T0-24: Fundamental Symmetries - Formal Description
T0-24: 基本对称性 - 形式化描述
Core Definitions
D24.1: Self-Reference Invariance
SRI := {I ⊆ Prop(S) | ∀t: S(t) ⊢ φ ⟺ S(t+dt) ⊢ φ, φ ∈ I}
Properties preserved during self-referential evolution.
D24.2: φ-Scale Transformation
T_φ: ℝ → ℝ
T_φ(x) := φⁿ · x, n ∈ ℤ
Scaling transformation by golden ratio powers.
D24.3: Symmetry Group
G_φ := {g: M → M | L_φ(g·x) = L_φ(x)}
Group of transformations preserving the φ-Lagrangian.
D24.4: No-11 Invariant Transformation
N11_inv := {T | Z(T(x)) maintains No-11 ∀x}
Transformations preserving the No-11 constraint.
Fundamental Theorems
T24.1: Invariance Necessity Theorem
SelfRef(S) ∧ A1 → ∃I ≠ ∅: I ⊆ SRI
Proof:
- Self-referential completeness requires persistent identity
- Without invariances, self-reference is lost
- Therefore non-empty invariance set exists □
T24.2: φ-Scale Invariance Theorem
∀x ∈ ℝ⁺: No11(Z(x)) ⟺ No11(Z(φⁿ·x))
Proof:
- Fibonacci scaling property: F_{k+n}/F_k → φⁿ
- Scaling preserves relative gaps in sequence
- No-11 pattern unchanged under φ-scaling □
T24.3: CPT Theorem
∀S: [C ∘ P ∘ T](S) preserves dS/dt > 0
Proof:
- C: |1⟩ ↔ |0⟩ reverses information flow
- P: x⃗ → -x⃗ reverses spatial gradients
- T: t → -t reverses time direction
- Combined effect: (−1)³ = −1 on each term
- But entropy has even number of factors
- Net result: dS/dt → dS/dt unchanged □
Conservation Laws
T24.4: φ-Noether Theorem
δS/δg = 0 → ∂_μJ^μ + φ^(-n)J^μ = 0
For continuous symmetry g, with φ-correction term.
T24.5: Energy Conservation
∂_tE + ∇·S_E + φ^(-n)E = 0
From time translation invariance.
T24.6: Momentum Conservation
∂_tP_i + ∂_jT_{ij} + φ^(-n)P_i = 0
From spatial translation invariance.
T24.7: Angular Momentum Conservation
∂_tL_i + ε_{ijk}∂_jM_k + φ^(-n)L_i = 0
From rotation invariance.
T24.8: φ-Charge Conservation
∂_tQ_φ + ∇·J_φ = 0
Exact conservation from φ-scale symmetry.
Gauge Symmetries
T24.9: Local Gauge Invariance
ψ(x) → exp(iθ(x)/φ)·ψ(x) requires A_μ → A_μ + ∂_μθ/φ
Local phase symmetry necessitates gauge fields.
T24.10: Yang-Mills Structure
[D_μ, D_ν] = (i/φ)F_μν
F_μν = ∂_μA_ν - ∂_νA_μ + [A_μ, A_ν]/φ
Non-abelian gauge field strength tensor.
Symmetry Breaking
T24.11: Spontaneous Breaking Criterion
S[symmetric] < S[broken] → ⟨φ⟩ ≠ 0
Entropy maximization drives symmetry breaking.
T24.12: Higgs Mechanism
m²_gauge = g²⟨I⟩²/φ²
Gauge boson mass from information condensate.
T24.13: Explicit No-11 Breaking
Symmetric state → "11" pattern → Forced asymmetry
No-11 constraint explicitly breaks certain symmetries.
Discrete Symmetries
T24.14: Charge Conjugation
C: |1⟩ ↔ |0⟩, Z(x) → Z̄(x)
Binary state exchange symmetry.
T24.15: Parity Transformation
P: x⃗ → -x⃗, Z(|x⃗|) = Z(|-x⃗|)
Spatial inversion preserves Zeckendorf encoding.
T24.16: Time Reversal
T: t → -t, dS/dt → -dS/dt (violates A1)
Time reversal violates entropy increase.
Supersymmetry
T24.17: φ-Supersymmetry
Q: |n·φ⟩ ↔ |(n+1/2)·φ⟩
{Q_α, Q̄_β} = 2φ·P_μ·(γ^μ)_αβ
Relates integer and half-integer φ-spins.
T24.18: SUSY Breaking Scale
M_SUSY = M_Planck/φⁿ
Entropy-driven breaking scale.
Anomalies
T24.19: Anomaly Cancellation
∑_i A_i = ∑_i Tr[T_i³] = 0
Total anomaly must vanish for consistency.
T24.20: Gravitational Anomaly Freedom
∂_μJ^μ_grav = 0 (automatic)
No-11 constraint ensures gravitational anomaly cancellation.
Scale-Dependent Symmetries
T24.21: Effective Symmetry Group
G_eff(n) = {g ∈ G | violations < φ^(-n)}
Symmetries valid at scale φⁿ.
T24.22: Asymptotic Symmetry
lim_{n→0} G_eff(n) = G_unified
lim_{n→∞} G_eff(n) = ∏_i G_i
Symmetry enhancement at extreme scales.
Master Equations
T24.23: Universal Conservation
∂_μT^μν_total + φ^(-n)T^μν = 0
T^μν_total = T^μν_matter + T^μν_gauge + T^μν_info
Complete stress-energy-information conservation.
T24.24: Symmetry Group Structure
G_total = [SO(3,1) × U(1)_φ × SU(3) × SU(2) × U(1)] ⋊ CPT
Complete symmetry group with semi-direct product structure.
T24.25: Breaking Pattern
G_total →^{φⁿ} G_SM × U(1)_dark
Symmetry breaking at scale φⁿ.
Topological Invariants
T24.26: Topological φ-Charge
Q_top = ∮ J_φ = n·φ, n ∈ ℤ
Quantized topological charge.
T24.27: Winding Number
W = (1/2π) ∮ dθ ∈ ℤ
Topological winding preserved by No-11.
Predictions
T24.28: Critical Exponents
ν = 1/φ², β = (φ-1)/2, γ = φ
Universal critical exponents from φ-symmetry.
T24.29: CPT Violation Bound
|δCPT|/CPT < exp(-φⁿ), n = log(E/E_P)/log(φ)
Exponentially suppressed CPT violation.
T24.30: Dark Matter Stability
Q_dark = n·φᵐ (conserved)
Topological protection of dark matter.
Formal System Properties
Consistency:
∀T ∈ G_total: No11(T(Z)) = true
All symmetries preserve No-11 constraint.
Completeness:
∀ conservation law ∃ symmetry: Noether(symmetry) = law
Every conservation law has corresponding symmetry.
Minimality:
G_total is the minimal group preserving SelfRef(S)
No smaller group maintains self-referential completeness.
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