Breaking the registry: ne electrons over ni ions
The same sliding experiment, but with
different numbers on the two rings. At 6 electrons over
6 ions the sea was pinned at
0.01186 Ha = 323 meV. A metal's sea is
incommensurate with
its lattice, and that is part of why it slides. This asks whether the framework knows the difference.
Keeping it neutral. ne electrons need +ne of ion charge, spread over
ni sites, so each ion carries ne/ni — a fractional
kernel. That is the usual device for putting a lattice of one period against a charge distribution of
another without charging the system.
The expected effect, in standard terms. Pinning of a charge-density wave falls off rapidly with
the order of the commensurability. 1:1 is order 1, the strongest case — which is what was
measured. 7:6 is order 6, and should be far weaker. If the framework reproduces that, it has the
physics that separates a pinned wave from a sliding one.
Built-in check, and it is sharper here. Rotating the electron ring by one electron
spacing returns the same configuration, and so does rotating by one ion spacing. Together
those force a period of gcd(ni,ne)/ni electron spacings
— 1/6 for 7 over 6, not 1. So E must repeat six times faster than in the commensurate
run, which is a much more demanding test than before: the scan below is one full period.
The 2×2 this makes possible. Two things change at once between 7 electrons and 6, so
they have to be separated. neutral=1 gives fractional kernels ne/ni
and an uncharged system, so only the registry changes. neutral=0 keeps the kernels
at +1 so the 7th electron is a genuine surplus, and the filling changes too. Comparing the two
isolates which of the two is doing the work — the question the corona-vs-sea contrast left open.
URL: ?ni=6 ions, &ne=7 electrons, &neutral=1, &a=2.2 ion arc
spacing, &dr=1.2 outer-ring offset, &phi=0 rotation in
electron spacings, &bs=0.25, &reset clears.
relaxing…