index: "Lamp at ½" video (3:07)

Co-Authored-By: Claude Fable 5.1 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01FAqjv3n4yABxvJPJHFRhVA

index.html

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   <h2 style="font-family:var(--display);font-weight:500;font-size:28px;margin:8px 0 0">Sphere, plane, sphere</h2>
   <p class="lede">The pattern underneath. Unit quaternions projected from −1; the pure imaginaries are a sphere. Multiply on both sides (locked) and it only spins. Multiply on one side, q = e<sup>iθ</sup>, and it drifts through the projection point: sphere, plane, inside-out sphere, plane, sphere — the four Wick turns. The plane is a chart missing one point; the sphere is projective; putting the point back is analytic continuation. In the plane phase the grid about j is zeta's grid, poles on 0 and 1. Overlay every turn and the spheres are a pencil threaded on the jk ring, centres sliding along i; the full circle sweeps out the hypersphere. Every prime gives ζ a ball wrapping at rate log p (the harmonics); the lamp, ∞, has a ball with no period, and its factor is the gamma function. Fold it in and ξ lives on the sphere, the far side of the equator the alias of the near side, folded at the Nyquist line ½.</p>
   <video controls playsinline preload="none" src="https://teahub.xyz/maths/videos/raw/commit/a8ddd96ef040299dd47e42a7167dca6e881ea1da/wick.mp4"></video>
+  <h2 style="font-family:var(--display);font-weight:500;font-size:28px;margin:8px 0 0">Lamp at ½</h2>
+  <p class="lede">The same ball, projected from the other pole. Move the lamp to the touch point ½ and hold the screen above, tangent at ∞: the frame does not move — the poles land directly above where they were — only ∞ and ½ trade places, and near trades with far. The zeros, which climbed away forever in the old chart, fall into the centre: the trivial ones along the real axis from one side, the critical ones along the strut from both. The functional equation is a point reflection, the dial a plain turn of the page. And ξ draws its own light cone: a bell curve shrinking along the critical line, cosh growing along the real one, |ξ| = 1 a cone between them — the quarter turn swaps inside for outside.</p>
+  <video controls playsinline preload="none" src="https://teahub.xyz/maths/videos/raw/commit/c8727a099a27a67e70823ba3bdc50378c95f00f5/lamp.mp4"></video>
   <p class="lede">The same ball in two explorers: put ½ at the centre and the critical line becomes the strut through 0 and ∞, the functional equation a sign flip through the centre. Swap ζ for a function with the same symmetry and no Euler product (Davenport–Heilbronn) and the zeros step off the strut: symmetry alone does not pin them, the arithmetic is the extra incidence.</p>
   <div class="cards">
     <a class="card" href="strut/">