Motion of the Third Kind (II) Notes on kinematics, dynamics, and relativity in Semantic Spacetime

Abstract

In part I of these notes, it was shown how to give meaning to the concept of virtual motion, based on position and velocity, from the more fundamental perspective of autonomous agents and promises. In these follow up notes, we examine how to scale mechanical assessments like energy, position and momentum. These may be translated, with the addition of contextual semantics, into richer semantic processes at scale. The virtualization of process by Motion Of The Third Kind thus allows us to identify a causally predictive basis in terms of local promises, assessments, impositions. As in physics, the coarser the scale, the less deterministic predictions can be, but the richer the semantics of the representations can be. This approach has immediate explanatory applications to quantum computing, socio-economic systems, and large scale causal models that have previously lacked a formal method of prediction.

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(WORKING DRAFT 0.2)
Motion of the Third Kind (II)
Notes on kinematics, dynamics, and relativity in Semantic Spacetime
Mark Burgess
June 10, 2022
Abstract
In part I of these notes, it was shown how to give meaning to the concept of virtual motion,
based on position and velocity, from the more fundamental perspective of autonomous agents and
promises. In these follow up notes, we examine how to scale mechanical assessments like energy,
position and momentum. These may be translated, with the addition of contextual semantics, into
richer semantic processes at scale. The virtualization of process by Motion Of The Third Kind thus
allows us to identify a causally predictive basis in terms of local promises, assessments, impositions.
As in physics, the coarser the scale, the less deterministic predictions can be, but the richer the
semantics of the representations can be. This approach has immediate explanatory applications to
quantum computing, socio-economic systems, and large scale causal models that have previously
lacked a formal method of prediction.
Contents
1 Introduction 2
1.1 Notationandterminology.................................. 3
1.2 Relationship between assessments and promises . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Process relativity: how agents observe one another . . . . . . . . . . . . . . . . . . . . 6
1.4 Covariance between observers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Kinematics and dynamics of promises and assessments 9
2.1 Qualitative and quantitative description . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Embedded spacetime view, configuration and phase space . . . . . . . . . . . . . . . . . 10
2.3 Derivatives on a static graph topology . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 Separation of boundary value information and guiderails . . . . . . . . . . . . . . . . . 12
2.5 Equations of motion and constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.6 Canonical and derived currencies, e.g. energy, money, etc . . . . . . . . . . . . . . . . . 13
2.7 Promise dynamical manifesto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Laws of behaviour with fixed boundary conditions 14
3.1 First order static equilibrium solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Second order flow from promises with channel diffusion . . . . . . . . . . . . . . . . . 15
3.3 Currency landscapes vs process semantics . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.4 Directionandcurrent .................................... 17
3.5 The meaning of momentum in physics . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.6 The meaning of momentum in Promise Theory . . . . . . . . . . . . . . . . . . . . . . 20
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4 The semantics of force, momentum, and energy in physics 21
4.1 Quantifying gradients and exchange in Newtonian physics . . . . . . . . . . . . . . . . 21
4.2 Quantifying gradients and exchange in Quantum Mechanics . . . . . . . . . . . . . . . 22
4.3 Quantifying gradients and exchange in Promise Theory . . . . . . . . . . . . . . . . . . 23
4.4 Currency accounting patterns in an agent viewpoint . . . . . . . . . . . . . . . . . . . . 25
4.5 Example: Coupled oscillators in Newtonian mechanics and Promise Theory . . . . . . . 25
4.6 Coupled agent interpretation in Semantic Spacetime . . . . . . . . . . . . . . . . . . . . 27
4.7 Some examples from different semantic scales . . . . . . . . . . . . . . . . . . . . . . . 31
4.8 The variational energy (currency) formulation in physics . . . . . . . . . . . . . . . . . 32
4.9 A variational formulation for agents? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.10 Reinterpreting the action principle in terms of locality . . . . . . . . . . . . . . . . . . . 34
4.11 Conservation requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.12 Ballistic or impulsive change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.13 Loops, sampling processes, and fixed ‘currency price’ and energy levels . . . . . . . . . 36
5 Promise lifecycle and boundary dynamics 37
6 Summary and discussion 39
1 Introduction
In part I of these notes [1], we didn’t write down a comprehensive dynamical picture of promise graphs,
when defining virtual motion. Typical uses of promise theory have involved the solution of only static
collections of agents and promises—as preordained circuitry on a low level. However, some hints about
the dynamical evolution of the promise graph were provided in [2–6]. In these follow up notes, I want to
explain how we can use Promise Theory to expose the semantics latent in physical law and construct a
causal dynamical account of virtual processes by appealing to the same quantitative techniques that have
proven successful in physics.
Promise Theory is founded on the assumption that the autonomy of local agents is primary, i.e. that
all motive impetus originates from the inside out. The agents suggest the concept of ‘aether’, except that
our agent substrate is the active party, not a passive medium. The agents are more akin to the cellular
substrate of a cellular automaton, but without a lattice topology. This contrasts with the fully exterior
viewpoint taken by Newtonian mechanics of rigid bodies, where activity was essentially ballistic and
measured by continuous variables for position and momentum. In an agent view, we need to expose
the assumptions that make such a point of view possible. In our model of Motion Of The Third Kind,
the world is made of agents that promise to share interior properties with one another. These sharing
channels are the result of promise bindings between an offer or donor (+) and an acceptance or receptor
(-) by causally independent agents. As the agents coordinate their autonomous activities, processes are
formed between them. The simplest processes are transactions, then oscillations, waves, and then the
formation of large scale structures that can move relative to one another.
The nature of agents’ promises is undefined. Signals may be passed over any kind of channel,
stigmergic (publish-subscribe), ballistic (push), telescopic (pull), etc. Rather, we are interested in under-
standing the resulting semantics and dynamics of changes that result from them. Agents do not move
in any sense—indeed, we don’t consider agents to be bodies embedded in some theatrical spacetime
as Newton did—only their promised attributes move, by changing host, like musical chairs. It turns
out that this model of motion is more like the model of state evolution in Quantum Mechanics, which is
surely interesting in its own right. The symmetries and objective relativistic of Euclidean and Minkowski
spacetime are not presumed in advance. Such assumptions of continuity and global uniformity are to be
determined as virtual constructs built on top of the basic agent to agent interactions. In the semantic
spacetime, space and time are not a passive theatrical backdrop but an active causal circuitry by which
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agents interact and larger processes can form on a collective scale. Process first, embedding for conve-
nience.
Promise theory features a separation of concerns formally into promises, impositions, and assess-
ments. In building a theory of cooperation, assessments (including observations and measurements) are
of crucial importance, since they are the origin of relativity. Without assessment an agent would not have
the ability to measure external promises and alter its own state in response to external influence. Without
assessment, we would have to assume that agents were altered ballistically by imposition. The process
by which assessments are made—on whatever level an agent has the capability to perform—has to play
a central role in determining its response. The default model of response in physics is a direct driving,
like a contact potential, but this is known to be simplistic even in elementary physical systems.
In the Promise Theory view of virtual motion, the ground state of all agents is to be fully autonomous,
and each agent’s assessments inform only the receiving agent itself. We thus need to explain how vir-
tual processes can embody and propagate across large numbers of agents, when agents are themselves
causally independent.
In this picture, it’s natural to ask whether fully autonomous agents might not simply refuse to com-
ply with information offered to them by neighbours. Indeed, that’s entirely possible—and we see this
behaviour on scales at which agents become more connected and more sophisticated. However, on an el-
ementary scale agents will not necessarily have the interior capacity to ‘know enough’ about the process
they partake in to resist or oppose the general flow (think of birds in a flock, or so-called emergent swarm
intelligence). Separation of scales in a pattern of behaviour will tend to lead to mean field behaviours
that are collective in nature [7].
1.1 Notation and terminology
For completeness, let’s summarize the assumed nomenclature. The notation of Promise Theory follows
that described in [8]. There are many indices in the expressions that follow. Readers used to vector
labels should take care. Subscripts like Aido not refer to coordinate components of spacetime vectors
but rather refer to agents, which are effectively point locations. In that sense, a label Aisignifies a
complete location, not a component of a pointer to one. This gives primacy to locally active sites rather
than to passive symmetries invoked for simplicity. I’ll also avoid using the Einstein convention over
repeated indices in order to make summations explicit and avoid confusion with coordinate systems.
We may recall that agents Aimake promises to offer (+) or accept (-) information independently.
Promises have a type and magnitude which are contained in the body b:
Ai
±b
−→ Aj,(1)
and that the aggregation of all such promises forms a graph, whose matrix may be written Π(±)
ij .
•Since any graph is equivalent to an adjacency matrix, we may consider a quantity αito be part of a
column vector on the total graph. The rows and columns label agent identity or effective ‘position’,
as discussed already. We must not assume that positions form a regular lattice as in a Cartesian
basis. The promise graph could be fluctuating in any fashion, e.g. the structure of the Internet is a
diverse graph with varying dimensionality [9]. When we write
•Continuum derivatives have their usual meanings on coordinatized spaces, when we refer to con-
ventional physics.
•Partial derivative ∂trefers to an interior change, i.e. a change at the same agent location Aior
Euclidean location x. The clock referred to by tis to be explained in the semantics of its usage.
•−→
∇jrefers to an exterior change, between agents, starting from the current agent location Aiin the
direction of Aj, if that direction exists.
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•−→
∇without an index refers to the sum over all directions leading from the agent on which it acts,
like a divergence.
•ψirefers to the interior state of an agent Ai, i.e. the self-graph of interior promises Πi
ii.
•A quantity irefers to a exterior promised partial state belonging to Ai, i.e. is equivalent to stating
Ai
+
−→ ∗. Thus, when any promised quantity Ai
+b
−→ ∗ is promised by a domain of many agents
we can write it as an effective function of the position b(Ai), as from a third party observer view.
The closest analogue of the adjacency matrix from traditional graph theory for a promise graph is the
matrix of combined ±promises:
Π(±)
ij ∼Aij .(2)
Note that the matrix labelled (±)is not a matrix product: Π(±)
ij is unrelated to the usual matrix product
over agent indices (Π(+)Π(−))ij . We can use the wedge notation to represent the logical semantics of
‘AND’ (not geometrical forms), and the product notation
Π(±)
ij =Π(+) ∧Π(−)ij (3)
thus represents the element-by-element product Π(±)
ij = Π(+)
ij Π(−)
ij , not the usual matrix product. This
comes about because linearity is applied over an exterior scale, whereas matching occurs locally at what
we associate with ‘point locations’.
Π(±)
ij (b) = Ai
+b+
−−→ Aj
Ai
−b−
←−−− Aj!=Ai +b+
−−→
−b−
←−−− !Aj.(4)
We can define the assessment of what degree of this promise binding b+∩b−is kept, by the capability
matrix:
Ck
ij (b) = αkΠ(±)
ij (b),(5)
i.e. an assessment, by some agent Akthat Aiand Ajkeep their promises about bto one another with
expected magnitude b+∩b−. This matrix Ck
ij (b)is what we must use instead of a plain adjacency matrix,
as it contains crucial information about the inhomogeneity, or intrinsic local characteristics of the agents
involved in our society.
Ck
ij is the adjacency matrix for a directed graph. Readers used to reasoning by fiat relationships (by
imposition like behaviour), e.g. in Category Theoretic bindings, should take care. The symmetrization
over ±promises does not render a binding undirected, in the sense of graphs. An undirected link would
require four promises, with ±in both directions. Any promise, which is not complemented by its oppo-
site sign, is null potent and can be ignored. This is somewhat analogous to the way quantum transitions
require both a ψand ψ†or bra and ket to complete a channel.
An assessment, in Promise Theory, is an interior property of an agent. Since agents may have dif-
ferent interior properties and state resources, we can’t make a general rule about the resolution of as-
sessments. An assessment is some mapping from set-values promise bodies bto a supportable internal
representation α(b):
α() : Set 7→ Rn,(6)
which is assumed only to be some general internal vector real values. In practice, assessments will be
constrained by interior and exterior boundary conditions. This is the most general scalar mapping. At
this stage we needn’t assume more than this1.
1Some readers might be tempted to delve into Category Theory to express some of the relations, however Category Theory’s
goal is abstract rather than operational, which would distract from the mechanical view I want to focus on here.
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In a graph, topology and continuity of direction are separate ideas. In a continuum, the coincidence
limit in derivatives essentially removes this distinction and average directions are normalized in terms of
regional or global basis vectors. Thus, in a continuum, one can have concepts like velocity of a pointlike
object, and position itself is a vector. In a graph, position is only a name or label, and velocity is only
promised along point-to-point channels. We’ll makes extensive use of these facts in what follows.
1.2 Relationship between assessments and promises
The ‘measurement problem’, or the separation of a process of measurement sampling from evolution of
a dynamical system, is an issue that only surfaced in the twentieth century—first in Special Relativity
and later in Quantum Mechanics. Before this, it was assumed that the equations for dynamical variables
would yield a precise causal answer that decoupled from direct observation. In other words, the process
of observation was assumed to be ubiquitous and instantaneous and was taken for granted.
Questions like ‘how does one body observe the properties of another’ didn’t need to be answered
because equations represented all behaviours as expressions of universal law, which bodies had no choice
but to obey. This decoupling between the agents of study and the processes of interaction to trace them
(also referred to as decoherence in quantum theory) becomes less and less tenable as the agents in a
system probe smaller scales.
Information theory introduced the idea of communication channels in [10]. Promise Theory brings
these ideas together so that every agent is expected to observer every other by an explicit process.
The interactions are not assumed to be independent of the agents’ ability to observe one another, as it
is in most physical equations of motion. Instead, we have to make the interactions of agents compatible
with Information Theory: agents sample one another’s states, something like the dynamics of a game
between players. For every exchange of one agent sampled by another, there has to be a local commu-
nication of state between them. This implies that agents have interior subprocesses (interior timescales)
that are unobservable but faster than observable exterior changes.
In Promise Theory, measurements are replaced by ‘assessments’, which are process valuations made
within agents, in the role of observer. Spacetime is discrete, and each agent forms its own assessments, so
the detection of gradients can only be made from these assessments by each single agent. The observation
of gradients thus relies on information shared between agents, and thus forces are always non-local
properties2. By contrast with the Newtonian calculus, where limits are used to define all properties at a
point, the observation of a force (as an external party) requires the assessment of at least two successive
agents (e.g. by a third party) to form a concept of a gradient. The assessment of gradients by third parties
is how we calibrate our agent view to form a universal Newtonian kind of description.
All models of spacetime play a basic role in discriminating change. In simple mechanical formula-
tions of physics, the equilibration of forces is treated as instantaneous, which really means ‘much faster
than any timescale in the resulting behaviour’, e.g. force is transmitted instantaneously throughout a rigid
body, or by a connecting spring or a potential V(x), but the same does not extend to the displacements
of normal modes. These assumptions are usually left implicit and are built into the semantics of the
formulae.
Example 1 (Schr¨
odinger equation) The outcome of the Schrodinger equation is a representation of
an ‘instantaneous’ equilibration of changes leading to a state we call the wavefunction. The name
refers to the deliberate injection of wave semantics into the representation for dynamics, based on the
observation that p=h/λ for waves. Wave semantics for momentum lead directly to the apparent non-
locality of predictions at different locations. The idea of momentum as a time derivative m˙xis replaced
with momentum as a gradient of a distributed process over space (which may be static or moving).
2In physics the term ‘information’ is used to imply a bulk measure of possible configurations, represented by an entropy.
We need to distinguish the measure of bulk information from specific information (a particular state configuration or message).
Information channels are thus things that can transmit specific messages, not just thermodynamic bulk.
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Example 2 (Game theory) An instantaneous response is not allowed in Game Theory. Each player
takes their turn in a synchronous manner. The cycle of assessment intervenes in the causal flow through
the Nyquist sampling law. This game proceeds in rounds of interaction [11–13].
Finally, it’s worth remarking that, in elementary physics, we typically deal with (memoryless) Markov
processes, but as we increase the sophistication or ‘complexity’ of agents at a given semantic scale, the
role of interior process memory (represented by the interior degrees of freedom of agents) becomes more
important. This begins with phenomena like hysteresis, and extends to phenomena in biology, where
agents like DNA encode complex processes. It also clearly applies to cloud computing where agents are
virtual computers.
1.3 Process relativity: how agents observe one another
Observing agent processes is a surprisingly difficult problem. It suggests the need for apparent con-
straints on the homogeneity of systems. Suppose we have an agent Akthat wishes to observe a process
P, as it moved between agents A1and A2(see figure 1).
1
AA2
Ak
relay
guiderail
Ij
Figure 1: Trying to observe a remote process in agent spacetime. To help discriminate paths, an agent can
cooperate with others along different routes, like a grating (black circles) that label different routes. An autonomous
agent can’t promise that an observation would follow the same path on repeated measurement, so we can never
expect to measure velocity remotely—unless the measurement involves so many agents and paths, over a consistent
equilibrium guiderail, that the differences are irrelevant (such as in a dense lattice or manifold).
The first possibility for observing the motion is for agents A1and A2promise to voluntarily signal
Akwhen they are involved with the process. In order to receive such a signal, Akhas to be tuned in to
the signals i.e.
A1, A2
+signal|P
−−−−−→ Ak(7)
Ak
−signal
−−−−→ A1, A2.(8)
In this case, the agents either have to know to observer’s location or they need to broadcast the signal.
It’s impossible for Akto distinguish these case, or to know the time it took for the signal to arrive. The
observer is a passive receiver. Ultimately, Akis beholden to the uncertainties of the arrangement.
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A second possibility is that the agent Akprobes the agents, assuming that they will respond with a
reply, i.e.
Ak
+probe
−−−−−−→ A1, A2(9)
A1, A2
−probe
−−−−→ Ak(10)
A1, A2
+reply|probe ∧P
−−−−−−−−−→ Ak(11)
Ak
−reply
−−−−−−→ A1, A2.(12)
This relies on Ak’s ability to send and sample messages fast enough (at the Nyquist frequency) to capture
a frame in which the process is passing through the agents and measure the round-trip time. This relies
on a number of key assumptions.
The intermediate agent theorem [8] tells us that we can’t take any agent’s cooperation for granted,
and we clearly can’t guarantee the density of intermediate agents in order to provide a sufficient number
of unique paths by which Akmight distinguish A1from A2. In the Newtonian view, we simply take this
for granted based on the continuum of points in Euclidean geometry.
To go beyond the simple description of velocity, from a local information channel to a collaborative
network is impossible without some high degree of order. Such long range order would amount to being
a homogenous phase of the agents forming spacetime. This can only occur above a certain density of
paths. Without some kinds of agent model, such a claim would have to be speculative.
The departure from thinking of spacetime as a continuum to a dense, homogeneous and isotropic
medium leads us to model an unreliable network of observational locations (see figure 1). Path variations
can lead to insurmountable distortions of the observational medium. We know this kind of trouble occurs
in solid state media. The phenomenon of Anderson localization is an example in which random impurity
sites, acting as holes, confound transport by trapping waves in local wells [14]. Relativity plays a role in
two ways:
•There are observational delays in observing remote phenomena due to the need to pass information
along some intermediate process, through paths that may or may not be inert, e.g. light or sound
signals can be absorbed, refracted, etc.
•We may need to be able to transform the view of one agent into the view of another. This can
only be imagined on a large coherent scale, since one isn’t even assured of equivalent information
being available to all agents on a fundamental level. We have to assume certain equivalences and
homogeneities between observers in order to satisfy the requirements of a theory like Galilean or
Lorentz invariance, for instance. Symmetry is not a guarantee.
There is only one experiment a process can perform to try to observe remote motion of another
process, without basing measurement on the assumption of a regular measure: the observer Akcan
offer (+ promise) repeated signals at a constant time intervals, and hope that these will be accepted and
returned by the remote process along a consistent path. The observer can measure the round trip time
and the time interval between repeated measurements. If the paths taken by probes are random (as they
are in Internet traffic, for example), or in random vibrating lattices, then the round-trip time can vary.
All this assumes that the agents that are intermediate between observer and observed are cooperative,
reliable, indeed invariant in their behaviour, so that repeated measurements can average away uncertain-
ties and detect patterns and trends. We have to confront the idea that we simply don’t know this, and
it may well be that we have to just take it as an assumption and hope for its consistency. Note partic-
ularly that, the geometry such paths cannot be determined, since one is completely dependent on them
for cooperation. In fact, not all agents may participate. It is only the formation of a stable guiderail
that can allow observation to take place at all. For example, the stabilization of a quantum wavefunction
weights to chance of certain paths participating in change due to the cooperative distribution of local
energy density.
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Longitudinal motion could be inferred by a temporal dilation of the intervals (red shift). Transverse
motion relies on there being sufficient number of different paths for the observer to be able to distin-
guish position, by an angle θobserved to change with lateral motion. If we assume a consistent and
fixed geometry with constant speed of signalling, then the anglular change observed could be estimated
repeatedly by taking a longer time as ∆t2,
cos θ=h∆t1i ± ∆(∆t1)
h∆t2i ± ∆(∆t2),(13)
where ∆t1and ∆t2are the round-trip times between Akand A1,A2(see figure 1), with statistical
uncertainties. One can try to measure the drift over long times and large distances and find an average
for motion. Then, by geometric arguments and the belief in a constant speed, the answer has to converge
on the usual Newtonian and Lorentzian results.
In the long time limit, measurement could be approached statistically and might converge to a con-
stant answer, However, over timescales that are short compared to the changing of paths (e.g. in quantum
systems or the Internet) there is effectively no way of obtaining the necessary information to calculate
motion. The assumption of a regular path and consistent velocity relative to the observer’s interior clock
is not a reliable one. This problem faces everyone trying to measure data flows on the Internet, for exam-
ple. The observation of a suitably cooperative propagating process may in the worst case be impossible.
In the best case, it might appear random and Brownian in nature, as in field path integrals3.
Assuming control over a local region, one might try to use assemblies of agents in cooperation to
form gratings that label different paths (the dark agents in figure 1). These can act as alternative satellite
receptors to discriminate between different directions. However, there is no guarantee that the paths
taken by the observational signals will not join up later. We need to make some assumptions in order to
be able to observer:
•The distance or path from Akto Aiand back should be single valued, on average, but may fluctuate.
This is known to be untrue for the Internet, whose dimensionality is not even constant over the short
range [9].
•The agents or their observational paths have to be distinguishable by Ak. However, paths that
begin separate can still cross later in their paths, with topological irregularities. This is directly
observable in the Internet, and in lattices with holes.
•One has to assume that there is a consistent field of partially ordered intermediate agents that.
Again, if the number of possible paths is not infinite, then a guiderail can route several paths, like
a lens, through common agents, leading to indistinguishability.
These are essentially the conditions one applies to solutions of constraint equations when modelling a
dynamical problem, e.g. in a diffusion distribution, or for the wavefunction in Quantum Mechanics.
1.4 Covariance between observers
We normally think of the transformation of a spacetime viewpoint as a tensor transformation. We can
try to imagine such an object for autonomous agents—though the idea that one would have sufficient
information to construct this is far from clear. Suppose we use agent labels k, ` to refer to a neutral third
party agents, whereas i, j will refer to the active agents in a pair process. The capabilities Ck
ij (b)of an
agent are assumed to be intrinsic to the agent, but the assessment of those capabilities is determined by
the agent making an assessment Ak. Thus, different agents will make different assessments and we can
postulate some transformation matrix Ωto transform these:
Ck
ij (b)7→ C`
ij (b) = X
k
Ω`
kCk
ij (b).(14)
What kind of transformation group would this represent? How might this change of perspective be
transported around a loop, as parallel transport? These questions remain to be answered.
3See for example the discussion by [15] on Brownian support for path integral measures for a quantum field.
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2 Kinematics and dynamics of promises and assessments
Let’s review some of the kinematic and dynamic concepts used to describe change in physics, and relate
them to agents and promises. This should inform the accounting procedures that are applied to dynamics
in a familiar light. Concepts like force and rate of change are ubiquitous in classical physics, and are
based on gradients within a differential description. These smooth functions apply over scales that can
be considered sufficiently differentiable4.
Partial order is a discrete concept Gradients are the effective differential characterization of partial
ordering of locations within a field. Thus the existence of a changing concentration of some property
over ‘space’ allows us to observe that space. Without such labels, the points would be indistinguishable
and the idea of space and distance would be a purely theoretical construct. In physics, we typically ignore
this and justify the existence of empty space via the processes that move through it, or introduce fields
and potentials. In Promise Theory, general gradients are effectively replaced by chains of conditional
promises that prescribe an effective order in space and time over the processes that satisfy them as
boundary conditions. Spacetime has no invariant meaning outside of processes in an agent model.
As we’ll see, classical mechanics favours smooth behaviours because it assumes there is an in-built
rigidity of the ‘causal channel’ acting between the entities embedded within spacetime. Such a rigid
guarantee of transmitted influence is not reasonable in general. indeed, as noted in part I, there is no
obvious reason why a transition from one location to another in a trajectory would continue from the
new position in the same direction. Such an effect must be a large scale collaborative property of agents.
Without large scale coherence, changes could appear at apparently random locations over a guiderail
(as is the case in quantum mechanics), because interior causality may not be directional or even be
determined by what can be observed on the exterior between local entities. Solutions for agent behaviour
may therefore appear to fluctuate at disparate locations at different times, appearing disordered or even
‘random’.
2.1 Qualitative and quantitative description
Computer Science is strongly focused on the semantics of processes. Physics focuses on mainly on
the dynamics of processes, and embeds semantics only by assumption, through the properties of alge-
bra. One of the goals of Promise Theory is to unify semantics (qualitative functional behaviours) with
dynamics (quantitative behaviours). The suppression of semantics leads to famous confusions when
interpreting different theories. In the general case (and thus for virtual motion) we need to separate qual-
itative and quantitative issues carefully. Both can be represented as promises. The price we pay is for
the equations of a dynamical system to be supplemented with additional expressions—as we’ll see in the
coupled oscillator example to follow.
In the general case, we need equations to determine exterior influences as well as interior states
for every active location in spacetime. This includes the connectivity of agents via possibly multiple
independent channels of influence. These channels are the equivalent of forces and the promises they
make and they reflect their semantic roles. This is closer to the situation in modern particle physics.
Remark 1 (Dynamical variables) The principal kinematic variables in classical systems stem from bal-
listic origins: they are positions ~x and momenta ~p. Directionality is built into vector spaces, so the vector
representation compactifies a lot of reasoning by submerging it as part of the vector infrastructure. From
these, the concept of energy emerges as a link between properties over space and time through the ve-
hicle of ‘forces’ Fand potential energy V, where the forces arise from gradients of some underlying
landscape function of ‘potential’. Both forces and potentials appear in Newtonian mechanics as fields
over space itself, providing a surrogate labelling. The equations of change in physics refer to material
4The continuum limit uses the limit dx, dt 7→ 0, which seems to refer to the very small. In truth, this artificial limit is a
representation of ‘zooming out’ of system to a low degree of resolution. Thus it makes most sense in the limit of the very large
systems, where the rough small scale details are negligible.
9

