Cosmology Rs Cosmo Module 010 Rscosmo010 Cert

What the certificate proves

In the Recognition Science framework, a ledger (a discrete record of events) assigns a cost (a forced price) to any ratio of two positive quantities. The declaration RSCosmo010Cert packages three proved facts about that cost function. First, when the two quantities are equal, the cost is zero. Second, for any two positive inputs, the cost is never negative. Third, a particular threshold value, the golden ratio minus 1.5, is greater than zero. These are general properties of the cost function, not claims about any specific physical system.

The certificate's name suggests a connection to cosmology, specifically to solar metallicity, the fraction of the Sun's mass that is not hydrogen or helium. A research note attached to the module records an intended match: the square of the cost function at the golden ratio, about 0.01393, is close to the accepted solar value of about 0.014. That note is explicitly not a result. The machine-checked part of the module defines the cost as a function of a ratio m over e, but never defines what m and e mean in astronomical terms. Without such definitions, the certificate proves nothing about the Sun or about metallicity.

The three proved facts hold for any positive real numbers m and e. The certificate is a template, shared verbatim across 2383 sibling modules, each with a different subject in its name but the same generic content. The universal version of these facts lives in a single shared module, and this cosmology module simply instantiates it. What would turn this certificate into a theorem about solar metallicity is a definition of m and e in the Sun's own terms, for instance as measured masses of hydrogen and heavier elements. That definition does not appear here.

In plain terms, the certificate is a small, correct, and entirely general piece of mathematics wearing a cosmology-themed label. It proves that a certain cost function behaves sensibly: zero at equality, nonnegative everywhere, and with a positive threshold. It does not, and cannot, establish anything about the actual Sun. The research note is an aspiration, not an achievement. A reader who wants a result about solar metallicity must look for a module where the physical quantities are defined, not just named.

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]

structure RSCosmo010Cert 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

This certificate does not prove any property of the Sun's actual metallicity. The numerical match with Z_sun is a research note, not a theorem. The certificate does not define what m and e mean in astronomical terms.

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_010.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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