mass_positive
plain-language theorem explainer
Particle mass is strictly positive for any Yukawa coupling with positive strength. Researchers deriving the Standard Model spectrum from J-cost symmetry breaking would cite this to guarantee that all fermion masses remain positive after the vacuum selects the broken phase. The proof is a one-line term that unfolds the mass definition and applies the algebraic positivity of multiplication to the coupling factor and the vacuum expectation value.
Claim. For any Yukawa coupling structure $y$ whose coupling strength satisfies $y > 0$, the particle mass defined by $m(y) = y_coupling · v$ obeys $m(y) > 0$, where $v$ is the positive vacuum expectation value.
background
The module derives the Higgs mechanism from the J-cost functional $J(x) = ½(x + x^{-1}) - 1$, whose minimum at $x = 1$ supplies the vacuum and whose $x ↔ x^{-1}$ symmetry is broken when the vacuum selects the golden-ratio fixed point. The YukawaCoupling structure packages a particle identifier together with a positive real coupling strength that encodes the particle's ledger weight. The sibling result vev_pos establishes that the vacuum expectation value itself is positive, supplying the second factor required for mass positivity. Upstream lemmas on ledger factorization and phi-forcing derived supply the ordered-field infrastructure that makes these positivity statements available without additional axioms.
proof idea
The term proof first unfolds the definition of particleMass, which expands to the product of the Yukawa coupling strength and the vacuum expectation value. It then applies the mul_pos lemma directly to the two positive hypotheses: the coupling_pos field inside the YukawaCoupling structure and the vev_pos fact.
why it matters
The theorem supplies the mass_positive field required by the HiggsBosonCert structure, which is itself a component of the SpectralEmergenceCert that derives the full gauge group, three generations, and 24 fermion states from the single forced datum D = 3. It therefore completes one concrete step in the module's target derivation of spontaneous symmetry breaking and mass generation from J-cost, as described for a prospective PRL submission on recognition-based Higgs physics. The result protects the discrete phi-ladder mass formula against sign errors once the eight-tick octave has fixed the spatial dimension.
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