Unity as Closure Capacity: The Borwein Threshold

The Borwein integrals demonstrate a sharp closure phenomenon: a product-of-sincs integral equals π/2 precisely while the sum of reciprocal denominators stays below unity, and breaks the moment unity is exceeded. We give the Fourier and probabilistic proofs (the latter from Majumdar and Trizac, 2019), articulate the closure-capacity-equals-unity principle as the durable structural contribution, and then show that for the canonical odd-reciprocal sequence aₖ = 2k+1, exactly six terms fit before overflow. A pre-registered null test (this paper, §8) confirms that the count six is sequence-dependent — random monotone sequences with comparable density give counts spanning 2 to 12 — so the "dimensional signature"…