Understanding how we remember, break, and rebuild through life
A simplified introduction to the Coherence-Rupture-Regeneration framework and how it helps to explore how any system (including you) maintain identity through change
Have you ever noticed how life seems to follow a pattern? You build up knowledge and habits over time, then something disrupts everything, and you have to rebuild yourself somehow different than before. This isn't random; it's a fundamental pattern that appears everywhere from how we learn to how forests grow to how artificial intelligence works.
CRR stands for Coherence-Rupture-Regeneration. It's a way of understanding how any system (including you!) maintains its identity while changing over time. Think of it as the mathematics of becoming.
The Building Phase
Everything you've learned, experienced, and integrated. Your accumulated memory and understanding. Like building up knowledge for an exam or developing a skill through practice.
The Breaking Point
Those moments when your current way of being can't handle what's happening. A crisis, a revelation, a contradiction that forces change. Like failing an exam and realising you need a new study approach.
The Rebuilding Phase
How you reconstruct yourself using the wisdom of your past. You don't start from scratch; you rebuild while keeping what was valuable. Like developing a better study method based on what worked before.
The mathematics may appear deceptively simple, but these three equations capture something fundamental about how systems maintain identity through change:
1. Coherence Integration:
C(x,t) = ∫₀ᵗ L(x,τ) dτ
What this means: Coherence (C) is the integral; the accumulated sum; of all your memory density (L) over time. Think of L as the rate at which you're building understanding at each moment. The integral adds up every moment from the beginning (0) to now (t).
In everyday terms: You are the sum of all your experiences, weighted by how much coherence each one built. A transformative year contributes more than a year of routine.
2. Rupture Detection:
δ(t-t₀) = Dirac delta at rupture time
What this means: The delta function (δ) is a mathematical way of representing an instantaneous event. It's zero everywhere except at exactly time t₀, where it "fires." This captures the discontinuous, all-at-once nature of rupture moments.
In everyday terms: Some changes aren't gradual; they happen in a moment. The instant you understand a concept. The moment a relationship ends. These aren't smooth transitions; they're discontinuous jumps.
3. Regeneration Operator:
R[χ](x,t) = ∫₀ᵗ φ(x,τ)·e^(C(τ)/Ω)·Θ(t-τ) dτ
What this means: Regeneration rebuilds using your historical field signal (φ) weighted exponentially by how much coherence you had at each past moment (e^C/Ω). The Heaviside function (Θ) ensures causality; only the past influences the future, never the reverse.
In everyday terms: When you rebuild after a crisis, you don't use all past experiences equally. Moments when you had high coherence (deep understanding, strong connection) influence your regeneration exponentially more than confused or chaotic moments.
Key Parameters:
These three operations; integrate, rupture, regenerate; may look mathematically simple, but they enable non-Markovian temporal modeling. That's a fancy way of saying: systems whose future depends on their entire history, not just their current state.
This single formalism helps to explore:
The universality comes from the fact that any system navigating time, memory, and change can be described by these three operations.
Past (Coherence): All your attempts, falls, and little victories accumulate. Each try builds up your body's "memory" of balance. Mathematically: C = ∫L(τ)dτ where each practice session contributes to the integral.
Present (Rupture): That moment when you realise "I'm doing it!" or when you fall hard. These are discrete moments where everything crystallises or breaks apart. The δ-function captures this instantaneous quality; not gradual, but sudden.
Future (Regeneration): Once you can ride, that accumulated experience guides you. Years later, you can still ride because your body regenerates that pattern using its deeply stored memory, weighted by how coherent those original experiences were.
People, organisations, and even ecosystems develop characteristic patterns; we call these memory signatures. Here are the five main types:
Builds up forever without breaking, then collapses catastrophically
Mathematical form: L > 0 accumulates monotonically, rupture thresholds too high
Example: Someone who never processes trauma until they have a breakdown
Has small crises regularly, prevents catastrophe, stays flexible
Mathematical form: Moderate L, rupture at intermediate thresholds, efficient regeneration
Example: Someone who goes to therapy and processes emotions regularly
Rhythmic building and releasing, like breathing
Mathematical form: L alternates sign (±), rupture times periodic
Example: Creative people who cycle between intense work and necessary rest
Breaks apart so often that nothing builds up
Mathematical form: Rupture thresholds extremely low, frequent δ impulses
Example: Someone constantly starting over, never building depth
Holds contradictions, synthesises opposites
Mathematical form: Multiple coherence fields L₁, L₂ interfere constructively and destructively
Example: Integrating different cultural identities into something new
There's no "best" signature; each serves different purposes. Understanding your pattern might help you navigate it intentionally rather than being swept along by it.
