Masses Anchor B Pow Up Quark Eq

The up quark yardstick

The up quark is the lightest of the six quarks, with a measured mass near 2.2 MeV/c². In the Recognition Science framework, each particle sector carries a yardstick, a real number built from a few fixed integers and the golden ratio. The declaration B_pow_UpQuark_eq establishes that one integer in that yardstick, the exponent B_pow for the up quark sector, equals -1. The proof is a short computation: the definition of B_pow for the up quark is the negative of A, and A is defined as one active edge per tick, so B_pow .UpQuark = -1 follows by simplifying the definitions. The machine-checked library of formal theorems records this as a theorem, not as an assumption.

The number -1 is not a mass. It is one factor in a product. The full yardstick for a sector is (2 ^ B_pow) times a coherence energy E_coh times the golden ratio raised to the r0 power. For the up quark, B_pow = -1 and r0 = 35, so the yardstick is 2⁻¹ · φ⁻⁵ · φ³⁵ = 2⁻¹ · φ³⁰ in the framework's units. The framework models particle masses as sitting on a phi-power ladder; the scaling law is proved, but the comparison against measured PDG values is an empirical check, not a theorem. This declaration fixes the exponent, it does not derive the mass itself.

What the declaration does not claim is as important as what it proves. The docstring for the module is explicit: the values are fixed, but the sector assignment is not yet forced. The theorem B_pow_UpQuark_eq proves that if you define B_pow for the up quark sector as -A, then the value is -1. It does not prove that the up quark sector must have this exponent, nor that the yardstick formula correctly predicts the measured up quark mass. The sector assignment, the choice of which formula belongs to which particle, is a model choice, not a forced consequence of the framework's axioms.

The distinction matters for reading the framework honestly. The theorem is real and machine-checked, but it lives in the Model layer, not in the forcing chain that derives constants like hbar and G. The framework's own documentation says no proofs claim experimental agreement. So the up quark exponent is a defined quantity with a verified arithmetic value, and the connection to the measured mass remains a hypothesis to be tested, not a result to be cited as established fact.

theorem B_pow_UpQuark_eq : B_pow .UpQuark = -1 := by
  simp only [B_pow, A, active_edges_per_tick]
  norm_num

/-- Sector yardstick `A_s = 2^{B_pow} * E_coh * φ^{r0}`. -/
@[simp] noncomputable def yardstick (s : Sector) : ℝ :=
  (2 : ℝ) ^ (B_pow s) * E_coh * Constants.phi ^ (r0 s)

/-- Derived powers of two for each sector.
    These are NOT arbitrary—they come from cube edge counting. -/
@[simp] def B_pow : Sector → ℤ
  | .Lepton      => -(2 * (E_passive : ℤ))     -- = -(2 × 11) = -22
  | .UpQuark     => -(A : ℤ)                    -- = -1
  | .DownQuark   => 2 * (E_total : ℤ) - 1       -- = 2 × 12 - 1 = 23
  | .Electroweak => (A : ℤ)                     -- = 1

What this page does not claim

The up quark mass is derived from first principles. The sector assignment of B_pow = -1 is forced by the framework's axioms. The yardstick formula has been verified against experimental data.

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/Masses/Anchor.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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