Cosmology Rs Cosmo Module 006 Rscosmo006 Cert

A certificate for a template

In mathematics, a certificate is a compact, machine-checkable record that a certain statement is true. The Recognition Science declaration RSCosmo006Cert is one such record. It packages three facts about a function called domainCost, which is defined for any two positive real numbers as Jcost (m / e). The three facts are: the function returns zero when its two inputs are equal; it never returns a negative number when both inputs are positive; and the number phi minus 3/2 is greater than zero, where phi is the golden ratio.

These facts are proved in the framework's machine-checked library of formal theorems. The first fact, that the cost is zero when the two inputs match, follows directly from the definition of Jcost. The second, that the cost is nonnegative for positive inputs, is a property of Jcost applied to a positive ratio. The third, that phi minus 3/2 is positive, is a simple numerical inequality that holds because phi is approximately 1.618. The certificate bundles these three proofs into a single object, and a further theorem confirms that this object exists.

In Recognition Science, this certificate appears in a module labeled "Cosmology RS Module 6." That module's research note states a hypothesis: that a dark matter mass of 1.787 GeV, equal to M_W/45, could be tested by the XENONnT experiment in 2026. The certificate itself proves nothing about this hypothesis. The definition of domainCost does not mention mass, dark matter, or any specific physical quantity. The three proved facts are general properties of the cost function, true for any positive inputs whatsoever.

The same three facts are stated once, universally, in a shared template used by 2383 sibling modules. What would turn this certificate into a theorem about its named subject is a definition of m and e in that subject's own terms, a definition the module does not provide. The certificate is therefore a proof about the cost function, not about cosmology.

What the certificate does establish is a clean, verified foundation for future work: any model that defines its inputs in physical terms can rely on these three properties without re-proving them. The dark matter hypothesis remains a prediction with a named falsifier, not a result.

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]

noncomputable def cert : RSCosmo006Cert where
  cost_at_eq := domainCost_at_eq
  cost_nonneg := domainCost_nonneg
  threshold_pos := canonicalThreshold_pos

theorem cert_inhabited : Nonempty RSCosmo006Cert := ⟨cert⟩

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

What this page does not claim

The certificate does not prove the dark matter mass prediction. The certificate does not identify the inputs m and e with any physical quantity. The certificate does not claim the 1.787 GeV value is derived from the framework.

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_006.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.

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