Ready to go deeper? The other tabs explore how CRR helps us understand time, surprise, learning, and the patterns that shape your life. Each section builds on the formalism introduced here while maintaining accessibility.
Physics treats time like a ruler; each moment independent, flowing uniformly forward. However, that isn't necessarily how we actually experience time. In CRR, experience is shaped by your accumulated past, punctuated by sudden moments of clarity or crisis, and projected into a future that's weighted by your history.
In CRR, past experience is not separate snapshots filed away. Instead, everything you've lived through accumulates into a continuous field that shapes who you are now.
C[x](t) = ∫₀ᵗ L(x(τ), ẋ(τ), τ) dτ
What this captures:
Key insight: L(x,ẋ,τ) is the rate at which experience accumulates at moment τ. High activity (large ẋ) or far-from-equilibrium states (unusual x) contribute more to coherence buildup.
You don't consciously recall each time you've been there. Instead, all those experiences have accumulated into an immediate feeling of familiarity. That's coherence; the integrated weight of history.
Mathematically: Each visit contributed some L(τ) to the integral. A dramatic event in that room contributes more (higher |ẋ|) than routine visits.
This is why trauma doesn't just "go away with time." It accumulates into your coherence field, shaping how you respond to the present. Healing isn't forgetting; it's reorganising that accumulated experience through rupture and regeneration.
Yet we also experience radical "nowness"; moments when everything crystallises. The moment you understand a concept. The instant you decide to speak. The snap of recognition.
δ(t-t₀) with ∫ δ(t-t₀) dt = 1
What this captures:
Key insight: This isn't smoothing or averaging; it's genuine discontinuity. Like water freezing at exactly 0°C, not gradually becoming "ice-ish."
Often there's a specific moment when months or years of accumulated doubts crystallise into a clear decision. That's a rupture; not gradual change but an instantaneous reconfiguration of your entire situation.
Mathematically: The coherence C had been building, then at time t₀, a rupture δ(t-t₀) fires, discontinuously changing your state.
These aren't smooth transitions. They're discontinuous jumps. The delta function is the only mathematical structure that captures this quality of lived experience.
You don't experience the future as blank. It's conditioned by your accumulated history, with recent high-coherence moments mattering more than distant ones.
R[χ](x,t) = ∫₀ᵗ φ(x,τ)·e^(C(τ)/Ω)·Θ(t-τ) dτ
What this captures:
Key insight: If you had C = 10 at τ₁ and C = 1 at τ₂, the influence from τ₁ is e^10/e^1 ≈ 12,000 times stronger! This is why formative experiences shape you so profoundly.
When facing a difficult piece, the pianist's fingers seem to "know what to do" based on accumulated practice. That's regeneration; the weighted projection of integrated past experience into present action.
Mathematically: Years of high-coherence practice (large C) are exponentially weighted in the regeneration integral, making those patterns much more influential than recent casual playing.
Classical Physics (Markovian):
P(x_t | x_{t-1}, x_{t-2}, ...) = P(x_t | x_{t-1})
Only the current moment matters. F=ma depends just on position and velocity now.
Lived Experience (Non-Markovian):
P(x_t | history) depends on ∫₀ᵗ L(τ) dτ
Your entire history matters, weighted by how coherent those experiences were. You're carrying the past forward.
This is the gap CRR bridges. It gives us mathematics for systems that:
When C ≈ 0: System is approximately Markovian
e^(C/Ω) ≈ e^0 = 1 → no exponential weighting → all history matters equally (i.e., not much)
When C >> Ω: System becomes strongly non-Markovian
e^(C/Ω) >> 1 → past high-coherence states dominate present behaviour → deep memory effects
After rupture: C resets to pC (partial preservation)
System can shift from non-Markovian back towards Markovian, then rebuild non-Markovian character
This reveals something profound: memory itself is a form of agency; the active construction of temporal structure rather than passive storage. You're not just recording history; you're constructing your relationship to time.
The CRR conjecture is that you are not just "in" time; you construct your temporality through how you accumulate, rupture, and regenerate.
Two people can experience the "same" ten years completely differently:
Your relationship with time isn't fixed; it's a skill you can develop by understanding and modulating your L, δ, and R patterns.
Traditional views treat memory like a hard drive; storing and retrieving fixed files. CRR shows memory is active construction:
This could explain why therapy works: it doesn't erase traumatic memories but creates opportunities to rupture and regenerate them into a different coherent form. The integral is recalculated with different parameters.
The deepest philosophical question: How do you remain "you" while constantly changing?
