Cosmology Rs Cosmo Module 004

A template, not a cosmology

Cosmology RS Module 004 is a formal file in the Recognition Science library that defines a cost function and proves three general facts about it. The cost function takes two real numbers, m and e, and returns J(m/e), where J is the framework's forced cost function. The module proves that this cost is zero when m equals e, that it is never negative for positive inputs, and that the number phi minus 3/2 is positive. All three statements are proved in the machine-checked library of formal theorems, so they are certain within the framework's axioms.

The surprise is what the module does not do. Its own documentation says the file is shared verbatim with 2,383 sibling modules, and that the content is a template, not a result about cosmology. The file defines m and e only as real numbers; it never defines them as, say, matter density and expansion rate. Without a subject-specific definition of m and e, the theorems are true but empty of physical content. The module records where the idea was meant to go, not a conclusion about the universe.

One research note sits at the top of the file: a value for the scalar spectral index n_s of 0.9556, compared with the Planck measured value of 0.9649, a difference the note calls a 2.2 sigma tension. That comparison is not a theorem. It is a handwritten aspiration, explicitly marked OPEN, and the module proves nothing that would connect its cost function to that number.

What the module does establish, in plain language, is a small piece of mathematical hygiene. It shows that the framework's cost function behaves sensibly at the point where its two inputs agree, that it never reports a negative cost, and that a particular threshold built from the golden ratio is positive. Those facts are real and checked. What would turn this file into a cosmology result is a definition of m and e in the subject's own terms, and that definition does not exist here.

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]

structure RSCosmo004Cert 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 module proves any statement about the scalar spectral index n_s or its tension with Planck data. This module establishes a cosmological model or any physical prediction about the universe. The value 0.9556 for n_s is a derived result of the framework; it is an OPEN research note.

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_004.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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