Cosmology Hubble Constant Precise2 From Jcost

What the module proves

The Hubble constant H_0 measures how fast the universe expands, usually given as about 67.4 kilometers per second per megaparsec from Planck satellite data. The module HubbleConstantPrecise2FromJCost carries that number in its name, but the formal content does not derive it. The file defines a cost function domainCost, a measure of mismatch between two positive quantities, as J(m/e), where J is the framework's forced cost function. It then proves three general facts: the cost is zero when the two quantities are equal, the cost is never negative for positive inputs, and the golden-ratio threshold phi minus 3/2 is positive. These are true for any positive numbers m and e; they say nothing specific about the expansion rate.

The research note inside the file records a plan: write H_0 as phi^k divided by the age of the universe, 13.8 billion years, and match the Planck value at some rung k. That note is a statement of intent, not a proved result. The formal theorems do not define m as a measured distance scale or e as a theoretical time scale, so the module cannot certify any particular number for H_0. What the machine-checked library does establish is a small, clean set of properties about the cost function itself, reusable wherever a cost between two positive quantities appears.

In plain language: the file proves that the cost function behaves sensibly, zero only at agreement, nonnegative elsewhere, with a positive threshold. It does not prove that the universe expands at 67.4, or that phi raised to any power equals the Hubble constant. The gap between the name and the content is intentional in the framework's audit: the module is a template, shared with many siblings, waiting for a definition of m and e in cosmological terms. Until that definition exists, the Hubble-specific claim remains a research note, not a theorem.

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 cert_inhabited : Nonempty HubblePrecise2Cert := ⟨cert⟩

What this page does not claim

Not claimed: the module derives the value 67.4 km/s/Mpc for the Hubble constant. Not claimed: the module proves that H_0 equals phi^k divided by the age of the universe. Not claimed: the module establishes any empirical match between the framework and cosmological measurements.

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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