Foundation Qrft Fermion Kinetic Cert Fermion Mass At Pos

Mass at a rung

The declaration fermionMassAt_pos is a theorem in the framework's machine-checked library of formal theorems. It states a simple positivity fact: if the starting mass m0 is a positive real number, then the mass at every rung k of the ladder is also positive. The mass at rung k is defined as m0 * phi^k, where phi is the golden ratio, approximately 1.618. Because m0 is positive and any power of the golden ratio is positive, their product is positive. The theorem is proved by a short chain of arithmetic steps in the Lean proof assistant, with no axioms beyond the standard three and no gaps marked as unfinished.

The fact matters because it underpins the framework's structural claim about the Standard Model. The framework models the fermion mass term m ψ̄ψ as a cost on a recognition ratio, and it predicts that fermion masses sit on a ladder where each step multiplies the previous mass by the golden ratio. The positivity theorem is a necessary sanity check for that ladder: a mass that could turn negative at some rung would break the physical interpretation of mass as a positive quantity. The theorem also feeds into the certificate structure FermionKineticCert, which bundles this positivity fact with the ratio property that adjacent rungs differ by exactly phi.

What the theorem does not claim is just as important. It does not say that any particular fermion mass equals m0 * phi^k for some measured m0 and k. That comparison against measured particle masses is an empirical check, not part of the theorem. The theorem also does not derive the value of m0 itself; it takes m0 as an input and proves a property of the sequence it generates. Finally, the theorem says nothing about the kinetic term ψ̄ γ^μ ∂_μ ψ or about the 15 Weyl fermions per generation; those are separate structural claims in the same module, not consequences of this positivity result.

theorem fermionMassAt_pos {m0 : ℝ} (hm : 0 < m0) (k : ℕ) :
    0 < fermionMassAt m0 k := by
  unfold fermionMassAt
  exact mul_pos hm (pow_pos Constants.phi_pos k)

/-- Fermion mass at φ-ladder rung `k`. -/
def fermionMassAt (m0 : ℝ) (k : ℕ) : ℝ := m0 * phi ^ k

theorem fermionMassAt_pos {m0 : ℝ} (hm : 0 < m0) (k : ℕ) :
    0 < fermionMassAt m0 k := by
  unfold fermionMassAt
  exact mul_pos hm (pow_pos Constants.phi_pos k)

What this page does not claim

No measured fermion mass is asserted to equal m0 * phi^k for any specific m0 and k. The theorem does not derive the value of the starting mass m0. The theorem says nothing about the kinetic term or the count of Weyl fermions per generation.

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

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