Cosmology Rs Cosmo Module 002 Rscosmo002 Cert

A certified template

The declaration RSCosmo002Cert is a formal certificate, a machine-checked bundle of three mathematical facts. It is built from a single definition: domainCost, a cost function that measures the price of recognition when a system moves from one state to another. The certificate proves that this cost is zero when the two states are identical, that it is never negative for positive inputs, and that a certain threshold number, phi minus 1.5, is greater than zero. These are the only claims the certificate establishes.

The certificate's name suggests a statement about cosmology, and the module header records a research note about a cosmological constant. That note is not a result. The definition of domainCost uses the cost function J with a ratio of two real numbers, m and e, but the certificate never defines what m and e mean. Without a definition of those quantities in physical terms, the certificate proves nothing about the universe. It is a template, shared verbatim with 2,383 sibling modules, each of which names a different subject but proves the same three general facts.

In Recognition Science, the framework's library of formal theorems provides the underlying cost function J and its properties. The certificate assembles those properties into a structure, a named bundle that can be passed to other proofs. The threshold phi minus 1.5 is positive because the golden ratio phi is greater than 1.5, a fact the library proves. The certificate's value is organizational: it packages three known facts under one name, ready for reuse. It does not add new mathematics, and it does not connect those facts to any physical model.

The distinction matters for anyone reading the module list. A certificate named for cosmology that proves no cosmology is a placeholder, not a result. What would make it a theorem about its subject is a definition of m and e in that subject's own terms, for example as a mass and an energy scale. Until that definition exists, the certificate is a clean example of what the framework can certify and a clear statement of what it cannot yet claim.

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)

structure RSCosmo002Cert 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

What this page does not claim

The certificate does not prove any value for the cosmological constant. The certificate does not define mass or energy in physical terms. The certificate does not establish that the research note's numerical agreement is a 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/RS_Cosmo_Module_002.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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