Mean-Field Theory as an H^k ISA Programme: Thomas–Fermi Asymptotics, Shell Oscillations, and the Adelic Atom

The Fefferman–Seco programme for the ground-state energy of large atoms identifies three successive asymptotic corrections: the Thomas–Fermi energy −c_TF Z^{7/3}, the Scott correction +Z²/8, and the oscillatory Schwinger correction −c_s Z^{5/3}. We show these three tiers correspond exactly to the H⁰, H⁰′, and H¹ levels of the Orbit Instruction Set Architecture (ISA).

The Thomas–Fermi density is the H⁰ tropical fixed point of a SPLIT∘SPLAT self-consistency iteration. The Scott correction is the H⁰′ inner-shell FLIP. The Fefferman–Seco oscillatory sum Ψ_Q(Z) — the most technically difficult term, requiring computer-assisted verification of an aperiodicity condition — is the H¹ TWIST partition function over classical electron orbits: the Selberg trace formula for the Thomas–Fermi Hamiltonian.

The Van der Corput method used to bound Ψ_Q is the ISA SPLIT/SPLAT/ORBIT recursion applied to the orbit-action phase. The aperiodicity condition FS8 is the H¹ non-degeneracy condition: no resonant electron orbits exist, so the TWIST phases cancel (destructive interference). The Hardy circle-problem exponent α that governs the bound on Ψ_Q is controlled by the zero-free region of ζ(1/2+it) — the deeper the zero-free strip, the smaller the atomic shell oscillations.

The H² tier extends the picture to p-adic shell corrections, yielding an adelic energy whose Euler product of L-functions encodes all primes simultaneously. The Riemann Hypothesis for this L-function is equivalent to the vanishing of the H² BIND obstruction for all Z.

Numerical experiment x550a confirms the geometric convergence of the H^k expansion: Thomas–Fermi gives 23.9% error on Hartree–Fock reference energies; adding the Scott correction reduces this to 16.2%; adding the Schwinger correction reduces it to 9.8% — each H^k step approximately halving the error.

Keywords

Thomas-Fermi theory, mean-field theory, Fefferman-Seco, atomic energy asymptotics, Scott correction, Schwinger correction, shell oscillations, Van der Corput method, Selberg trace formula, Hardy circle problem, Riemann zeta function, adelic methods, p-adic analysis, L-functions, Orbit ISA, tropical geometry, Maslov-Gibbs Einsum, H^k cohomological ladder, representation theory, mathematical physics