Law 23 — Goldstone's theorem from broken U(1) (Đợt 2 · 10/05/2026 v3.3)

For each spontaneously broken continuous symmetry of the SPT Action, exactly one massless scalar (Goldstone boson) appears. Yin-yang U(1) breaking with Mexican-hat V(|φ|²) produces m_θ² ≡ 0 EXACT for the phase mode. SM electroweak: 3 Goldstones eaten by W^± and Z.

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

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⚡ Law 23 (Goldstone · Tier-B EXACT): Yin-yang U(1) breaking via Mexican-hat V(|φ|²) = −μ²|φ|² + ½λ|φ|⁴, minimized at |φ| = μ/√λ ≠ 0. Phase mode mass: m_θ² ≡ 0 EXACT. SM electroweak: 3 Goldstones eaten → W/Z mass.

§1 Cách verify hoạt động (5 bước)

Step 1 — Mexican-hat V

V(|φ|²) = −μ²|φ|² + ½λ|φ|⁴. SymPy minimizes: dV/d|φ| = 0 ⇒ |φ| = μ/√λ = v.

Step 2 — Polar decomposition

φ = (v + h) · exp(iθ/v) where h = radial fluctuation, θ = phase mode.

Step 3 — Compute mass²

m_h² = ∂²V/∂h² = 2μ² (Higgs mode, MASSIVE). m_θ² = ∂²V/∂θ² = 0 EXACT (Goldstone).

Step 4 — Ward identity

⟨0|J^μ(x)·φ(0)|0⟩ has a pole at p² = 0 ⇒ massless mode required.

Step 5 — SM EW count

SU(2)_L × U(1)_Y → U(1)_em: dim(G/H) = 4 − 1 = 3 Goldstones eaten by W^±, Z.

§2 Dẫn chứng SymPy

SymPy verify — download for offline testSYMPY ✓

Reproduce Goldstone theorem with SymPy

Mexican-hat polar expansion → m_θ² = 0 EXACT. ~150 LOC.

scripts/spt_goldstone.py

spt_goldstone.py — verifies m_θ² ≡ 0 for U(1) broken Mexican-hat

150 LOCDownload

Reproduce in 30 seconds

pip install sympy numpy && python3 scripts/spt_goldstone.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 EXACT (m_θ² = 0 algebraic identity). Tier-B.

§4 So sánh với học thuyết hiện đại

Goldstone-Nambu (1960-61): separate theorem in QFT, requires Mexican-hat input. Higgs mechanism (1964): same input. SPT: Mexican-hat V derives from U(1) yin-yang invariance (Law 19) — Goldstone is automatic.

§5 Tầm quan trọng

🌟 HIGH — Goldstone's theorem underlies the Higgs mechanism, all of EWSB, and pion mass scales (chiral perturbation theory). SPT derives it as an automatic consequence of yin-yang Z₂.

§6 Falsifiable claim

📣 SPT claim: every spontaneously broken continuous symmetry yields one Goldstone. Falsifier: SSB with NO accompanying massless mode (other than via gauging).

§7 Kết luận

✅ Goldstone's theorem is a corollary of yin-yang U(1) invariance — no separate axiom needed.