Distributed Field Processing V: Field Closure and the Topology of Gravity

The number four pervades gravitational physics. Textbook frameworks establish this independently: the ADM formalism requires four constraints (one Hamiltonian, three momentum); tetrad gravity requires four orthonormal basis vectors; Loop Quantum Gravity identifies the tetrahedron (four faces) as the quantum of space; Regge calculus builds discrete spacetime from four-vertex simplices. LIGO's measurement of exactly two graviton polarizations confirms this structure observationally: 10 - 2k = 2 implies k = 4. These are textbook results. What textbooks leave unexplained is why four.      
   
   
The simplex theorem provides the answer: d+1 points are required to enclose a region in d dimensions. For three-dimensional space, k = 3 + 1 = 4. The tetrahedron is not merely convenient; it is the minimal solid, geometrically necessary to enclose volume. This paper bridges textbook tetrad gravity and Loop Quantum Gravity with the Distributed Field Processing (DFP) framework, demonstrating that the tetrahedron appearing in both is the same geometry that emerges from field closure analysis: the Field Chamber, bounded by k=4 Coupling Ports through which field configurations settle to consistency.      
   
   
The Port Ratio sqrt(k/pi) = 1.128 converts between continuous geometry (the spherical Field Envelope, characterized by pi) and discrete topology (the tetrahedral Field Chamber, characterized by k=4). Three independent routes converge on the characteristic scale r_e = 2.82 fm: electromagnetic stability, cosmic topology, and the k^2 = 16 coupling test. Paper IV established this convergence through a bidirectional proof: electron parameters predict H = 69.92 km/s/Mpc; observed H recovers the electron mass to 0.10% of CODATA values. The prediction matches the GW170817 gravitational-wave standard siren measurement (70.0 ± 12 km/s/Mpc) to 0.11%.       The bridge is the tetrahedron itself: the quantum of space in Loop Quantum Gravity, the local frame in tetrad gravity, the Field Chamber in DFP, the minimal solid in geometry. Different vocabularies, same geometry, same physics.