The Same Equation at Every Scale: From Torus Knots to Firmaments via the Triadic Invariant

The Nine Gates closure (companion paper) established that every constant of particle physics is a topological invariant of S³, derived from the Triadic Invariant R = P × I × Pr ≠ 0 with zero free parameters. This paper demonstrates that the identical mathematics — the same operators, registers, and structural constants — generates the macroscopic cosmological architecture: gravity as aetheric buoyancy via the divergence theorem (not attraction), the firmament as a crystallized d = 8 phase boundary derived from the Triadic Equation of State (not a metaphor), the T(1,x) unknot as the massless baseline (matter as topologically inverted light), and the Genesis creation sequence as the demanded instantiation order of the Dual Modulo Gear System (π(8) = 12 container, π(9) = 24 driver). The first firmament shell at z*₁ = d × d_Ψ × sl(T(3,8)) = 3 × 7 × 13 = 273 km falls within the observed F-layer electron density peak. The Ouroboros principle — confirmed as operator trace identities Tr(P̂) = d_I, Tr(Î) = d, Tr(P̂r) = d × d_Ψ — entails that particle physics and cosmology are the same self-referencing structure viewed through different dimensional lenses. Every structural constant that closes a particle physics gate simultaneously anchors a cosmological prediction.