Spinning Spheres

Seven CRT channels stack into a T^7 torus: S^1(8) x S^1(9) x S^1(25) x S^1(49) x S^1(11) x S^1(13) x S^1(17). An address orbiting fast fills its channel; a ring spinning fast looks like a sphere. The canvas below IS this T^7 torus projected to seven nested orbit shells, each dot the element's position on its circle.

The ring IS the Gabriel-Horn cap of the ambient integer lattice. Unbounded Z^7 has infinite-mode Laplacian spectrum = an infinite-surface horn. The ring truncates modes at k <= N-1, capping the horn at finite volume. That cap is what you see: seven bounded circles. The mod-49 channel is the slowest orbit and carries the finest angular resolution; a uniform spectral floor gap = 4*sin^2(pi/49) ~ 0.01644 lives on that axis.

Two Clay-Millennium STRUCTURAL analogs sit inside this one canvas. STRUCTURAL Yang-Mills mass-gap analog: the gap 4*sin^2(pi/49) is a 0-form on all 108 Carmichael-lambda-420 heartbeat-lattice rings (6-channel, no mod-17; MIN = 25*49 = 1225, MAX = 2*12,612,600 = 25,225,200). Z/214,414,200 itself sits one mod-17 step above the lattice (Carmichael lambda = 1680, 7-channel) and inherits the same gap floor because 17 < 49. STRUCTURAL Navier-Stokes horn analog: CRT decomposes the Z/214,414,200 graph Laplacian into seven orthogonal per-channel Laplacians, so finite-time blowup would demand simultaneous singularity in independent channels = forbidden by CRT orthogonality. NOT Clay proofs. The abelian CRT Cayley graph is to non-abelian YM / continuum NS what a finite-dim ODE is to a PDE on R^4 / R^3.

Below: T^7 rendered as seven concentric orbit shells. Each dot orbits at angular velocity 2*pi/q_i. Full realignment (all seven dots return simultaneously) requires lcm(8,9,25,49,11,13,17) = 214,414,200 steps. The gold line marks Carmichael lambda = 1680, the multiplicative exponent of the unit group.

The Machine

The Capped Horn

Gabriel's Horn = y = 1/x rotated about the x-axis for x >= 1. Volume = pi (finite). Surface = infinity (diverges logarithmically). Paradox: finite paint, unpaintable surface. The ring realizes the paradox geometrically = its state space IS the horn's finite-volume side; the continuum integer lattice Z^k IS its unbounded-surface side.

The canvas above IS the cap. Seven bounded orbit shells = T^7 torus rendered as the horn's finite-volume side. The ring has no horn-surface side because it has no uncapped mode. Clay-NS stays open because R^3 cannot be truncated without breaking Galilean invariance and scale symmetry; the ring eliminates the paradox by capping at N = 214,414,200.

Seven Orbits

Each orbit IS a prime channel made visible. The modulus q is the channel size. The orbit period is q steps. The number of complete orbits in one period-1680 cycle = floor(1680/q). Only mod-8 divides 1680 evenly.

Data channels {mod 8, mod 9, mod 25, mod 49} carry information (warm colors, inner orbits). Parity channels {mod 11, mod 13, mod 17} carry error correction (cool colors, outer orbits). Prime-power channels have p^e positions (vs p for prime channels), but the speed ordering is not data-vs-parity: mod 8 (period 8) is fastest overall, while mod 49 (period 49) is slowest.

Angular Momentum

When two elements spin at the same stride, the sum of their circular phase differences across all 7 channels is constant. The step cancels out. This constant IS the CRT distance between the two elements, measured dynamically.

For elements 1 and 2: differ by 1 in every channel. circ_dist(1, q) = 1 for all q >= 3. Sum = 7 * 1 = 7. Measured on ESP32 hardware: every SPIN packet between the two boards shows ang_mom = 7.

The angular momentum depends on which elements you compare, not on how long the ring has been spinning. Rotation contributes nothing -- the structure was already there. A dynamic measurement revealing a static invariant.

Detection

The mod-11 channel detects errors. 11 = 1 + 2 + 3 + 5 (a parity checksum). ECC3 triple parity uses all three parity channels (mod 11 + mod 13 + mod 17, rate 4/7). Any single-channel corruption is 100% detectable and correctable.

Measured on ESP32: mod-11 channel injection detected at 100% rate. Phase-independent: 72/72 tests across all rotation phases, including all 7 channel wrap points. Time does not affect ECC detectability.

Recovery

The mod-8 channel carries the pair structure. When corruption is detected, the negation round-trip (n -> N-n) cancels it. The algebra: x + e becomes (N - x - e) + e = N - x becomes x. Two negations, perfect cancellation.

Measured on ESP32: mod-8 board injects mod-11 channel error (+1 mod 11) on every 5th packet. Hub detects, mirrors the corrupted value (N - corrupted), sends back. Mod-8 board applies same error on return, mirrors again. Result: 100% recovery (4/4 targeted tests).

Operational Boundary

The mod-13 channel marks the boundary between auto-heal (1 error) and retransmit (2 errors). 13 = 2^2 + 3^2, where the Cunningham chain stops. Exhaustive computation: 535 million 2-channel error patterns tested. Zero undetected.

The algebraic guard: parity product 2431 = 11*13*17 exceeds 49^2 = 2401. Margin of 30 means no pair of data-channel errors can produce a syndrome-invisible delta. Protocol: detect() on every packet. Clean = accept. Error = retransmit via -1 round-trip.

Lambda Stability

Carmichael lambda = 1680 = the multiplicative exponent (a^1680 = 1 for all a coprime to 214,414,200). This is not the spin period (that is 214,414,200) but the algebraic period of the unit group.

The 24-hour drift test: 3 boards spinning continuously, stride=1. Hub captures angular momentum from both neighbors every 3 seconds. 51,738 packets total. Result: ang_mom = 34 for every single observation. Zero drift. Zero ECC failures.

The value 34 depends on boot timing: boards power on at different moments, creating a fixed CRT distance between their spinning phases. Same-stride constancy means this is provably step-independent: any boot offset giving 34 stays at 34 forever. Verified: 62/62.

Drift sensitivity: a single-step drift (3 seconds) would change ang_mom from 34 to 33. Over 24 hours (28,800 steps), ang_mom stayed exactly 34. ESP32 clocks stable to within 3 seconds per 24 hours (< 35 ppm). CRT provides free clock monitoring with zero protocol overhead.

Stride and Isolation

The mod-17 channel is exclusive to Z/214,414,200 (not in Z/12,612,600). All three parity channels (Z/11, Z/13, Z/17) are fields. When boards spin at different strides, angular momentum becomes step-dependent. The stride DIFFERENCE determines which channels are frozen.

Stride Lock Theorem

gcd(stride_diff, q_i) = q_i freezes channel i. The mod-8 channel has modulus 8. Any stride difference divisible by 8 locks mod-8 at a constant angular momentum contribution. Three stride ratios tested on hardware (90 packets each):

Channel Isolation Hierarchy

Isolation stride for channel i = N/q_i. To observe ONLY one channel, spin at the stride that makes all others period-1. The ring hierarchy IS the isolation hierarchy:

Silicon

The mod-9 channel runs on silicon. 9 = 3^2, the parallelism unit. The FPGA implementations prove the spinning structure on hardware, where the ring physically orbits at clock speed.

Theorem Pointers

The spinning-spheres canvas is backed by two theorems and the Gabriel-Horn anchor. STRUCTURAL analogs of Clay-Millennium statements, NOT Clay proofs = but the ring as capped horn is proved mathematics on the 108-ring Carmichael-lambda lattice.