Condensed Matter

b=7 -> D*b=14 -> D^5=32 -> D*E*23=230

The crystal counting hierarchy: b=7 systems, D*b=14 lattices, D^5=32 point groups, D*E*23=230 space groups. Every count factors over axiom primes. Quantum Hall filling fractions walk the chain. Bott periodicity = D^3 = spider's legs.

Crystal Counting Hierarchy

Quantity	Value	Axiom	Detail
Crystal systems (3D)	7	b	Depth = boundary creates structure
Bravais lattices (3D)	14	D*b	Duality doubles systems
Point groups (3D)	32	D^5	Five powers of duality
Space groups (3D)	230	DE23	23=CRT palindrome prime
Wallpaper groups (2D)	17	ESCAPE	D+K+E+b = first chain-breaking prime
Sohncke (chiral)	65	E*GATE	Life uses only chiral structures
Magnetic space groups	1651	GATE*M(b)	13*127 = gate times Mersenne(depth)

Quantum Hall: Cunningham Denominators

Jain composite fermion filling fractions nu = n/(2n+1). The denominator 2n+1 IS the Cunningham map. Shadow(Cunningham(n)) = n:

n	nu	Axiom
1	1/3	sigma/K
2	2/5	D/E
3	3/7	K/b
4	4/9	D^2/K^2
5	5/11	E/L
6	6/13	D*K/GATE

Laughlin-Cunningham Theorem

Laughlin states nu = 1/(2p+1) have denominators {K, E, b, K^2} = {3, 5, 7, 9}. The Jain series walks the full axiom chain: numerators {sigma,D,K,D^2,E,D*K} and denominators {K,E,b,K^2,L,GATE}. At n=6, the GATE appears. The quantum Hall effect IS the axiom chain in electron gas.

Topological Classification

AZ symmetry classes

D*E = 10

Tenfold way. Same count as spacetime generators.

Bott period (real)

D^3 = 8

Spider's legs. Topology repeats every D^3 dimensions.

Bott period (complex)

D = 2

Duality itself. Real/complex = D^3/D = D^2 = quaternionic.

TI invariants (3D)

D^2 = 4

1 strong + K=3 weak Z_2 invariants.

Scaling Laws

Law	Exponent	Axiom
Debye (phonons)	T^3	T^K
Bloch (magnons)	T^(3/2)	T^(K/D)
Dulong-Petit	3R	K*R
Wiedemann-Franz	pi^2/3	pi^2/K
Orbital sequence	{1,3,5,7}	{sigma,K,E,b}

What Others See

7 crystal systemsEmpirical enumerationb = depth. Boundary creates structure32 point groupsCrystallographic restrictionD^5: five powers of dualityFilling fractionsComposite fermion theoryCunningham map = shadow function = axiom chainBott periodicity = 8Deep K-theoryD^3 = spider's legs. Topology IS duality cubed10 AZ classesTenfold way classificationD*E: observation times dualityOrbital sequence s,p,d,fAngular momentum accident{sigma,K,E,b}: axiom chain IS orbitals

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