Mathematically: Your identity is the trajectory x(t) that satisfies the CRR dynamics. Change the content (the specific x values) but maintain the signature (how C, δ, R relate), and you remain yourself.
The goal isn't to avoid rupture (impossible) or maximise coherence (leads to fragility). It's to metabolise rupture; make breaking and rebuilding part of your growth process rather than something that happens to you.
This is the difference between being a passive object in time versus an active agent constructing your temporality.
Imagine you're walking down a familiar street and suddenly see a purple elephant. Your brain would experience enormous "surprise"; a huge mismatch between what you expected and what you observed.
The Free Energy Principle (FEP) says that all living things are fundamentally trying to minimise this kind of surprise. Not just psychological surprise, but physical surprise; the difference between what you expect and what happens.
F = D_KL[q(s|o)||p(s)] - 𝔼_q[ln p(o|s)]
Breaking this down:
The goal: Minimise F by making your internal model q match reality p, and making good predictions.
Change q to match reality
Reduce D_KL by making your beliefs more accurate
Example: "Oh, that's not a person; it's a coat rack in the dark"
Change observations to match q
Make the world conform to your predictions
Example: You expect to have coffee, so you make coffee
You're constantly doing both; perceiving the world to update your model, and acting on the world to fulfill your predictions. This is called Active Inference.
Here's where it gets beautiful; CRR and FEP are describing the same thing from different angles:
Coherence ↔ Negative Free Energy:
C(t) = -∫₀ᵗ F(τ) dτ
High coherence = successfully minimised surprise over time. When things "make sense," C grows and F shrinks.
Memory Density ↔ Surprise Reduction Rate:
L(x,τ) ∝ -dF/dt
When you're learning successfully, F decreases (surprise drops), so L is positive (coherence builds).
Rupture ↔ Free Energy Crisis:
δ(t-t₀) fires when F >> F_typical
When observations become so surprising that your model can't accommodate them, rupture occurs.
Regeneration ↔ Model Reconstruction:
R[χ] = new generative model using ∫φ·e^(-F/Ω) dτ
Rebuild using states where F was low (successful predictions), weighted exponentially.
CRR provides the temporal dynamics for how F changes over time through accumulation, crisis, and reconstruction. FEP tells us what is being minimised (surprise). CRR tells us how that minimisation unfolds through cycles of coherence-rupture-regeneration.
A young child (age 4-5) sees two identical glasses with equal amounts of water. You pour one into a taller, thinner glass. They insist there's now "more water" in the tall glass.
Why? Their current model of the world is "taller means more." This has been working fine:
The rupture moment: Around age 6-7, they encounter the conservation task. This creates massive surprise:
Regeneration: They rebuild their understanding:
The child will never go back to thinking tall = more. The rupture created a permanent phase transition to a higher level of understanding with lower baseline free energy.
CRR Version:
Δt = C_crit / ⟨L⟩
Time to accumulate critical coherence threshold
FEP Version:
Δt = -ln(F_crit/F₀) / λ
Time to minimise free energy from F₀ to F_crit at learning rate λ
Example: Sensorimotor Stage (Birth to 2 years)
Not all surprises cause ruptures. You're surprised when you drop your keys, but it doesn't reorganise your worldview. Why?
This is why first-time experiences often don't change you (low C; you have no model yet) but repeated anomalies eventually force rupture (high C model encountering systematic contradiction).
Mathematical criterion: Rupture when C ≥ C_crit AND F > θ_critical (typically 3σ_F)
People have different patterns of how they minimise surprise:
Avoid surprise at all costs, build very rigid models
Math: High C_crit, low F tolerance → rare ruptures
Strength: Very efficient in stable environments (minimal F)
Weakness: Catastrophic failure when environment changes
Regularly test predictions, comfortable with moderate surprise
Math: Moderate C_crit, accepts F fluctuations
Strength: Adapt well to changing conditions
Weakness: May be less efficient than specialists
So much surprise they can't build stable models
Math: Very low C_crit, constant F fluctuations
Strength: Highly exploratory, creative
Weakness: Never develop deep expertise (low C)
Mental health challenges often involve dysfunctional surprise-minimisation patterns:
Anxiety:
Depression:
PTSD:
Organisms don't just minimise F passively; they act to sample observations that reduce expected free energy G:
G = 𝔼_q[D_KL[q(s|o,π)||q(s|π)]] - 𝔼_q[ln p(o|s)]
where π represents available policies (action sequences).
Key insight: Development is active inference. Children actively sample their environment to minimise expected free energy across increasingly complex policy spaces.
Rupture condition in Active Inference: Stage transitions occur when no available policy π can minimise G below threshold, given current model constraints. This forces model restructuring (rupture + regeneration).
