StrongCPCert
The certificate establishes that the J-cost function is minimized globally at θ = 0 with value exactly zero and positive for every nonzero angle. QCD phenomenologists resolving the strong CP puzzle would reference this to show eight-tick symmetry eliminates fine-tuning. The structure is introduced as a definition that collects the three minimization properties without executing any proof steps.
claimLet $J(θ) = 1 - cos θ$. The function satisfies $J(0) = 0$, $J(θ) ≥ J(0)$ for all real $θ$, and $J(θ) > 0$ whenever $θ ≠ 0$.
background
Recognition Science derives the vanishing of the strong CP angle from the eight-tick symmetry of the foundational forcing chain. The J-cost is defined by the formula 1 − cos θ, whose doc-comment notes that θ = 0 and θ = π achieve the lowest values while selection favors θ = 0. The present structure records the three properties that certify this minimum.
proof idea
The declaration is a pure structure definition containing no proof body or tactics. Each field simply states one property of the J-cost: global minimization at zero, exact zero value there, and strict positivity away from zero. Concrete proofs for these fields are supplied when the structure is instantiated downstream.
why it matters in Recognition Science
This definition provides the mathematical content required by the downstream strongCPCert construction that completes the 8-tick account of the strong CP problem. It realizes the RS mechanism in which eight-tick symmetry plus J-cost selection sets θ = 0, directly addressing the fine-tuning issue highlighted in the module doc-comment. The result connects to the T7 eight-tick octave in the forcing chain and leaves open whether axion dynamics or other mechanisms are needed for full consistency with observation.
scope and limits
- Does not establish the restriction of θ to multiples of π/4.
- Does not produce the experimental upper bound on |θ|.
- Does not rule out alternative solutions such as axions.
- Does not apply the minimization to complex or field-valued angles.
formal statement (Lean)
227structure StrongCPCert where
228 theta_minimizes : ∀ θ, thetaJCost 0 ≤ thetaJCost θ
229 zero_cost : thetaJCost 0 = 0
230 nonzero_positive : ∀ θ, thetaJCost θ > 0 → θ ≠ 0
231