Ω is the boundary permeability (characteristic variance). Small Ω → rigid, crystalline forms (only highest-coherence memories accessible). Large Ω → fluid transformation (all history equally weighted).
CV = Ω/2 follows from the Dirac delta distributing unit mass equally across the Markov blanket boundary at rupture: σ(C*) = 1/2 universally.
Certain spatial configurations are stable under CRR dynamics—once particles arrive and align with neighbours, their coherence increases, which strengthens their position via exp(C/Ω). The geometry becomes self-stabilising.
The forms in this simulation are either constitutive of CRR's mathematical structure, or emerge from applying CRR's optimisation framework to classical mathematics. Each is annotated with its derivation status:
§CRR Follows from CRR's equations or symmetry classification
§SOLVED CRR poses the optimisation; external mathematics solves it
§PARTIAL CRR provides genuine structure; geometry is classical
Circle §CRR
S¹ is axiomatic to CRR. All temporal processes map to the circle. This is not
derived—it is the space CRR lives on.
Yin Yang §CRR
Z₂ partitions S¹ into two complementary halves. The S-curve boundary sits where
coherence of the two phases is equal: Cyin = Cyang. For circular
coherence fields at ±R/2, this traces two semicircles—the characteristic
embracing curve. The seeds (dots) are coherence nucleation points where
exp(C/Ω) for the minority phase is maximal.
Lemniscate (∞) §CRR
A Z₂ system traversed continuously. The crossing point is the rupture—phase
inverts, coherence flips sign. CRR predicts the topology (figure-8); the
specific metric (Bernoulli's lemniscate) is one realisation.
Ouroboros §CRR
The C → δ → R cycle closing on itself. Body = accumulated coherence.
Mouth = rupture. Eating = regeneration feeding back. The body thickens as
θ increases because coherence accumulates along the cycle: thickness
∝ exp(θ/2π). This is a process isomorphism, not a derivation.
Torus §CRR
Two independent SO(2) processes give S¹ × S¹ = T², the torus.
CRR's symmetry classification produces this topology directly.
Fibonacci Spiral §SOLVED
Sequential coherence events placed on S¹. To maximise events before rupture
(C·Ω = 1), distribute them as uniformly as possible. By Weyl's
equidistribution theorem, the unique optimal angle has continued fraction
[1,1,1,…] = 1/φ, giving 2π(2−φ) ≈ 137.508°.
CRR provides the optimisation criterion; number theory solves it.
Flower of Life §PARTIAL
Multiple SO(2) coherence basins. Overlap regions (vesicae) have elevated
exp(C/Ω), making them preferential regeneration sites. Packing maximum
basins without exceeding the rupture threshold gives the 2D kissing number
(6 = hexagonal). CRR motivates the optimisation and gives the vesicae
physical meaning; the answer (hexagonal) is classical.
Vesica Piscis §PARTIAL
Two equal SO(2) basins overlapping. The intersection has CA + CB
> max(CA, CB), creating elevated exp(C/Ω)—a preferential
site for regeneration. For small Ω (rigid), the peak is sharp and focused;
for large Ω (fluid), the entire overlap contributes. The geometry is
Euclid; CRR gives the overlap physical meaning.
In a standard particle system, particles follow instantaneous forces. Here, particles carry coherence history. Their influence on neighbours scales as exp(C/Ω)—highly coherent particles have exponentially more pull. This creates organic formation: aligned particles arrive first and draw others through the coherence field. The geometry becomes self-stabilising because arriving at the target increases alignment, which increases coherence, which strengthens attraction. This feedback loop is CRR's genuine contribution to the visual dynamics.
Framework by Alexander Sabine · temporalgrammar.ai