CRR

Coherence is the temporal integral of all past states. It represents the accumulated history of a system, a non-Markovian memory that weights the present with everything that came before.

Here, $L(x,\tau)$ is the memory density at position $x$ and time $\tau$. The integral runs over all past time, meaning coherence carries the full temporal signature of the system. This is fundamentally different from Markovian dynamics, where only the immediately preceding state matters.

Omega ($\Omega$) is the system temperature parameter, a normalisation constant that sets the scale of the Markov blanket. It determines how much coherence a system can hold before reaching saturation.

When $C = \Omega$, the system reaches Markov blanket saturation. At this critical point, the exponential term $e^{C/\Omega}$ in the Regeneration operator equals Euler's number $e \approx 2.718$:

This is the inflection point where the accumulated past has filled the system's capacity to integrate more history. The Markov blanket, the statistical boundary separating system from environment, becomes saturated. The system can no longer absorb new information without structural change.

Omega controls the porosity of the Markov blanket, determining how easily information flows between system and environment:

Higher Omega means a more porous, liquid system. The blanket is permeable. The system can integrate more history before reaching saturation, and the exponential weighting in Regeneration remains moderate even at high coherence. Such systems are adaptable, responsive, perhaps unstable.

Lower Omega means a more rigid, crystalline system. The blanket is impermeable. Even small amounts of accumulated coherence push the exponential term toward large values, meaning recent priors dominate heavily. Such systems are stable, perhaps brittle, locked into their recent histories.

In the simulation below, adjusting Omega lets you see this directly: low Omega systems rupture quickly and regenerate with heavy weighting on recent states. High Omega systems can accumulate vast coherence before transition.

Rupture is the phase transition, the discrete moment when coherence saturates and the system reorganises. Formally, it is represented by a Dirac delta function at the rupture time:

The Dirac delta is infinitely concentrated at a single instant: infinite intensity, zero duration, unit integral. This captures the scale-invariant nature of rupture. Whether the system is a synapse or a civilisation, the rupture event is singular, a point of discontinuity in the otherwise continuous accumulation of coherence.

Rupture can be endogenous (arising from internal dynamics reaching threshold) or exogenous (imposed by external perturbation). At any spatial or temporal scale, rupture marks the moment when the past is metabolised, when accumulated history is processed and the system prepares for reconstitution.

Regeneration is the reconstruction of future states from the historical field, weighted by the coherence accumulated before rupture:

The exponential term $e^{C/\Omega}$ is the key: it weights the reconstruction by how much coherence was present. Systems that rupture at high coherence regenerate with stronger weighting on the historical field. Systems that rupture at low coherence are freer to explore new configurations.

The Heaviside function $\Theta(t-\tau)$ ensures causality: only past states contribute to regeneration. The reconstruction field $\phi$ represents the available material for rebuilding, whether that's synaptic weights, social ties, or metabolic pathways.

The CRR is deliberately coarse grained. It describes patterns that emerge at many scales without specifying microscopic mechanisms. This is intentional: the framework seeks universality rather than precision.

This is a grammar in the linguistic sense: a formal structure that generates valid descriptions without specifying content. Just as grammar tells you how to combine words without telling you what to say, the CRR tells you how temporal dynamics unfold without specifying what the system is. A neuron, a forest, a financial market, a human life: all can be parsed through this grammar.

The coarseness is the point. By abstracting away particulars, the grammar becomes applicable across scales and domains. The price of this generality is that specific predictions require additional domain knowledge. The CRR provides the temporal skeleton; flesh must be added by understanding the particular system in question.

Euler's number $e$ emerges naturally in processes involving continuous growth and decay. It is the base of the natural logarithm, the eigenvalue of differentiation, the limit of compound interest as compounding becomes continuous.

In the CRR, $e$ appears in the Regeneration operator as the weighting factor when $C = \Omega$. This is not arbitrary. The exponential function $e^x$ is the unique function equal to its own derivative: it describes processes where the rate of change is proportional to the current value. This is precisely the behaviour of coherence accumulation and its influence on regeneration.

When $C/\Omega = 1$, the weighting factor $e^1 = e$ represents a precise balance point. Below this, the historical field is underweighted. Above this, it becomes increasingly dominant. The appearance of $e$ at the saturation threshold is not coincidental; it reflects the deep mathematical structure of continuous integration and exponential dynamics.