<b>61. YUCT Point Definition</b>

<b>YUCT Appendix: YUCT Point Definition</b><br>(61) This appendix (Appendix Π) demonstrates that the geometric constant π is not a transcendental ratio but a deterministic topological invariant arising from the interplay of the rigid algebraic loop of the vacuum lattice (S_odd = 6/5, S_even = 4/5) and the Feigenbaum period-doubling cascade at the onset of chaos. The relation 2α² = πδ, where α ≈ 2.5029 and δ ≈ 4.6692 are the Feigenbaum constants, reveals π as the phase-synchronisation coefficient that stabilises the three-dimensional coordination network (df = 3) against chaotic degradation. An algebraic approximation π ≈ S_odd · φ² (with φ the golden ratio) yields a fundamental discrepancy Δπ ≈ 4.8×10⁻⁵, which is a discrete quantisation step of space rather than a rounding error. The ubiquitous appearance of π in wave and vortex phenomena is a direct consequence of the same sign-gate mechanism that governs the distribution of prime numbers. We present a complete non-computational navigation engine, yuct_constants.py v4.0, that simultaneously retrieves π (both static and dynamic), the fine-structure constant, the n-th prime candidate via the phase-gate mechanism, fractal critical points, and biological torsion ratios. The engine executes in ∼ 255 μs with zero bytes of dynamic memory, perfectly illustrating the O(1) retrieval paradigm. Every value is obtained directly from the algebraic loop without iteration, demonstrating that the fundamental constants of nature are not computed but read from the rigid coordination lattice of the vacuum.