Graph Energy Density
Graph Energy Density (IV.D77) is the local-density form of the τ-Energy Index: the CR-tension integrand evaluated pointwise on the τ-graph (the discrete approximant of T²). It is the bridge between continuum E_CR (IV.D76) and the discrete defect-functional formulas (IV.D274).
τ-Definition
Graph Energy Density (IV.D77) is the local-density form of the τ-Energy Index: the CR-tension integrand evaluated pointwise on the τ-graph (the discrete approximant of T²). It is the bridge between continuum E_CR (IV.D76) and the discrete defect-functional formulas (IV.D274).
Categorical invariant. ρ_E(x) := |∂̄ Φ(x)|² evaluated at vertex x of the τ-graph; an E2 local invariant whose graph-sum recovers the Energy Index.
Primary registry anchor:
IV.D77
τ-Derivation Chain
-
I.K0— Universe Postulate establishes τ -
IV.D21— Energy Index — the abstract energy invariant -
IV.D76— Energy as CR-Tension — continuum operational form -
IV.D77— Graph Energy Density — pointwise / per-vertex form on the τ-graph -
IV.D274— Defect functional uses ρ_E summed over the graph
Lean modules referenced:
TauLib.BookIV.Physics.DefectFunctional
SI Translation
Calibration anchor: PG-P01-neutron
Calibration chain:
- m_n (anchor)
- c² via IV.D23 → energy scale
- τ-volume → m³ via IV.D293 (length scale)
Manuscript reference: manuscript-sources/book-04/part04-quantum-mechanics/ch-cr-tension.tex