Law 20 — Noether's theorem from membrane Action (Đợt 2 · 10/05/2026 v3.3)

Every continuous symmetry of the SPT Action gives a conserved current J^μ with ∂_μ J^μ = 0 on-shell. Time-translation ⇒ energy; space-translation ⇒ momentum; yin-yang U(1) phase ⇒ electric charge. Noether 1918 becomes a direct corollary of the membrane variational principle.

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

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✨ Law 20 (Noether's theorem · Tier-B EXACT): Every continuous symmetry of the SPT membrane Action S = ∫dτ[½Ẋ² + iψ̄γψ + ½Tr(J·Ṙ) − V(φ)] generates a conserved current. Energy, momentum, and electric charge conservation are not three separate laws — they are three corollaries of one variational principle.

§1 Cách verify hoạt động (5 bước)

Step 1 — Lagrangian density

L = ½(∂_t φ)² − ½(∂_x φ)² − V(φ) from the SPT scalar sector. SymPy constructs L symbolically.

Step 2 — Euler-Lagrange

∂_μ(∂L/∂(∂_μφ)) − ∂L/∂φ = 0 derived from δS = 0. Equations of motion on shell.

Step 3 — Time-translation

δt: φ(t) → φ(t + ε). Stress-energy T^00 = (∂L/∂φ_t)·φ_t − L is conserved. ⇒ Energy E = const.

Step 4 — Space-translation

δx: T^0x = (∂L/∂φ_t)·φ_x is the momentum density. ⇒ Momentum p = const.

Step 5 — Yin-yang U(1)

φ → e^{iα}φ: Noether current J^μ = i(φ∂^μφ − φ∂^μφ). ⇒ Electric charge Q = const.

§2 Dẫn chứng SymPy

SymPy verify — download for offline testSYMPY ✓

Reproduce Noether's theorem with SymPy

Symbolic Euler-Lagrange + 3 conservation laws derived end-to-end. ~150 LOC.

scripts/spt_noether.py

spt_noether.py — verifies every continuous symmetry → conserved current J^μ

150 LOCDownload

Reproduce in 30 seconds

pip install sympy numpy && python3 scripts/spt_noether.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 (Euler-Lagrange algebraic identity). Tier-B.

§4 So sánh với học thuyết hiện đại

Noether (1918) is a stand-alone theorem in classical + quantum field theory; standard textbook treatment requires it as an independent ingredient. SPT (this Law): derived directly from the membrane Action's variational principle — no external theorem needed.

§5 Tầm quan trọng

🌟 VERY HIGH — Noether's theorem is one of the most-used results in 20th-century physics. By deriving it inside the SPT framework, conservation laws (energy, momentum, charge) become INTERNAL consequences, not external axioms.

§6 Falsifiable claim

📣 SPT claim: every continuous symmetry of the membrane Action gives a conserved current. Falsifier: any observation of a continuous symmetry of the Action with NO associated conservation law.

§7 Kết luận

✅ Noether's theorem is not an axiom — it is a corollary of the SPT membrane variational principle. Conservation laws emerge automatically.