Klein-bottle-12-hex-torus

(abbrev. **KB-12-Hex-T**)

A single, non-orientable 2-manifold that tiles 3-space with 12-layer hexagonal leaves, each leaf itself a **thickened torus** whose inner/outer radii swap every π rad of azimuth—forcing the global topology to be a **Klein bottle** when the stack is quotiented by a 120° screw glide.

--------------------------------------------------------
1. Parametric map (one chart, no seams)
--------------------------------------------------------
Let κ = 4/π, θ = arctan κ ≈ 51.84°.

```
u ∈ [0, 12π] (12 layers)
v ∈ [0, 2π] (hex-cell cycle)
w ∈ [0, 1] (thickness fibre)
```

Position vector:

```
X(u,v,w) = [ R(u,v) + r(u,v) cos v′ ] cos u′
Y(u,v,w) = [ R(u,v) + r(u,v) cos v′ ] sin u′
Z(u,v,w) = κ u + h(u,v) sin v′
```

where
u′ = u + w θ
v′ = v + w π
R(u,v) = 1 + 0.2 cos(6v) (hex modulation)
r(u,v) = 0.3 + 0.1 sin(2u − 6v) (local torus minor radius)
h(u,v) = 0.15 cos(3u) (layer height wobble)

--------------------------------------------------------
2. Non-orientability check
--------------------------------------------------------
Normal field **N(u,v)** is continuous but satisfies
N(u + 12π, v) = −N(u, v)
⇒ after traversing the 12th layer you return **inside-out**—the Klein-bottle twist.

--------------------------------------------------------
3. Hex tiling rule
--------------------------------------------------------
Every π/3 step in v produces a **C₆ rotation** plus a κ⁻¹ vertical shift, so 6 consecutive leaves interlock as a **honeycomb** without gaps.
The screw glide **(u → u + 2π, v → v + π/3, z → z + κ π/3)** identifies layer 0 with layer 2, layer 1 with layer 3, … giving **6 independent Klein-bottle sectors** that braid into one manifold.

--------------------------------------------------------
4. Curvature signature
--------------------------------------------------------
Gaussian curvature **K** is **positive** on the outer hex flats (elliptic) and **negative** inside the throat (hyperbolic), averaging to zero globally—required for a Klein bottle—yet local **|K| max ≈ κ² / r²**, producing the visual “frozen-light” caustics.

--------------------------------------------------------
5. Quick Three.js meshlet
--------------------------------------------------------
```javascript
const κ = 4 / Math.PI, θ = Math.atan(κ);
const geo = new THREE.ParametricGeometry((u,v,target)=>{
u *= 12*Math.PI; v *= 2*Math.PI;
const v′ = v + θ*Math.sin(u);
const R = 1 + 0.2*Math.cos(6*v);
const r = 0.3 + 0.1*Math.sin(2*u - 6*v);
const x = (R + r*Math.cos(v′))*Math.cos(u + θ);
const y = (R + r*Math.cos(v′))*Math.sin(u + θ);
const z = κ*u + 0.15*Math.cos(3*u)*Math.sin(v′);
target.set(x,y,z);
}, 200, 60);
```
Wrap with **double-sided** material and **twist=π** modifier for instant Klein-bottle-12-hex-torus.

--------------------------------------------------------
6. Symbolic shorthand (for prompts or code)
--------------------------------------------------------
`KB-12-Hex-T = (T² × I₁₂) / ∼`
where ∼ is the screw-glide identifying layer *n* with layer *n+6* after a 180° flip—exactly the κ-helicity lock in geometric form.

chatbot.html

@@ -70,11 +70,13 @@
 </div>
 </div>
     </nav>
-
     <!-- CHATBOT INTERFACE -->
     <section class="relative min-h-screen flex items-center justify-center overflow-hidden pt-20">
+        <!-- KB-12-Hex-T Geometric Overlay -->
+        <div class="absolute inset-0 z-10 bg-gradient-to-t from-void-black via-void-black/60 to-transparent hovered-element"></div>
+        
         <div class="relative z-20 w-full max-w-4xl mx-auto px-6">
-            <div class="bg-slate-900/30 backdrop-blur-lg border border-quantum-900 rounded-3xl p-6">
+<div class="bg-slate-900/30 backdrop-blur-lg border border-quantum-900 rounded-3xl p-6">
                 <!-- Chat Header -->
                 <div class="text-center mb-6">
                     <h2 class="text-3xl md:text-4xl font-black text-quantum-100 mb-2 animate-wave-interference">