‘bodies’ that are acted on by these forces. The bodies are the only active parts of space in the Newtonian
scheme. This changes in Einstein’s General Relativity.
Example 3 (Music) Music is an example of how virtual processes take place on many levels. If one
considers a piano to be a semantic spacetime, then the playing of music amounts to virtual motion of
the third kind across the keys. In frequency space, melody is a virtual promise encoded by dimensionless
ratios of the string frequencies, not directly by the actual physical strings—but denying the existence of
the ‘aether’ of physical strings would be foolish. There are lessons here about how we should not take
dogmatic positions of ideology when describing phenomena.
2.2 Embedded spacetime view, configuration and phase space
The principal function of spacetime is to chart causality. In a Promise Theory model, exterior spacetime
‘exists’ only in terms of the sequences of interacting agents and their connective channels, through which
a process passes. Snapshots of all agents form spacelike hypersurfaces. Processes can only access one
or more cones of connected agents, through promised channels of different kinds, analogous to the light
cone in physics.
In classical physics, the assumption of trajectories as the solutions of equations of motion within a
system of coordinates suggests the first confusion of notational semantics. When we write a trajectory as
x(t)or a force field as F(x, t), the quantities labelled ‘x’ in these two cases have very different semantics.
The former is a specific association representing an ordered sequential orbit of positions taking its values
from the possible x, t. The latter is an ordered field with no preferred locations or times.
The spacetime notion of a continuous path trajectory is firmly planted in our expectations by Newto-
nian physics, thus we find behaviours that seem to jump from location to location to be ‘weird’. However,
this is perfectly normal where behaviour arises from within agent rather than by transport between them.
In Promise Theory, our goal is the maintain clear semantics at all times. Agents connect together
to form guiderails rather than trajectories, by virtue of the explicit promises they offer and accept5.
The interior states of agents become causally dependent only if they promise to receive and respond to
promised signals from one another. This is not a quantum idea, indeed rigid entanglement is built into
classical mechanics as an axiom, and we are usually looking at ways to relax this assumption. In other
words, Promise Theory makes minimal assumptions about interaction, and consequently looks more
similar to quantum than classical mechanics. Thus, when we talk about equations of ‘motion’, we really
mean equations about changes to a process that involves and spans multiple agents.
2.3 Derivatives on a static graph topology
Rates of change are an intrinsic part of processes: without change there is no process. Change in time
is the fundamental or intrinsic change (at a single agent or location). Change over space (at constant
time) refers to state which is imprinted in the local memory of agents. Virtual processes on graphs
muddle spacetime concepts, since a single hop represents both forward motion and a tick of a clock,
with effectively constant velocity, so we need to take care in defining rates over spacetime intervals with
semantics that are appropriate for the process6.
In the Promise Theory, with its successive causal boundaries, we are always faced with a distinction
between interior time and exterior time. Interior state may be accumulated from the change processes
over past times (memory processes)—and may be externalized as an instantaneous configuration, as in
centre of mass motion. This induces purely exterior timelike changes (i.e. Markov processes), such as
when a force acts on some rigid body.
5This is similar to particle physics, where messenger particles of different types distinguish different meanings. It contrasts
with the idea that meaning comes from the order and continuity of a set of points alone.
6This subtlety is responsible for much confusion in quantum mechanics, and is responsible for the inability for a quantum
oscillator to come to rest.
10

