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Made of Maths

Coherence, Rupture, Regeneration

A coarse-grain mathematical framework for exploring transformation, and renewal in complex systems

  • How do complex systems maintain coherence through change in the passage of time?
  • How can mathematical systems help us to understand the nature of intelligence and awareness in a shared context?
  • How might we better understand and adapt to the psychological, sociological and ecological impacts of Technology on human and non-human species?

What is CRR?

CRR (Coherence, Rupture, Regeneration) is a three-part mathematical formalism aiming to describe how systems maintain themselves, undergo transformations, and emerge in new configurations across time. This approach captures the dynamic interplay between stability, disruption, and reconstruction that appears in many natural and artificial systems.

Coherence

The mathematical representation of how systems maintain organised patterns over time. Coherence captures the accumulated memory and structural relationships that maintain a system's identity as it changes state through time (Rupture).

$$C(x,t) = \int_0^t L(x,\tau) \, d\tau$$
Where L(x,tau) represents information density accumulated at position x over time tau

Rupture

Represents critical thresholds where existing patterns undergo scale-invariant transitions. These events create opportunities for system reorganisation and the emergence of novel structural arrangements (Regeneration).

$$\delta(t-t_0)$$
A Dirac delta function encoding sudden disruptions at time t-zero

Regeneration

The reconstruction process that builds new stable patterns whilst drawing upon the historical information, C(x,t), which is never lost, only transformed. This phase enables systems to maintain continuity whilst expressing novel configurations.

$$R[\chi](x,t) = \int_{-\infty}^t \phi(x,\tau) \cdot e^{C(x,\tau)/\Omega} \cdot \Theta(t-\tau) \, d\tau$$
Where phi(x,tau) is the field function and Omega is the coupling constant

This mathematical structure provides a way to study systems that exhibit memory-dependent behaviour; where past configurations influence present dynamics in ways that Markovian models might miss. The toy simulations on this site realise this three-part formalism in code, a playful way to explore the deeper mathematical, philosophical and phenomenological concept of how identity persists through change.

The Bee Journey: CRR in Action

Watch the same bees travel through all three phases: collecting nectar from flowers, surviving predator attacks, and creating honey in the hive. Each bee maintains memory of its journey, demonstrating non-Markovian dynamics.

Coherence

Bees pollinate flowers, building memory traces and collecting nectar through organised foraging behaviour

$$C(x,t) = \int_0^t L(x,\tau) \, d\tau$$

Rupture

Bees encounter a predator whilst travelling to the hive, causing sudden dispersal and flight behaviour

$$\delta(t-t_0)$$

Regeneration

Surviving bees reach the hive and create honey, using accumulated memory from their journey

$$R[\chi](x,t) = \int_{-\infty}^t \phi(x,\tau) \cdot e^{C(x,\tau)/\Omega} \cdot \Theta(t-\tau) \, d\tau$$

Why CRR?

Coherence

Systems, from atoms to technological networks to whole ecosystems, exhibit organised patterns that persist through time. Understanding how coherence resists entropy over time offers potential insight for maintaining and creating more stable and resilient systems in various domains.

Rupture

Complex systems frequently undergo periods of reorganisation at various spatial and temporal scales. Rather than viewing these transitions as purely disruptive, the CRR framework examines how such events create space for system adaptation and novel emergent properties. The 'scale-invariance' of the Dirac Delta makes it a good candidate for modelling the ever-present moment of "now" as a temporal 'rupture' where agentic choice occurs, as well as for modelling any discrete or prolonged shock to a system, causing change and subsequent Regeneration.

Regeneration

Natural systems demonstrate remarkable abilities to recover and adapt following disruptions. The Regeneration component of CRR explores how systems use accumulated information to guide reconstruction processes; informing approaches to resilience in engineered systems.

NB.

The CRR formalism was approximated and simplified to model every working simulation showcased on this website. These simulations are mathematically driven, and do not use any external textures, hard-coded animations, or dependencies. It should also be noted, that once an algorithm has been isolated using the CRR derivations, it can be implemented without the CRR pipeline. For example, pearlescence values were derived from Reaction-Diffusion (with memory) experiments, using the CRR, but enabling pearlescence in a simulation does not require the full CRR pipeline. This, perhaps, says something about how algorithms in separate systems can be locally implemented, but are all compatible, or 'coherent', between each other

Research Interests

This work may be of interest to researchers exploring mathematical approaches to complex systems. Potential areas of collaboration include:

Computational Biology

Mathematical models of development, adaptation, and ecosystem dynamics

Dynamical Systems

Non-Markovian processes and memory-dependent dynamics

Machine Learning

Continual learning and memory-augmented neural networks

Pattern Formation

Reaction-diffusion systems with memory effects

Network Science

Adaptive networks and resilience mechanisms

Cognitive Science

Mathematical approaches to Psychology and Development

Systems Biology

Mathematical frameworks for biological complexity

Applied Mathematics

Novel mathematical structures and computational methods

Philosophy

Philosophical conjecture and ontological lens

Interested in collaborative research? Contact: alexander@cohere.org.uk