Imagine teaching a robot to play chess. It gets really good at chess. Then you try to teach it checkers. Suddenly, it completely forgets how to play chess. This isn't a bug; it's a fundamental problem called catastrophic forgetting.
Current AI systems are the textbook example of a fragile signature:
1. Monotonic coherence buildup:
L(θ, t) > 0 for single task
C(θ, t) = ∫₀ᵗ L(θ, τ) dτ grows unbounded
The network accumulates information about Task 1 with no mechanism to "hold space" for future tasks.
2. No intermediate ruptures:
C_crit → ∞ (rupture threshold effectively infinite)
No δ(t-t_i) until forced by new task
No mechanism for controlled forgetting or reorganisation before catastrophe.
3. Catastrophic collapse:
New task → massive uncontrolled δ
Preservation p ≈ 0 (total overwrite)
Task 2 completely overwrites Task 1's learning rather than integrating with it.
1. Elastic Weight Consolidation (EWC)
Protect weights ∝ Fisher information: I_i = 𝔼[(∂log p/∂w_i)²]
Like putting locks on important memories. Better than nothing, but eventually you run out of room for new learning. CRR perspective: Partial preservation (p > 0) but no metabolised rupture; brittleness still accumulates.
2. Progressive Networks
Add new capacity: θ_new = θ_old ∪ θ_task
Like hiring a new person for each task. Works, but your system becomes enormous. CRR perspective: Avoids rupture entirely (δ never fires); no regeneration, no synthesis.
3. Memory Replay
Rehearse old data: 𝒟_train = 𝒟_new ∪ 𝒟_old
Like constantly reviewing old lessons. Helps maintain C, but doesn't transform. CRR perspective: Prevents forgetting but no regeneration; memory is repeated, not reorganised.
None of these approaches metabolise forgetting; they all try to prevent it. That's the wrong goal.
Instead of trying to prevent forgetting, CRR says: make forgetting part of the learning process.
1. Monitor Coherence:
L(θ, θ̇, t) = integration_quality(θ) - interference_cost(θ̇)
C(θ, t) = ∫₀ᵗ L(θ, τ) dτ
Track how well-integrated current knowledge is and how much new learning interferes.
2. Detect Interference:
If |∇_θ ℒ_new · ∇_θ ℒ_old| < -θ_conflict → interference detected
Notice when gradients point in opposite directions before catastrophe.
3. Trigger Selective Rupture:
ρ_i(θ) · δ(t-t_i) = targeted_reset(weights, I_i)
where I_i = Σ_tasks e^(C_task/Ω) · |∂ℒ/∂w_i|²
Selectively "forget" low-importance connections while protecting high-coherence ones.
4. Regenerate with Synthesis:
R[θ](t) = ∫₀ᵗ ∇ℒ(τ) · e^(C(τ)/Ω) · exp(-(t-τ)/λ) dτ
Δw = -η∇ℒ_new + β·R[θ](t)
Rebuild using weighted memory of past gradients. High-coherence moments exponentially influence current learning.
Current AI (Fragile Signature):
CRR-Based AI (Resilient Signature):
Just like people, AI systems could develop different signatures depending on what they're learning:
Best for: Continual learning across related tasks
Pattern: Regular small ruptures, good preservation
Example: A robot learning multiple household tasks
Best for: Tasks with natural phases
Pattern: Build expertise, consolidate, switch context
Example: Alternating between learning and sleep-like consolidation
Best for: Creative synthesis
Pattern: Hold contradictions, create novel solutions
Example: Combining chess strategy with Go tactics to invent new game strategies
Best for: Rapid exploration
Pattern: Frequent ruptures, low preservation
Example: Early-stage architecture search
Interestingly, biological brains already do something like CRR:
Memory Reconsolidation: When you recall a memory, it becomes temporarily unstable (rupture opportunity), then reconsolidates with new information
Sleep: During sleep, your brain selectively strengthens some connections (high coherence) and weakens others (controlled forgetting)
Neurogenesis: Creating new neurons in certain brain regions provides fresh capacity while preserving important patterns
Multiple Memory Systems: Different types of memory (episodic, semantic, procedural) interfere less with each other
CRR doesn't just store information or protect it; it transforms memory through controlled rupture and regeneration cycles.
Old Approach: Learning Task B should not disturb Task A
CRR Approach: Learning Task B creates an opportunity to reorganise both A and B into something better
This is genuine learning; not just accumulation, but transformation.
