Astrophysics Rs Astro Module 004

Module 004 and its scope

Astrophysics RS Module 004 is one of 2,384 sibling modules in the Recognition Science library, and its main text is a research note, not a result. The note says that the recognition cost, a measure of how expensive it is for one quantity to stand in for another, applied to the ratio of Jupiter's mass to Earth's mass gives a period of about 11.09 years, close to Jupiter's observed orbital period of 11.86 years, a 6.5 percent gap. That comparison is a structural guess, a placeholder for where the idea was meant to go.

What the module actually proves in the machine-checked library of formal theorems is much narrower. It defines a function domainCost that takes two real numbers, m and e, and returns the recognition cost of their ratio. The library proves three general facts about this function: it equals zero when the two inputs are equal, it never goes negative for positive inputs, and a certain threshold value, phi minus 1.5, is positive. These are properties of the cost function itself, not of Jupiter or of any astronomical system.

The gap between the note and the theorems is the whole story. The module defines domainCost as the cost of m divided by e, but it never defines what m and e mean for Jupiter or Earth. Without those definitions, the period claim has no formal support. The theorems hold for any positive numbers, so they would apply just as well to the mass of a pencil and the mass of a moon. The module is a template, not a finished astrophysical derivation.

In plain language, Module 004 establishes that the cost function has three basic, expected properties: zero at equality, nonnegativity, and a positive threshold. It does not establish that Jupiter's period follows from the cost function. The research note is a statement of intent, and the formal content is a universal fact about a function, shared verbatim across thousands of sibling modules. What would turn this into a theorem about Jupiter is a definition of m and e in astronomical terms, which the module does not provide.

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

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]

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)

What this page does not claim

This module proves the Jupiter period match. This module defines m and e in astronomical terms. The 6.5 percent gap is a measured or derived result. The module's theorems are specific to astrophysics.

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_004.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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