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" reading cannot rest on statistical anomaly of the count. It rests instead on a structural identity: D = 3 is the unique non-trivial integer satisfying triangular(D) = D!, i.e., 1 + 2 + 3 = 1 · 2 · 3 = 6. This identity is structurally analogous to φ² = φ + 1 — both are unique non-trivial solutions of equations equating an additive operation to a multiplicative operation, at their respective scales (integer for D = 3, continuous for φ) — though not identical in mathematical shape, as the two equations live at different levels of abstraction. The Borwein integral and the Tower series (Keeble, 2026) are conjugate operational witnesses to this closure through shared atomic content: §6.1 establishes that the canonical sequence aₖ = 2k+1 is the Leibniz atomic content of π/4, which IS Tower's base L. Borwein registers the closure at the additive face (1+2+3) by keeping the atoms distinguished; Tower registers it at the multiplicative face (powers of π/4 raised through D dimensions terminate at D=3) by aggregating them via Leibniz. The π/2 result of the Borwein integral is, by §3's argument, the value the Borwein integrand specifically gives via the standard normalisation of sinc. Interpreted on the achiral substrate of S³ / 2I (the even projection of the manifold; Paper 176): the integral is built in Gaussian coordinates and so measures, in the interpretive sense supplied by Paper 176, the achiral projection of self-referential content from S³ / 2I. The specific value π/2 is the Borwein integrand's value (sinc's normalisation), not a canonical measure on the substrate. We acknowledge that earlier framings of "Eisenstein vs Gaussian coordinates" as alternative integral measures were malformed; the resolution is projection-theoretic, with Eisenstein carrying the self-referential structure on S³ / 2I and Gaussian denoting the algebra of the achiral (even) projection that is the real domain. Keywords: Borwein integrals, closure capacity, unity, sinc function, addition-multiplication duality, φ, achiral projection, S³ / 2I.