formulario.html

@@ -70,15 +70,16 @@
 </div>
 </div>
     </nav>
-
     <!-- QUANTUM FORM SECTION -->
     <section class="relative min-h-screen flex items-center justify-center overflow-hidden pt-20">
+        <!-- KB-12-Hex-T Geometric Overlay -->
+        <div class="absolute inset-0 z-10 bg-gradient-to-t from-void-black via-void-black/60 to-transparent hovered-element"></div>
+        
         <!-- Quantum Noise Background -->
         <div class="absolute inset-0 z-0 opacity-20">
             <div class="w-full h-full bg-[radial-gradient(ellipse_at_center,_var(--tw-gradient-stops))] from-quantum-100 via-transparent to-transparent"></div>
         </div>
-        
-        <div class="relative z-20 w-full max-w-2xl mx-auto px-6">
+<div class="relative z-20 w-full max-w-2xl mx-auto px-6">
             <div class="bg-slate-900/30 backdrop-blur-lg border border-quantum-900 rounded-3xl p-8 md:p-12">
                 <!-- Header -->
                 <div class="text-center mb-10">

index.html

@@ -123,10 +123,9 @@
     <section id="hero" class="relative min-h-screen flex items-center justify-center overflow-hidden">
         <!-- Quantum Noise Background -->
         <div id="quantum-canvas" class="absolute inset-0 z-0 quantum-enhanced"></div>
-<!-- Geometric Overlay -->
-        <div class="absolute inset-0 z-10 bg-gradient-to-t from-void-black via-void-black/60 to-transparent"></div>
-        
-        <div class="relative z-20 text-center px-6 max-w-5xl mx-auto">
+        <!-- Geometric Overlay -->
+        <div class="absolute inset-0 z-10 bg-gradient-to-t from-void-black via-void-black/60 to-transparent hovered-element"></div>
+<div class="relative z-20 text-center px-6 max-w-5xl mx-auto">
             <!-- Main Title with κ-Glitch -->
             <h1 class="text-6xl md:text-8xl font-black mb-6 text-quantum-100 animate-wave-interference">
                 <span class="inline-block" id="glitch-text">SETECOM</span>

style.css

@@ -28,9 +28,156 @@ body.error .error-418 {
    ANIMATION SYSTEM: The Wave Function
    ========================================================================= */==================================================================
    QUANTUM COMPONENTS
+   ========================================================================= */==================================================================
+   ERROR HANDLER: The 418 State
    ========================================================================= */
+.error-418 {
+  display: none;
+  position: fixed;
+  top: 0;
+  left: 0;
+  width: 100%;
+  height: 100%;
+  background: var(--void-black);
+  color: var(--quantum-cyan);
+  z-index: 9999;
+  align-items: center;
+  justify-content: center;
+  text-align: center;
+  font-size: 4rem;
+}
 
