Law 22 — Wigner classification of particles (Đợt 2 · 10/05/2026 v3.3)

Particles = unitary irreducible representations of the Poincaré group. Two physical classes: massive (spin s) and massless (helicity h). In SPT, yao count per particle determines max spin via SU(2) tensor product. All 17 SM particles match their Wigner class with SPT yao count.

Created 05/14/2026, 01:28 GMT+7Updated 05/14/2026, 01:28 GMT+7

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🎭 Law 22 (Wigner · Tier-B EXACT): Particles partition into Poincaré irreps. Massive (P² > 0) with spin s, massless (P² = 0) with helicity h. SPT yao count gives spin/helicity via SU(2) tensor product. All 17 SM particles match.

§1 Cách verify hoạt động (5 bước)

Step 1 — Poincaré algebra

10 generators: 4 translations P^μ + 6 Lorentz M^{μν}. SymPy verifies the Lie commutation relations.

Step 2 — Casimir operators

P² (mass²) and W² (Pauli-Lubanski, spin) commute with all 10 generators ⇒ label irreps.

Step 3 — 4 classes

P² > 0 (massive, spin s); P² = 0, P ≠ 0 (massless, helicity h); P² < 0 (tachyon — excluded); P² = 0, P = 0 (vacuum).

Step 4 — Yao SU(2) tensor

Each yao = SU(2) doublet. n-yao composite = (1/2)^⊗n. Decompose to find max spin n/2.

Step 5 — SM particle match

Electron (1 yao, spin 1/2), photon (0 yao, helicity ±1), Higgs (0 yao, spin 0), pion (2 yao, spin 0). All 17 SM particles match.

§2 Dẫn chứng SymPy

SymPy verify — download for offline testSYMPY ✓

Reproduce Wigner classification with SymPy

Symbolic Poincaré algebra + yao tensor product + SM matching. ~170 LOC.

scripts/spt_wigner.py

spt_wigner.py — verifies 4 irrep classes + 17/17 SM particle matches

170 LOCDownload

Reproduce in 30 seconds

pip install sympy numpy && python3 scripts/spt_wigner.py

Or quick-verify with AI (Grok / Claude / ChatGPT)

Don't want to install Python? Paste the prompt straight into Grok / Claude / ChatGPT / Gemini — the AI fetches the public script URL below and independently verifies each assertion in ~30 s. Open grok.com or claude.ai , paste, send.

⚠️ AI can be wrong — running the Python above is the only 100% certain check. Full AI guide →

Inputs: Bagua integers + π/√ only — no CODATA, no PDG, no calibration (Tier B). SymPy-verified as exact fractions (not floating-point). See full context at /theory/sympy-breakthrough-2026.

§3 Độ chính xác

📊 Δ ≡ 0 for all 17 SM particles. Tier-B EXACT.

§4 So sánh với học thuyết hiện đại

Wigner (1939): stand-alone group-theoretic classification, requires Poincaré invariance + spin postulate. SPT: derives the classification from yao SU(2) tensor product + Lorentz invariance (Law 3).

§5 Tầm quan trọng

🌟 HIGH — Wigner classification underpins all of particle physics taxonomy. SPT delivers it from internal Bagua structure rather than as a separate axiom.

§6 Falsifiable claim

📣 SPT claim: every particle's spin/helicity matches its yao count via SU(2) tensor. Falsifier: discovery of a particle whose spin contradicts yao-count prediction.

§7 Kết luận

✅ Wigner's particle classification is internal to SPT: yao tensor product determines spin; Lorentz invariance is automatic.