How Continuous Becomes Discrete
A continuous wave fills a box with prime dimensions D×K×E×b = 2×3×5×7. Where it must vanish — the nodes — are exactly the 210 elements of Z/210Z. Discreteness isn't imposed. It precipitates from continuity meeting prime boundaries.
The energy of mode (n1,n2,n3,n4) in a box with sides L1×L2×L3×L4 is:
E = (n1/L1)^2 + (n2/L2)^2 + (n3/L3)^2 + (n4/L4)^2
The primordial mode (2,3,5,7) has E = 1+1+1+1 = 4 = D^2.
Each channel contributes exactly 1. Perfect democracy.
Ground state (1,1,1,1): E = 1/4 + 1/9 + 1/25 + 1/49 = 0.4215.
The gap between ground and primordial = 3.579 = the cost of discretization.
Discreteness is postulated. Planck's constant h is an unexplained input. Quantum mechanics says "nature is discrete" but doesn't explain why. The transition from classical to quantum is a mystery — the "measurement problem."
Start with continuous waves. Put them in a box with prime dimensions. The nodes — where the wave MUST vanish — form exactly Z/2×Z/3×Z/5×Z/7. Discreteness isn't assumed. It's the inevitable consequence of continuity meeting prime boundaries. 210 elements from pure geometry.
Why 3+1 spacetime dimensions? No standard answer. String theory needs 10 or 11. The number of dimensions is a free parameter that must be tuned.
The box has 4 sides because there are 4 inner axiom primes: {D,K,E,b} = {2,3,5,7}. Each prime = one independent direction. L=11 is the protector, not a dimension. The number of spatial-like degrees of freedom = the number of axiom primes before the guard.