CRR proposes that systems accumulate history (Coherence), undergo discrete phase transitions when constraints reach threshold (Rupture), and reconstitute through exponentially-weighted memory selection (Regeneration). Built from information geometry and the geometry of a circle, CRR describes temporal structure as a mathematical grammar by which past becomes future. The framework is deliberately parsimonious: three equations.
Coherence
How systems accumulate historical constraint over time. Coherence represents the temporal integration of structure: the past becoming present as accumulated pattern. When the distance from where it began reaches the bound (C*·Ω = 1), rupture follows.
Rupture (Now)
The Dirac delta encodes the dimensionless present: the first moment the system reaches the state opposite to where it began (C*·Ω = 1). Any wandering on the way is surplus. At rupture, local coherence resets while historical coherence values remain accessible through regeneration, enabling continuity through discontinuity.
The line through the empty centre joins now (P0) to the state the occasion completes in (P1): the projection, which costs nothing. The walk around the circle is what it costs. Each step back is marked on the rim and adds to the surplus S. On arrival, the walked arc settles into the past, and the rings of memory keep what each passage cost, weighted by recency × salience.
Regeneration
The reconstruction process that builds new stable patterns from what the system has inherited: its own completed occasions, and the wider past it stands on. Crucially, history is never lost, only selectively weighted, by recency and by salience, enabling continuity through transformation.
Technical Notes
The Rupture Condition: C* · Ω = 1
Two lengths are kept apart. C is the whole path walked: accumulated arc length on the statistical manifold (Fisher–Rao speed integrated over time). C* is how far the system now is from where the occasion began. The cut uses C*, so wandering adds to C without bringing the cut closer; the difference is the surplus S = C − C*.
CRR proposes a bounded system accumulating coherence until inside matches outside.
Symmetry
Čencov's theorem fixes the metric as the Fisher information metric, up to an overall constant. Ω = 1/a, where a is the manifold's diameter: the half-turn, the distance from a state to its opposite.
Z₂ (bistable) systems exhaust their configuration by crossing from one basin to the other. The Bernoulli manifold has geodesic diameter π, giving Ω = 1/π ≈ 0.318 and C* = π at the cut. Candidate examples (proposed, not tested) include the heartbeat (systole-diastole), the breath turning (inhale-exhale), a neuron firing, the pupil dilating, and sensory edges.
SO(2) (rotational) systems. On a circle the cut is the half-turn, so the diameter is half the circumference. For a circle of circumference 2π that gives Ω = 1/π, the same as Z₂: the two classes differ in shape, not in Ω. A circle's Fisher circumference depends on the statistical carrier, not on the topology alone. Candidate examples include the circadian rhythm, the gait cycle, the menstrual cycle, the solar magnetic cycle, and the continuous revision of prior beliefs.
Substrate symmetry (Z₂ or SO(2)) describes the manifold on which coherence accumulates. The rupture event itself is Z₂: a commitment to "this regime / next regime". The recurrent cycle that carries the system to the cut can be SO(2), but the cut is binary.
Two Channels: an Active Inference Proposal
A modelling proposal (Sabine, 2026): in a network of Markov-blanketed agents, an edge is a boundary alternating between two regimes of influence (Z₂), while a node is an internal model traversing a continuous cycle of belief updating (SO(2)). This assigns Z₂ to sensory (likelihood) precision and SO(2) to prior (transition) precision. The dynamics are visible in the network simulation: nodes glow slowly toward large belief updates while edges flicker rapidly between regimes.
Phase-Gating
If two channels share one evidence stream but sit on carriers of different Fisher size, they reach their cuts at different times, and the phase of one at the other's cut could carry information: whether an update drives learning or action. This is a hypothesis, not a result. It is structurally compatible with work on neuromodulatory timing (Jang et al., 2026), where the relative timing of dopamine and acetylcholine signals, not their magnitude, shapes what a signal does. For AI alignment, a phase mismatch between human and AI could be read as a form of temporal misalignment: updates arriving at the wrong phase of a person's cycle. The Epistemic Drift simulation explores how LLM-generated signals propagate through human networks when the receiver cannot tell the precision class of the source.
The Beauty Function (a toy for poise)
Count C* and the half-turn a = 1/Ω in units of the resolution σ, so both are plain numbers. Then B = eC* · (a − C*): progress toward the cut, rising exponentially, times the distance still to go, falling linearly. It peaks at C* = a − 1, one just-distinguishable step before the cut. On this picture the system is most poised at the edge, not at the transition itself, and we like to think of this as where agency might live: close enough to the turn that history is fully in play, far enough that choice remains. It is an illustration, not part of CRR's core. In the self-evidencing tree, each fork can be read as a Z₂ cut, and the tree's accumulated shape as coherence made visible.
What is chosen, and what was withdrawn
CRR is not parameter-free. The resolution σ is calibrated separately; the Fisher metric is fixed only up to a constant scale (scaling it and σ together changes nothing); and recency q has no universal value. Earlier versions of this page predicted timing variability as CV = Ω/2. That was not a derivation: how variable the time between cuts is depends on a system's drift and noise, which is a first-passage question, and the claim has been withdrawn.
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 may miss. The following exemplars: CRR across sensory physiology, physical systems, and human-AI networks.