  We describe a novel geometric methodology for analyzing free-energy and
kinetics of assembly driven by short-range pair-potentials in an implicit
solvent, and provides illustrations of its unique capabilities. An atlas is a
labeled partition of the assembly landscape into a topological roadmap of
maximal, contiguous, nearly-equipotential-energy conformational regions or
macrostates, together with their neighborhood relationships. The new
methodology decouples the roadmap generation from sampling and produces: (1) a
query-able atlas of local potential energy minima, their basin structure,
energy barriers, and neighboring basins; (2) paths between a specified pair of
basins; and (3) approximations of relative path lengths, basin volumes
(configurational entropy), and path probabilities. Results demonstrating the
core algorithm's capabilities have been generated by a resource-light,
opensource software implementation EASAL. EASAL atlases several hundred
thousand macrostates in minutes on a standard laptop. Subsequent path and basin
computations each take seconds. The core algorithm's correctness, time
complexity, and efficiency-accuracy tradeoffs are formally guaranteed using
modern geometric constraint systems. The methodology further links geometric
variables of the input assembling units to a type of intuitive topological
bar-code of the output atlas, which in turn determine stable assembled
structures and kinetics. This succinct input-output relationship facilitates
reverse analysis, and control towards design. We use the novel convex Cayley
(distance-based) parametrization that is unique to assembly, as opposed to
folding. Sampling microstates with macrostate-specific Cayley parameters avoids
gradient-descent search used by all prevailing methods. This increases sampling
efficiency, significantly reduces the number of repeated and discarded samples.
