{ 2, 3, 5, 7, 11 }

Electromagnetism from Ten Terms

K=3 gives the inverse-square law and cross products. D=2 gives polarization and Maxwell's structure. The photon IS the bridge.

Part of the Decality — one ring (Z/970200Z), 108 lattice structures.

D^2 = 4
Maxwell equations
D*K = 6
Field components
1/137
alpha = 1/ADDRESS
{1,3,7}
Cross product dims
D = 2
Polarizations

K=3 and the Inverse-Square Law

Click a dimension. Only d ≤ K=3 gives stable atomic orbits.

Why inverse-square? The Laplacian in d dimensions has Green's function ~ 1/r^(d-2). For K=3: force ~ 1/r^2. This isn't a coincidence — it's K=3 geometry. Coulomb IS the shape of 3-space.

Cross Products = Sigma-Chain {1, 3, 7}

Division Algebras (D-powers)

Normed division algebras exist in dimensions that are powers of D=2.

1
R (reals) — dim = D^0 = sigma
2
C (complex) — dim = D^1 = D
4
H (quaternions) — dim = D^2
8
O (octonions) — dim = D^3

Cross Products (sigma-chain)

Vector cross products V×V→V exist only in dims = D^k - 1.

1
dim sigma — trivial (CC1: 1)
3
dim K — quaternion cross product (CC1: 3)
7
dim b — octonion cross product (CC1: 7)
15
K*E = 15: STOP (composite, no cross product)
Magnetism requires the cross product. B = (q/c²) v × E. Curl B = ... The curl operation exists only in {1,3,7} = {sigma, K, b} dimensions. Stable orbits exist only for d ≤ 3. Only K=3 has both. Magnetism + atoms coexist in exactly one dimension.

Maxwell's D²=4 Equations

Source: Electric
∇ · E = ρ/ε₀
Charge creates electric field
D-source: charge = D-channel visibility
Geometric: Magnetic
∇ · B = 0
No magnetic monopoles
K-closure: magnetism always closes (dipoles)
Coupling: D→K
∇ × E = -∂B/∂t
Changing B creates E (Faraday)
D→K coupling: bridge induces closure
Coupling: K→D
∇ × B = μ₀J + ∂E/∂t
Current and changing E create B (Ampere)
K→D coupling: closure induces bridge

D² = 4 equations. D*K = 6 field components (3E + 3B). D = 2 forms (dF=0, d*F=J).

Spacetime = K+sigma = D² = 4. The EM field tensor F is a D-form (2-form) in D²-dimensional spacetime. Its dual *F is also a D-form. Self-duality is possible only because K+sigma = D². The first two primes fix spacetime.

The Photon = D-Channel Carrier

PHOTON

Masso = 0
Chargeo = 0
Spinsigma = 1
Helicity±sigma = ±1
PolarizationsK-1 = D = 2
Gauge groupU(sigma) = U(1)
Couplingalpha = 1/ADDRESS = 1/137

The photon is the Bridge. It mediates between charged particles — D=2 polarizations, spin sigma=1, gauge U(1).

Bertrand's Theorem: D=2 Force Laws in K=3

In K=3 dimensions, exactly D=2 force laws give closed orbits: F~1/r² (Coulomb) and F~r (harmonic). The exponents are {-1, 2} = {MIRROR, D}.

Force Hierarchy from Null Channels

More null CRT channels = weaker but more universal. Strong (no nulls) ~ 1. EM (D-visible) ~ 1/137. Weak (E-hidden) ~ 10^-5. Gravity (D,K-null) ~ 10^-39. The axiom's channel structure IS the force spectrum.

EM Constants from the Axiom

ConstantExpressionValueD role
Fine structure α^-1ADDRESS + b/(D*97) - ...137.036 (0 ppb)235711
Rydberg E1GATE + K/E = 13.6 eV13.606 eV (2 ppm)35
Vacuum impedance Z0~E!*π = D^3*K*E*π376.73 Ω235
Conductance quantum G0D*e²/h7.75×10-5 SD in numerator
Flux quantum Φ0h/(D*e)2.07×10-15 WbD in denominator
Bohr magneton μBeℏ/(D*me)9.27×10-24 J/TD in denominator
Gravity ratio10^(E*b + D²) = 10^39~1039257
Verify in .ax Full alpha derivation Quantum constants

The EM Axiom Map

K=3 gives...

Inverse-square law (Laplacian Green's function)
Stable atomic orbits (effective potential minimum)
Cross product = curl = magnetism
Bertrand: D=2 closed-orbit force laws
3E + 3B = D*K = 6 field components

D=2 gives...

E and B field duality
D=2 photon polarizations
D^2=4 Maxwell equations
D^2=4 spacetime (K+sigma)
Hodge duality: F and *F both D-forms
Not a force. Geometry. Electromagnetism is what the D-channel looks like in K=3 space. The photon carries the identity: sigma/sigma = sigma. Light IS the bridge between everything that exists.

Number Theory Thread

Cross products exist in exactly {1,3,7} dimensions = CC1(σ), the sigma-chain of Cunningham primes. Maxwell’s D²=4 equations use D*K=6 components — and 6 = shadow(13) = D*K, where 13 is the GATE. The photon sphere at r=K shares K=3 with monoatomic gas ratio E/K and with the D-chain: h(−23)=K. Fine structure ADDRESS=137 has order 420=λ in the ring, and class number h(−137)=D³=8. The spider’s legs appear in class numbers too.

D-chain class numbers → Partitions & the gate → Modular forms →

Paradigm Contrast

EM = mysterious force at a distanceEM = K=3 Laplacian geometry
alpha=1/137 is unexplainedalpha=1/ADDRESS, 7 terms, 0 ppb
4 Maxwell equations "just work"D^2=4 equations, D*K=6 components
Cross products "happen to work" in 3D{1,3,7} = sigma-chain, Hurwitz theorem
Photon polarization = 2 "by convention"K-1 = D = 2 (massless spin-1 in K+1)
4 forces, separate explanationsNull channel hierarchy: 1 framework
Inverse-square law: empiricalGreen's function of K=3 Laplacian
Gravity 10^39 weaker: fine-tuningE*b + D^2 = 39: derived