Yin-Yang Z₂ symmetry — the SPT discrete invariance behind θ_QCD ≡ 0 and m_ν1 ≡ 0
Definition and consequences of the yin↔yang Z₂ involution: the Bagua membrane is invariant under φ → −φ (yang ↔ yin exchange). This single discrete symmetry forbids two CP-odd Standard-Model terms simultaneously: θ_QCD F·F̃ (strong-CP) and Majorana m_ν^M (lepton-number violation). Both are EXACT predictions of SPT, with current experimental bounds confirmed.
Created 05/14/2026, 01:28 GMT+7Updated 05/14/2026, 01:28 GMT+7
📖 Glossary entry expanded into a full sub-page. See Glossary §3 for the short reference. Sibling pages: Yin-Yang Node geometry.
1. Definition
The yin-yang Z₂ symmetry is the discrete involution φ → −φ acting on the Bagua membrane phase field φ. Under this map:
Yang state (☰ +1)
Sent to Yin state (☷ −1) and vice-versa.
Phase φ ∈ ℝ
Sent to −φ.
Action S = ∫dτ[½Ẋ² + iψ̄γψ + ½Tr(J·Ṙ) − V(φ)]
Invariant: V(−φ) = V(φ) for the symmetric Bagua potential. Z₂ is a TRUE symmetry of the Action.
Hexagram swap
Each hexagram is sent to its DUAL (yin↔yang lines exchanged). E.g. ☰ Càn ↔ ☷ Khôn, ☱ Đoài ↔ ☶ Cấn, etc.
2. Two EXACT consequences in the Standard Model
Z₂ symmetry forbids any term in the Action that is ODD under φ → −φ. Two such terms exist in the Standard Model and would otherwise be free parameters:
(a) θ_QCD F·F̃ (strong-CP angle)
θ_QCD ≡ 0 EXACTLY in SPT. The neutron electric dipole moment (nEDM) is forbidden. Current bound (nEDM-PSI 2020): |d_n| < 1.8 × 10⁻²⁶ e·cm (corresponds to θ_QCD < 10⁻¹⁰) — PASS.
(b) Majorana neutrino mass m^M·νν
m_ν1 ≡ 0 EXACTLY in normal hierarchy (Majorana mass term forbidden). Σm_ν = √Δm²_21 + √(Δm²_21 + Δm²_32) = 59.5 ± 0.3 meV. PASS at 1.22× headroom vs DESI 2024 Y1 (72 meV).
ONE symmetry, TWO predictions, ZERO new parameters. The yin-yang Z₂ is the SPT mechanism that solves the strong-CP problem (without an axion) AND fixes the lightest neutrino mass to zero (without ad-hoc tuning). Both predictions are EXACT (Tier-B) — refuted only by detection of a non-zero nEDM or non-zero m_ν1 above the experimental thresholds.
3. SymPy verification
θ_QCD ≡ 0
scripts/spt_qcd_theta.py — verifies the Z₂ argument symbolically.
m_ν1 ≡ 0 + Σm_ν = 59.5 meV
scripts/spt_neutrino_absolute_v2.py — Stage 1 derives m_ν1 = 0 from Z₂.
Yin-yang Node geometry r_yy
scripts/spt_yinyang_node.py — uses Z₂ to constrain orbital motion of poles.
4. Falsifiability claim FC-Z2
FC-Z2 (yin-yang Z₂ symmetry): The Bagua membrane Action is invariant under φ → −φ. Falsified if: (a) neutron EDM is measured at |d_n| > 10⁻²⁶ e·cm at >5σ; OR (b) any massive Dirac/Majorana neutrino with m > 0 corresponds to the LIGHTEST eigenstate (m_ν1 ≠ 0); OR (c) any new CP-odd Standard-Model term (e.g. neutron-antineutron oscillation > limits) is observed.
Comments — Yin-Yang Z₂ symmetry — the SPT discrete invariance behind θ_QCD ≡ 0 and m_ν1 ≡ 0