Mathematical Foundations for Non-Markovian Dynamics with Threshold-Triggered Discontinuities
A. Sabine · Version 5.2 · January 2026
Imagine how your experience of the present moment works. You don't experience time as a smooth flow—you experience distinct "nows," each one feeling complete before giving way to the next. William James called these "perchings" and "flights" of consciousness.
CRR proposes that this pattern—build up, break, rebuild—is a universal signature of how complex systems navigate time.
Think of coherence as "accumulating constraint." As you read this sentence, your brain is integrating information—words build on words, meaning accumulates. Mathematically, coherence is like filling a bucket until it reaches capacity.
When the bucket is full, it tips over. This is rupture—the discrete moment when accumulated coherence reaches threshold and the system must reorganise. The key insight: Rupture is when the "inside" finally matches the "outside." At this moment, you can write to memory—this is the write window.
After rupture, the system rebuilds using its history—but not all history equally. Moments of high coherence contribute more; moments of low coherence fade away. This is why significant experiences shape you more than forgettable ones. High-coherence moments matter most, regardless of when they occurred.
The framework has one central parameter: Ω (omega). Think of it as a dial between "rigid" and "flexible":
If CRR is correct, it provides: a unified language for dynamics across scales, testable predictions about timing and variability, a bridge between physics and psychology, and a formal account of what it feels like to be in time.
For each $x$ and $t$, the function $\tau \mapsto \mathcal{L}(x,\tau)$ is locally integrable on $[0,t]$.
$\mathcal{L}(x,\tau) \geq 0$ for all $x, \tau$. History accumulates; it does not spontaneously dissipate.
There exists $M > 0$ such that $\mathcal{L}(x,t) \leq M$ for all $x, t$. This ensures each coherence cycle has duration at least $\Omega/M$, preventing infinitely many ruptures in finite time.
The mnemonic entanglement rate is a function $\mathcal{L} : \mathcal{X} \times [0,T] \to \mathbb{R}_{\geq 0}$ that assigns to each state-time pair $(x,\tau)$ a rate of coherence accumulation.
Dimensions: $[\mathcal{L}] = [T^{-1}]$ (a rate, so that $\int \mathcal{L}\, d\tau$ is dimensionless).
Let $t_j$ denote the most recent rupture time before $t$ (with $t_0 = 0$). The coherence at state $x$ and time $t$ is the accumulated constraint since the last rupture:
Properties: Dimensionless, monotone non-decreasing within each cycle, resets to 0 at each rupture.
The threshold parameter $\Omega > 0$ is a positive dimensionless constant characterising the system's capacity for coherence accumulation before structural change becomes necessary.
A rupture occurs at time $t_*$ when coherence reaches threshold:
The rupture event is represented by a Dirac delta $\delta(t - t_*)$. Following rupture: $C(x, t_*^+) = 0$.
The regeneration operator reconstructs the system state by weighting historical field values $\Phi(x,\tau)$ by the coherence at each historical moment:
where $C(x,\tau)$ is the coherence value at moment $\tau$—how far into its cycle the system was at that historical moment.
The regeneration integral uses $C(x,\tau)$—the coherence value at each historical moment—not a cumulative sum. Since $C$ cycles between $0$ and $\Omega$, moments just before rupture (when $C \approx \Omega$) have weight $\approx e$, regardless of when they occurred.
This means a high-coherence moment from 1000 cycles ago contributes equally to regeneration as one from the last cycle. History is weighted by significance (coherence), not by recency. This is consonant with Bergson's insight that memory is "the continuous presence of history," not retrieval from storage.
Under (A3), each coherence cycle has duration at least $\Omega/M$. Hence the number of ruptures in any finite interval $[0,T]$ is at most $\lfloor TM/\Omega \rfloor + 1 < \infty$.
At each rupture, $C(x,t_*) = \Omega$, so the rupture moment contributes with weight:
This is the maximum weight any moment can have, and it occurs at every rupture regardless of when it happened.
| Quantity | Symbol | Dimensions |
|---|---|---|
| Mnemonic rate | $\mathcal{L}$ | $[T^{-1}]$ |
| Coherence | $C$ | dimensionless |
| Threshold | $\Omega$ | dimensionless |
| Historical field | $\Phi$ | [F] |
| Regeneration | $\mathcal{R}$ | [F] |
When a system regenerates after rupture, it doesn't treat all of history equally. Important moments (high coherence) contribute more; forgettable moments (low coherence) fade away.
Crucially, this weighting is based on how coherent each moment was in its own context—not on how recent it was. A significant experience from years ago can shape you as much as one from yesterday, if both reached high coherence.
Define the normalised weight function:
Here $C(x,\tau)$ is the coherence at moment $\tau$—how far through its cycle the system was at that time.
$\Omega$ plays the role of "temperature" in the Boltzmann weighting:
But unlike a recency-biased model, all high-coherence moments contribute equally, whether ancient or recent.
At rupture, coherence equals threshold: $C(x, t_*) = \Omega$. Therefore, every rupture moment contributes with weight proportional to:
$$e^{C(x,t_*)/\Omega} = e^{\Omega/\Omega} = e \approx 2.718$$This is the fundamental constant of CRR—the weight ratio between a rupture moment and a moment of zero coherence is always $e$.
Two systems with identical current coherence but different histories will in general have different regenerations.
History matters, not just its summary. Two people at the same point in life but with different histories will respond differently to the same challenge. Significant experiences persist in their influence regardless of how long ago they occurred—muscle memory doesn't fade just because it's old.
| Property | CRR (Coherence-Weighted) | Recency-Weighted |
|---|---|---|
| What determines weight? | Coherence at each moment: $e^{C(\tau)/\Omega}$ | Time since event: $e^{-\lambda(t-\tau)}$ |
| Ancient high-coherence moments | Fully preserved (weight $\approx e$) | Exponentially forgotten |
| Recent low-coherence moments | Low weight (near 1) | High weight (recent) |
| Philosophical alignment | Bergson: memory as continuous presence | Standard decay models |
| Empirical match | Muscle memory, trauma, skill retention | Short-term forgetting curves |
Your brain constantly balances two opposing forces: excitation (E) and inhibition (I). Tucker, Luu, and Friston (2025) show that consciousness emerges when these forces are perfectly balanced—at criticality.
CRR proposes that E and I collapse into a single parameter Ω. When C = Ω, the inside matches the outside—this is the write window, where early LTP can occur and experiences become memories.
Oscillatory: Theta and gamma/beta meet → standing wave resonance.
Plasticity: Resonance plateau lasts ~100-200ms—optimal for LTP. This is the write window.
CRR: Coherence C has reached threshold Ω. The rupture δ(now) fires. The pattern is inscribed.
Phenomenology: James's "perchings"—the felt now, coherently given, then reorganised.
| System | Dorsal (Papez) | Ventral (Yakovlev) |
|---|---|---|
| Control Mode | Excitatory feedforward | Inhibitory feedback |
| Sleep Consolidation | REM | NREM |
| CRR Mapping | High Ω | Low Ω |
At rupture: E and I are balanced, a standing wave resonance forms (~100-200ms), LTP can occur, the pattern is inscribed. This is the neural mechanism for James's "perchings."
The FEP proposes that living systems survive by minimising "free energy"—the mismatch between what they expect and experience. CRR provides the temporal dynamics that FEP presupposes: when do beliefs update? How does history shape reconstitution?
| Question | FEP Answer | CRR Extension |
|---|---|---|
| What do beliefs update to? | Free energy minimum | (Presupposed) |
| When does updating occur? | (Not explicit) | When $C(x,t) \geq \Omega$ |
| How does history shape reconstitution? | (Not explicit) | Via $\mathcal{R} = \mathbb{E}_w[\Phi]$ |
| What triggers discrete insight? | Bayesian Model Reduction | Rupture $\delta(t - t_*)$ |
In Friston et al. (2025) "Active Inference and Artificial Reasoning," an "aha moment" occurs when evidence accumulates until confidence exceeds a threshold, triggering Bayesian Model Reduction. CRR's Rupture is the same phenomenon.
CRR Rupture and Bayesian Model Reduction share: (1) Accumulation of evidence/coherence, (2) Threshold criterion, (3) Discrete transition, (4) Reconstitution based on history.
| CRR Concept | FEP Concept | Mapping |
|---|---|---|
| Coherence $C(x,t)$ | Evidence since last update | $C \leftrightarrow \log p(D_{\text{new}}|m)$ |
| Threshold $\Omega$ | Model selection threshold | $\Omega \leftrightarrow \theta$ |
| Rupture $\delta(t-t_*)$ | BMR / Occam's Razor | $\{C \geq \Omega\} \leftrightarrow \{\max_m p(m|D) > \theta\}$ |
The threshold parameter corresponds to inverse precision:
This creates a three-way bridge: geometric ($1/\varphi$), statistical ($\sigma^2$), and inferential ($1/\pi$).
Rupture IS the "aha moment." Both frameworks describe the discrete transition from uncertainty to commitment when accumulated evidence warrants model selection.
Friston, K., et al. (2025). Active inference and artificial reasoning. arXiv:2512.21129.
Tucker, D. M., Luu, P., & Friston, K. J. (2025). The Criticality of Consciousness. Entropy, 27(8), 829.
Parr, T., Pezzulo, G., & Friston, K. J. (2022). Active Inference. MIT Press.
For systems whose coherence-rupture cycle has underlying phase structure:
where $\varphi$ is the phase (in radians) traversed during one coherence accumulation cycle.
| Symmetry | Phase to Rupture | $\Omega$ Value | CV Prediction |
|---|---|---|---|
| $\mathbb{Z}_2$ (flip) | $\pi$ (half-cycle) | $1/\pi \approx 0.318$ | $\approx 0.159$ |
| $SO(2)$ (rotation) | $2\pi$ (full-cycle) | $1/2\pi \approx 0.159$ | $\approx 0.080$ |
Saltatory growth (Lampl & Johnson, 1998): Prediction $CV \approx 0.159$, Observed $CV \approx 0.15$–$0.16$. Status: Confirmed (11/11 individual predictions).
| Domain | Test Type | Status |
|---|---|---|
| Saltatory growth | Prospective | Confirmed |
| Wound healing | Post-hoc fit | Consistent ($R^2 = 0.9989$) |
| Muscle hypertrophy | Mixed | Consistent (10/10) |
The framework would be challenged if: $\mathbb{Z}_2$ systems showed $CV$ significantly different from $\sim 0.16$, $SO(2)$ systems showed $CV$ significantly different from $\sim 0.08$, or regeneration proved Markovian (no path dependence).
Current assessment: Preliminary evidence is encouraging but not conclusive. The framework generates specific, testable predictions that distinguish it from generic curve-fitting.