The Dynamics of Geometric Computation: A Universal Equation Family for Geometric Phase Evolution Across Transformer Architectures

When a transformer processes text, its internal representations pass through a sequence of layers, each performing a different kind of geometric computation. Geometric Contrast Imaging (GCI) is a measurement method that classifies the computational geometry at each layer by analyzing patterns in how attention heads route information. The resulting "geometric trace" reveals that inference is not a uniform process: models consistently organize their layers into distinct geometric phases, with structured transitions between them. This paper asks: what forces govern how the geometric character of computation changes from one layer to the next? We present a five-force Lagrangian equation family that models the acceleration of geometric change through depth. The five forces are: (1) geometry-curvature coupling, which accelerates computation when the geometric mode changes; (2) exponential information pressure, which increases toward the output layer as the model commits to a prediction; (3) spherical-phase oscillation, capturing periodic modulation during integrative processing; (4) phase boundary effects at transitions between geometric modes; and (5) a baseline depth trend reflecting the model's overall computational rhythm. The equation family is validated across eight decoder-only transformer models spanning five architecture families (GPT-2 Small/Medium/Large/XL, Llama 3 8B, Mistral 7B, OPT-6.7B, Pythia6.9B), parameter counts from 124M to 8B, and layer counts from 12 to 48. All eight models achieve R² = 0.83–0.97 with residual autocorrelation below 0.20. Three architecture-dependentcorrection classes emerge from residual diagnostics: (A) no correction needed, (B) localized spline knots for depth-specific dynamics, and (C) autoregressive closure for structured phase-specific dynamics unique to the sole parallel-block architecture. The paper also identifies an entropy-speed anti-correlation law (r = -0.84 to -0.87) linking routing weight uncertainty to representational velocity, and resolves a Simpson's paradox in the coupling force via geometry-conditional regression. An honest prospective failure — four of six pre-registered predictions confirmed for Pythia, leading to the discovery of the third correction class — is reported in full. This is the fifth and final paper in the GCI program: instrument (Paper 1) → emergence mechanism (Paper 2) → statistical mechanics (Paper 3) → normative principle via the Free Energy Principle (Paper 4) → dynamics (this paper). Paper 5 is self-contained and does not require the earlier papers, which remain unpublished.