E8 operator-completeness gate

Date: 26 October 2026

Status: exact truncation correction; inverse branch requires shell 15

Parent: restarted nonlinear eigensolver

Evidence: operator-completeness ledger

Certificate: step-energy and theta-series verifier

Complete states do not guarantee complete operators

The common E8-shell basis retains every charge state below a kinetic-energy cutoff. That alone does not ensure that every declared cosine generator acts nontrivially on the retained vacuum.

For the (K)-kernel branch, the eight steps are (ei), and each has kinetic energy (ei^TKei/2=1). All eight cosine directions are present from shell one. For the inverse branch the common-basis steps are (K^{-1}ei), with exact energies

The inverse branch therefore cannot contain the full vacuum action of all eight cosines until kinetic cutoff 15.

The norm fraction follows because the sixteen signed vacuum steps are orthogonal and each carries amplitude one half.

Correction to the convergence narrative

The inverse-kernel shell-eight eigenvalues remain valid eigenvalues of the projected Hamiltonian, and they still demonstrate that the two sparse graphs differ. They are not yet an admissible approximation to the full eight-cosine Hamiltonian: two entire vacuum channels are projected away. Even shell twelve misses one.

This does not affect the (K)-kernel convergence ladder, whose operator set was complete from shell one. It bifurcates the next computation:

• continue the (K)-kernel at shells 10 and 12 with leakage and transitions;
• treat the inverse branch below shell 15 as reconnaissance only, and use a distributed/reduced-memory run or a better representation before its first operator-complete convergence point.

Operator completeness at shell 15 is necessary but still not sufficient. Further shells must show that the spectrum and detector weights stabilize.