Cosmology Inflation Parameters5 Inflation Param5 Cert

A certificate's limits

In the Recognition Science framework, a ledger, a discrete record of recognition events, assigns a cost, a forced price, to comparing two quantities. The declaration InflationParam5Cert is a machine-checked bundle of three such facts. First, when the two quantities are equal, the cost is zero. Second, for any two positive quantities, the cost is never negative. Third, the number phi minus 3/2, where phi is the golden ratio, is positive. These are general properties of the cost function, not claims about any particular physical process.

The certificate's construction is direct. It packages three results already proved about the cost function J(x) = (x + 1/x)/2 - 1. The first result states that J(1) = 0, which gives the zero-cost-at-equality property. The second states that J is nonnegative for positive inputs, which gives the nonnegativity property. The third is a simple arithmetic fact about the golden ratio. The certificate simply collects these three results into one structure, and a companion result confirms that such a certificate exists.

What the certificate does not do is connect these facts to inflation. The module defines a quantity domainCost as J(m/e), a cost of comparing two real numbers m and e, but it never defines what m and e mean in cosmology. The research note attached to the module mentions spectral index n_s = 0.9649 and tensor-to-scalar ratio r < 0.036, with RS-derived values n_s = 1 - 2/45 = 0.9556 and r = 2/(45*phi^2) = 0.0169. The note claims these pass at 2.1 sigma for n_s and within bounds for r. But this note is explicitly marked as a research note, not a formal result. The formal results prove nothing specific to inflation, because the quantities m and e are undefined in that context.

In Recognition Science, this certificate is a template. The same three general facts appear verbatim in 2383 sibling modules, each with a different subject name but the same formal content. What would turn this template into a statement about inflation is a definition of m and e in cosmological terms, for instance as a measured scale versus a predicted scale. Without that definition, the certificate is a statement about the cost function alone, and the inflation connection remains a research aspiration, not a proved result.

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]

noncomputable def cert : InflationParam5Cert where
  cost_at_eq := domainCost_at_eq
  cost_nonneg := domainCost_nonneg
  threshold_pos := canonicalThreshold_pos

What this page does not claim

The certificate does not prove that the spectral index n_s equals 0.9556 or that the tensor-to-scalar ratio r equals 0.0169. The certificate does not establish that inflation parameters match observation; the research note is not a formal result. The certificate does not define what m and e mean in cosmology, so it makes no physical prediction.

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/Inflation_Parameters5.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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