Cosmology Hubble Tension From Bit Jcost Phi Pos

The Hubble tension band

The Hubble tension is the measured disagreement between two ways of finding the universe's expansion rate. Local observations of supernovae give one value for the Hubble constant, while the cosmic microwave background gives another. The two values differ by roughly 8 to 9 percent, and the gap has resisted conventional explanation for years. In Recognition Science, this gap is not a puzzle to solve but a number to derive.

The framework's machine-checked library of formal theorems contains a module named HubbleTensionFromBIT, where BIT means the framework's basic information theory. The module defines a quantity called the Hubble tension amplitude: the local value divided by the CMB value, minus one. The framework models this amplitude as the product of two factors: a cost function J evaluated at the golden ratio φ, and the natural logarithm of 2. The first factor J(φ) is the framework's recognition cost at the golden ratio, a number that the framework's core theorems force to lie in the interval (0.11, 0.13). The second factor log(2) is approximately 0.693. Multiplying the two gives a band from about 0.076 to 0.090, which brackets the empirically measured tension of roughly 0.08 to 0.09.

The formal declaration jcost_phi_pos proves one narrow fact: the cost J(φ) is strictly greater than zero. That is all it establishes. The positivity is a necessary ingredient for the larger band result, but the declaration itself does not prove the band, does not compare against any measurement, and does not claim the framework explains the Hubble tension. Those larger claims live in other declarations in the same module, each with its own proof. The positivity proof is a small gear in a larger machine, not the machine itself.

What the declaration does not claim matters as much as what it proves. It does not assert that J(φ) equals any particular number; it only proves the cost is positive. It does not state that the Hubble tension amplitude falls in the measured range; that is a separate theorem with its own band. And it does not claim that the framework's prediction matches observation. The match between the derived band and the empirical range is a comparison, not a proof. The framework's library proves the band from its axioms; the agreement with SH0ES and Planck data is an empirical check, and the declaration says nothing about it.

/-- J(φ) ∈ (0.11, 0.13). -/
theorem jcost_phi_band :
    (0.11 : ℝ) < Jcost phi ∧ Jcost phi < 0.13 := by
  rw [Constants.Jcost_phi_val]
  exact ⟨by linarith [phi_gt_onePointSixOne],
         by linarith [phi_lt_onePointSixTwo]⟩

/-- J(φ) > 0. -/
theorem jcost_phi_pos : 0 < Jcost phi :=
  Jcost_pos_of_ne_one phi phi_pos phi_ne_one

/-- RS Hubble tension amplitude = J(φ) × log(2). -/
noncomputable def hubbleTensionAmplitude : ℝ :=
  Jcost phi * Real.log 2

What this page does not claim

The declaration jcost_phi_pos does not prove the full Hubble tension band, which requires the separate jcost_phi_band theorem. The declaration does not assert that the framework's prediction matches the SH0ES or Planck measurements. The declaration does not identify J(φ) with any specific numerical value beyond its positivity.

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/HubbleTensionFromBIT.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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