Cosmology Horizon Problem3 From Jcost Horizon Prob3 Cert

The certificate

The horizon problem in cosmology asks why regions of the universe so far apart that light has never had time to reach between them nevertheless look nearly identical in temperature. The standard answer is cosmic inflation, a brief period of extremely rapid expansion in the first fraction of a second. In the Recognition Science framework, the declaration HorizonProb3Cert packages three proved facts about a particular cost function, but it does not, by itself, establish that inflation happened or that the framework explains the horizon problem.

The cost function in question, domainCost, takes two positive real numbers, m and e, and returns J(m/e), where J is the framework's central cost function. The three proved facts are: first, when m equals e, the cost is zero; second, for any positive m and e, the cost is never negative; third, the number phi minus 1.5 is positive, where phi is the golden ratio. These are general facts about the cost function, not specific to cosmology. The certificate simply collects them into a single structure.

The research note attached to the module goes further, suggesting that inflation with about 44 e-folds of expansion, starting at a temperature related to phi, could resolve the horizon problem. But that note is explicitly marked as a research note, not a result. The formal theorems prove nothing about the actual universe, because the quantities m and e are not defined in terms of any physical measurement. The certificate is a template, shared verbatim with 2383 sibling modules, each of which would need its own definitions to become a theorem about its subject.

What the certificate does establish is a small piece of mathematical hygiene: the cost function behaves sensibly at equality, never returns a negative cost for positive inputs, and the golden ratio exceeds 1.5. These are necessary conditions for any further physical claim, but they are far from sufficient. A reader should take the certificate as a building block, not as a solution to the horizon problem.

structure HorizonProb3Cert 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]

What this page does not claim

The certificate does not prove that inflation occurred or that the horizon problem is resolved. The certificate does not define m and e in physical terms. The research note's inflation scenario is 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/HorizonProblem3_FromJCost.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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