Cosmology Cmbpolarization3 From Jcost

A formal shell for a cosmic ratio

The cosmic microwave background (CMB) is the oldest light in the universe, and its polarization carries a signature of the conditions just after the Big Bang. When light scatters off free electrons in the hot early plasma, it becomes linearly polarized, and the pattern of that polarization splits into two parts: E-modes, which are curl-free, and B-modes, which have handedness. E-mode polarization is the stronger signal, roughly a tenth of the temperature anisotropy, and it traces the density fluctuations that seeded galaxies.

One module in the Recognition Science framework, CMBPolarization3_FromJCost, builds a formal object around this ratio. The framework's central tool is a cost function, a measure of how expensive it is for a system to recognize one state as another. The module defines a cost for a ratio of two quantities, m and e, as J(m/e), where J is the framework's unique forced cost function. The machine-checked library of formal theorems proves three general facts about this cost: it is zero when m equals e, it is never negative for positive inputs, and a threshold value built from the golden ratio is positive.

These three facts are true, but they are true for any positive numbers, not for anything specific to cosmology. The module defines domainCost without saying what m and e physically are; they could be masses, energies, or any two positive real numbers. The name of the module records a research intention, that m and e should become the E-mode polarization and the temperature anisotropy, but the formal proof does not connect the cost to that physics. The research note inside the module sketches a hoped-for ratio: J(phi) times phi gives about 55.7 percent, while J(phi) alone gives 11.8 percent, close to the observed 10 percent. That sketch is a hypothesis, not a proved result.

What the module does establish, with machine-checked certainty, is the arithmetic shell: the cost function behaves as the framework requires, vanishing at equality and staying nonnegative, and the golden-ratio threshold is a positive number. What it does not establish is that this cost describes CMB polarization. The bridge from the formal cost to the physical E/T ratio remains open; the module is a template waiting for a definition of m and e in cosmological terms.

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]

What this page does not claim

The module does not prove that the E/T ratio equals J(phi) or any specific value; that remains a research note. The module does not connect its cost function to any physical quantity in cosmology; m and e are undefined real numbers. The module does not establish the framework's cost function as the unique description of CMB polarization.

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/CMBPolarization3_FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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