## Modelling the Phase Shift

(first cut)

 The basic bud/planet phenomena are these: A bud undergoes a change of shape ( a "bud event") twice per lunar circuit, most species becoming measurably more elliptical (a few become less) for an interval of generally 1 to 3, sometimes 4, very occasionally 5, days, before reverting to a relatively steady, (most frequently "sharp-upwards") egg shape. The frequencies, or rhythms, of these changes closely match the frequencies of the geocentric alignment (opposition or conjunction) of the Moon with bodies of the Solar System. They are specific. That is to say, the rhythm of the form-change of a particular bud kind is that of a particular body's lunar alignment, always the same body per bud kind. Bud Events appear unrelated to: The Distances between associated bodies. The "Sense" of these bodies—that is, their order in any given line-up.
 The features specific to the so-called "Phase Shift" are: The timings of the bud changes, relative to the timings of associated alignments, vary, but by the same amount (within the usual limits of observational error) for all bud/body pairings. The timings of the bud changes slip consistently ahead of the timings of the corresponding alignments. If T is a shift interval, and t denotes time, then the "slip rate", dT/dt, is always negative or zero, and varies very approximately as αsin(γt - φ) - β, where α,β,γ,φ are constants. In other words, the rate shows cyclic variation on a secular mean.
 If we seek an agent for these phenomena, what mathematical entity can best represent it? Vectors are said to represent quantitities having magnitude and direction in a convenient, geometric way. That is, it is convenient because their geometric additions and subtractions are equivalent to the actual additions and subtractions of the (sometimes non-geometric) quantities they stand for. In fact, vectors usually also represent the sense of the "action" or "effect" of the quantities they otherwise represent: Newton's Action and Reaction are instances. For these, a force on a body is countered by another (equal) force from the body (i.e., in the opposite sense), but in identically the same direction. So a vector that does not take account of sense, but only of direction, is not quite a vector! Let us call it a "quasi-vector." If the bud phase shift is the result of some interaction between the bodies of the Solar System, then the agent—"agent" in the way that 'force' may be thought of as the agent of Gravity, for example, that 'causes' a thing to happen that would not without it—the agent of the shift is not likely to be one that can be represented by a "proper" vector, since, while direction is definitely retained, the quality of sense seems not to be, and it is not obvious how to assign magnitudes. Hence the agent will, at most, have a 'quasi-vector' [One may discover the general direction and amount of motion (speed) of, say, all the bodies of the Solar System, without taking regard to the senses (velocities) of the component motions, and draw a line of appropriate length and direction to represent it. But it would not have a head or tail. This would be a quasi-vector, having direction and magnitude, but not sense.]