDeep4Cert is exactly that: a formal structure that bundles three proved facts about a quantity called domainCost, which the framework defines as J(m/e), where J is its fundamental cost function and m and e are positive real numbers. The three facts are simple and general. First, when the two inputs are equal, the cost is zero. Second, for any positive inputs, the cost is never negative. Third, a particular constant built from the golden ratio, phi minus 3/2, is positive. Each of these is proved in the framework's machine-checked library of formal theorems, and the certificate simply collects them into one reusable object.

The name of the declaration suggests a connection to cosmology, and the research note attached to the source file records the intent: the author meant to use this certificate as a step toward a statement about the dark energy equation of state, the parameter w that relates pressure to density in cosmology. The note goes further and states a concrete prediction: the framework's vacuum model forces w = -1 exactly, and any deviation at two sigma in the DESI survey would falsify that prediction. Current DESI data give w0 = -0.95 and wa = -0.27 from baryon acoustic oscillations, a mild tension with w = -1 that has not yet reached two sigma. That prediction, however, is not part of the certificate. The certificate itself proves nothing specific to cosmology, because domainCost is defined as J(m/e) without any reference to a physical system. The three proved facts hold for any positive real numbers m and e, and the leap from that abstract statement to a claim about dark energy would require a separate definition of m and e in cosmological terms. That definition is absent from the file.

What the certificate does establish is a small, solid foundation. The equality case, where cost vanishes when the two arguments match, is the kind of property one expects of a sensible measure of difference. Nonnegativity is the same expectation in weaker form. The positivity of phi minus 3/2 is a numerical fact about the golden ratio, roughly 0.118, that the framework's earlier work proves. Together, these three facts are the entire content of EOSDeep4Cert. The declaration is inhabited, meaning the framework's library demonstrates that such a certificate exists, and that existence is itself a proved theorem. A reader who wants to know what the framework has actually checked about this object can point to exactly these three statements and nothing more.

The gap between the certificate and its name is the honest boundary of the claim. The framework has not derived the dark energy equation of state, has not defined what m and e would mean for the universe, and has not produced a theorem about w. The research note records a target, not a result. Anyone citing EOSDeep4Cert as evidence for w = -1 would be overreading the source. The certificate is a general-purpose tool, and the cosmological application remains an open problem, waiting for a definition that connects the abstract cost function to the physics of dark energy.

structure EOSDeep4Cert where
  cost_at_eq : ∀ r : ℝ, r ≠ 0 → domainCost r r = 0
  cost_nonneg : ∀ m e : ℝ, 0 < m → 0 < e → 0 ≤ domainCost m e
  threshold_pos : 0 < canonicalThreshold

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]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

The certificate does not prove w = -1 or any statement about the dark energy equation of state. The certificate does not define what m and e mean for the universe; that definition is absent from the source file. The research note's prediction about DESI is a recorded intent, not a proved theorem.

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/Cosmology.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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