Cosmology Hubble Constant Precise2 From Jcost Hubble Precise2 Cert

The certificate's scope

The Hubble constant H₀ measures the rate at which distant galaxies recede, roughly 67.4 kilometers per second per megaparsec according to Planck satellite data. In the Recognition Science framework, a ledger, a discrete record of recognition events, assigns a cost, a forced penalty, to any ratio of two quantities. The declaration HubblePrecise2Cert packages three proved facts about that cost function when it is applied to a ratio m/e.

The three facts are general and modest. First, the cost vanishes when m equals e, meaning there is no penalty for matching a quantity to itself. Second, the cost is never negative for positive inputs, so it never rewards a mismatch. Third, the golden ratio minus 1.5 is positive, a small numerical threshold. Each is a theorem in the machine-checked library of formal theorems, and together they form a structure called a certificate.

The certificate's name suggests a precise value for the Hubble constant, but the formal content does not deliver one. The definition of domainCost uses the ratio m/e without connecting m to a measured recession velocity or e to an age of the universe. The docstring records an intended research direction, where H₀ might equal phi^k divided by a 13.8 billion year timescale, but that paragraph is a note, not a result. What the certificate establishes is that the cost function behaves sensibly at equality and positivity, and that a threshold involving phi is positive.

In plain terms, the certificate proves a small piece of mathematical hygiene. It says the cost function has no kink at the identity, no negative values, and a specific constant is positive. It does not measure the Hubble constant, derive its value from first principles, or compare against any observational data. The gap between the name and the content is deliberate: the formal proof stops where the physics would have to begin, at a definition of m and e in cosmological terms.

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 HubblePrecise2Cert 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

The certificate does not measure or derive the Hubble constant's value. The certificate does not compare against any observational cosmology data. The certificate does not establish that phi to a power equals any specific cosmological constant.

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/HubbleConstantPrecise2FromJCost.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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