The definition of a derivative on a graph is thus more closely connected with flow systems, and
the definition of advanced, retarded, and Feynman propagators for fields. We need to be careful about
the semantics of derivatives for graphs, since the coincidence limits δ(x, t)→0cannot be taken. A
difference over some set of value, distributed across multiple agents ε(Ai), distributed over all agents Ai
may be written
−→
∇jεi≡−→
∇jε(Ai) = vj−vi,(15)
=Aij vj−vi.(16)
It refers effectively to an arrow, anchored (by one end) at the agent Ai. The question here is what values
of jare available at a given location Ai? The only directions javailable to form a difference of values
for εiare constrained by the adjacency matrix (Aij ) for the graph of agents Ai. However, this is not a
homogeneous matrix, so Aij = 0 for a large range of i, j. For this reason, we need to be more careful
about defining the channels that connect agents, by using the promise matrices defined in (4) [16].
Taking the effective adjacency to be Aij ↔Ck
ik, as viewed by some agent k, we account for the
channel capacities of the flows from past or future into the local state.
There are now three choices for the proper time derivatives, depending on how we anchor the arrow
to the root agent Ai. We define the total derivatives, in basis kby:
−→
∇εi=
X
j
Ck
ij εj−εi
(retarded)(17)
←−
∇εi=
εi−X
j
Ck
ji εj
(advanced)(18)
←→
∇εi=
X
j
Ck
ij εj−X
j
Ck
ji εj
(Feynman mixed),(19)
where the repeated jindices are summed over. If we don’t sum over repeated indices, we can define the
partial derivatives
−→
∇jεi=Ck
ij εj−εi(20)
←−
∇jεi=εi−Ck
ji εj(21)
←→
∇jεi=Ck
ij εj−Ck
ji εjj(not summed).(22)
The mixed derivative is analogous to the Feynman propagator in electrodynamics, and also to the Wigner
function, since it is anchored at the mid-point of a two-hop arrow conjunction.
Notice that if the matrix Ck
ij is sufficiently smooth and we consider paths that are sufficiently large,
then the gradient will scale so that
−→
∇j∼ −←−
∇j,(23)
since this is the limit in which the Newtonian limit dx 7→ 0emulates the scaling properties of the
promise graph. This leads, in turn, to the conservation of momentum or conservation of scaled promise
alignments. On a smaller scale, we should not take this precisely for granted.
Finally, there is a purely exterior view derivative one could define, as a scaled instantaneous snapshot
of the graph:
Djεi=Ck
ij (εj−εi).(24)
Each interval is scaled by its channel width, but this leads to no constraint on the flows, so we discard
this.
11

2.4 Separation of boundary value information and guiderails
Boundary information is the complement of spacetime trajectories on an interior region. The purpose of
boundary values is to separate a space into what one knows with certainty, and what can be predicted
based on causality. In a differential model, with continuum parameters, a boundary consists of spacetime
points at which we have definite information. This is normally at the edge of the system, at the perimeter,
at the start or at the end.
Causal equations, which predict interpolated behaviour, use some notion of continuity in the changes,
and rely on gradients of properties available to the observer. The semantics of these representations may
be subtle. Are the gradients those measured by the observer or by the neighbouring entities? Whose point
of view is the right one? In the Newtonian realm, there is no difference between viewpoints since space
and time are universal. However, in Einsteinian relativity, in quantum theory, and in Promise Theory, the
role of observer is crucial—and every agent can form its own assessment, even a random one.
We showed in part I that there is often a natural decomposition of a system into the formation of a
‘guiderail’ or map of resources supporting a process, and thus revealing where is can take place, and the
execution of the process itself as a measurable phenomenon. The hierarchy of information describing
processes thus begins with the separation of boundary conditions from equations of motion, and then
subsequent separation of measurement from the equations of motion. For example, in Quantum Me-
chanics one has a boundary condition described by a static potential, equations of motion describing
the distribution of energy throughout the space (guiderail) and a separate process of measurement of
dynamic variables. We return to this issue in this second part since it plays a key role in the non-locality
of agent processes.
2.5 Equations of motion and constraint
Equations of motion and constraint take on a variety of forms. The most common representation for these
is through the algebra of rings and fields, in which addition and multiplication represent superposition and
modulation of values directly and instantaneously. The local response in configuration space, analogous
to Newton’s F=ma must take the form:
Fij =−→
∇jαkΠ(±)
ij =−→
∇jCk
ij (jnot summed) (25)
where α() is an assessment function and Π(±)ij is the promise graph. In a system with many promises
at play, it will be convenient to use the derivative notation to look at a dynamic variable as a local state
ρiat each agent Ai, i.e. simply writing
Fij =−→
∇jρi(26)
Delayed responses can be incorporated using Green functions, as needed in electrodynamics and material
physics, for example, but these are a last resort where simpler equations simply won’t do. Fourier trans-
forms to wave space or momentum space provide other techniques for simplifying non-local behaviours.
We have dispersion relations, which describe the composition of dynamics in terms of complementary
superpositions of dynamical processes—waves being the simplest distributed process.
Remark 2 (Static systems) The dynamics expressed in the diffusive equations above should not mislead
us into thinking that all systems must diffuse and equilibrate. A document is a valid semantic spacetime,
and thus is should obey the equations we write down here. In a static system, which has no interior
degrees of freedom to change, the self-assessments must all vanish, since there is no interior time, so
αi7→ 0, and thus nothing changes in the frame of the document. An exterior observer Akcould still
observe some changes as a result of its own receiver promises changing, so αk6= 0, and it may therefore
observe changes as artifacts of the channel connecting it to the document (perhaps failing eyesight).
12

2.6 Canonical and derived currencies, e.g. energy, money, etc
Newton’s great accomplishment in formulating change, was to describe local behaviour as being guided
(and even determined) by the distribution of an underlying energy currency. Yet, hidden in the expres-
sions, was an implicit algebraic formulation to imply the semantics of these interaction to make it true.
Dynamical currencies are an artifice or accounting device associated with counting activity. Physics’
ballistic origins and economic analogies, lead us to think of energy and money, whimsically, as being
like the fuel for activity within a system. This view is not strictly correct as we’ll see from a promise
theoretic analysis, and can distract from the autonomy of processes.
In Promise Theory, currencies like energy can only be changed by assessments, i.e. ‘valuations’ of
an agent’s promises, made by some other agent’s promise. Nothing material or even manifest needs to
change hands in order for this to happen, the changes can be understood in terms of internal accounting
of information alone—at least for autonomous agents. Currency can be counted by different kinds of
measure, some semantic or symbolic, some dynamic or quantitative (see table 1). In quantitative physics,
one tends to focus on numerical counters that describe average state in terms of flows. This owes its origin
to the history of essentially ballistic descriptions of behaviour, in what all changes were externalized as
motion in space and time. However, in more advanced descriptions, labels like force-charge types and
biochemical signatures are needed to account for the multi-dimensional interactions.
GEN ER AL PHYSICS SO CIO ECONOMIC COMPUTING
Graph Configuration Society Economy Software agent
Agent AiPosition ~x Person/Tool Account Processor
Promise Π(±)
ij Momentum ~p Promise Money Data
Interior promise Hidden variables Hidden state Account balance Internal state
Exterior promise Tensor property Character Wealth? API
Offer (+) Emission rate Capability Supply Capability
Acceptance (-) Affinity / absorption rate Availability Demand Availability
Imposition (+) Collision Immigration Innovation Write/Operation
Proxy agent Particle Token/credential Coin / IOU Packet/Transaction
Assessment α() Measurement/sample Judgement Valuation Read
Currency Energy Trust Money Protocol codes
Interior currency Potential VSelf trust Balance MSubroutine
Exterior currency Kinetic energy TEngagement/risk Payment PAPI function
Binding Bound state Relationship Relationship Session
Currency Gradient Force / field Bias Incentive Preference/policy
Table 1: Approximate correspondences between different promise concepts at a range of scales, grouped by
similar semantics. Semantic complexity increases from left to right.
In the promise model, we favour an interpretation in which changes and activity come entirely from
within agents. This view is apparently the opposite of the Newtonian ballistic approach, though we’ll be
able to compare them in section 4. The difference of viewpoint doesn’t invalidate the need for currency
counters. Indeed, quantitative predictions still need explicit counting. Any agent Ak, with access to the
information about a promised process, can evaluate an assessment in its currency αk:
αk(π) = αkAi
+b
−→ Aj(27)
These assessments belong to Rn, so they can be added. Agents Aican also assess their own states αi():
αi=αi(Π(+)
ij ) = αiAi
+b
−→ Aj.(28)
2.7 Promise dynamical manifesto
Based on the foregoing observations, our manifesto for discerning quantitative motion is thus as follows:
13

•In a given system of agents, identify the dominant exchange currencies that count outcomes.
•Capture the semantics of the exchange processes to some level of approximation. What shall we
consider to be fixed (slowly varying) or fluid, stochastic, etc (quickly varying) over a timescale that
relates to the intrinsic scales of the observer’s cognitive process (sampling rate, internal memory,
etc).
•Identify the semantics of prediction: what do we consider to be meaningful measures? This is
usually based on invariances.
3 Laws of behaviour with fixed boundary conditions
The assumption that there are laws of nature that are the same everywhere is reflected in the existence of
‘global symmetries’ in physical law, which—in turn—directs us to a high level top-down view view of
causation7. Even when the details of laws are local (e.g. in gauge symmetries), certain aspects are still
assumed to be global: e.g. the charge of the electron8. There are two possibilities here: either there are
rigid non-local symmetries that violate causality, or whatever local differences there may be are simply
unobservable, hidden by the nature of measurement.
In [2, 18] Burgess and Fagernes described how promises can be developed as a scalable system
of state, in the manner of statistical or Quantum Mechanics. This was further refined in [5] with the
separation of interior and exterior degrees of freedom.
3.1 First order static equilibrium solutions
Let’s define an intrinsic process of a promise graph, over the whole connected region, defined by some
promise graph Π(±)
ij that leads to a landscape potential ρi. The promises that are part of this graph
of bindings form a matrix of channel availability for that subset of promise types. Since the gradient
incorporates an implicit adjacency matrix Ck
ij , the eigenvector ρiis an implicit function of Ck
ij too. The
keeping of those promises thus has an equilibrium distribution:
−→
∇ρi=−→
∇Ck
ij ρi(Ck
ij )=0,(29)
where ρπ
iare the components of the principal eigenvector of the matrix Ck
ij for the promises concerned.
The ρifor each agent is an shared property, which becomes intrinsic for it’s own degree of involvement
with its neighbours. We can label ρithe mass, encumbrance, inertia, drag, or connectedness of agent Ai
(see section 3.6).
At equilibrium, we consider there to be no net force on the graph to first order in the gradient:
Fj(Ai) = −→
∇jρi=Ck
ij ρj−ρi= 0,(30)
for partial derivative along the direction j(i), so rearranging gives
Ck
ij ρj=ρi.(31)
If we sum over the j, this is an eigenvector equation to first order. By the Frobenius-Perron theorem, the
normalized eigenvector of the promise binding matrix is thus the equilibrium distribution of exchange
currency in the system [3, 19]. We know that, for an asymmetric directed graph, this network cannot be
sustained, and trust will end up at the sources or sinks of the graph. An external forcing term (called
pumping in [3]) is needed to balance an equilibrium.
7One sometimes talks about the existence of godlike observers with access to all information instantaneously.
8Feynman and Wheeler recounted their story about why there is only one universal charge on all electrons. Feynman
recalled that John Wheeler said it’s because there is only one electron! If all electrons appear in pairs, by a strict tree process,
this is assured [17].
14