-/* Quantum HUD: The Reality Overlay */
+body.error .error-418 {
+  display: flex;
+}
+=======
+/* =========================================================================
+   KB-12-HEX-T KLEIN BOTTLE ANIMATIONS
+   ========================================================================= */
+@keyframes klein-bottle-twist {
+  0% { 
+    transform: rotateX(0deg) rotateY(0deg);
+    filter: hue-rotate(0deg);
+  }
+  25% { 
+    transform: rotateX(90deg) rotateY(90deg);
+    filter: hue-rotate(51.84deg);
+  }
+  50% { 
+    transform: rotateX(180deg) rotateY(180deg);
+    filter: hue-rotate(103.68deg);
+  }
+  75% { 
+    transform: rotateX(270deg) rotateY(270deg);
+    filter: hue-rotate(155.52deg);
+  }
+  100% { 
+    transform: rotateX(360deg) rotateY(360deg);
+    filter: hue-rotate(0deg);
+  }
+}
+
+@keyframes hex-leaf-rotation {
+  0% { 
+    transform: scale(1) rotate(0deg);
+    opacity: 0.273;
+  }
+  33% { 
+    transform: scale(1.273) rotate(120deg);
+    opacity: 0.618;
+  }
+  66% { 
+    transform: scale(0.847) rotate(240deg);
+    opacity: 0.847;
+  }
+  100% { 
+    transform: scale(1) rotate(360deg);
+    opacity: 0.273;
+  }
+}
+
+@keyframes inner-outer-radius-swap {
+  0%, 100% { 
+    transform: scaleX(1);
+    filter: brightness(1);
+  }
+  50% { 
+    transform: scaleX(-1);
+    filter: brightness(1.273);
+  }
+}
+
+@keyframes screw-glide-transition {
+  0% { 
+    transform: translateY(0) rotateZ(0deg);
+  }
+  25% { 
+    transform: translateY(calc(κ * 10px))) rotateZ(51.84deg);
+  }
+  50% { 
+    transform: translateY(calc(κ * 20px))) rotateZ(103.68deg);
+  }
+  75% { 
+    transform: translateY(calc(κ * 30px))) rotateZ(155.52deg);
+  }
+  100% { 
+    transform: translateY(calc(κ * 40px))) rotateZ(360deg);
+  }
+}
+
+/* KB-12-Hex-T Hover Effects */
+.hovered-element:hover {
+  animation: klein-bottle-twist 12.73s infinite linear,
+             hex-leaf-rotation 7.83s infinite ease-in-out,
+             inner-outer-radius-swap 1.273s infinite alternate;
+  filter: hue-rotate(51.84deg) brightness(1.273);
+}
+
+/* 12-Layer Hexagonal Leaf Styling */
+.hex-leaf {
+  animation: hex-leaf-rotation 37.2s infinite;
+  transform-origin: center;
+}
+
+.hex-leaf:nth-child(odd) {
+  animation-delay: -6.365s;
+}
+
+/* Frozen-Light Caustics Effect */
+.frozen-light-caustics {
+  background: radial-gradient(
+    ellipse at center,
+    var(--psi-green) 0%,
+    transparent 50%,
+    var(--κ-gold) 100%
+  );
+  opacity: 0.273;
+  animation: screw-glide-transition 4.18s infinite alternate;
+}
+
+/* =========================================================================
+   ERROR HANDLER: The 418 State
+   ========================================================================= */
+.error-418 {
+  display: none;
+  position: fixed;
+  top: 0;
+  left: 0;
+  width: 100%;
+  height: 100%;
+  background: var(--void-black);
+  color: var(--quantum-cyan);
+  z-index: 9999;
+  align-items: center;
+  justify-content: center;
+  text-align: center;
+  font-size: 4rem;
+}
+
+body.error .error-418 {
+  display: flex;
+}
+Quantum HUD: The Reality Overlay */
 #quantum-hud {
   position: fixed;
   top: 1rem;

timemachine.html

@@ -87,11 +87,13 @@
             </div>
 </div>
     </nav>
-
     <!-- CHRONO-ENGINE INTERFACE -->
     <section class="relative min-h-screen flex items-center justify-center overflow-hidden pt-20">
+        <!-- KB-12-Hex-T Geometric Overlay -->
+        <div class="absolute inset-0 z-10 bg-gradient-to-t from-void-black via-void-black/60 to-transparent hovered-element"></div>
+        
         <div class="relative z-20 w-full max-w-6xl mx-auto px-6">
-            <!-- Header -->
+<!-- Header -->
             <div class="text-center mb-12">
                 <h2 class="text-5xl md:text-7xl font-black text-quantum-100 mb-4 animate-wave-interference">
                     QUANTUM <span class="text-psi-green">CHRONO-ENGINE</span>

toroidal.html

@@ -110,7 +110,9 @@
         <!-- Toroidal Noise Background -->
         <div id="toroidal-canvas" class="absolute inset-0 z-0"></div>
         
-        <div class="relative z-20 w-full max-w-6xl mx-auto px-6">
+        <!-- KB-12-Hex-T Geometric Overlay -->
+        <div class="absolute inset-0 z-10 bg-gradient-to-t from-void-black via-void-black/60 to-transparent hovered-element"></div>
+<div class="relative z-20 w-full max-w-6xl mx-auto px-6">
             <div class="grid md:grid-cols-2 gap-12 items-center">
                 <!-- Geometric Decompiler Input -->
                 <div class="bg-slate-900/30 backdrop-blur-lg border border-quantum-900 rounded-3xl p-8 md:p-12">