[I revised and expanded this post since first posting it last night, so you roving rabid readers of mine may want to take it for a second spin.]
The soul is only as accidental to the living organism as the number one is to the number three. One suffuses three, yet is not identical to it. The soul suffuses the organism, yet is not identical to it. Three is not "made up of" three ones, since it does not instantiate the form of one three times. If it were of the form of one, it would be one.
Nor, however, is three made up of some brute substance––call it triplex matter––that could possibly exist without the primal act of one's existence. The act of three's existence integrally instantiates the act of one's existence, and in a way that renders them mutually concrete. The difference is that three is an expression of the powers of one, whereas one is an expression of a more potent mode of being which suffuses into the other integers. Three exists by the power of one but not as the power of one, exists in its own unique mode by the power of one. Three is something one can do, yet not something one can do on its own, qua concrete singularity, apart from the dynamic complexity of three. Likewise, the organism as a whole is an expression of the powers of the soul: it is something the soul can do, yet not something the soul qua form can do on its own, apart from the dynamic complexity of the organs.
If you cut apart three, you will find only one(s). Three cannot be observed apart from its integral act of existence, and every attempt to 'reduce' it to something more ontologically 'basic' is simply to destroy three qua something that exists in a formally unique (proper) way. By the same token, one can never truly be observed in three, since every investigation of three will manifest three's proper (unique) act of existing, not as one but as three. The two 'elements' of an integer, then, must be combined, like lenses, in order to get a clear picture of actual being. The same goes for an organism: its primal act of being (the soul) must be focused together with--notionally superimposed on--its secondary, dynamic mode of being (the body).
A different avenue: The surface of a contained fluid is no more accidental to the fluid than the soul is to the organism. There is no actual "film" or "plane" that can exist abstracted from the actual volume of the fluid,* yet the fluid can only exist within the confines of its volume, which is to say, within the (formally reified) enclosing film. No one can remove the "outermost layer" of the fluid to scrutinize it. First, not only is a material plane infinitely divisible into further "outermost layers," but also, second, shaving off the fluid's outer surface alters its volume, which would then require a newly measured shaving, et cetera, ad infinitum. In the same vein, no one can vivisect the organism to excise the soul (as the "innermost layer") for empirical scrutiny. The fluid exists as its outer surface but not in its outer surface, for the outer surface is not an independent container of an equally independent substance. Rather, the fluid can only be the fluid that it actually is by virtue of its being contained in a specific way, and yet it is not identical to this abstractly specific, pure way of containment.
By the same token, the outer surface exists as the boundary of the fluid but not in any boundary of the fluid: it is immeasurable--indeed, intangible--precisely because it is that by which the fluid is measurable in the first place. If the fluid lacked a measurable boundary, it would not be a distinct substance: it would be literally amorphous (i.e. formless). By analogy, the soul exists as the functions of the body but not in (any of) them. For that reason the soul is imperceptible and immeasurable--literally im-mense, as Scholastic theology has it--outside of the integrated life of the organism. The life of the organism, as an integral act of specific being, points to its own unity qua the soul precisely because it is one act of being, one organized entity, one organism.
* Even so, there does seem to be a form of the fluid's contours by which such a pure film could be modeled, as an intentional object of existence, and which could later be used to replicate the contours of the fluid itself under the so to speak "formalized form" of the film. This is to suggest how the soul can exist, albeit only in an intentional way, apart from the body, and ultimately suffice to actualize the same organism at the Eschaton.