We should be cautious in interpreting this equation as a representation of currency. Which currency
is represented by this αi()? It is a purely local quantity based on conserved flow, so it satisfies the
conditions to be a currency. However, what is it an assessment of? The condition of zero derivative
is more a constraint on homogeneity than an expression of dynamics. We see that the significance is
equivalent to the existence of an equilibrium eigenvector. If trust is purely positive, this corresponds
to the eigenvector centrality of the effective adjacency matrix. The principal eigenvector is then a well
understood importance rank based on social connectivity. On the other hand, if the graph is directed,
or its assessment can be negative, then with its associated issues associated with the Perron-Frobenius
theorem [19].
In this formulation there is no global exterior time. The interior time for arriving at an equilibrium
solution is represented by iterations of the eigenvalue matrix operator to yield the steady state solution,
until it becomes stable and the time effectively stops at equilibrium. In practice this is an idealization
of many exchange interactions between connected agents. This whole discussion of autonomous agents
has some interesting analogues to procedures used in effective field theory [20]—the methods are framed
rather differently in the guise of externalized continuum physics, but the methods find a way to represent
interior processes through algebraic representations.
Remark 3 (Mass localizes non-local effects) The essential feature of this mass is that it’s equilibrium
decouples from the other processes once the promise graph topology (long range order) has settled. So
it becomes just a local parameter representing the degree of influence that an agent has on neighbours
and vice versa.
If the promises behind ρiare co-dependent, i.e. symmetrically bi-directional, then the process is locked
in phase step by entanglement over the region, and we can interpret ρi∼mias the effective mass of
Ai, with respect to those processes. The magnitude of midetermines an effective local momentum for
processes that pass through the agent.
The mass concept for agents is thus a kind of detailed balance equilibrium. Local dynamics, e.g. in
statistical systems and queueing theory, are based on instantaneous detailed-balance conditions for junc-
tions occur at the level of vertex rules Π(±)
ij . A major distinction between graph circuitry and continuum
trajectories is that vertex conditions act as effective boundary conditions that are distributed throughout
the system, at every agent. These can’t easily be transplanted by a homogeneous equation of motion,
since the agents might themselves be inhomogeneous. Caution is required. As an effective equilibrium
interaction, mihas an instantaneous value and a stable equilibrium value. The relaxation time is fairly
quick, requiring only a few neighbour exchanges to stabilize. Nevertheless, this effective mass is defined
over the same co-time timescale as entanglement, so it’s consistent with a classical limit.
3.2 Second order flow from promises with channel diffusion
Diffusion is a process that occurs relative to a fixed coordinate system. While the processes of diffusion
must reflect the symmetries of translation, these are broken by the boundary conditions, just as a the
temporal invariance of a thermal system is broken by the selection of a rest frame for relative motion.
Without changing the conclusions in a significant way, we can introduce and explicit exterior time t
and make the trust a function of it α(Ai, t) = αi(t). The diffusion equation in (41) for a homogeneous
network
−→
∇jαi=Ck
ij αj−αi, j not summed (32)
∇2αi=X
j
−→
∇jCk
ij αj−αijsummed (33)
=Ck
ij −δij 2αj(34)
∝∂αi
∂t .(35)
15

In the last line, I wrote αito indicate that the equation is solving for some kind of consistent assessment
of the ambient state. It combines interior state with the state of neighbouring agents probed through
kinetic processes. It relates the kinetic process to something that can drive the system from the outside,
as a boundary condition.
The similarity to the heat-diffusion and the Schr¨
odinger’s equation is both intentional and suggestive.
Since this is postulated, like the Schr¨
odginer equation, based on the structural conditions of the problem,
we are free to suggest the constant of proportionality in the last line. The difference between diffusion
and Schr¨
odinger mechanics is that the latter includes a factor of iin the right hand side, turning relaxation
into waves. Physicists will tend to argue that this is a fundamental difference—indeed it’s a fundamental
change of semantics. A relaxation phenomenon comes to a halt when it reaches equilibrium, but a wave
process doesn’t. The complex factor is more natural for autonomous agents, since they have no reason
to ‘stop’ due to exterior changes: their action comes from within. Relaxation is an effective response to
externalized change. Thus, it’s easy to argue that the Schr¨
odinger complex form is more natural for local
autonomous agents.
Apart from looking like a classic diffusion or Schr¨
odinger wave equation, there is another interpreta-
tion in graph dynamics. The time derivative is a diagonal part of the graph matrix. This is associated with
‘pumping’ of the system [19], or the injection of currency αifrom a source Ai. Thus, we might postulate
this type of equation to calculate the dynamic equilibrium of currency from source to sink along different
promise channels:
Wave-Diffusion process ←Pumping source (36)
This aligns with the role of the time derivative in Schr¨
odinger’s equation too: as a total energy, whose
time variable is for the whole composite system, not for the interior dynamics, where interior velocities
are replaced by implicit waves.
3.3 Currency landscapes vs process semantics
To preserve the causal independence or locality of the agents, we can only relate this to each agent’s
assessments of one another, so the force can be represented as a gradient of an assessment of the promises
αi(Vj)in some currency:
~
F(ij)≡−→
∇ij αi(Vj).(37)
Now α(V)becomes the fitness landscape or potential surface that we’re familiar with in smooth classical
systems. This assessment maps straightforwardly to concepts like utility in von Neumann’s economic
game theory [11], and to statistical expectation values for processes that are algorithmically simple in
nature. As agents become more complex in their interior processes and interactions, assessments can be
based on an increasing number of issues, and the simple algorithmics of Newtonian thinking will not
represent all the interior details.
At every stage in this formulation, agents are behaving independently, but signalling one another
at a certain rate with information, through channels formed from promises. How could this lead to
phenomena in which momentum is conserved in collisions? Recall that, in motion of the third kind, it
isn’t the agents that are moving but what they promise. This is more like the way we think about a wave
in classical mechanics than an extended body or abstract centre of mass.
Example 4 (Process semantics) Waves interfere are are absorbed. The collision of bodies is a distinct
concept, where the abstraction of a ‘body’ has clear boundaries of causal independence. Bodies are
themselves separate collective processes (more like wave packets) that become temporarily entangled
with each other. When bodies collide and become conjoined, or when they split, the relative proportions
of the promises are not directly related to the process that runs on top of them: the effective mass and
velocities are virtual things, are determined by the underlying collective processes, but can behave as
virtual agents. It’s natural to express these fractions using a shared currency (energy, trust, money, etc)
in order the describe ratios and directions of the split.
16

Example 5 (Attraction and repulson) The energy-currency concept is also useful for describing the
tendency for agents to create and destroy processes (see figure 2). In classical mechanics, an attractive
potential well tends to induce an attraction, e.g. in gravity or electrostatics. In this kind of directional
landscape view, gradients guide behaviour by setting up an effective guiderail. In economics, rather
than saying that successful economic agents make a lot of money, we would say that successful agents
are those that can attract money, since they cannot create or destroy it themselves. In sociology, we
would say that collaboration is attracted by reputational trustworthiness. Counter-concepts like mass
represent resistance to such forces due to orthogonal interactions (obligations).
A1A2
stored
exchange currency
V =
T =
LOCATION
SHARED PROCESSES
kinetics/risk
force
trustworthiness
currency potential
mass
mass
p = mv −> ‘intent’
Figure 2: Currency is the generalization of energy in mechanics. It’s manifestation on different scales comes
with new semantics, such as the emergence of ‘intent’ from momentum. One should be careful when interpreting
these classical mechanics-style illustrations of energy as continuous landscapes, as in this mnemonic. Energy is
an interior assessment of an agent or a process. Moreover, Promise Theory predicts that stored currency (potential
energy) is accounted on the interior of agents, while exchange currency accounted in links between agents, i.e.
a property of shared processes formed by agent interactions. In motion of the third kind, there are no bodies or
exterior potentials that store energy in the Newtonian sense.
3.4 Direction and current
At the simplest level, there are two orthogonal kinds of promise involved in motion of the third kind. In
the semantic spacetime model, these are referred to as EXPRESS and FOLLOWS promises. Expression
promises refer to interior scalar properties that are exposed for observation. Following promises form
partially ordered trajectories of agents along which motion can occur (these are guiderails, in the lan-
guage of part I). The collaboration between interior and exterior promises leads to a continuity equation
for the interior states ρ∼bEXPRESS, for each agent Ai, of the form:
∂tαi(interior) + −→
∇ · ~αi(exterior) = 0 (38)
Assessments of agents, by one another, behaves as a kind of local density ρand current ~
J, measured
in currency units α:
•αi(interior)∼αi(bexpress)7→ ρi, and the exchange of influence between agents
•~αi(exterior)∼~αi(bfollows)∼~
Ji∝−→
∇iVwould be measured by interior assessments also, leading
to the equivalent of Fick’s law Ji=Ji(−→
∇α(b)).
17

Locally, this then satisfies a continuity equation that becomes a heat-diffusion equation subject to bound-
ary conditions and forces. The continuity equation for the currency locally at a point
∂tρ(x, t) + −→
∇ · ~
J(x, t)=0,(39)
which becomes the diffusion equation on use of Fick’s law:
~
J(x, t) = −D(ρ, x)−→
∇ρ(x, t).(40)
These combine to give a diffusion equation:
∂tρ=−→
∇ · (D−→
∇ρ)(41)
If we translate the variables to the present case of agents and promises:
∂αi
∂t +X
j
−→
∇j~
Jij =F(Ai, t),(42)
and by analogy with Fick’s law
~
Jij =−D(Ai, t)−→
∇jρi,(43)
for some D(Ai, t), which we would expect to be constant for the most elementary agents with few
degrees of freedom internally. Kirchoff’s laws at each agent junction [21] tell us that the flow part of
trust would be conserved:
∂α
∂t =incoming −outgoing +F(44)
=Aji δαj−Aij δαj+Fi,(45)
as seen from the perspective of any single agent.
Remark 4 (Autonomy or not?) We need a reasonable answer to the question, why would an agent
voluntarily reduce its currency and under what circumstances? A simple answer is that the reduction is
not always voluntary, because promises may be conditional on some shared property between the agents.
How this comes about is a separate question that we can try to discuss towards the end. We should be
careful not to take the fluid dynamical or monetary analogy too far. The pictures we use to describe
flows and ballistic processes have become ubiquitous and are misleading in the general case. A picture
based on internal states looks somewhat different to the effective picture we might perceive. A quantity
corresponding to potential or internal energy, is an interior resource like a savings account which can
be used to ‘finance’ activity. This is not directly observable (as potential energy is not observable in
physics). It can only be taken as a hypothesis for predicting behaviour.
3.5 The meaning of momentum in physics
In physics, we also use the term momentum for a number of different process characteristic with inequiv-
alent semantics. The common feature of these is that they are locally directional. Momentum represents
a measure of ‘alignment’ or similarity between the interior states of dynamical entities.
For point particles, momentum summarizes an instantaneous guiderail for the direction of motion—
where am I going next, based on where I am now (essentially like a train following a track). The
Newtonian momentum ~p =m
~
˙xis a surprisingly subtle quantity, involving a time derivative of a vector.
It’s an oddity in dynamicsm but it dominates our idea of momentum, because calculus has established an
unreasonably idealized expectation for what momentum is—and that leads to much confusion.
The concept of a trajectory, or vector field, relies on the continuity of this alignment concept too, as
measured relative to a fixed coordinate system in which ~p can be described globally. The coincidence
18

t
∆
semantic
complexity
Agents
Interior assessment layer
Semantic coupling layer
+
b.c.
STATE
GAMES
ACTION POTENTIAL
ψ
−
Π
α() τ
ε
Figure 3: The semantic layers (scales) of action for a general virtual phenomena. At the fundamental level, there
are agents with interior activity (time). The agents can modify one another’s activity by effectively exchanging
a canonical currency through promised channels. Their activity level is an assessment that affects the frequency
of their interactions sampling each others’ state. Thus there’s a layer of state, which is driven by (dependent on)
the activity level of each agent, but which controls the direction and type of state-changing interactions. Exterior
promises represent exterior states, which are channels of interaction. Interior promises represent internal states
that compute the processes within each agent, responsible for sampling and delivering on the promises.
limit, used in the calculus of derivative gradients, means that we can define momentum as a point quantity,
involving only a time derivative. We transform something non-local into something that appears to
be purely local! This sleight of hand means that it now looks as though every entity ‘contains’ its
momentum. In fact, this is only true in this limit. No such thing is possible in an agent model, and indeed
it isn’t true in Quantum Mechanics either. Think of a game of musical chairs. The chairs cannot contain
momentum. Momentum lies only in the process that takes place between them.
Momentum is carried inside an agent in Newtonian thinking. How could this be? In terms of agents,
this is only possible if all the agents hosting the trajectory have sufficient interior resources for the
collective process to remember its own trajectory. How, for instance, could a dimensionless point ‘know’
about coordinate systems and directions? A pointlike object has no degrees of freedom to align with
and thus encode or ‘remember’ any direction. The locality of Newtonian momentum is an artifact of
the idealizations of Newtonian mechanics that doesn’t survive generalization. We know that it doesn’t
generalize:
•In the presence of electromagnetic fields, the momentum has to become m~v −e~
A(x, t), where
A(x, t)is a non-local field. The point momentum doesn’t even survive a simple coupling! An
extended field cannot be represented by a purely local change at a single point.
•In Quantum Mechanics, a similar generalization is needed. The time derivative momentum re-
placed by a non-local gradient over space that inserts explicit wave semantics to momentum, based
on the observation from experiment that p=h/λ. It doesn’t play the same role as the Newtonian
momentum, since it’s carried by the wave, whereas the Newtonian momentum is carried by the
centre of mass process. We usually extrapolate the momentum to be something carried by the
entire body by virtue of the linearity of momentum under composition.
In the differential prescription, momentum can exist instantaneously at a point. However this is impos-
sible for either a wave or a promise, since the latter are non-local processes. The fact that we can take
a differential limit of δx 7→ 0or δt 7→ 0is an artifact that was heavily disputed in Newton’s time, but
which has become accepted without discussion in modern times.
19

