Cosmology Inflaton Mass3 From Phi Ladder Inflaton Mass3 Cert

What the certificate proves

In cosmology, the inflaton is the hypothetical field thought to drive the rapid exponential expansion of the early universe. Its mass is often estimated around 10^13 GeV, a scale far above what any collider can reach. The Recognition Science framework keeps a discrete record of recognition events, called a ledger, and assigns each event a cost. The declaration InflatonMass3Cert is a formal structure in the framework's machine-checked library of formal theorems. It bundles three proven facts about the cost function J(x) = (x + 1/x)/2 - 1 when applied to a ratio of two positive numbers.

The first fact is that the cost vanishes when the two numbers are equal: if m and e are the same positive value, then J(m/e) = 0. The second is that the cost is never negative for positive inputs. The third is that the golden ratio phi, approximately 1.618, exceeds 1.5 by a positive amount. These three statements are general properties of the cost function itself. The certificate assembles them into a single object and proves that such an object exists.

The name suggests a connection to the inflaton mass, and a research note in the source file records an intended path: write the inflaton mass as phi^k times a coherence energy E_coh, then choose k so the product lands near 10^13 GeV. With E_coh = 0.121 MeV and k = 57, the product phi^57 times 0.121 MeV gives about 10^13 GeV. That arithmetic is structural, meaning it is a definitional choice about how to express the mass, not a derived result.

What the certificate itself does not do is define m and e in terms of the inflaton. The cost function is defined as J(m/e) without any reference to a physical field. The three proven facts hold for any positive real numbers m and e. The certificate proves nothing specific to cosmology or to the inflaton mass. The research note is a paragraph recording where the idea was meant to go, not a result.

In Recognition Science, the framework models the inflaton mass as a point on a phi-power ladder, and the scaling law that places masses on that ladder is proved elsewhere. The certificate here is a smaller piece: it establishes the cost function behaves as expected for any positive ratio. A reader should see the certificate as a clean formal lemma about J, and the inflaton mass estimate as a separate, unformalized identification. The two are not yet connected by a definition of m and e in cosmological terms.

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 certificate does not derive the inflaton mass value. The certificate does not define m or e in terms of any physical field. The phi^57 arithmetic is a structural identification, not a proved 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/InflatonMass3_FromPhiLadder.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.

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