Examine the existence of a given singularity, and a perception of this singularity.
Fundamental Equivalence - Conceiving of the perception is equivalent to stating the higher singularity necessarily exists.
The proof for the downward part of this theorem is fairly self-evident, to understand the higher singularity is to understand the totality of all its form as an infinitely generalized state. The understanding takes place as a conception from which all such perceptions stem forth. It is as such necessary that one is able to conceive of the perception otherwise the sum totality would not be realised.
The most useful aspect of this realisation is the construction of higher singularity from the conception of its perception. The perception itself must satisfy each higher existence property analogous to the upward Lowenheim-Skolem theorem; though each extension may be ontologically null. The 'higher existence property' is specifically perceiving of the aspect satisfying an order of existence of arbitrarily large size, while this may seem counter-intuitive and indeed paradoxical it is a necessary entailment from the iteration of the aspect to the degree of the cardinality. This is necessary since the singularity by definition is a state of infinite generalisation, and for a perception of this to fully satisfy the sum totality it must satisfy each indexed order of magnitude. This extension is a reflection of the singularity, and not an innate projection from the perception - so it is indeed that the extension from the point of the perception is 'ontologically null', i.e. no new meaning is derived from the construction. From what principle is the meaning ordained? The answer is simple, the singularity.
This aforementioned discussion is now sufficient for the proof of the second direction of the fundamental equivalence. To say that one has knowledge of the higher singularity is to say that one understands the totality of all its form as an infinitely generalised state. The conception of a perception stipulates an infinite generalisation of that aspect, we stated before that the extension is a projection of the singularity rather than the form since arbitrary extension does not generate meaning. To elaborate on this idea consider earlier discussions where it was explained that the 'shadow' as perception exists as that perception only because we relate it to the higher conception. The shadow without knowledge of the conception is meaningless. But the perception itself must be meaningful as being generalised, the generalisation reflects the identity of the higher singularity which henceforth guarantees its existence.
In fact an alternative proof of this account can be given. The conception of a perception lies outside of the immediate field of understanding. To attain this level of generalisation and totality it is clear that one must transcend the immediate circumstance to a higher plane of understanding. The externalisation of this aspect is already apparent, in generalising a perception to conception it is analogous to and a reflective understanding of the generalisation of form to singularity. We can in fact, conceive of the generalisation by abstracting each form. The part cannot exist independent of the whole, and a particular generalisation cannot exist without the embodying conception of totality itself realised as singularity. The subtlety in this argument is not apparent, but we are essentially generalising the generalisation.
One may ask as to why we do not initially invoke the premise that the part cannot exist without the whole to derive the fundamental equivalence. Here one would be making an error to equate the 'whole' with the higher singularity, when the 'whole' could embody the primary conception. That is to say we may have immediate knowledge of the form but nothing higher than that. To prove that the higher singularity exists we could conceive of the form, but it may only produce ontologically null extensions. However the fundamental equivalence should still allow us to guarantee the existence of the higher singularity even though one cannot conceive of it. In practice there is only one situation in which this principle comes into affect, and that is in proving the existence of greater structure to the self.
