The Closed Bridge: Do Zero and Infinity Meet? The Loop, the Anti-Spectrum, and the Shadow Boundary

The bridge of Purple Mathematics was drawn with two ends and no answer to the oldest question it raises: do Zero and Infinity ever meet, or is the structure a straight line forever? This paper answers it. The classical resolution — Möbius's projective line and Riemann's sphere, fully attributed — closes the line into a loop: the two poles become antipodes of one connected world, exchanged by the framework's own founding involution x↦1/x, whose fixed points are 1 and −1; the Balance Boundary thereby acquires a partner, the Shadow Boundary, on the half of the loop the framework had never drawn. So the poles meet: they are joined into one closed structure. But they never merge, and this is the paper's theorem: identifying 0 with ∞ contradicts the balance form (the product formula requires the archimedean and non-archimedean poles distinct, by Ostrowski's dichotomy) and the framework's own equilibrium axiom (a balance requires two populated sides); the poles are joinable, never identifiable. The meeting also has a signal: since valuations negate under reciprocals, the Infinity pole carries the anti-spectrum Ω̃ = Πp^(−∞) — every amplitude the exact mirror of Zero's fullness — sharpening the framework's founding assertion that infinity, too, is composed. The classical indeterminacy of 0·∞ becomes, in signal language, the undetermined sum of a spectrum and its mirror; the balance-symmetric reading assigns it silence: the poles' only meeting value is the Balance Boundary itself (stated explicitly as a reading of a convention, not a theorem). Finally, the closed bridge is not symmetric to walk: the arrow of anticipation of the Mirror Wall orients the loop, with legibility flowing toward Zero. The founding picture completes: not a road with two ends, but one world with two poles, two boundaries, one law of balance, and a direction of sight.