A field of pulls, and the lights moving through it. Watch it; then disturb it. The mathematics is in the methods.1
tip: drag any node in the field to move it (basin test)
click node = lock (role-lock) · drag node = move in-plane (basin test) · drag timeline = scrub · space pause · R reset
A field of pulls, and the souls moving through it.
The glow is what we keep returning to.
The lights are people, or the parts of one person.
The lines are who moves whom.
Watch it breathe. Then disturb it, and watch everyone change.
Wake the Third.
Toward Moloch, peace curdles into competition.
Toward the Buddha, nothing is allowed to harden.
Either way it begins to grow.
What grows, grows through friction.
The mathematics beneath is exact. You don’t need it. Watch the movement, not the numbers.
skipComputational instrument — interactive specification, v0.4 “the Third.” · Called woo to turn away those who would only count. The mathematics below is exact.
Each of N actors carries a latent state zi(t)∈ℝ10 over interpretable affective–relational coordinates [security, threat, desire, status, intimacy, ambiguity, control, shame, curiosity, trust], a circumplex-style decomposition of interpersonal affect.14 The display is a 2-D coordinate projection pi=Π(zi); the field is the corresponding planar slice of the full potential.
States evolve by gradient descent on a continuous potential, plus directed coupling, perturbation, and stochastic forcing — a Langevin flow on an attractor landscape in the tradition of associative-memory1 and continuous neural-field models,2 over a Waddington-style landscape metaphor.3
Attractor strengths Ak(t) = Ak(1 + b·sin(ωkt + φk)) “breathe,” a slow periodic modulation analogous to coupled-oscillator entrainment.8 Coupling strength κ is the order parameter of an interaction phase transition: above a critical κ the constellation collapses into a single basin — a saddle-node/catastrophe-type bifurcation.456 Coordination-dynamics models of dyadic coupling motivate the directed term.7
The directed graph is not fixed. Weights relax toward a structural prior W₀ and are driven by a lagged, motion-gated directional alignment — a Hebbian9 proxy for directed causal influence in the spirit of Granger causality10 and transfer entropy.11 If i’s motion at t−1 predicts j’s motion at t, Wij grows.
U = ΣiU(zi) is the system potential energy. The Kuramoto-style order parameter R = ‖N⁻¹Σi vi/‖vi‖‖∈[0,1] measures directional coherence of motion (R→1 at collapse).8 Coupling asymmetry Δ = ⟨|Wij−Wji|⟩ quantifies net directedness. Basin occupancy assigns each actor to argmink‖zi−ck‖.
Perturbation is treated as measurement: a controlled input Pi(t) deforms the landscape and the response reveals coupling — an active-inference reading of probing.15 The framework follows the dynamical-systems program for social and dyadic psychology.1213
A two-body system — a dyad, or a single actor in a fixed landscape — is effectively integrable: it settles. Sociologically, Simmel argued the triad is categorically distinct from the dyad; the third party transforms every relation it enters (mediator, tertius gaudens, divide-and-rule).16 Dynamically, the analogue is exact: adding a third gravitating body destroys integrability and admits chaos — Poincaré’s three-body problem18 and its sensitive dependence on initial conditions, quantified by a positive largest Lyapunov exponent.19 We add a third body with mutual coupling to the actor cloud and measure λ online with the Benettin twin-trajectory method. With the Third off, λ ≈ 0 (the dyad settles); switch it on and λ > 0 — the system becomes perpetually generative.
The Third has two archetypes along an axis μ∈[−1,+1]:
Moloch (μ > 0): the coordination-failure attractor. Its pull scales with the group’s own coherence — the more the actors harmonize, the harder it drags them toward the “sacrifice” well, inflaming competition. This is the multipolar trap: the tragedy of the commons,20 Schelling’s perverse macro-behaviour,21 the god to whom cooperation is sacrificed.27 Girard’s mimetic third — the model-rival who generates desire and rivalry — is the interpersonal face of the same force.17
The hidden Buddha (μ < 0): the empty witness. It applies a rotational, de-grasping force — weakening attractor capture and inducing orbit rather than collapse. It is “hidden” (low amplitude, never dominating) yet decisive: by refusing to fix the system it keeps it in continual becoming — the non-substantial third of Nāgārjuna’s śūnyatā and dependent origination.24
Either way the moral is one: adversity is the engine of complexity. A frustrated system explores; order arises through fluctuation far from equilibrium (Prigogine’s dissipative structures22) and richness peaks at the edge of chaos.23 The dialectic needs its negation: the third moment that sublates the pair into a higher unity.25 We report this as complexity C, the rate of basin reorganization. The directed graph is the Symbolic order through which the Third becomes legible.26
| object | contemplative reading | formal object |
|---|---|---|
| attractor well | a samskara / habitual refuge | local minimum of U(z) |
| perturbation | a koan, a rupture, grace | impulse input P(t); a measurement |
| breathing | the field inhaling/exhaling | periodic modulation Aₖ(t) |
| Moloch | the demon of sacrificed cooperation | coherence-coupled adversarial attractor |
| hidden Buddha | śūnyatā, the witness, non-grasping | rotational, attractor-attenuating operator |
| λ > 0 | irreducible aliveness / freedom | positive Lyapunov exponent (chaos) |
| complexity C | growth through adversity | basin-transition rate (order-through-fluctuation) |
Presets reconfigure the whole instrument — forces, the Third, projection, and (for scenarios) the cast and the landscape itself. The attachment-theory set recasts the four wells as felt-security basins (secure base · hyperactivation · deactivation · disorganization) and the actors as attachment styles, after Bowlby,28 Ainsworth,29 and the anxiety×avoidance four-category model;3031 the default plane is intimacy (avoidance) × threat (anxiety). Sub-scenarios encode the well-documented interaction patterns: the anxious–avoidant pursue–withdraw trap, the securely-anchored base, anxious–anxious escalation, avoidant–avoidant parallel distance, and disorganized strategy — the last coupled to the Third so the “no coherent strategy” of fearful-avoidance shows up as genuine chaos (λ>0). Further scenarios recast the landscape for inner parts (IFS — the Self as the witnessing Third),32 Bowen family systems (triangulation, differentiation),33 and the therapeutic dyad (transference held within the frame).34
The fit tools invert the model: given observed trajectories zi(t) — either the current recording or an uploaded CSV in the export schema — they recover a generative model. Attractor centers are estimated by k-means over visited states35 (the wells are where the system dwells); directed couplings W are estimated by the same lag-1 gated alignment used online, batched over the trajectory (a heuristic Granger/transfer-entropy proxy1011); the display plane is the top-variance pair. The fitted model is loaded back into the instrument, so a recovered structure can be replayed and extrapolated — and, by fitting a run you generated, checked against its known generator.
Goodness of fit. We score the model-consistent one-step law Δz(t) = φ·Δz(t−1) + β·(attractor) + γ·(coupling) by ordinary least squares, reporting R² on a held-out final 30% of frames (typically 0.8–0.98 on self-generated runs — the dynamical form is well specified). Directed structure is assessed by the stability of each inferred edge across five time blocks (mean > threshold, low block-variance); this recovers which couplings exist robustly. An honest caveat: at one-step resolution the momentum term φ dominates the displacement variance, so the coupling magnitude γ is only weakly identified — structure is recoverable, scalar strength is not. Recovering magnitudes would need multi-step prediction or explicit noise modelling.