M = 299 (13×23) (V28)

COLS Torus — M = 299 (13×23), 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: 299 = 13×23. Base: a=2. Periods: M=299=13×23, base a=2. Orders: λ_13=12, λ_23=11 ⇒ 2λ=264.

What are we looking at? A discrete CRT torus. Horizontal: x = n mod 299, vertical: y = t mod 264. 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 13 or 23.
Orange points (bilateral): times when 2^t ≈ ±1 (mod 13) and (mod 23); 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.