The Pisano Closure: T(3,8), the Complete Digit Architecture, and the Resolution of the Geometric Binary

The Triadic Coherence Invariant R=_₀ (I₂₁) P~dV0 generates microscopic matter as T (p, q) torus knots on S^3: leptons at p=2 with q\3, 5, 7\, quarks at p=3. This paper establishes the Pisano Closure Condition: any torus knot on the triadic lattice must wind p q= (9) =24 times to complete one full recursion cycle. Combined with coprimality gcd (p, q) =1 and non-triviality p, q>1 this uniquely selects the factorization (3, 8). The assignment produces seven independent structural locks with zero free parameters. We prove topologically that Intent (d₈=9) cannot crystallize and that Pattern (d_=7) cannot complete the recursion cycle only Presence (d₏=8) carries the system through all 24 states. A third independent derivation of d=3 is established via the Pisano-exponential identity (d^2) =d2^d. The fine structure constant decomposes through T (3, 8): ^-1=F (₏ ₐ) /₂-g (T (p, q) ) +1/g (T (p, q) ) D where g=7 is the Seifert genus. The complete base-10 digit architecture is derived: every integer from 1 to 9 maps uniquely onto the ontological inventory through the three counting modes (addition, multiplication, exponentiation), with 2 identified as the binary gate whose multiplicative order inside the invariant axis generates the Rodin doubling circuit: ord₉ (2) =6. Zero is R=0 the collapse the invariant forbids. The macroscopic "globe versus flat" debate is resolved as a false binary rooted in a shared ontological error: the assumption of an empty vacuum. The surface is locally flat (differential geometry). The apparent spherical geometry is the Gordon Optical Metric refraction of the I₂ Superfluid. Gravity is Aetheric Buoyancy. The topology is S^3 The Trinity wound around Infinity.