Proposed Canon
Erf, possibly derived from the word "Earth", tends to deal with the physical, tangible, and practical. Findamancy locates a unit, object, or location that exists. Turnamancy affects a unit's Loyalty, the timing of a Side's Turn or a city's unit production. Dirtamancy creates and heals golems from available material, and can move and restructure the Erf itself. Lookamancy allows you to see a distant location as it is at that moment. Flower Power and Shockmancy have direct effects on units. Hat Magic creates magic hats and transports physical matter. Least clearly physical is Luckamancy, which seems to directly affect reality.
Erf is the practical things of the world, not only the physical, but some of the intangibles as well. It seems that a lot of the “Natural” magics can be manipulated by magics in the Erf axis. Loyalty is Natural Thinkamancy, but it can be altered by Turnamancy as well. Turn order also falls under the Erf axis in Turnamancy. Luckamancy works with Numbers, but it merely rearranges them to change results, it doesn’t actually modify the numbers. Hat Magic has a little bit of everything, but it all appears to be very practical: sending messages, creating items and creatures (e.g. Bun Bun). Everything in the Erf axis deals in broad strokes. Erf magic is not subtle in its effects, even if the consequences of those effects are a little more so. Shockamancy is the premier example of this. It takes Matter and Motion and uses it to destroy, ripping apart what is there as it damages and stuns with overt blasts of magic.
 Speculation on Erf in Magic
Fate controls the paths of units and sides. Numbers deals in likelihoods and abstracts. Erf is the world around you, the terrain, reality itself. Looking at each possibility of Erf + Life + Motion + Matter
Luckamancy - Erf alone. Devoid of the influences of Fate and Numbers (Predictamancy and Mathamancy), untied to Life, Motion, or Matter (which seek to change or control), luckamancy is free to do the impossible. Or not.
Lookamancy - Erf, Life, and Motion (no Matter)- Watching the motion of life on erf. Observing its changes.
Flower Power - Erf, Life, and Matter (no Motion)- Maintaining life, preserving matter... the absence of motion. No change. Flaking. Time standing still.
Shockmancy - Erf, Motion, and Matter (no Life) - Change or movement of matter, absence of Life. Shock and awe.
Hat Magic - Erf, Life, Motion, and Matter - Transmission of matter and knowledge (life) across the erf.
augmented upon the notes on the discussion page by Commander I. Heartly Noah June 3 2009