Cosmology Rs Cosmo Module 004 Rscosmo004 Cert

The certificate's scope

The Recognition Science framework builds physical structure from a single starting point: reality keeps a ledger, a discrete record of recognition events, and the cost of recognition is forced, not chosen. A machine-checked library of formal theorems has proved that any cost function satisfying five plain conditions must equal J(x) = (x + 1/x)/2 - 1. The declaration RSCosmo004Cert is a small certificate inside that library. It packages three general facts about this cost function J when applied to a ratio of two positive real numbers, m and e.

The three facts are these. First, when the two numbers are equal, the cost is zero: J(m/m) = 0. Second, for any two positive inputs, the cost is never negative: J(m/e) ≥ 0. Third, the golden ratio phi minus 3/2 is a positive number, which works out to about 0.118. Each fact is proved from the definition of J and from previously established theorems about it. The certificate bundles these three proofs into a single structure, and a companion proof shows that such a certificate exists. These are general mathematical facts about the cost function; they hold for any positive real numbers, with no further conditions.

The certificate's own documentation states plainly what it does not do. It proves nothing specific to cosmology, because the cost is defined simply as J(m/e) without any definition of what m and e mean in cosmological terms. The spectral index n_s = 1 - 2/45 = 0.9556, compared with the Planck measured value 0.9649, is a research note recording where the idea was meant to go, not a result. The gap between those numbers is about 2.2 sigma, and the certificate labels the status as OPEN. The certificate itself is a template, shared verbatim with 2383 sibling modules, each one waiting for a definition of its own m and e in that subject's own terms.

What the certificate does establish is a reusable mathematical skeleton. Any future module that defines m and e for a specific physical situation can inherit these three proved facts about the cost of the ratio. The certificate guarantees that the cost is zero at equality, nonnegative everywhere, and that the golden-ratio threshold is positive. That is a modest but solid foundation. The cosmological claim, the spectral index comparison, is not part of the certificate; it is a stated target for future work, and the certificate is explicit that the work remains open.

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 claim that the spectral index of the cosmic microwave background is 0.9556. The certificate does not claim that the Planck measured value of 0.9649 is wrong or in tension with a derived result. The certificate does not claim that phi - 3/2 is a physically meaningful threshold in cosmology.

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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: