The Zero Delusion v0.99x

unit size (ad − bc = 1) form the modular group, the group of linear fractional
transformations (LFT) (az+b) /(cz+d) acting on the upper half of the Ǎ plane
(z ∈ Ǎ and I [z] > 0), where {a, b, c, d} ∈ Ž [162]. Instead, if {a, b, c, d} ∈ Ǎ,
we obtain the Möbius group [157], the group of LFT that maps circles to circles
preserving angles between crossing or touching circles throughout Ǎ. Specifically,
for any three distinct points {A, B, C} ∈ Ǎ, there is a Möbius map f ∠ that takes
z ∈ Ǎ to f ∠ (z) ∈ Ǎ, A to −1, B to 1, and C to ∞.
Möbius transformations are the most straightforward examples of conformal
transformations [130], mappings or diffeomorphisms (smooth deformations) that
do not alter angles within a point’s neighborhood but possibly distort extent or
curvature. Although many conformal functions are not Möbius in two dimensions,
it turns out that an exponential map is locally a Möbius map conformal
at any point of Ǎ. Moreover, the exponential’s inverse function log z is also a conformal
map within the principal branch. The give-and-take between the base-α
logarithm log α z and power α z maps, with α ≠ 0, provides a universal method of
information encoding-decoding. The Lie group-algebra correspondence and the
Laplace direct-inverse transform undertake the same two-way procedure.
Conformality is principally a local property generalizable over rings; all conformal
groups are local Lie groups represented by a class of LFT. Conformal
maps preserve conic forms and angles between intersecting conics through the
cross-ratio, another double ratio of differences between four points or vectors.
This rational construct is extremely powerful; the paradigm of regulating zero
and infinity in the same framework and a building block of conformal maps. Taking
the cross-ratio’s logarithm, a subset of the ring’s domain becomes a coding
space, a region of negative curvature characterized by a coupling factor between
distance and distortion of angles. Within a coding space, a geodesic line is a
conic, the calculation of distances uses hyperbolic geometry, and the zeroes and
poles of an LFT are limiting values of (vanishing and diverging) sequences, all
of which allows relaxing the notion of proximity, opening the door to quantum
nonlocality [153]. Physically, conformality is closely related to randomness in
two dimensions and causality and scale invariance in all dimensions. Because
of the fundamental curvature factor that blocks flatness, i.e., zero, conformally
compactified spaces are at the heart of Quantum Field Theory (QFT) and many
gravitational theories, representing one of the avenues to a robust theory of QG.
The following sections delve into zero from different stances. First, we introduce
the main trouble with zero; it is inseparable from infinity because both
comprise the same fundamental duality. Then, we explain how zero causes havoc
on General Relativity (GR) and QFT, setting off an unsolved crisis that QG will
someday overcome. Zero fictitiousness leads to the prospect of a universal minimal
length. We analyze the role of zero in IT; physicality is computability, and
zero is uncomputable. We posit that change is equivalent to information flow,
both unstoppable. Because information coding must be efficient, the universe
likely uses PN; unlike standard PN, bijective PN creates zero-free and unique
numeral representations that boost productivity. We scrutinize Set Theory (ST)
concerning the signification of emptiness, finding that the null class is a waste