## COLS Torus — M = 275 (5^2×11), V28

Surface = E_uni (default red→blue). You can invert the palette below. Thick red/blue lines are native dynamic cross‑sections (no hover JS).
Iso‑lines (E_uni)
Show overlays unilateral/bilateral
Torus transparency semi‑transparent
Reverse palette (blue→red)
Dynamic line thickness (8–64)
Overlay point size (1–6)
🧭 Quick reader’s guide (2 min)

Number: 275 = 5×11. Base: a=2. Periods: M=275=5×11, base a=2. Orders: λ_5=4, λ_11=10 ⇒ 2λ=40.

- What are we looking at? A discrete CRT torus. Horizontal: x = n mod 275, vertical: y = t mod 40. Each dot is one state \((n,t)\).
- Surface colours: the colour encodes unilateral energy \(E_{\mathrm{uni}}\in[0,1]\). Default red=low, blue=high (switch can reverse).
- Green points (unilateral): times when \(2^t\) is “close to ±1” for one modulus 5 or 11.
- Orange points (bilateral): times when 2^t ≈ ±1 (mod 5) and (mod 11); resonance rings.
- Red points (GCD bilateral): exact events where \(2^t\equiv\pm1\) mod both moduli (strict CRT synchrony).
- Iso‑lines: colour‑coded — 0.30 yellow, 0.50 orange, 0.70 red.

❓ What is the CRT torus?

With two periodic variables, \(x = n \bmod M\) and \(y = t \bmod 2\lambda\), the state space is \(\mathbb{Z}_M \times \mathbb{Z}_{2\lambda}\) — a discrete torus. Over a full mixed period \(L = \mathrm{lcm}(M,2\lambda)\), the map \(s\mapsto (s\bmod M, s\bmod 2\lambda)\) visits all pairs (up to \(\gcd(M,2\lambda)\)) as guaranteed by CRT.

We embed this grid in 3D using \((\theta,\phi)=(2\pi x/M, 2\pi y/(2\lambda))\). The coloured surface is \(E_{\mathrm{uni}}(x,y)\); green/orange/red points mark unilateral/bilateral/GCD‑bilateral resonances.