Acceleration Allocation Segment Defining Order for Density Functional Theory (DFT) of a) By Element Sawtooth Step-Function b) Distance Scaling to r_e, c) LDA, and then d) Gradient (MetaGGA).

An acceleration allocation segment determines the order of operation from the underlying math for the Perdew Ladder for DFT. For DFT, and more generally the mass (1/m) in all equations, . . . in restated order:

- Scale must be net-dimensionless, (d/d_Base)^V/W. LDA requires particle-edge radius (r_e) base, versus standard force scaled hc at Bohr-Hydrogen(a₀), so converting hc@a₀ via fine α=(r_e/a₀)^1/2.

- That base updates for each Element (based upon a sawtooth step-function for each element (S_N)*r_e: odd (8/7), even (1/2)^1/3, then with outer subshell refinements, refining Okun, Burke’s presentation last year.

- LDA Localization, mass, gets replaced by position-in-field using the known 1/x^3/4, so (r)^4/3, transformation as rescaled above.

- The relative position-in-field inverse, interactive for both particles (1+PiF₁/PiF₂) by 1/x^3/4, refining MetaGGA, gradient for Conservation of Force by acceleration allocation. So, force per particle become acceleration that allocates by gradient from 2-particle, 2-level (acceleration, velocity) 4x4 matrix.

One integrated segment has all four, importantly the relative position-in-field inverse, interactive for both particles (1+PiF₁/PiF₂) gradient for Conservation of Force by acceleration allocation. So, force per particle allocates by 2-particle, 2-level (acceleration, velocity) 4x4 matrix. Critically, then (Σa_n=F).

That math defines order of operation: 1) r_e scaling, 2) sawtooth step-function by Element . . . and eventually outer subshell refined, 3) LDA, 4) position-in-field double gradient, inverse and interactive. Now, with two steps before today's Perdew Ladder.