Dynamics

Raise every element to a power and watch what happens. At k = 7, the universal period is λ = 240: every unit satisfies x²⁴⁰ = 1. Lambda is tower-parametric — at each rung, λ = lcm(p₁−1, …, p_k−1). The tower splits cleanly (proved, k = 3–14): geometric quantities (curvature, spectral gap) depend on the primes themselves; dynamical quantities (lambda, orbits) depend on the primes minus one. The two are orthogonal — geometry cannot see lambda. Everything on this page is the tower's prime lens at work: the dynamics read the multiplicative structure of the shifted primes p−1, one factor per sieve step.

Power map

The map x → x^e on Z/210 = 2·3·5·7 — small enough to draw every element. At e = 2 (squaring), the fixed points are exactly the 16 idempotents of Z/210 (x² = x is the definition; one per subset of the four primes); 16 more elements sit on eight 2-cycles, and the remaining 178 lie on tails flowing into them. Move the slider to see how the exponent transforms the ring.

The carousel

Addition as rotation on 7 independent circles. Each channel has its own clock: +1 advances every dial one tick, and each wraps on its own period. For example, 209 sits at (1, 2, 4, 6, 0, 1, 5) on the dials mod (2, 3, 5, 7, 11, 13, 17); one step to 210 = 2·3·5·7 wraps the first four dials to zero at once: (0, 0, 0, 0, 1, 2, 6). Enter a number to see its position on all 7 dials.

Eigenvalue cost

Each CRT channel has a spectral activation cost: c(p) = 4sin²(π/p) for p ≥ 3, and c(2) = 2 — the mod-2 channel has only one direction (1 = −1 mod 2), so its cost is halved. From 3 onward, the larger the prime, the cheaper the channel:

Total cost = 7.816, and the full ordering is 3 > 2 > 5 > 7 > 11 > 13 > 17: the tower builds its most expensive channels first. The cheapest channel sets the spectral gap — 4sin²(π/17) = 0.135 at k = 7.

Plateaus: how the spectrum grows

Adding a prime to the tower either jumps lambda or leaves it fixed. If p_k−1 brings a new factor to the lcm, lambda jumps; if p_k−1 already divides lambda, nothing dynamical changes — a plateau. Rungs k = 11 through 14 sit on one: 30, 36, 40, and 42 all divide λ = 55,440.

The ring's frequency response tells the two apart. The normalized Ramanujan sum factors into per-channel gates — factor 1 when the channel's prime divides the frequency n, factor −1/(p−1) otherwise — and the distinct values it takes over one period n = 1…λ form the rung's spectrum. At a jump the period extends and the spectrum gains many new values; at a plateau the period is fixed and the new gate merely rescales the old spectrum, which by itself gains nothing (computed, k = 3–22). Jumps widen the spectrum; plateaus refine it.

A plateau's entire gain comes through a window: old spectrum values newly seen at frequencies divisible by p_k. A window value survives unless it collides with a rescaled old value, and the collision condition is exact (proved): a multiplicative relation ∏_q∈A(q−1) = (p_k−1)·∏_q∈B(q−1) over disjoint sets A, B of older tower primes — together holding an odd number of them, so the negative gates match sign — inside the lambda budget. Whether the spectrum grows at a plateau is pure multiplicative arithmetic of the shifted primes q−1. At k = 11–14 the windows are nearly equal, so the collision rate alone sets the gain:

The collision rate is computable before enumerating any spectrum: counting patterns (the set of tower primes dividing n) instead of distinct values, the rate is a closed sum of subset-counting functions read off the multiplicative relations of p_k−1 — exact in pattern measure, verified k ≤ 22. As a predictor of the distinct-value rate it over-predicts one-sidedly (measured gaps 0.00–0.24 across the census), because collided values carry more representing patterns than survivors. Census across all 18 plateau rungs up to k = 37: collision rates 0.50–1.00. The formula's first out-of-sample test: it predicted 74–98% of the k = 37 window would collide, before that spectrum was ever enumerated; the measured rate is 92%.