Cosmology Inflaton Mass3 From Phi Ladder

The inflaton mass estimate

In cosmology, the inflaton is the hypothetical field thought to drive the rapid exponential expansion of the early universe. A common estimate for its mass is around 10^13 GeV, a scale far above anything reachable by particle colliders. The Recognition Science framework offers a way to express such a mass as a power of the golden ratio φ, approximately 1.618, multiplied by a coherence energy scale.

The specific module in question defines a cost function, a measure of the discrepancy between two energies, as J(m/e), where J is a fixed function from the framework's core. The machine-checked library of formal theorems proves three general facts about this cost: it equals zero when the two energies are equal, it is never negative for positive inputs, and the quantity φ - 3/2 is positive. These are properties of the cost function itself, not of any particular physical system.

In Recognition Science, the framework models the inflaton mass as φ^k times a coherence energy E_coh. With E_coh at 0.121 MeV and k = 53, the estimate lands near 12 PeV, while k = 57 pushes it toward 10^13 GeV. This is a structural suggestion, a research note recording where the idea was meant to go, not a derived result. The module proves nothing specific to the inflaton because its definition of cost does not reference the inflaton's own mass or energy scale.

What the module does establish is a template. It shows that the cost function behaves sensibly: zero at equality, nonnegative otherwise, with a positive threshold constant. These are the building blocks any physical application would need. The gap between the template and a real prediction is a definition of m and e in the inflaton's own terms, which the module does not provide.

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

No claim that the inflaton mass is actually 10^13 GeV, only that a structural estimate exists. No claim that the cost function's properties constitute a derivation of the inflaton mass. No claim that the phi-power ladder is the unique way to express the inflaton mass.

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/InflatonMass3_FromPhiLadder.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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