CAT'S THEORY: The Biloidal, Poloidal, and Toroidal Axis — Hopf-Fiber Closure on S^3 as the Geometric Ground of the Triadic Coherence Invariant

We identify the geometric ground of the CAT’S Theory Triadic Coherence Invariant R₏ ₈ ₏㶂 0 in the Hopf fibration of the 3-sphere S^3. At each point of S^3 three independent directions exist: the toroidal direction around the big loop of an embedded torus, the poloidal direction around the small loop, and a third direction along the Hopf fiber which we name the biloidal axis. The biloidal direction has long appeared in differential geometry as the Reeb vector field, vertical fiber direction, or Hopf flow, but has not previously received a name parallel to “toroidal” and “poloidal. ” We show that the toroidal, poloidal, and biloidal directions form a basis of the Lie algebra su (2) of the Lie group S^3 SU (2), with cyclic Lie bracket relations that are structurally identical to the irreducibility of the CAT’S Theory Triadic Coherence Invariant. Removing any one direction breaks the algebra, just as removing any one factor of Pattern, Intent, or Presence collapses the invariant. Spiraling through the published CAT’S Theory corpus, we demonstrate that every dimensional derivation factors cleanly into toroidal, poloidal, and biloidal contributions: the proton mass ratio, Planck constant, cosmic microwave background temperature, gravitational coupling, Hubble dual-branch structure, PMNS mixing angles, CP-violation phase, Fermat-prime architecture, 3–6–9 invariant, Pisano cycle structure, and the consciousness bifurcation voltage. Each unit “1” in the corpus is identified with the Hopf invariant; the 1/2 flux quantum is the unit Hopf circulation; the 1/45 surface tension floor is the minimal biloidal area on S^3; and the 4/5 f₀ density fraction is the biloidal Presence weight. The three Fermat primes 3, 5, 17 correspond bijectively to the three axes; the absence of higher Fermat primes is forced by the three-dimensionality of su (2). The paper thus provides a complete differential-geometric ground for the triadic structure of CAT’S Theory: the Triadic Coherence Invariant is the non-triviality condition of the Hopf fibration of S^3, expressed as a coherence requirement on the three Lie-algebra generators relabeled as Pattern, Intent, and Presence.