Foundation Ckmhierarchy From Phi Ladder Mass Ratio Top Up Pos Band

The top-to-up mass ratio

The six quarks, the building blocks of protons and neutrons, have masses that span an enormous range. The heaviest, the top quark, weighs about 172.7 GeV, while the lightest, the up quark, weighs about 2.16 MeV. Their ratio is roughly 80,000. In the standard model of particle physics, this hierarchy is an unexplained input, not a derived consequence. The Recognition Science (RS) framework attempts to derive it from a discrete ladder of masses, where each rung is the golden ratio φ ≈ 1.618 times the previous one.

Within RS, the mass of a quark at rung k is defined as m_unit · φ^k, with the top quark at rung 30 and the up quark at rung 8. The framework's library, a machine-checked collection of formal theorems, proves that the ratio of these two masses is exactly φ^22, which is approximately 39,089. The specific declaration mass_ratio_top_up_pos_band establishes a narrow, formal fact: this ratio is a positive number. It is a small piece of a larger proof, confirming that the predicted ratio lies above zero, a necessary check for any meaningful physical quantity.

The theorem's proof is trivial: it derives from the positivity of the golden ratio and the properties of exponents. The declaration itself does not assert that the ratio is close to the measured value. The empirical ratio of about 80,000 is roughly a factor of two larger than φ^22. The framework's own documentation acknowledges this gap, attributing it to a scale-running correction, but that correction is not part of the formal theorem. The theorem's scope is the abstract ladder, not the experimental data.

In Recognition Science, this formal result is a rung on a larger ladder of claims. It supports the framework's structural prediction for the quark mass hierarchy, but it does not, by itself, validate that prediction against nature. The declaration is a guarantee of internal consistency, not a statement of empirical agreement. The measured ratio remains a target for the framework, not a consequence of this particular theorem.

/-- The top-to-up mass ratio: `φ^(top_rung - up_rung) = φ^22`. -/
def mass_ratio_top_up : ℝ := phi ^ 22

/-- The ratio is positive (used downstream). -/
theorem mass_ratio_top_up_pos_band : 0 < mass_ratio_top_up :=
  mass_ratio_top_up_pos

/-- The ratio is positive (used downstream). -/
theorem mass_ratio_top_up_pos_band : 0 < mass_ratio_top_up :=
  mass_ratio_top_up_pos

What this page does not claim

The theorem does not claim the predicted ratio matches the measured value. The theorem does not derive the rung assignments for the quarks; they are definitional choices. The theorem does not provide a physical mechanism for the mass ladder.

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/CKMHierarchyFromPhiLadder.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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