CRR

Coherence

How systems accumulate historical constraint over time. Coherence represents the temporal integration of structure: the past becoming present as accumulated pattern. When coherence approaches the information-theoretic bound (C·Ω = 1), the system has traversed all distinguishable states available to its current regime, and rupture follows.

Rupture

The Dirac delta encodes the dimensionless present: the moment where C·Ω = 1 and phase transition occurs. This is structurally analogous to Cramér-Rao saturation: the system has exhausted the distinguishable states available to its current regime. At rupture, local coherence resets while historical coherence values remain accessible through the regeneration integral, enabling continuity through discontinuity.

Regeneration

The reconstruction process that builds new stable patterns by drawing upon accumulated historical information. Crucially, history is never lost, only selectively weighted. The exponential term exp(C/Ω) determines which past moments reconstitute, enabling continuity through transformation.

Technical Notes

The Rupture Condition: C · Ω = 1

C is accumulated arc length on the statistical manifold (Fisher-Rao speed integrated over time). Ω = σ² is the characteristic variance, fixed by the topology of the state space. Their product saturates the Cramér-Rao bound at rupture: the system has traversed all distinguishable states available to its current regime and must reorganise.

This bound appears independently across fields: the Heisenberg uncertainty principle (ΔE·Δt ≥ ℏ/2), the Gabor limit (Δt·Δf ≥ 1/4π), the precision-variance trade-off in Bayesian inference, and the thermodynamic speed limits established by Ito and Dechant (2020). CRR proposes that these share a common geometric origin: a bounded system accumulating coherence until inside matches outside.

Symmetry Determines Ω

Čencov's uniqueness theorem constrains the metric on any statistical manifold to be the Fisher information metric. The geodesic structure of that metric fixes the maximum arc length a system can traverse. Ω = 1/ℓ, where ℓ is the geodesic extent.

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. Examples include the heartbeat (systole-diastole), the breath turning (inhale-exhale), a neuron firing, the pupil dilating, and sensory edges.

SO(2) (rotational) systems exhaust their configuration by completing one full cycle. The circular manifold S¹ has circumference 2π, giving Ω = 1/2π ≈ 0.159 and C* = 2π. Examples include the circadian rhythm, the gait cycle, the menstrual cycle, the solar magnetic cycle, and the continuous revision of prior beliefs.

The ratio between classes is exactly 2, a topological invariant independent of any physical parameters.

Two Channels: FEP Correspondence

In any network of Markov-blanketed agents, an edge is a boundary alternating between two regimes of influence (Z₂ dynamics), while a node is an internal model traversing a continuous cycle of belief updating (SO(2) dynamics). This assigns Z₂ to sensory (likelihood) precision and SO(2) to prior (transition) precision, not by analogy but from the graph topology of Active Inference itself (Sabine, 2026). The dynamics are directly visible in the network simulation: nodes glow slowly toward large belief updates while edges flicker rapidly between regimes.

Precision corresponds to (1/Ω)·exp(C/Ω), where 1/Ω is the topological contribution (fixed by geometry) and exp(C/Ω) is the dynamical contribution (growing with accumulated evidence). Variational free energy maps to coherence C (inversely). The FEP Markov blanket maps to the temporal boundary delineated by δ between past states (C) and future states (R).

Phase-Gating

Because the two channels share the same evidence stream but accumulate on manifolds with different geodesic extents, they rupture at different rates. The Z₂ phase at the moment of SO(2) rupture follows a non-uniform distribution (χ² = 8,041 in simulation), suggesting that the temporal relationship between channels may determine whether each update drives learning or action. This finding is structurally compatible with empirical work on neuromodulatory timing (Jang et al., 2026), where the relative timing of dopamine and acetylcholine signals, not their magnitude, determines functional output.

Both channels process equal total precision-gain per unit time (ratio 1.003, CI [1.000, 1.005]). The Z₂ channel makes many small updates; the SO(2) channel makes few large ones. The factor of 2 sets the exchange rate between frequency and depth, not the total throughput. For AI alignment, this implies that a phase mismatch between human and AI constitutes a measurable form of temporal misalignment: the AI's updates may arrive at the wrong phase of the human's coherence cycle, promoting action when learning is needed or learning when action is called for. The Epistemic Drift simulation explores how LLM-generated signals propagate through human networks when the receiver cannot distinguish the precision class of the source.

The Beauty Function

B(C) = exp(C/Ω) · (C* − C). The product of accumulated coherence (rising exponentially) and remaining capacity (falling linearly). It peaks at C* − Ω: one capacity-unit before rupture. The system is most responsive, most poised, at the edge, not at the transition itself. This is where agency lives: close enough to rupture that history is fully weighted, far enough that choice remains. In the self-evidencing tree, each fork is a Z₂ rupture; the accumulated shape of the tree is the coherence integral made visible.

Parameter-Free Predictions

CV = Ω/2, derived from Wijsman attainment and Jaynes' maximum entropy principle: for Z₂ systems, CV = 1/(2π) ≈ 0.159; for SO(2) systems, CV = 1/(4π) ≈ 0.080. The ratio is exactly 2. The free-energy-optimal precision allocation converges to π_p/π_s = √2, investing √2 times more confidence in priors than in sensory data, because transformation (SO(2)) requires twice the coherence of absorption (Z₂).