3.6 The meaning of momentum in Promise Theory
In Promise Theory, it is promise bindings that describe alignment. The analogue of momentum belongs
not to agents, which are stationary, but to virtual processes that use agents and their promises to propagate
information between them. These are virtual processes. Two promises Ai
+b
−→ Aj, Ai
−b
←− Ajare
partially or fully aligned if and only if b∩b06=∅. Note that this alignment doesn’t refer to a universal
direction in a Euclidean meaning. The concept of direction only exists between pairs of agents on a small
scale. The existence of a long range order, and long range direction are expected to emerge at scale.
The graph matrix Π(±)
ij refers to transfers over a channel, whereas the momentum in the sense of
classical kinematics is a property of a body, transmitted by coincidence or contact. However, it’s also a
vector, meaning that it points in a certain direction. These facts can only be reconciled in a Euclidean
embedding. For graph circuitry, a vector is a channel or link between communicating agents. So the only
corresponding possibility is for the momentum to be linked to the magnitude of what one agent can pass
on to another:
Π(±)
ij :Ai
+~p+
−−−→
−~p−
←−−−
Aj,(46)
which we denote by the causal overlap Ck
ij (p+∩p−), when assessed by a third party Akin scope. The
momentum is thus a guiderail for a process kept as a series of impulse events between the agents.
In Promise Theory, one distinguishes impositions from promises:
•An imposition S+b+
−−−−−→ Ris an offer of influence that is not aligned with a sampling process
R−b−
−−−→ Sin co-time.
•A promise S+b+
−−→ Ris an offer of influence that is aligned with the sampling process R−b−
−−−→ S
of the receiver in co-time:
These two modes of interaction correspond roughly to the semantics of ballistic and force field transfers.
Impositions are like collisions. When they hit, an imposition arrives to meet a pre-existing promise, but
with no particular phase alignment to the receiver. Thus, impositions are often ineffective.
A pair of agents can exchange influence through an interplay between interior agent promises (and
the processes that keep them) and exterior promises. In collisions, momenta may be aligned or anti-
aligned. When bodies split and move off in opposite directions (opposite promise alignments), the ratio
of the split involves another parameter in Newtonian thinking: the mass. Thus, the semantics of the
express may be taken as the co-modulation of interior and exterior promises:
momentum ∼interior promises AND exterior promises (47)
∼m×v(48)
The mass is a property that encodes the relative encumbrance of the process ‘body’, in making changes
due to other interactions or commitments. When the encumbrances that lead to effective mass are de-
coupled from those that lead to motion, we can write momentum as a product of independent variables
p=mv. The mass refers to effectively interior properties codifying purely local encumbrances, and the
velocity refers to non-local exterior properties between agents.
These points tell us something about what momentum must correspond to in an agent theory. A
momentum difference p1−p2can be positive or negative. The promise bodies b1∩b2cannot be negative
unless we include their orientation relative to the graph of oriented connections Ck
ij . If we allow Ck
ij to
depend on a proper time (either Ak’s time or Aiand Aj’s co-time, for global versus relative formulations)
then we can make the identification:
~p(t)∼mi~
Ck(49)
20

Or from the perspective of agent Ak, over a direction from Ai
±
−→ Aj.
pk
ij (t) = miCk
ij .(50)
The force acting to change this momentum is then relative to the co-time t(ij)
~
F∼dp
dt =∂t(ij)miC(ij)
ij =F(ij)(51)
Finally, the assessed contribution to changes in some promise body bcontribute to bas a memory func-
tion. Just as the momentum ‘remembers’ additively the effects of past interactions in a phase space state
with direction and magnitude, so the overlap bij =b+
i∩b−
jis the route by which we can accumulate
kinetic contributions as a locally stored potential. In general, this will not be possible. Each agent can
only affect its own changes.
4 The semantics of force, momentum, and energy in physics
To better understand how to bring quantitative analysis into Promise Theory, let’s compare the strategies
that emerged for describing physics with what makes sense for an agent view. Physics has gone through
three main paradigms: I’ll call them the Newtonian, the Einsteinian, and the Quantum Mechanical. They
join up in certain limits, but they don’t form a seamless whole. We want to extend their ideas to a more
neutral or ‘unifying’ description within semantic spacetime, so as to apply a similar method to agent
systems—such as we would meet in biology, computing, sociology, and economics.
4.1 Quantifying gradients and exchange in Newtonian physics
In the Newtonian description of elementary mechanics9, influence is transmitted by direct modulation
of variables, i.e. through the addition and multiplication of dynamical quantities. This leads to an in-
stantaneous response. In more complicated scenarios, this direct modulation doesn’t represent the nature
of responses, and ‘response functions’ typically involve time delay and spatial shifts that introduce non-
local aspects into the description. This is worth mentioning since, in discussions of Quantum Mechanics,
one often attributes non-local effects to something uniquely odd about Quantum Mechanics, and we
should out such ideas out of our minds. Non-locality is a necessary part of interaction on an elementary
level—however, it can sometimes be scaled away in certain effective descriptions over a sufficient scale.
Since the agent point of view aims to expose semantics (or qualitative assumptions) on an equal
footing to the dynamics (quantitative assumptions), let’s start by examining the semantics of a local
Newtonian description and expressing these in terms of agents and their promises. In this way, we can
make explicit the hidden assumptions for comparison with other scenarios. There are essentially two
parts to each decomposition:
•An accounting principle defining flow as an exchange of currency, using the concepts for ‘energy
and momentum’ (or equivalents).
•A precision of the semantics of momentum for each process, e.g. pointlike phase space for classical
mechanics, wave delocalization for quantum mechanics, diffusion for hydrodynamics, etc. These
are the in-out association functions for each process type.
Force is defined to be that influence, which appears to extend over space in order to accelerate or
retard a body’s motion. On a high level, we can interpret force as an incentive to change. The anthro-
pomorphism is frowned upon in physics (though generally harmless), but it’s exactly what we want in
describing socio-economics! Similarly, the concept rendered impartial as momentum has a directionality
that associates with intentional behaviours at scale. No concept of free will—or of human uniqueness—is
9We should acknowledge that many thinkers contributed over time to the views that now bear Newton’s name.
21

Role E Physical Social Economic
Kinetic T Energy Risk Investment
Potential V Well Trust Savings
Table 2: Heuristic roles for canonical currency at different scales of semantic complexity. Ultimately one might
be able to reduce all counting to energy, but it will not reflect the semantics of different scales in a convincing way.
needed to make this or other semantic identifications, as they provide no more explanation for phenomena
than physics does. We’re merely looking for dynamical representations for the relevant semantics.
A mechanical ‘body’ is represented by a point-like proxy agent, which is the seat of the centre of
mass. We define the work done as an investment of energy, where ~v ≡~
dx
dt and adding the mass (or
encumbrance) scale for momentum
~
F=~
dp
dt ,(52)
we can identify the relative rates of change by using the collective notion of a trajectory or process in
spacetime to relate these ideas:
~
F·~
dx =~
F·~v dt (53)
=~
dp
dt ·~v dt (54)
=~v ·~
dp (55)
=m~v ·~
dv (56)
=1
2md(~v ·~v)(57)
=d1
2mv2(58)
=dT. (59)
Since this is a total derivative, it depends only on the initial and final states. It’s not path dependent. We
can integrate it along any path and it will appear to be conserved, by continuity.
4.2 Quantifying gradients and exchange in Quantum Mechanics
In Quantum Mechanics, the reasoning about an energy currency is similar but the reasoning about force
is completely different (see the correspondence in table 2). In particular, the momentum goes from being
a scaled time derivative, i.e. a velocity or time rate of change in some parameter, at a fixed location,
to being a spatial gradient of such a state displacement, with the stipulation that spatial gradient and
temporal evolution are linked by a wave process rather than a uniform motion in a straight line.
Thus we have the peculiar idea that momentum is no longer carried with the location of the centre
of mass, but is rather something like a potential or field whose bias deliberately smears the concept of
motion over a wave process. The justification for this was, of course, the empirical fact of interference
phenomena. The effect is to replace interior state changing over time with a diffusion of state over space.
This leads to many of the confusions concerning the so-called non-intuitive behaviours in Quantum
Mechanics.
22

~
F·~
dx =dh~pi
dt ·~
dx, (60)
=d
dthψ|~p|ψi · ~
dx, (61)
=dhψ| − i~−→
∇|ψi · ~
dx
dt ,(62)
7→ −~2hψ|
−→
∇
m|ψi · dhψ|←→
∇ |ψi,(63)
=d−~
2mhψ|−→
∇|ψi·hψ|−→
∇|ψi(64)
=d−~2
2mhψ|∇2|ψi.(65)
Notice how the mnemonic dx/dt is replaced by a current ~
J∼←→
∇, which is the only process in Quan-
tum Mechanics that represents a time ordered relationship between general xand general t, and which
embodies the wave assumption again by virtue of the use of the gradient operator. The result is the
equivalent, but the implicit velocity (here represented by the quantum probability current, which is the
only travelling quantity) is no longer the velocity of the interior state process but the relative velocity
of propagation of the exterior propagation process. This is because we no longer identify state ψas
being related to a local displacement within the spacetime coordinate x. It has become encoded as the
phases of some superposition of wave process. In other words, we’ve shifted from a purely local process
to one where state is externalized and always moves as a wave, exchanging its energy currency with
neighbouring locations. Schr ¨
odinger mechanics is clearly not even defined for point particles.
Minimizing energy locally now implicates spacetime around the location, which is why the simple
harmonic oscillator can’t ever have a stationary state. It’s ground state has to be a wave and thus is
must carry some non-zero energy. Autonomy is gradually eroded at scale in order to maximize future
potential.
4.3 Quantifying gradients and exchange in Promise Theory
For a general promise theory, about an isolated process with promise type ρ, we define the state memory
at agent Aito be ρi, and we can let this consist of a slowly varying background value ρiand a fluctuating
flow process ˜ρi. We ask: what is the equivalent chain of reasoning for kinetic currency transactions
between agents in Promise Theory? Let the change in potential value of a transaction be realized by a
promised process Π(±)
ij . The effective momentum pfor the process is defined in terms of a current J
pij =miJij ,(66)
where miis some effective mass, encumbrance, drag, inertia, connectedness, etc, see equation (66).
Then the effective work done by a force dp/dt at Aimay be expressed both as a gradient of a guiderail
potential V, or as the rate of change of the momentum over some proper timescale intrinsic to the process.
The work is:
~
F·dAi≡(~
∂iV)·∆i≡d~pij
dt ·∆i. (67)
Now, we define the momentum in terms of the current to obtain the trivial likeness for the kinetic cur-
rency:
Fij dAi=dpij
dτ ∆i(68)
=d
dτ (miJij ) ∆i(69)
=d(miJij )×di
dτ .(70)
23

where we use the proper time τto underline that this is an interior process co-time for spatial exchange
interactions, not the externalized global time of Newtonian-Galilean physics. We can compute the effec-
tive value of di/dτ simple to be the velocity or current Jij . So,
Fij dAi=Jij d(miJij )(71)
=d(1
2miJ2
ij ).(72)
The latter is the change of kinetic process currency for the virtual process, corresponding to the promise
ρi. To complete this, we need to specify the interaction semantics, or the nature of the current. One
possibility is to invoke the analogy of Fick’s law in diffusion, or the quantum wave momentum (which
turns a time derviative into a change over space, as waves couple space and proper time in their process).
For an arbitrary cooperative graph process, we have to observe the non-locality of interaction. There
is no way of taking a dx 7→ 0type of limit as Newton could for the continuum, so we’ll inevitably take
on flow semantics analogous to Fick’s law. We can take
Jij 7→ −Λ−→
∇j˜ρi,(73)
for some scale Λ, giving for (67),
Vi=1
2miΛ2˜ρ2
i,(74)
for the currency constraint. For a specific potential V=Vi˜ρi, the guidrail potential can modulate
the changes of state, as in the quantum formulation, for convenience. Finally, we can choose causal
semantics for the gradient, giving the naturally retarded proper time interpretation (setting observer k7→
i:
Vi=1
2miΛ2X
ijk
(˜ρi−Cij ˜ρj) (˜ρi−Cij ˜ρj),(75)
which is to be understood as an equation for ρi, alongside the first order
−→
∇jρi= 0,(76)
which implies that ρiis an eigenvector (Cij/λ)ρi=ρiresulting from self-consistent steady state flow.
Note that the assessments implies by Cij are those of the participating agents now (proper time process),
not of a godlike exterior observer.
When Cij >0, the Frobenius-Perron theorem comes into play, and the principal eigenvector is real
and position, with a probabilistic interpretation. In other cases, potentially negative regions lead to more
than a single relevant eigenvector, and the solution is more akin to a Wigner function decomposition
ρi∼b+b−. The currency constraints (75) and (76) are analogous to the Cauchy momentum equation for
a stress tensor Cij in hydrodynamics. This is not unexpected, since we’re basically describing a graph
as a confluence of forward flowing channels scaled by mutual channel strength. This steady state flow
approximation, a result of the continuum assessment values and derivatives used, implicitly assumes
an average flow behaviour. Thus, we derive the analogous utility of a currency concept for large scale
behaviours guided by slow (adiabatic) boundary conditions for an isolated process. What this helps
to underline is that the sometimes mysterious looking results T=1
2mv2,p=mv, etc, are simply
accounting practices that conceal assumed interaction semantics. If the counting is similar, the results
have to be similar in structure too.
We should, of course, ask: what happens when these conditions of continuity break down, for in-
dividual transactions between agents over small timescales, where external conditions are not adiabatic,
and so forth. In that case, we need to develop new rules sets to encompass these semantics. The result
might look more like a cellular automaton, and the usefulness of counting with a conserved currency
may become less clear.
24

4.4 Currency accounting patterns in an agent viewpoint
The general pattern of all these currency accounting formulae assumes a similar form:
∆(reservoir of influence)∝(interior interactions)×(∆exterior)2,(77)
e.g. mv2, I 2R, CV 2, etc. This is the ‘work’ done by a force. Why would such a force be experienced
by an autonomous agent? Why would an autonomous agent even ‘agree’ to accept or react to an exterior
influence10? This may not be the correct question to ask in every case, as the promises linking agents
may be co-dependent and thus unavoidable. In the broader case, on socio-economic scales, obstinate
refusal might seem more likely, though this too depends on an assumed absence or mutual dependency.
A full answer requires us to understand how promises are made in the first place (see section 5).
When all descriptions are externalized, we can only imagine transfer by collision. However, rec-
ognizing interior properties of spacetime too, distinctions can be distinctions of local state. The field
becomes a kind of fabric of agents, as in solid state physics. An autonomous agent needs to accept
promises and form its assessments in advance of the process moving into new territory—and thus accept
a currency transaction. This must involve non-locality at the edge, or equivalently, the existence of an
existing guiderail that primes the space for the potential transactions.
The guiderail is natural when we think of Motion Of The Third Kind (also of the second kind).
However, in a material interpretation (which is still fixated in most classical physics) it stumbles into
problems of interpretation—which is surely the reason why classical thinking make Quantum Mechanics
seem ‘weird’.
If potential energy (or currency) is a store of assessed wealth, what is kinetic energy? In physics, this
is carried by moving bodies, but in virtual motion there are no moving bodies, only moving promises11.
So the counting of transferred currency has to be promised as part of the transfer. In order for the
accounting to be fair and conserved, the promises need to be kept homogeneously. This is Noether’s
theorem. Kinetic energy is the outcome of exchanging locally stored energy currency. An intrinsic
policy of seeking the most potential, trusted, hoarded positions would explain that. Risk taking would be
the opposite that would allow processes to tunnel through apparent barriers.
4.5 Example: Coupled oscillators in Newtonian mechanics and Promise Theory
To better understand how promises and assessments take on the roles of dynamical variables, it’s helpful
to consider an example from the repertoire of basic dynamical systems—the familiar case of coupled
harmonic oscillators. This exhibits aspects of transmission and cooperation that need to be explained in
the context of autonomous agents, as well as accounting of position, momentum, and energy. We can
look at counterparts at other semantic scales.
In the classical view, interactions are based on the forces transmitted by direct contact of idealized
springs. A body with mass m1is connected to an immovable wall by a spring of stiffness k1; another
body with mass m2is connected to another immovable wall by a spring of stiffness k2, and the two
masses are connected by a third spring of stiffness k12 (see figure 4). The springs are representations of
force and shouldn’t be taken too literally. Most potentials are contactless; what matters is that the force is
approximately linear in relation to the separation of the two bodies—and we treat the length of a spring
as a proxy for that promise.
It appears superficially, from the use of a name xirooted in an association to Euclidean coordinates
(x, t), that we are embedding the system in a single Euclidean space, but this is misleading. The param-
eters entering the equations are displacements from equilibrium positions, which I’ll call ∆xito clearly
distinguish them from the actual coordinate positions. The distinction is important, because each agent
10This anthropomorphism applies directly on a large scale, but we’re not implying any kind of ‘anthropic principle’ for
reverse inference.
11Note that kinetic energy is only apparent by relative velocity, so it can’t be carried as an intrinsic property. It can only be
assessed as a quantity moving inside something relative to the observer. So stored potential becomes moving stored potential
of the proxy agent.
25

