E8 adaptive-shell nonlinear solver

Date: 26 October 2026

Status: exact basis and operator compilation; nonlinear eigenvalues open

Parent: constitutive-metric gate

Evidence: shell and solver ledger

Certificate: theta-series and integral-equivalence verifier

One kinetic carrier, two nonlinear graphs

The two conditional quadratic forms are integrally isospectral. If (m=K^{-1}n), unimodularity gives a bijection of (mathbb Z^8) and

Both branches can therefore be diagonalized on one basis ordered by complete E8 kinetic shells. They are still different Hamiltonians. In the common (m)-basis, the (M=K) cosine moves by (pm ei), while the (M=K^{-1}) cosine moves by the dense integral columns (pm K^{-1}ei). The charging energies agree as multisets; the potential hopping graphs do not.

This is the nonlinear version of the constitutive firewall. It also gives a clean typed implementation: carrier = E8 shell; operator = declared step set; detector = converged low spectrum and transition weights.

Exact resource forecast

The E8 theta series gives every shell multiplicity without enumerating a rectangular eight-dimensional box:

With kinetic energy (m=Q/2), complete-shell cutoffs have the following dimensions.

Each row has at most sixteen potential neighbors, so the Hamiltonian should be applied matrix-free. The difficult inverse-kernel branch is not dense as an operator: its individual steps are long in coordinates, but there are still only sixteen of them.

Convergence contract

At cutoffs 4, 6, 8, 10 and 12, record the ground energy, first three gaps, parity, eight local-drive transition weights and the eigensolver residual. Promote a level only when successive complete-shell cutoffs agree inside the declared tolerance and boundary weight or residual decreases. A Rayleigh--Ritz level in one finite shell is an upper bound, not the continuum gap.

This construction avoids the misleading coordinate-box cost created by the smallest Cartan eigenvalue and preserves every state at a given kinetic energy. It is now concrete enough to implement as a hash-indexed shell enumerator plus a matrix-free block Lanczos or Davidson solve.

The operator-completeness audit adds a necessary qualification: the dense inverse-branch step with the largest vacuum kinetic cost first enters at shell 15. Lower inverse-branch rungs are useful reconnaissance but not complete eight-cosine approximations.