Robots: Could learn new tasks without forgetting old ones, and get better at old tasks when learning new ones
Personal Assistants: Could adapt to your changing needs without losing personality coherence
Medical Diagnosis: Could update with new medical knowledge while preserving what worked in the past
Creative AI: Could synthesise across different training data to create genuinely novel combinations
Catastrophic forgetting isn't just a technical problem; it reveals something fundamental about intelligence.
Current AI has memory but not metabolised rupture. That's why it can't truly learn continually; it can only accumulate until catastrophe.
CRR offers a path towards AI systems that learn more like biological systems; through cycles of building, breaking, and rebuilding that create genuine transformation rather than just accumulation or replacement.
Your memory signature isn't about what you remember; it's about how you navigate the cycle of building up experience, encountering disruptions, and rebuilding yourself. It's your characteristic pattern of coherence, rupture, and regeneration.
Think of it as your "temporal fingerprint"; the unique way you move through time, accumulate experience, handle crises, and transform yourself.
How It Works: You build up coherence for a long time without releasing pressure. Everything seems fine, then suddenly everything collapses.
Everyday Examples:
Strengths: Highly efficient in stable environments; deep expertise in familiar domains
Weaknesses: Catastrophic when faced with unavoidable change; rigid thinking; difficulty adapting
Path to Resilience: Practice small ruptures regularly (therapy, trying new things, honest conversations before crises)
How It Works: You build coherence but regularly process small disruptions before they become catastrophic. You bend but don't break.
Everyday Examples:
Strengths: Adaptable; maintains identity through change; learns from mistakes without shattering
Weaknesses: Can seem less decisive than fragile types; may not specialise as deeply
Path to Growth: Stay curious; don't become complacent; keep challenging yourself
How It Works: You naturally cycle between building up and releasing, like breathing. Intensity followed by integration, work followed by rest.
Everyday Examples:
Strengths: Sustainable long-term; natural rhythm prevents burnout; deep-rest deep-work balance
Weaknesses: May struggle with constant-demand environments; others might not understand your rhythm
Path to Growth: Honour your rhythm; build environments that support it; don't try to be "always on"
How It Works: Ruptures happen so frequently that coherence never builds up. You're constantly starting over, never building depth.
Everyday Examples:
Strengths: Highly exploratory; creative; open to novelty; sees connections others miss
Weaknesses: Never develops deep expertise; others see you as flaky; hard to build on past success
Path to Resilience: Create containers for coherence (commitments, routines, practices that deepen over time)
How It Works: You hold contradictions and allow them to interact until something new emerges. Multiple coherence streams interfere constructively.
Everyday Examples:
Strengths: Creative synthesis; handles complexity; sees beyond false dichotomies; integrative wisdom
Weaknesses: Can seem indecisive; others may want clearer positions; requires high cognitive load
Path to Growth: Trust the synthesis process; don't force premature resolution; cultivate tolerance for ambiguity
Your signature often changes as you age:
Early Childhood (0-5): Oscillatory; rapid cycling between building and breaking, high plasticity
Childhood/Adolescence (5-18): Transitioning to Resilient; learning to handle disruptions more gracefully
Young Adulthood (18-40): Risk of becoming Fragile if you stop growing, or deepening Resilience if you keep learning
Middle Age (40-65): Either rigid Fragility or wise Dialectical capacity
Later Life (65+): Either crystallised wisdom (Dialectical) or brittle fragility
The key insight: signatures aren't fixed. You can shift your pattern intentionally.
How do you handle disruptions?
How do you build expertise?
When you face a crisis, what happens?
Fragile → Resilient:
Chaotic → Oscillatory:
Resilient → Dialectical:
Any → Fragile (Warning Signs):
Understanding signatures can illuminate relationship dynamics:
Fragile + Fragile: Stable until crisis, then both collapse
Fragile + Resilient: Resilient partner carries emotional load, may eventually resent it
Resilient + Resilient: Healthy growth together, but may become complacent
Oscillatory + Chaotic: Rhythm vs chaos; can complement or clash
Dialectical + Any: Can integrate different patterns but requires patience from both
You might have different signatures in different areas:
This is normal! The goal is awareness and intentional development where it matters most to you.
1. Pick a domain (career, relationships, health, learning)
2. Track over the past year:
3. Ask yourself:
4. Experiment:
Your signature is not your destiny; it's your current strategy for navigating time and change.
Understanding it gives you agency. You can:
The goal isn't to have the "best" signature; it's to consciously navigate your cycles of coherence, rupture, and regeneration rather than being unconsciously swept through them.
Remember: CRR isn't about avoiding rupture or maximising coherence. It's about metabolising change; making breaking and rebuilding part of your growth process rather than something that happens to you. That's the path from victim to agent, from passive to active, from stuck to evolving.