A
L
A
R
A2
A
1
1
x2
x
1
k12
k
1
m2
m2
k
Figure 4: Classical representation of coupled oscillators as materialized bodies with internal properties and exter-
nalized dynamics within a Euclidean embedding, alongside an agent view of coupled oscillators with completely
interior state, communicating via promise channels.
formally has its own internal space, but assumes a common time. It makes the relativity of the entities in
the system play a latent role via the magnitude of their interaction via the spring. We take this sleight of
hand entirely for granted for physical springs and ballistic forces, but the origin of these effects in virtual
systems is far from obvious.
The displacement is negative if to the left, and positive if to the right. The signs of the forces are
aligned with the signs of the displacements, To find the equations of motion, we can consider either the
forces acting on each body locally, or the total energy expressed in terms of the local bodies. Let’s begin
with the forces. The force on body with mass m1
•Force on entity 1 at equilibrium position x1due to moving the mass is:
F11 =−k1∆x1−k12∆x1(78)
The sign of the restoring force due to the springs is against the direction of the displacement from
both springs even though one is distended and the other is compressed.
•The force on entity 1 due to the movement of entity 2, with position x2and displacement ∆x2is:
F12 =−k12∆x2.(79)
The sign is now opposite since it pulls to the right and there is no local back-reaction on entity 1
from the spring with k2.
By symmetry, with some signs reversed, we obtain two similar equations for the two entities:
F1=m1¨x1=−(k1+k12)∆x1−k12 ∆x2(80)
F2=m1¨x2=−(k2+k12)∆x2−k12 ∆x1.(81)
There are no equations for the walls, since we assume that they are fixed and therefore experience no
deformation. Note first how these equations express the essential locality of the processes taking place at
each of the entities, yet some force is effective transmitted from the other. Note also that we don’t need
to know the initial equilibrium positions xi, i = 1,2in order to write these equations in terms of the
displacements ∆xi. It only matters that there exists such a state, which could be considered a boundary
or an initial condition on the configuration.
An interesting outcome of the solutions of these equations is how the behaviour doesn’t depend on
the actual entities at all. This is easiest to see if we simplify away the details, by making k1=k2
26

and m1=m2, and introduce average (slow) and fluctuation (fast) coordinates x=1
2(x1+x2), and
˜x= (x1−x2).
Adding and subtracting the equations now gives simply an equation for the average centre of mass
location and their relative position, a location that doesn’t actually exist except implicitly in the embed-
ding:
mƬ
x=−k∆x, (82)
mƬ
˜x=−(k+ 2k12)∆˜x, (83)
which has natural harmonic solutions for the centre of mass and total system of the combined ‘molecule’—
not for the individual entities. This means the system naturally behaves like a single entity not two
separate ones. Why would this be? These facts are all well known and stated in any elementary text-
book, though one does not usually draw attention to the assumptions or question the outcome. However,
this has special significance when we come to think of a system representation in terms of agents and
promises.
4.6 Coupled agent interpretation in Semantic Spacetime
Now let’s re-frame this problem as one of communicating agents Ai, i = 1,2without any springs or
physical paraphernalia. This is an unfamiliar exercise for most readers, since we are programmed to
accept the tenets of Newtonian mechanics from an early age.
We need to explain how force can be transmitted (over what channel), why independent agents would
accept information from other agents and how intermediate agents would pass on forces they experience
to others when connected by springs. In particular, we remove any reference to an embedding space
containing the agents, and expresses the displacements from equilibrium with states that are fully internal
to each agent—effectively drawing a boundary around the formerly externalized motion. We expect a
careful decomposition to expose the difference between interior and exterior processes, at the scale of
description. Indeed, this gives some insight into the nature of the common kinematic variables.
How do agents decide what their equilibrium states should be in the first place? In the formulation
above, the equilibrium configuration is considered to be an initial state, and is therefore taken for granted.
We can’t compute the initial positions of the masses in an exterior spacetime, because that information
is not contained in the formulation: the coordinates are displacements! This might come as a surprise
to the reader, since Newtonian mechanics is generally formulated in a single Euclidean theatre. In this
case, the coupled oscillators are formulated each in its own Euclidean theatre; these are coupled through
Hooke’s law of springs and the assumption of transmitted force. What we have to do now is ask: how
could we simulate the same behaviour using messages passed between independent agents, e.g. by a
series of agents writing letters to one another?
We begin by postulating two agents A1and A2in the role of the masses (see figure 4) that make
promises to one another. Next, since the position of the agents is undefined, we ask what corresponds
to ‘displacement’ for agents. There are no positions, so the displacements must refer to some change
of the interior states of the agents. Suppose we postulate a vector ψi, with various components, some
combination of which represents the position. This might be extracted with a filter, say by some operator
ˆx(we are always free to choose such a representation, without knowing to precise nature of ψ), as in
Quantum Mechanics. We still have to explain what initial value the agents would have for their position,
i.e. corresponding to displacement ∆xi= 0, and how that comes about.
The role of the springs is even more subtle than the question of displacement, and forms a much
longer discussion, if we’re going to be able to tick all the boxes. In a Newtonian view, force is a property
that extends in actual space between the agents and transmits an influence non-locally, i.e. it implicates
a process in the spacetime continuum, realized by the spring, which therefore cannot exist outside the
agents of Promise Theory. Reaction and back-reaction are assumed equal from the start. That’s an
external imposition, or obligation for the agents. Autonomous agents needn’t accept such an imposition,
so we can’t assume this to be true once we place all the control in the hands of local autonomous agents.
27

Each spring is a physical entity in the Newtonian model, which spans two massive entities, effectively
‘entangling’ their behaviours [22]—a material representation of the influences, i.e. noting exists without
a material entity to carry it. There need be no such physical entity when information is passed by message
alone, except insofar as information itself requires some medium to represent it. It would be ridiculous
to imagine a kind of classical force transmitted by the ballistic pressure of the message itself (as one
sometimes does with messenger particles in Quantum Field Theory).
Interactions between agents are now handled as the autonomous behaviours, promises, and assess-
ments of agents individually. So if an agent changes its internal displacement, it does so only because
it has independently promised to do so, e.g. when it accepts a certain message, and not because such a
change is imposed and involuntary. Promise Theory thus shifts the externalized magic of ballistic im-
position to an internal magic of localized interior change. This isn’t more explicable, only more locally
accountable, and divorced from a redundant embedding in spacetime.
The walls of the classical system are boundary conditions, assumed immovable and are thus ignored
in the Newtonian formulation, but we must introduce these as agents here because our formulation of
springs requires a ‘source’ location for the information channel replacing the spring to connect to, i.e.
the message of force has to come from somewhere. In Promise Theory that means there must exist an
agent to make that promise. The walls offer their part of the ‘spring’ force (+) that joins them, and the
adjacent agents accept this (-):
AL
+k1
−−→ A1(84)
AL
−k1
←−− A1(85)
AR
+k2
−−→ A2(86)
AR
−k2
←−− A2.(87)
The fact that we need to write four statements here is a consequence of strict autonomy or locality. Notice
that although there are arrows in both directions, the force is only transmitted one way from the walls
to Ai, i = 1,2. This encodes the immovability of the walls. The forces between Ai, i = 1,2are
bi-directional, by assumption of the semantics of springs. If we use the strict notion k(k)
ij to mean the
assessment by agent Akof the spring constant kij offered by Aito Aj, then we have:
A1
+k(1)
12
−−−→ A2(88)
A1
−k(2)
12
←−−− A2(89)
A2
+k(2)
21
−−−→ A1(90)
A2
−k(1)
21
←−−− A1.(91)
Can the agents here still be considered to have an independent choice? Without withdrawing from these
voluntary promises, they have indeed bound themselves into a condition of co-dependence. The effective
transmitted overlap values k12 =k(1)
12 ∩k(2)
12 and k21 =k(2)
21 ∩k(1)
21 . Without further constraint, there is
no need to assume that k12 =k21, however Newton’s third law tells us this must be true, so we shall
assume it here. This can be assured by promising conditional co-dependence or mutual calibration of the
interior values:
A1
±k12|k21
−−−−−→ A2
A2
±k21|k12
−−−−−→ A1,(92)
and the ±notation is a shorthand for mutual offer and acceptance. Now we have established that both
agents know and agree on the value of k12, due to some co-dependent entanglement relationship of
communicating and assessing one another. If either changes its value, the other will follow in step, though
28

several iterations of interior action may be needed to settle on this equilibrium, which has implications
for the local interior time rate at each agent. This equilibration may be considered the formation of the
guiderail, as discussed in part I [1]. It corresponds to a Nash equilibrium in game theory [12, 23].
The same argument, given for k12 also applies to the sharing of displacement state ∆xibetween the
agents, since the responses to one another depend on each agents assessment αiof one another’s state.
Let’s simplify the notation slightly to avoid carrying the assessment process α() around with us:
∆x(i)
i=αi(∆x1,∆x2).(93)
Thus we can write,
A1
+∆x(1)
1
−−−−→ A2
A1
−∆x(2)
1
←−−−− A2
A2
+∆x(2)
2
−−−−→ A1
A2
−∆x(1)
2
←−−−− A1.(94)
The effective overlaps are then ∆x1= ∆x(1)
1∩∆x(2)
1and ∆x2= ∆x(1)
2∩∆x(2)
2.
Given all these local promises, whose ‘decisions’ are made entirely autonomously, we can now form
a promise rule for the dynamics of the oscillators, by promising to obey Newton’s second law locally (as
an interior promise to self). This is just like a programmed algorithm that guides the process within the
agent’s interior. The promise is only observable by its outside effect, by assessing ∆xi. It requires no
communication outside the agents: all changes occur within the agent Ai, so we can write it like this:
Ai
+(F=m∂2
t(∆x))
−−−−−−−−−−→ Ai(95)
Thus the ‘law of nature’ is only a promise made by Ai. It doesn’t extend beyond its interior, except by
indirect coupling. That coupling is expressed by the rigid spatial channel equations (92), and what (95)
implies is the additional rigid coupling between space and time also. The question of time has so far
been suppressed. Autonomy implies that each agent will act independently, its own interior rate, thus
each agent has its own interior view of time. However, the entanglement or co-dependence between the
agents in (94) implies that both must wait for one another’s information in order to equilibrate:
A1
∆x1|∆x2
−−−−−→ A2(96)
A2
∆x2|∆x1
−−−−−→ A1.(97)
Thus processes advance at the shared rate of this feedback loop. We can call this co-time and denote it
by tand ∂t. All the pieces are now in place, and we can formulate each agent’s assessments by:
F1=m1∂2
t∆x(1)
1=−(k1+k12)∆x(1)
1(98)
F2=m2∂2
t∆x(2)
2=−(k2+k12)∆x(2)
2.(99)
Since we’ve hopefully made the point about locally determined values, for simplicity, let’s assume once
again that m1=m2and k1=k2, and suppress all the non-locality that implies, and quickly compare to
the Newtonian formulation. Adding these two equations, and defining complementary contra-oriented
non-local variables analogous to a Wigner formulation:
X=1
2∆x(1)
1+ ∆x(2)
2(100)
X†=1
2∆x(2)
1+ ∆x(1)
2,(101)
29

we have
m ∂2
tX=k12(X−X†)−kX. (102)
or with X−X†≡X12 representing the average equilibrium of the forward and backward processes
(like a Feynman propagator), them
m ∂2
tX=k12X12 −kX. (103)
Promise Theory thus predicts corrections to the standard model of oscillations, when viewed from the
perspective of uncorrelated agents involved. This looks familiar, but let’s remind ourselves that we sup-
pressed the local sampling and assessment process α() in these expressions, for notational convenience.
The variables X, X†are values that belong to the equilibrium entanglement or shared state assessments
of the agents, not to their actual state or to the assessment of either one of them alone. They are the
average self-assessments and co-assessments of the shared interior states. The subtlety here is that there
is a hidden loop of interior time involved in making these assessments, which reveals itself in the way
currency is transmitted (energy is counted). If the assessments can be assumed to be mutually calibrated
(say by a third part observer) then X7→ Xand this reduces to the Newtonian case of simple harmonic
oscillations over a common frequency:
m ∂2
tX=−kX. (104)
There are thus observer based corrections, which mean each observer has its own ‘world’ interpretation
of events. Collapsing these is not necessary for a shared interpretation as long as a single observer
calibrates the interpretation over some scale. Calibration of sender and receiver is built into the unitary
symmetry of Hilbert space in Quantum Mechanics12 . This is natural as long as observations are made
by a consistent agent. For quantum information pipelines, it might not hold. This could be an issue for
quantum computers in principle.
In closing, we might ask whether the purely unobservable interior states of different agents have any
external significance? From this, it seems that they are largely unimportant, at least in simple cases of
entanglement. Differences in interior time can be absorbed into local interpretations or assessments αi()
of the mass and spring constant variables. Thus, as long as we attempt to extract a common description
for all the components, the differences are effectively averaged away (analogous to quantum decoher-
ence). This is ultimately the reason why it makes sense to write down a shared view, at least over a range
at which the equilibration feedback of these promises states, by mutual Nyquist sampling, is negligible
compared to their rate of change. It’s known from the Fourier uncertainty principle—which becomes the
Heisenberg uncertainty in Quantum Mechanics [24, 25]—that this is the limit on observability for wave
decompositions of shared processes.
It’s interesting to speculate on how far this equivalence scales, when one considers agents with dif-
ferent interior resources at different scales. Representation of state and long range co-dependence, e.g.
over long chains of agents, requires an interior memory to capture and transmit and account for the cor-
related states. This is not something structureless point particles can reasonably do. As agents increase
in complexity, propagation can become less reliable, and may include more factors that compete for the
limited internal resources. So, if the agents were people and the promises were written letters, it seems
likely that the reliability of the promises could be compromised in more ways and the predictability of
the answer would fall within much wider margins.
12In Quantum Mechanics, one is used to the idea that all process agents, represented by bra and ket states, will behave
symmetrically, so that which is emitted must be absorbed, i.e. (b+)†=b−. This leads to the conserved norm of the Hilbert
space. This need not be true on a larger scale, where there are more open channels to account with. Conservation is only
a simplifying global convention. For each agent individually, the interior accounting has its own calibration. It’s natural to
assume that the accounting is lossless when assessments are made from the perspective of exterior, neutral third party agents.
30

