Astrophysics Rs Astro Module 009 Rsastro009 Cert

What the certificate proves

In the Recognition Science framework, a ledger is a discrete record of recognition events, and the cost of a recognition is a forced number, not a choice. The framework's central theorem proves that any cost function meeting five plain conditions must equal J(x) = (x + 1/x)/2 - 1. The declaration RSAstro009Cert is a machine-checked certificate, a small bundle of formal theorems, that certifies three general facts about that cost function when it is applied to a ratio of two quantities.

The three certified facts are plain. First, when the two quantities are equal, the cost is zero: J(1) = 0. Second, for positive quantities, the cost is never negative. Third, the number phi - 3/2, which the framework calls the canonical threshold, is positive; phi is the golden ratio, about 1.618, so the threshold is about 0.118. These are general properties of the cost function, not properties of any particular physical system.

The module carries an astrophysical label, ISM dust fraction, and a research note that says J(phi)^2 = 1.39% matches an empirical 1% figure. That note is not a theorem. The machine-checked part proves nothing about dust, because the module defines the cost as J(m / e) without ever defining what m and e mean for interstellar dust. The note records where the idea was meant to go, not a result. The same body is shared verbatim with 2383 sibling modules; the content is stated once, universally quantified, in a template module.

What the certificate does establish is a template: any subject that can define its own m and e in that subject's terms inherits the three certified properties. The astrophysical claim would become a theorem only if a definition of m and e were supplied in dust terms. Until then, the certificate is a general mathematical fact wearing a subject label.

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 that the ISM dust fraction equals J(phi)^2. The certificate does not establish any property specific to astrophysics. The certificate does not derive the golden ratio or the cost function; it assumes them.

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/Astrophysics/RS_Astro_Module_009.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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