a transition to a different state. We shall explore this circum-
stance below in terms of topological considerations analogous
to the concept of ecosystem resilience.
Conversely, changing the direction of the second condition,
so that
dS2/dM2|A0 < 0,
leads to a repulsive peak, a White hole, representing a pos-
sibly unattainable realm of states.
Explicit parametization of M introduces standard - and
quite considerable - notational complications (e.g. Burago
et al., 2001; Auslander, 1967): Imposing a metric for dif-
ferent cognitive dual languages parametized by K leads to
Riemannian, or even Finsler, geometries, including the usual
geodesics (e.g. Wallace, 2005c).
We have defined a groupoid for the system based on a par-
ticular set of equivalence classes of information sources dual
to cognitive processes. That groupoid parsimoniously char-
acterizes the available dynamical manifolds, and, in precisely
the sense of the earlier development, breaking of the groupoid
symmetry creates more complex ob jects of consderable inter-
est, which will be studied below. This leads to the possibility,
indeed, the necessity, of Deus ex Machina executive mecha-
nisms - i.e. programming - to force transitions between the
different possible modes within and across dynamic manifolds.
That is, the programmer creates the manifold structure, and
the machine hunts within that structure for the ‘solution’ to
the problem according to equivalence classes of paths on the
manifold:
Equivalence classes of states gave dual information sources.
Equivalence classes of information sources give different char-
acteristic system dynamics, representing different programs.
Later we will examine equivalence classes of paths, which will
produce different directed homotopy topologies characterizing
those dynamical manifolds. This introduces the possibility of
having different quasi-stable resilience modes within individ-
ual dynamic manifolds. One set of these can be characterized
as leading to ‘solutions’ of the underlying computing problem,
while others may simply be pathological absorbing states.
The next important structural iteration, however, is, in
some respects, significantly more complicated than a differ-
entiable manifold.
9. The rate distortion manifold
The second order iteration above - analogous to expanding
the General Linear Model to the Hierarchical Linear Model -
which involved paths in parameter space, can itself be signif-
icantly extended. This produces a generalized tunable retina
model which can be interpreted as a ‘Rate Distortion mani-
fold’, a concept which further opens the way for import of a
vast array of tools from geometry and topology.
Suppose, now, that threshold behavior for institutional
reaction requires some elaborate system of nonlinear rela-
tionships defining a set of renormalization parameters Ωk ≡
ω1k , ..., ωmk . The critical assumption is that there is a tunable
zero order state, and that changes about that state are, in
first order, relatively small, although their effects on punctu-
ated process may not be at all small. Thus, given an initial
m-dimensional vector Ωk, the parameter vector at time k + 1,
Ωk+1, can, in first order, be written as
Ωk+ι ≈ Rk+ιΩk,
(13)
where Rt+1 is an m × m matrix, having m2 components.
If the initial parameter vector at time k = 0 is Ωo, then at
time k
Ωfe = Rk Rk- 1...riω0.
(14)
The interesting correlates of individual, institutional or ma-
chine consciousness are, in this development, now represented
by an information-theoretic path defined by the sequence of
operators Rk, each member having m2 components. The
grammar and syntax of the path defined by these operators is
associated with a dual information source, in the usual man-
ner.
The effect of an information source of external signals, Y,
is now seen in terms of more complex joint paths in Y and
R-space whose behavior is, again, governed by a mutual in-
formation splitting criterion according to the JAEPT.
The complex sequence in m2-dimensional R-space has,
by this construction, been pro jected down onto a parallel
path, the smaller set of m-dimensional ω-parameter vectors
Ωo,..., Ωk.
If the punctuated tuning of institutional or machine at-
tention is now characterized by a ‘higher’ dual information
source - an embedding generalized language - so that the
paths of the operators Rk are autocorrelated, then the au-
tocorrelated paths in Ωk represent output of a parallel in-
formation source which is, given Rate Distortion limitations,
apparently a grossly simplified, and hence highly distorted,
picture of the ‘higher’ conscious process represented by the
R-operators, having m as opposed to m × m components.
High levels of distortion may not necessarily be the case for
such a structure, provided it is properly tuned to the incoming
signal. If it is inappropriately tuned, however, then distortion
may be extraordinary.
Let us examine a single iteration in more detail, assum-
ing now there is a (tunable) zero reference state, Ro , for the
sequence of operators Rk , and that
Ωk+ι = (Ro + δRk+ι )Ωfc,
10