E(n,l) = (2n+|l|+1)ħΩ − ½lħω_c · act I shells · act II the weave · act III the rebinding · √2 and √3 hiding in a semiconductor
verdict
⏳ conductor booting…
conductor RUNNING
probe-1 · shells & magic QUEUED
probe-2 · fold points QUEUED
probe-3 · Landau rebinding QUEUED
the levers · turn the knob, fill the atom
field ω_c/ω₀ 0.000electrons N 6
·
2D — the weave & the staircase
top: the Fock–Darwin weave — every (n,l) with 2n+|l|≤4, energy vs field. gold verticals = the exact fold points 1/√2 and 2/√3 where chapters overtake. green line = your slider; filled dots = the N occupied states at that field. bottom: the B=0 shell staircase — bar height = shell degeneracy 2(n+1), the running sum spelling out 2, 6, 12, 20, 30.
3D — the spectrum ribbon (soft-GL)
each rail is one (n,l) level drawn through (field, energy, angular momentum ℓ) — orange ℓ≥0, amber ℓ<0. the weave lives in the third dimension: crossings in 2D are misses in 3D, separated by ℓ. the bright plane is your slider's field. drag to rotate.
toddler corner
ELI5: the dot is a bookshelf. with no magnet the chapters sit in tidy rows — 2 books fit on the first shelf, 4 on the next, then 6, then 8 (that's why 2, 6, 12, 20 books make a "full" shelf feel). turn the magnet knob and the shelves start sliding past each other — and they trade places at EXACT clicks of the knob (the √2 click, the √3 click), not roughly. keep turning and every book slides into just a few huge bins at the end — same books, totally new shelving. one bookshelf, three stories, and the click positions are irrational numbers you can measure in a lab.
fold grammar · on the record
this kernel vs the ROOT0 kernel. The atom folds arithmetically: degeneracy 2(n+1), magic numbers n(n+1), fold points at 1/√2 and 2/√3, ending in exponential collapse onto Landau rails. The ROOT0 folded kernel as specified folds by powers of two with a linear damage rail and a seam at level 2. Different grammar, same genus — fold-story with a knob. The comparison is held open, not resolved; the bench measures its own side only.
method & provenance
double verification. fockdarwin.py bisected both crossings against closed forms (agreement 1e-16) and verified the Landau rails at r=100; this page re-runs the bisections, the magic-number ladder, and the rail collapse in independent JS. baked ≠ live ⇒ red monitors, dead verdict.
AMBER · anchors & scope. Shell structure and magic numbers measured in vertical-dot addition spectra (Tarucha et al., 1996) and Hund's-rule filling at half shells — training-memory anchors, unaudited live. Model is single-particle, spin-degenerate, no interactions (constant-interaction picture); real dots pay charging energy on top. The fold points are exact within the model.