As we have seen, the Euler Identity can be used in a fascinating manner to provide a holistic mathematical interpretation of the nature of contemplative spiritual union.
The deeper implication of this is that - properly understood - Mathematics entails a dynamic interaction between analytic and holistic aspects that lie at opposite extremes of each other.
Thus from the conventional analytic (Type 1) extreme, mathematical symbols are understood in terms of absolutely rigid symbols of form, interpreted in a linear (1-dimensional) manner.
However at the opposite holistic (Type 2) extreme, these "same" symbols are now understood in a completely fluid interactive sense, ultimately representing pure energy states in a circular (0-dimensional) fashion.
Thus the paradox inherent in the analytic, can then only be resolved through the corresponding holistic interpretation.
Therefore in order to achieve this, one must move from initial linear rational understanding (of the mathematical symbols in the fundamental Euler Identity) to a fully transformed circular intuitive appreciation (that appears paradoxical at the linear level).
So in the spiritual journey required, understanding - literally - comes full circle (deeply symbolised by the dimensional symbols 2iπ, which trace out the circumference of the "imaginary" i.e. holistic circumference of the unit circle).
However we can go even further here in demonstrating the relevance of the Euler Identity for the 3 levels of Band 4.
Once again, it may be helpful to initially explain - without holistic mathematical terminology - the precise nature of each of these three levels.
Now, again the 1st level (Band 4, Level 1) is characterised by the arrival at a transcendent form of union (representing the culmination of the development that took place during the previous stage (Band 3, Level 3).
This is identified as the attainment of a pure state of spiritual emptiness (i.e. as beyond all phenomenal form) at the "higher" level of personality.
However a certain imbalance is still likely to remain, whereby the "higher" spiritual still unconsciously dominates to an extent the corresponding "lower" physical aspect of personality.
Therefore the arrival at this transcendent state as beyond all physical form, remains to a degree unconsciously associated with a subtle repression of such form at the "lower" instinctive aspect of personality. This then results in the compensatory activity of the "lower" self to still involuntarily project residual unreformed physical material into consciousness.
Thus the subtle aversion to physical form (at the "higher" level) leads to the involuntary projection of this unrecognised form (with respect to the "lower" aspect of personality).
Therefore, once again we can only meaningfully conceive of such transcendent union in an approximate manner, which depends very much on personality and the intensity of the spiritual drive involved.
So in some "lesser" cases an attenuated degree of transcendent union could be reached with a substantial amount of residual "lower" level imperfection remaining.
However in other cases, especially where a gifted mystic - and perhaps prospective "saint" - is involved, a much greater degree of cleansing with respect to instinctive impulses would be required.
The second level (Band 4, Level 2) is then concerned with seriously addressing this remaining imbalance, so that the immanent aspect of spirituality (where spirit is seen as deeply inherent in all physical form) can ultimately be properly balanced with the corresponding transcendent aspect (where spirit is likewise seen as beyond all phenomenal form).
Now, I would suggest that especially for male mystics, this residual problem is likely to involve too much super ego control as exercised by the cognitive function.
Because in one's training one, has for long been conditioned to view the physical i.e. instinctive as inferior to the spiritual self (as mediated through one's refined rational capacity), there can be considerable difficulty experienced in sufficiently allowing this primitive emotional self speak for itself. However it is only when one can hear the voice of this self (with all its lustful impulses) freed from moral censorship, that the blind involuntary nature of such instinctive craving can be successfully dissolved.
So the second level here entails yet another withdrawal in the literal negation of such excess identification with one's higher self. This can equally entail a substantial withdrawal from all the social recognition that fosters undue identity with this self.
In my own case, this was to lead to the need for a further lengthy period of "intellectual isolation" free of the need for outside confirmation of my ideas.
The at the third level (Band 4, Level 3) one is then finally enabled to engage in the task of the balanced integration of both one's "higher" and "lower" selves (i.e. the transcendent and immanent aspects of personality).
Though this is commonly referred to in the Western spiritual tradition as the entry into "Spiritual Marriage", I would rather refer to it as the attainment of an "enhanced" normality where one no longer self consciously separates the spiritual from the material aspects of living. Put another way, the stage has now been reached, where one can begin to properly fulfil one's inherent potential through the actual circumstances of day to day activities.
Put more simply, this is true freedom to finally be oneself! Thus the longing to be (one's true self) finally culminates in a true state of be-longing (where this self is experienced as inseparable from God).