4.7 Some examples from different semantic scales
We are focusing on physics, given its position as the undisputed leader in process descriptions. However,
the aim is to carry these techniques on to more general interactions in semantic spacetime. Let’s pause
here to sketch some characteristics that we would expect to see in the semantics of different regimes over
a range of scales.
Example 6 (Physics) There are significant differences here between coherent field (wave) excitations
and an assembly of locally contained displacements ∆xiat agent locations Ai. In the former ψis
already a superposition of Fourier modes that assumes an equilibrium has been reached. In relativistic,
there’s a delay added for finite speed of propagation, but in Schrodinger it’s instantaneous. There is thus
a separation of fast and slow variables. In the latter case ∆xietc, long range response is calculated
from local response by looking for propagation modes.
•Local interioir responses ∆xiare different from distributed fields like ψ(x, t), in representation,
but their effect is effectively similar.
•Particles/waves flow away from contentious regions, resource sources and sinks (scattering), guided
by momentum.
•Particles follow internalized momenta, distributed processes like waves and collective phenomena
follow potential guiderails determined by energy/forces.
•Spins are interior state, depicted as a change of alignment in external spacetime.
Example 7 (Computing) •Workloads flow away from contention or resource sinks (scattering), in
response to promised policies.
•Workload deployment follows potential guiderails determined by space potential capacity.
•Internal configurations change in response to policy desired-state Π(±)
ij .
•Message interactions follow guiderails determined by neighbouring agent activity in BGP, IP.
Example 8 (Biology) •Cells have semantic ‘intent’ or relative purpose based in interior epigenetic
state.
•Cellular processes are attracted by biochemical gradients, e.g. nutrients, hormonal signals.
•Any agent can create its own proteins. These can only be used for exchange if the other cells
accepts them.
•Oganisms become wealthy relative to others by attracting cooperation from other agents.
Example 9 (Economics) •Investment/jobs flow away from contention/loss.
•Any agent can create its own money. This can only be used for exchange if the other party accepts
it.
•Agents become wealthy relative to others by attracting common currency from other agents.
Example 10 (Leadership/society) •Alignment around a seed leader or diffusion away.
•Firing accusations (impositions) leads to a reduction in trust (antitrust).
•Continued negative promises may bleed trust from agents and disconnect them from society.
•Authority is concentration of ‘trustedness’ or trustworthiness τi=αi7→ max
•Power is abilty to influence |ψ|?
31

4.8 The variational energy (currency) formulation in physics
The energy concept has taken on a deep philosphical meaning in physics. It provides a compelling
narrative as a basic ‘substance’ behind everything. It appeals to our penchents both for materialism and
mysticysm, with principles of minimization of energy and action! Ultimately, energy’s role lies in being a
means of counting. Agent models, however, can be applied to all kinds of problem so we need to explore
the role of such a measure in describing more general problems too. Energy has simple semantics, but
we also need to explore the idea that more sophisticated measures of realized and potential activity might
also be important.
The Hamilton or Lagrange formulation expose the deeper connection between spacetime and process.
Energy is introduced for a number of reasons: it is the complementary variable to time, and this has the
role of a proxy for counting temporal activity in the system. Assuming continuity of temporal change
is equivalent to assuming the local conservation of energy at a point. The latter follows from Noether’s
theorem and is exposed directly by the action principle [26]. It doesn’t imply global energy conservation,
which would require some magic to accomplish, and would seem to violate direct observation of an
expanding universe.
The action principle underlines the connection to spacetime through its variational approach, whereas
the equations of Hamilton and Lagrange are coordinate dependent. What these methods do is to embed
the variables of state within a spacetime framework in order to deal with rates of change over extended
scales, from the perspective of an external observer.
If we try to extend this thinking to other types of currency, such as trust and money, it’s not enough
to talk about trust and money alone13. We still need to build the structure and meaning of interactions
between agents, to make the equivalent of the T±Vexpressions in terms of dynamical variables. The
simplicity of this linear separation between saved and transactional energy is a misleading idealization.
A counter-example (such as the Lagrangian of the Standard Model) shows that the choice of independent
variables and their detailed balance conditions are not essential to explain dynamics in terms of energye
exchange.
In the Hamiltonian and Lagrangian formulations, and their equations of motion, dynamical variables
are not strongly related to physical manifestations of the system, i.e. as a configuration of observable
entities in spacetime. Extended structures are replaced by effective average coordinates (centre of mass,
etc) and associated conditions of rigidity which are separated as far as possible. Such idealizations
drive the major ‘laws of physics’ because they can be reduced to these detailed balance requirements for
conservation of currency.
As agents become larger and incorporate more complex semantics, the idea that agents can be treated
as generic (essentially indistinguishable) entities (particles) falls into disrepute, but there remains the
possibility that a single currency and interaction process might dominate the observed behaviour. So,
let’s consider that possibility. How should we write down kinetic and potential currency accounting for
something like generic agents and promises? Is it possible? We need both semantics and dynamics. How
might this apply to something like sociology?
Let’s consider oscillators again, from the perspective of a single third party assessment. What gen-
erating principle might replace the action principle for variables of state (∆xi, p 7→ Ai,Π(±)
ij )? The
classical formulation starts with the action:
S=Zdt X
i1
2m(∂t∆xi)2−1
2k∆x2
i−1
2kij (∆x2−∆x1)2,(105)
or the Hamiltonian
H=X
ip2
i
2m+1
2k∆x2
i+1
2kij (∆x2−∆x1)2.(106)
13It’s tempting to look at money alone in economics, since it’s apparently ubiquitous; but while useful for local accounting,
money is not conserved globally, and is only a proxy for a deeper issue: trust.
32

In Quantum Mechanics, we replace state ∆xiwith a generic vector ψicontaining this and other infor-
mation, which can be filtered with operators, and we replace a local kinetic encapsulation using a double
time differentiation from 1
2mv2by a spatial gradient over a wave-diffusion process ~p 7→ −i~−→
∇. This
explicit choice was motivated by experimental observations of wave behaviour and it leads to excellent
results.
Remark 5 (Classical and Quantum Mechanics) Let’s note in passing that while the quantization of
energy is sometimes highlighted as the important step in going from classical to quantum mechanics,
this is partly misleading. The quantization is not of energy, but of a transfer relative to a sampling
process with frequency ω, by n=n~ω. The confusion arises because we try to embed all the dynamics
within a single Euclidean theatre, in the usual way. It’s the same confusion that could be held in the
case of coupled oscillators. If we look at the centre of mass variables, which are natural positions
using coordinates x=1
2(x+x0),˜x= (x−x0)between each dynamical transition between entities
in the embedding space, e.g. using the Wigner function, we see that the expansion in terms of ~is
actually an expansion in ˜xor ∂/∂p around the average position x14. In other words, the quantum case
is a relaxing of the strict entanglement of neighbouring points in the classical scheme, allowing each
location to behave as if it were more independent, linked more loosely by wavelike currency processes.
The classical formulation, which uses a single Euclidean theatre, attempts to imply that exchanges are
implicitly immediate and localized without finite extent at every point xi, but ∆xiand xiare completely
independent variables. Their association is an implicit overreach of semantics. Indeed, it represents an
oversampling of spacetime, and the Fourier uncertainty theorem pushes back on that by pointing out that
wavelike transfer implies ∆p∆x≥i~/2[24, 25].
The action principle works well to generate the equations of motion in both classical and quantum
physics, but the energy interactions are idealized. There are more general equations of motion that
include ‘latency’ or finite response times:
m¨
φ=Zdt0R(t−t0)J(t0, φ),(107)
even ‘iterated game interactions’, where each step in the evolution is the outcome of an interior game
with a Nash equilibrium or fixed point outcome [23, 27]. The latter interaction has no simple algebraic
form, but is represented by cellular automata [28, 29]. In Effective Field Theories [20], these non-local
interactions can be rewritten as integrals over locally modulated variables. This is a redefinition of the
semantic boundary between interior and exterior for the effective agents. We call these ‘dressed particles’
in particle physics.
The relationships that come out of the Hamiltonian and Lagrangian formulations can be deceptively
simple, because they completely take for granted interaction semantics from the realm of classical bal-
listic thinking. Hence, when the semantics change, as they do in Quantum Mechanics, this leads to
understandable confusion.
We should remember that, in Promise Theory, currencies are like energy assessments, about the
state of a promise kept over time. It might be a self-assessment or the assessment of a neighbouring
relationship. It’s hard to get out of the habit of thinking about instantaneous responses and ballistic
physics. Imagine instead the regular relationship between your home and the garbage collection service.
If the currency is the amount of goodwill, then your assessment of the service changes according to
the rate at which the service keeps its promise, and the amount of goodwill you pass on in by feeding
the service. This ‘energy level’ is not an instantaneous judgement, it involves observing and learning
behaviour over time to summarize in a single potential.
14This procedure is what allows us to take the classical limit and recover the poisson bracket from the the quantum commu-
tator.
33

4.9 A variational formulation for agents?
How might we formulate the same kind of energy argument for agents, given that currency is an as-
sessment of individual agents. In the Lagrangian and Hamiltonian formulations of Newtonian and
Schr¨
odinger physics, the assumption that there is a single godlike observer with instantaneous access
to the state of the system at every point. All states at all locations advance in lockstep, by a rigid invisi-
ble hand. This third party observer forms assessments of every agent’s promise to externalize state and
constructs .
Variations are in the perception of the receiver.
∆xi7→ αk(∆x(Ai)) (108)
δ∆xi7→ δαk(∆x(Ai)).(109)
So that Ak’s understanding of the system is represented by the action principle:
S=X
i1
2m(∂tαk(∆x(Ai)))2−1
2k(αk(∆x(Ai))2−1
2kij (αk(∆x(A2)−αk(∆x(A1))2.(110)
The time derivative in this expression measures the single calibrated clock of the observer Ak, indepen-
dently of however time passes for the agents A1, A2.
4.10 Reinterpreting the action principle in terms of locality
The action principle feels rooted in ideas about the continuum and the exterior embeddings of Newtonian
physics, and yet it also generates the exterior equations of Quantum Mechanics, leaving the interior
aspects of measurement something of a controversial appendage. It’s natural to ask what the action
principle could mean for agent based systems and virtual processes. Is there an interpretation of the
action principle that applies to agent based systems?
If we recall that kinetic energy Tcorresponds to exterior exchange currency, while potential Vis
about interior accumulation of currency, then maximixing T−Vhas the semantics of maximizing
autonomy, or a principle of least dependency. The separation of interior and exterior processes and
compositionality of agents at different scales seems to preserve this.
Certainly this preference for an autonomous state is appealing on a number of levels, but it’s also
clearly not a panacea. Just as the action principle doesn’t always minimize energy, nor does Fermat’s
Principle of least time in optics necessarily minimize the time—we also have to take into account con-
straints on this from the pre-existing interactions with an ambient environment. We still haven’t dis-
cussed why agents might create or deprecate promises themeselves. How do particles in physics get their
charges? Why do they start responding to one kind of field or another? How and why do interactions get
switched on and off? This requires as much explanation as least action, but it aligns well with the most
basic axiom of Promise Theory, which is agent independence.
Part of the answers come from the semantics we derive at each level. Physics suppresses semantics to
appear impartial—a side effect of it’s manifesto for universality. When we reach biochemical and socio-
economic scales, agents are too rich in semantics to be neatly boxed as commodity particles. When we
minimize T−Vwe are trying to reduce the amount of kinetic activity (relative exchanges, risk, activity,
but also accusation and other impositions) and simulatenously increase the stored potential (reservoir
of savings, trusted potential, etc) from all the promises between agents. If we include semantics, then
part of the stored potential of high level agents includes encoded information, algorithmsm, personality
profiles, etc.
One of the weaknesses of the action principle and Hamilton’s exterior formulation lies in the use of
momenta as a key part of the construction for dynamics. This doesn’t naturally translate to systems such
as (anti)ferromagnetic spin systems, where alignment of interior quantum numbers only has a shadow
representation as exterior vectors. This has always felt like a confusion: spin is angular momentum, but
not translational—not exterior. Then where is the relevant boundary between interior and exterior? On
34

the other hand, spin waves and phonons are precisely the kind of virtual processes that we are studying
here. Agent models and virtual processes seem to have an interesting role of play here in exposing
more reasonable semantics for phenomena that have remained essentially mysterious by convention for
a century.
Remark 6 (Equilibrium rather than minimum) The variational principle seeks out a generic equilib-
rium rather than a minimum. An equilibrium outcome would make a steady state, but even a changing
state can minimize in stepwise (adiabatic) legs. There’s a difference between minimizing one’s depen-
dence on an external agent and doing nothing at all. Agents are constrained by certain promises built
into the description (Lagrangian or promise matrix). Minimizing dependence could mean trying to move
in step so as to reduce the tension between agents. Thus a ‘minimization’ of dependence could still come
about by redefining agents into entangled groups—molecules from atoms, etc.
4.11 Conservation requirements
In Euclidean and Minkowski spacetime, the conservation of quantities is explained by Noether’s theorem
as a necessary property of the homogeneity and continuity of spacetime. Under variations in space and
time, and functions that depend on these, conservation follows from
δS =ZdV δL= 0.(111)
In order for the variation to be nought, a change in one place must be compensation by a change in
another. If we vary with respect to ξ,
δS =ZdV Rδξ = 0 (112)
then implies conservation of the complementary quantity Racross any change contour for ξ. In Promise
Theory, a similar condition would be
X
ij
δξCk
ij = 0.(113)
Notice that this is continuity of assessment, which is the internalized process of agents, rather than the
externalized view of forces. Thus continuity doesn’t require us to sacrifice the principle of autonomy.
Conservation laws, or process continuity, are prominent features of processes that we rely of for
predicting outcome. The binding of ±promises in Promise Theory gives a way to account for such laws
without assuming material constraints. One can form as many agents as one likes without violating a
conservation principle as long as it relies on binding. Any agents created without a binding partner are
unobservable and may therefore be discounted. Thus conservation laws don’t rely on the counting of
agents, but rather of information channels.
4.12 Ballistic or impulsive change
Let’s return briefly to the question of Newton’s ballistic world. In a ballistic view, large scale processes
and forces rule the system from the exterior. We describe motion in terms of ‘rigid bodies’ of coherent
material, which behave as singular entities (by centre of mass) that move of their own right. The motion
of such a body isn’t usually described as a virtual phenomenon—indeed, during Einstein’s time, physi-
cists even denied the possibility of an absolute motion, associated with the idea of an aether, because
Einstein showed that one would not be able to measure such motion. But we should be careful here. The
fact that we are unable to measure something doesn’t imply its non-existence. It seems perfectly possible
that all phenomena we consider to by fundamental today are in fact virtual phenomena on some deeper
set of agents and nothing changes.
35

