higgs_mass_squared_pos
plain-language theorem explainer
The theorem establishes that the squared Higgs mass is positive in the Recognition Science derivation of spontaneous symmetry breaking. Researchers working on J-cost models of the Standard Model would cite it to confirm vacuum stability after symmetry selection at the golden ratio. The proof is a short term-mode reduction that unfolds the mass definition and invokes positivity of a power of phi together with the reciprocal of a positive quantity.
Claim. In the J-cost formulation of the Higgs mechanism, the squared mass of the Higgs boson satisfies $m_H^2 > 0$.
background
The module derives the Higgs mechanism from the J-cost functional $J(x) = ½(x + 1/x) - 1$, which is symmetric under $x ↔ 1/x$ and minimized at $x = 1$. Symmetry breaking occurs when the vacuum selects the golden ratio φ, so that particle masses arise proportional to the recognition cost at that point. Upstream results include the cost definition from ObserverForcing (non-negative J-cost of any recognition event) and the cost induced by multiplicative recognizers, together with ledger factorization that calibrates J.
proof idea
The term proof first unfolds the definition of higgsMassSquared. It then applies one_div_pos.mpr to reduce the claim to positivity of a power of the golden ratio, which follows directly from the lemma phi_pos.
why it matters
This positivity result is required for the physical viability of the J-cost derived Higgs mechanism, ruling out tachyonic instabilities after symmetry breaking. It supports the subsequent gauge-boson mass derivations in the same module and aligns with T5 J-uniqueness and T6 phi fixed point in the unified forcing chain. No downstream uses are recorded yet.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.