Crystals count in axiom primes. Quantum Hall filling factors ARE the shadow function.
Topological classification uses D=2 and D3=8. The solid state speaks five primes.
Every integer that counts crystal symmetries is expressible in {2, 3, 5, 7, 11, 13}.
Click any card for details.
Click a card to see why this number appears.
The hierarchy: 2D: D2 systems → E lattices → ESCAPE wallpapers.
3D: b systems → D*b lattices → D5 point groups → D*E*23 space groups.
Magnetic: D*61 point groups → GATE*M(b) space groups.
Coordination Numbers: 100% Axiom-Smooth
Every crystal coordination number is a product of {2, 3}. The set equals D times the GR orbit ladder.
Coordination = {D2, D*K, D3, D2*K} = D * {D, K, D2, D*K}.
The GR orbit zones (Schwarzschild radii) = coordination / D. Crystals and gravity share the same ladder.
Quantum Hall: The Shadow Function
ν = n / c(n) = n / (2n+1)
Jain composite fermion series. Denominators = {K, E, b, K2, L, GATE, K*E, ...}
The filling factor IS the inverse shadow. shadow(c(n)) = n.
Click a bar to see the filling factor and its axiom expression.
The fractional quantum Hall effect implements the axiom chain in a 2D electron gas.
Every observed plateau has an axiom-ratio filling factor.
Laughlin states: ν = 1/(2p+1) = σ/K, σ/E, σ/b, σ/K2.
Hole series: denominators = {σ, K, E, b, K2, L}. The full axiom chain.
Topological Classification: D*E = 10, D3 = 8
Click a card to explore topological structure.
The coincidence that isn't: D*E = 10 = string theory dimensions = AZ symmetry classes.
D3 = 8 = Bott period = spider's legs = real symmetry classes. D = 2 = complex period.
Orbital Hierarchy: {σ, K, E, b}
s=σ, p=K, d=E, f=b — THE axiom chain
With spin: D*{σ, K, E, b}. Shell capacities: {D, D3, ME, D5}.
Through d-orbitals: σ + K + E = K2 = 9. Through f: D4 = 16.
The angular momentum quantum numbers 2l+1 = {1, 3, 5, 7} are the axiom's odd primes plus sigma.
Spin doubles everything: 2(2l+1) = {2, 6, 10, 14} = D * {sigma, K, E, b}.
Scaling Laws of Solid State
Click a law to see its axiom expression.
Band Gaps: Suggestive Ratios
Band gaps are material-specific, not universal. These axiom ratios approximate measured values
but lack the rigor of SM or quantum derivations. Shown for completeness; the structural results above are the real gold.
Quantum Hall filling ν=n/c(n) uses the Cunningham map c(n)=2n+1 —
the same map that generates the axiom chain. Jain’s composite fermion denominators
ARE the shadow function s(p)=(p−1)/2 = inverse Cunningham.
b=7 crystal systems, D·b=14 Bravais lattices, D&sup5;=32 point groups.
Bott periodicity D³=8 mirrors the D-chain: h(−95)=D³=8 (spider’s legs in class numbers).
ESCAPE=17 wallpaper groups: same 17 that is the full axiom sum D+K+E+b=ESCAPE.