The Helix

Seven CRT channels are seven interlocking gears. Each channel's modulus sets its gear size: mod 8 is the smallest gear (fastest rotation), mod 49 is the largest (slowest). As element n advances through the ring, each gear turns at its own rate. The gear ratio between any two channels = the ratio of their moduli.

Below: a side view of seven helices winding around a shared axis. Each helix is one CRT channel. Its wavelength is its modulus. Where two helices cross at the same phase = their gears mesh. All seven gears return to start simultaneously after lcm(8,9,25,49,11,13,17) = 214,414,200 steps.

Gear Ratios

Each pair of CRT channels has a gear ratio = the ratio of their moduli. Since all moduli are pairwise coprime (distinct prime powers), every ratio is irreducible. Meshing period = lcm(q_i, q_j) = q_i * q_j.

Pair	Ratio	Mesh	Reading
mod-8 / mod-9	9/8	72	D^3*K^2 = ouroboros cycle
mod-9 / mod-11	11/9	99	K^2*L = closure + protector
mod-11 / mod-13	13/11	143	L*GATE = parity pair
mod-13 / mod-17	17/13	221	GATE*ESCAPE = boundary pair
mod-9 / mod-49	49/9	441=21^2	(Kb)^2 = closuredepth
mod-25 / mod-49	49/25	1225=35^2	(E*b)^2 = meta-meta-ring^2
mod-8 / mod-17	17/8	136	D^3*ESCAPE = bridge-transcend

Smallest mesh: 72 = 8*9 = D^3*K^2. The ouroboros cycle (3^2 * 2^3 = 72 transitions in the 3-bit state graph). The two innermost towers mesh first.

Largest mesh: 1225 = 25*49 = E^2*b^2 = 35^2. The two depth-2 quadratic towers take the longest to align. 35 = E*b is the meta-meta-ring in Tower B.

21 pairwise gear ratios

All irreducible

C(7,2) = 21 pairs. Each coprime pair has meshing period = product of moduli. The full ring 214,414,200 = product of all 7 = the period when all gears return simultaneously.

72 = ouroboros cycle

lcm(8,9) = 72 = 3^2 * 2^3

Smallest pairwise mesh. The D-K bridge is the tightest coupling. The mod-8 gear turns 9 times while mod-9 turns 8.

1225 = 35^2 = (E*b)^2

lcm(25,49)

Largest pairwise mesh. The two quadratic data channels are the most loosely coupled. 35 = E*b = meta-meta-ring.

When Gears Mesh

Gears mesh when both channels return to residue 0 simultaneously. For coprime moduli q_i and q_j, this happens every q_i * q_j steps. At mesh points, both helices cross the central axis at the same height.

The 490 split reappears. DEAD channels {mod 8, mod 25, mod 49} = {D^3, E^2, b^2} (perception). ALIVE channels {mod 9, mod 11, mod 13, mod 17} = {K^2, L, GATE, ESCAPE} (strategy).

Set	Mesh range	Count	Character
DEAD-DEAD	200 to 1225	3 pairs	Wide spread, loosely coupled
ALIVE-ALIVE	99 to 221	6 pairs	Narrow range, tightly coupled
DEAD-ALIVE	72 to 833	12 pairs	Cross-split coupling

DEAD channels span a 6x range in meshing periods (1225/200). ALIVE channels span only 2.2x (221/99). Strategy gears are more tightly synchronized than perception gears.

DEAD: wide frequency spread

Perception channels

Three DEAD-DEAD meshes: 200 (D^3*E^2), 392 (D^3*b^2), 1225 (E^2*b^2). The quadratic exponents create large mesh periods. Perception channels scan different timescales.

ALIVE: tight coordination

Strategy channels

Six ALIVE-ALIVE meshes: 99, 117, 143, 153, 187, 221. All under 225. Strategy channels mesh frequently = coordinated decision-making.

The Helix

The Machine

Gear Ratios

When Gears Mesh