Relational Realization Theory (RRT) III: Growth Rules, Volume Bounds, and Mesoscopic Transport in Discrete Causal Networks

Paper II of this series fixes the kinematics of RRT causal graphs but leaves the growth rule, which decides the large-scale geometry, as an input. This paper specifies that rule and analyzes it. Nodes are created in generations whose sizes follow a postulated quadratic law N_g = L_0 + \lfloor\beta g^2\rfloor starting from a two-dimensional simplicial seed; each new node handshakes with two parents in the previous generation and with neighbors in its own generation. We prove four results that hold for any locality kernel. Graph distance is bounded below by generation difference, and distance from the seed equals the generation exactly. The volume of every graph ball is bounded by the sizes of the generations it can reach, so cubic volume growth requires generations whose size grows quadratically, and any bounded active front forces asymptotically one-dimensional growth. Attaching each node to a uniformly chosen earlier node forces logarithmic diameter. Finally, every generation can be scheduled as a bounded number of Paper II ticks. The three-dimensional character of the resulting graph is therefore an input of the growth rule, not an emergent output. We report finite-size transport diagnostics: a heat-kernel spectral dimension near 3.3, above 3 by several standard errors at every size studied; a running spectral dimension that crosses 3 on its way to a finite-size crossover set by the shell circumference; and a Poisson exponent near 1 over a window of 3–8 hops, obtained with a conductance-balanced weight w = 1.5. All transport diagnostics are computed on the undirected handshake graph; Paper IV examines what such estimators can and cannot detect. The continuum limit is left open.