Waves are the most natural kind of elementary process that involves exterior motion. Ballistic in-
teractions are different from waves. A ballistic impulse transfer is a singular transaction, like a single
sample. In Promise Theory, ballistic interactions have the semantics of impositions (see section 3.6),
which translates into the extent to which timescales can be separated between sender and receiver pro-
cesses. This goes back to the Nyquist sampling law, where events that are undersampled can appear
as random fluctuations, out of the blue. The inexplicable nature of impostions makes them more like
boundary conditions than continuous evolution. The same is true for impulses in Newtonian mechanics.
To go deeper into this issue, we need to study a range of examples at different scales—in biology and
in socio-economic systems. That’s for another occasion.
4.13 Loops, sampling processes, and fixed ‘currency price’ and energy levels
In the agent description, agents are characterized by a boundary (whether physical or virtual). There are
interior processes and exterior processes with respect to this boundary. Since activity originates locally
and independently in all agents, it has to be interior processes that are the source of activity. Similarly,
interior processes must be responsible for interactions between agents. The most basic sampling loop is
the process that can detect changes in other agents. This cyclic process leads to a natural connection with
Fourier representations, and the Nyquist-Shannon theorem of sampling implies a fundamental limit on
observation between agents, which is the uncertainty principle associated with the average wavenumber
and position spectra.
Let’s suppose that an agent samples a promised process at a rate of ν=ω/2πsamples per second,
and that any proper change of configuration requires the reception of a whole number nof symbols.
The units of ‘action’ are energy ×time. The energy associated with a symbol is a reflection of the time
it takes to receive it. If symbols can be absorbed at a constant rate proportional to a sampling loop of
circular frequency ω, then we can read ω/2symbols per second (by the Shannon-Nyquist law). Now, if
the energy cost is proportional to the time taken to read a symbol then we can introduce a constant with
dimensions of action, and a fixed energy currency cost per unit of symbol encoding ∆.
H= 2∆×∆tsymbol.(114)
radian-energy seconds per unit length of symbol.
∆(ψ7→ ψ0) = nωH. (115)
Thus the rate symbolic information transferred (as opposed to entropy which is average information per
unit symbol), by a promise binding, results from a whole number of cycles. Thus energy is quantized
due to the cyclic nature of the interior time sampling process.
This relation effectively expresses the idea that a change of state requires a bounded cyclic process
to sample i.
Example 11 (Atomic transitions) Electron oribitals form cyclic processes, which have spectra that can
absorb or emit single photons, as long as the frequency of the transition matches
∆=hν (116)
with a high degree of accuracy in emission. The orbitals are a whole number of wavelengths? An atomic
orbital can accept a photon. Only agents with such interior processes will be able to absorb information
in this way. In usual Fourier variables, we use ω= 2πν, so we can define
∝∆ρ=n~ω, n = 1,2,3. . . , (117)
where H7→ ~. Only complete strings will be stable under absorption, since these correspond to stable
eigenstates.
36

Nomenclature is important here. In physics, information is associated with the number of degrees of
freedom in a system, or the entropy—not with a change of state (which is how one normally thinks of
information in computing). A data signal alters the existing states of that compose information. The
Shannon entropy is the information density or expected symbol frequency per unit length of message:
S=hI()i/L =−X

plog p,(118)
where piis the probability per unit path length L, of randomly finding an agent in state ψi={, mi, . . .},
i.e. expressing the symbol from the alphabet spanned by ψ. The alphabet is constrained as the outcome
of an equilibrium process, compatible with the boundary conditions, e.g. the usual eigenstates of the spa-
tial boundary dynamics in the Schr¨
odinger equation. If we constrain this with additional normalization:
X

p= 1 (119)
X

p=hi,(120)
then phas the Boltzmann form p= exp(−β). The Shannon entropy is is related to the intrepretion of
the von Neumann entropy
S=Tr ρlog ρ. (121)
The dimensions of entropy are related to energy and work by thermodynamic relations dU ∼dW ∼
P dV ∼T dS .
5 Promise lifecycle and boundary dynamics
Before ending, we need to (at least begin to) address the final elephant. There is more to promise
dynamics than the flow of currencies like energy and trust. There is configuration.
At the start we alluded to the idea that the separation of promises from agents leads to a separation of
underlying infrastructure and virtual processes on top. All the counting of the dynamics that we would
expect to see belongs to the virtual layer, but it relies on the precise matching of promises underneath.
How does the mutuality of promise bindings emerge?
How does the number of agents change in time, and how do the numbers and types of promises
between them change? What makes promises correlated into (+) and (-) that match? So far we’ve only
considered flows of activity concerned with the keeping of the promises—this is what corresponds to
dynamics in physical science. Fixed promises on existing agents sustain the activity with effectively
fixed boundary conditions. The missing piece in this picture is how such large scale changes in structure
comes about, with a degree of homogeneity that allows promises to be compatible, in the first place. This
is analogous to a scale on which the boundary conditions of classical equations of motion also change.
It leads us away from linearity.
As we rise up the scale hierarchy to increasingly complex or sophisticated agent scales, the roles of
semantics and boundary conditions become increasingly intertwined. In Promise Theory, the topology of
the promises, i.e. the boundary conditions can lead to the formation of promises on new scales through
combinatoric dependencies. This is how electronic circuits and biological cells operate. The missing
pieces in our story are: how are connections made, how new bodies are introduced, and how do the
boundary conditions change?
This is slightly paradox in these questions in a model of autonomous agents, since the formation of
components with correlated roles can’t be a purely autonomous activity. To yield a cooperative system,
change has to be mutual. Locks and keys have to be made together in order to fit. There are basically
two extreme answers for how agents get to know new agents and form promises with them:
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•It’s random–all agents can make make promises with others equally and one hopes for Darwinistic
selection. This approach seems to work in immunology for antibody epitope generation [30], but
it isn’t the only alternative.
•The types of promise could be preordained, with the topology given by some underlying process.
This leads to the turtle or the god problem of endless dependence.
A Promise Theory approach predicts two possible parts to the process. First, one needs to generate
generic agents with basic capabilities to represent states and form channels. These are process entities
like cells or computers, on some scale. Once we have a number of these agents, like stem cells, their
properties can be discriminated by fission and fusion.
Agent numbers can be altered by combinatoric processes—either spontaneously or deterministically
by fission and fusion. If agents appear spontaneously, then somehow they have to do so pre-entangled.
A simpler answer is that they arise from the fission of a single agent. Then the two agents split hand and
glove, lock and key, as do complex molecules in biochemistry (see figure 5).
b+ b−
AA + −
A
b+ | ξ
b−| ξ
Figure 5: Complementary pairs of promises are naturally calibrated when they emerge from the dissociation of a
single agent. In a hierarchy of such dissociation types, correlations between ±agent promises might be quite far
apart.
Example 12 (Molecular agents) At the level of chemistry, interaction channels are expressed through
electronic donor and receptor states in molecules embedded in three dimensional space. In biology,
larger molecular shapes form donor and receptor sites for biochemical interaction. On larger scales,
organisms can exchange smaller organisms or organelles, or respond to gradients of diffusing molecules
like nutrients or hormone signals. At each scale there are thus representations of information using
whatever proxies are available to carry it.
In promise notation, a stem cell making no particular promise, but having a common process origin ξ,
expresses:
A∅|ξ
−−→ ∗.(122)
After fission, this becomes
A+
+b1ξ
−−−→ A−,∗(123)
A−
−b1ξ
−−−→ A+,∗,(124)
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such that the sum is maintained. If the common origin dependency ξgoes away, or if one agent can
replace it by also promising ξ, perhaps by absorbing some other agent, then the entanglement of the two
agents also dissolves. This is how agents can become ‘standalone’ or entangled. These mechanisms are
known in industrial and biological terms, but they must also apply on any scale. ξcan include conditions
such as the sufficiency of interior resources, e.g. if kinetic energy or trust are below a certain threshold,
relative to an interior level potential certain behaviours may be suppressed. This is the case of tunnelling
in Quantum Mechanics. In our Promise Theory model, we find similarly that the concept of classically
forbidden processes T < V are not forbidden but are only less effective.
In Promise Theory, we have the general problem of understanding why some agents would have +b
and some would have −b, and how these come and go under scaling. The solution of the end to end
delivery problem in [8] shows how semantic scaling works by composition. The fusion of several agents
into a single agent with promises combined will only satisfy a strict algebra if there is some prequisitie
calibration of the agents by a common standard, i.e. what we label ξ. For independent agents to manage
this, they need to be able to recognize one another’s promises. Simply pouring transistors and other
components into a bag does not make a computer15. Logically, the prerequisites for promises to form at
an agent are:
•The capability to align with a certain intention (to form an arrow of the intended kind).
•The ability to distinguish promisees (to point to a specific recipient).
•Sufficient currency in the binding.
Scaling plays a role in computational distinguishability too [4–6]. No agent has enough memory to
recognize and remember the identities of every other agent in a system, so characteristic scalar promises
will tend to lead to there being agents that are indistinguishable, i.e. which promise only labels that fall
into generic classes. The usual algebras of group theory can represent the combinatorics of promises in
fission and fusion processes, but they cannot explain the processes or mechanisms responsible for their
implementation.
6 Summary and discussion
In these lecture notes, we’ve been looking at just a single aspect of agent systems, through the lens of
Promise Theory or the Semantic Spacetime model, namely processes and their representations of virtual
motion. Using one simple well-known example of coupled oscillators, we’ve expressed a physical system
in the alternative framework of Promise Theory to expose the layers of hidden assumptions that easily
bind us to a narrow and incorrect view of spacetime phenomena. In a Promise Theory view, what
appeared to be impartial dynamical quantities become individual local assessents of outcome, and forces
become promises expressing individual alignments over different channels of interaction. Energy and
momentum become surrogates for information and similarity of interior state.
Motion Of The Third Kind is something we experience all around us, from waves to biology, in
music, and even the changes we observe in a society. By formulating these ideas generally, we find a
hierarchy of processes to study in a common language of information. While learning a new language
won’t change the virtues of knowing the old, such a lingua franca helps us to see beyond a narrow field
of interest. The need for this expansive thinking has only grown in the Information Age, yet the most
important processes in our world remain poorly understood.
The exploration of more general phenomena requires us to combine both dynamics and semantics in
order to see patterns—if not to actually make quantitative predictions. The latter is likely too ambitous
in the short term, but even the characterization of phenomena based on flows of influence would be a
step forward in socio-economic sciences, which rely far too much on moral conjecture. Only a handful
15In physics, the particle-antiparticle process e+e−→γmay be viewed as a fusion of two opposite promise agents into a
single neutral agent.
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of authors have tried to formalize a physics of social scales [31]. Von Neumann was probably aware
of a deep connection between these areas in describing cellular automata and formal games, though he
made progress mainly in quantitative terms [28, 29, 32]. As a contemporary generation of researchers
seeks to create quantum computers, based on the analogy between virtual and physical processes, it has
never been more important to forge an understanding that unifies classical, quantum, and computational
behaviours [33]. Promise Theory can surely continue to play a role in this synthesis.
Where could this go next? Specialization could be the next step, in order to expose a number of
examples from the general approach. We need to pick a scale to go deeper. In modern physics, there
are four known promise channels for interactions: the electromagnetic, strong and weak nuclear, and the
gravitational channel that express influence. Then there is the Higgs field that contributes a uniform mass
‘encumbrance’ to the different types of agent process, at least for the Standard Model. Gravitation, on the
other hand, has been connected with the geometry of spacetime itself, which may actually underpin the
other three as a basic communication channel between points. As such it can form guiderails of its own
and influence generalized momenta (promises). In cloud computing, the natural currencies are identity
tokens and process counters, network protocols and process control blocks. In socioeconomic systems,
the currencies are things like trust, property, and money [34].
The interactions and potentials of general phenomena differ from the elementary configurations in
the world of physics, but we should be able to model them too. Proxy carriers feature prominently called
messenger particles explicitly keep the promises between agents, like coins in an economy. The idea of
an embedding space is convenient and comfortable, but is ultimately unhelpful in describing processes
on a detailed level. The beauty of continuum representations lies in the large scale world that Newton
and his forbears knew. The beauty of agent models is in directly understanding individual concerns.
Macroeconomics is an affront to the concerns of an individual’s microeconomics. There is nothing here
to suggest that only one kind of equations representing scenarios could be the last word in explaining all
behaviours.
It seems a reasonable view to treat all behaviours as virtual (as motion of the third kind) in some
space of agents if that is helpful too. In this age of computers and rich information, it is indeed very
useful, as this is a new realm of dynamical behaviour that we’ve only just begun to try to understand.
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