Euclid Levi Civita symbol contract zero
euclidLeviCivita_symbol_contract_zero
Plain-language statement
Full Euclidean Levi-Civita contraction ∑_b (ε4)_b · (ε4)_b = 24 at the symbol level: summing the square of every standard-basis component of ε4 over all four Fin 4 index slots, paired naively (no metric), counts the 4! = 24 permutations. The Lorentz contraction ε^{μνρσ} ε_{μνρσ} lowers one factor with η and equals -24 instead.
Exact Lean statement
lemma euclidLeviCivita_symbol_contract_zero :
∑ g : (Fin 4 → Fin 4), euclidLeviCivita g * euclidLeviCivita g = 24Formal artifact
Lean source
lemma euclidLeviCivita_symbol_contract_zero : ∑ g : (Fin 4 → Fin 4), euclidLeviCivita g * euclidLeviCivita g = 24 := by have hcast : ∀ g : Fin 4 → Fin 4, ((generalizedKroneckerDelta g id : ℝ)) * (generalizedKroneckerDelta g id : ℝ) = ((generalizedKroneckerDelta g id * generalizedKroneckerDelta g id : ℤ) : ℝ) := fun g => by push_cast; ring erw [Finset.sum_congr rfl fun g _ => hcast g, ← Int.cast_sum, sum_generalizedKroneckerDelta_mul_self] norm_num- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/Relativity/Tensors/LeviCivita/Contractions.lean:64-72
Reuse this declaration
Bring the exact result into your workflow
The import identifies the source module. Your project still needs the pinned package dependency shown on this page.
What this badge means
This completion status comes from the project or community source. It has not yet been represented here as an independent rebuild and axiom audit.
Continue in this project
Related declarations
Adiabatic relation log
adiabatic_relation_log
Plain-language statement
Adiabatic relation in logarithmic form: If S(Ua,Va,N) = S(Ub,Vb,N) with N fixed, then c * log (Ua/Ub) + log (Va/Vb) = 0.
Source project: Physlib
Person-level attribution pending.
Adiabatic relation Ua Ub Va Vb
adiabatic_relation_UaUbVaVb
Plain-language statement
Adiabatic relation in product form: If S(Ua,Va,N) = S(Ub,Vb,N) with N fixed, then (Ua/Ub)^c * (Va/Vb) = 1.
Source project: Physlib
Person-level attribution pending.
Deriv Within mean Energy Beta eq neg variance
CanonicalEnsemble.derivWithin_meanEnergy_Beta_eq_neg_variance
Plain-language statement
(∂U/∂β) = -Var(E) for finite systems.
Source project: Physlib
Person-level attribution pending.