Cosmology Baryogenesis Trajectory From Phi Ladder Eta B Pos

The positivity theorem

Baryogenesis is the process by which the early universe produced more matter than antimatter, leaving the surplus of protons and neutrons that fill galaxies today. Cosmologists measure this surplus as a dimensionless number, the baryon-to-photon ratio, usually written η_B. The standard model of particle physics cannot explain why this number is about six parts in ten billion; it must be put in by hand. A framework called Recognition Science (RS) attempts to derive such constants from a single forced cost function, and its library of machine-checked theorems contains a file that models how η_B might grow as the universe cools.

The declaration etaB_pos is one small theorem inside that file. It proves that the model's value for η_B is strictly greater than zero at every rung of the cooling ladder. The model defines a ladder of temperatures T_k = T_GUT · φ^(-k), where φ is the golden ratio, roughly 1.618, and k is a nonnegative integer counting downward from the grand unified theory scale. At each rung the asymmetry is defined as η_B(k) = φ^k / φ^44. The theorem states that for every k, this expression is positive, which follows because both numerator and denominator are powers of a positive number. The proof is a short chain of arithmetic facts: φ is positive, powers of positive numbers are positive, and the quotient of two positive numbers is positive.

The theorem also sits inside a larger certificate structure that bundles four properties of the trajectory: the ratio between successive rungs equals φ, the value at k = 44 equals 1, the value is always positive, and there are five named baryon-violation channels (sphaleron, electroweak, QCD, leptogenesis, neutrino mass). The certificate bundles these as a single object, but etaB_pos itself only establishes the positivity clause. It does not establish that the model matches the measured value, nor that the trajectory is physical, nor that the five channels are the correct ones. Those are separate claims, and the file's docstring describes the model as a prediction, not a derivation from first principles.

In plain terms, etaB_pos is a hygiene check: it guarantees that the model never produces a negative or zero asymmetry, which would be unphysical. It is the kind of lemma that a formal development needs before it can reason about ratios or logarithms later in the chain. The theorem is proved in the framework's machine-checked library with no axioms beyond the standard logical ones, but that proof only covers the arithmetic of the definition. Whether the definition corresponds to the real universe is a question the theorem does not touch.

What the declaration does not claim is worth stating sharply. It does not say that the universe's baryon asymmetry is φ^(-44), nor that the golden ratio appears in nature. It does not prove that the electroweak scale is the right endpoint, nor that the five channels are exhaustive. It only proves that a particular algebraic expression, chosen by the model, stays positive. The physical interpretation, the connection to measured values, and the choice of the φ-ladder itself all remain outside the scope of this single theorem.

/-- η_B is strictly positive on the whole trajectory. -/
theorem etaB_pos (k : ℕ) : 0 < etaB k := by
  unfold etaB
  exact div_pos (pow_pos phi_pos k) (pow_pos phi_pos 44)

noncomputable def etaB (k : ℕ) : ℝ := phi ^ k / phi ^ 44

noncomputable def baryogenesisCert : BaryogenesisCert where
  etaB_rung_ratio := etaB_ratio
  etaB_complete := etaB_at_gap45
  etaB_always_pos := etaB_pos
  five_channels := bViolationChannel_count

What this page does not claim

The theorem does not assert that the universe's baryon asymmetry equals φ^(-44). The theorem does not prove that the golden ratio appears in physical baryogenesis. The theorem does not establish that the five named channels are the correct or exhaustive set.

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

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