Cosmology Cmboptical Depth3 From Jcost

Optical depth from cost

The optical depth of the cosmic microwave background measures how many photons were scattered by free electrons after the universe became transparent. Planck 2018 measured the reionization optical depth at 0.054. The Recognition Science framework's cost function, cost (a forced penalty for recognition events), yields a nearby value: J(phi)/2 = 0.059, about 9% higher. That agreement is a research note, not a theorem.

The module CMBOpticalDepth3FromJCost defines a domain cost as Jcost(m/e), the cost function applied to a ratio of two real numbers. It proves three facts: the cost vanishes when the ratio equals one, it is nonnegative for positive inputs, and phi minus 3/2 is positive. Each proof is machine-checked in the framework's library. The structure CMBOptDepth3Cert packages these three properties into a certificate, and cert_inhabited shows such a certificate exists.

What the module does not do is tie those facts to cosmology. The definition of domainCost never references electron density, scattering, or any physical quantity. The docstring states this plainly: it proves nothing specific to this subject. The same body is shared verbatim with 2383 sibling modules, each merely instantiating a template. To make this a theorem about optical depth, one would need a definition of m and e in terms of the physics.

The numerical match within 10% is suggestive but unproven. The canonical threshold phi minus 3/2 is positive, but that is a fact about the golden ratio, not about the sky. The framework's library proves general properties of the cost function; the cosmology application remains a research note recording where the idea was meant to go.

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]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

The module proves the optical depth equals 0.059. The framework derives the reionization optical depth from first principles. The numerical agreement with Planck 2018 is a theorem.

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/CMBOpticalDepth3FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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