Theorems
Results
Subsystem Contraction
Gradient descent on an \( L \)-smooth, \( \mu \)-strongly convex objective with step \( \eta \le 2/(\mu + L) \) is a contraction in the parameter with rate \( \beta = 1 - \eta\mu < 1 \).
Cosine-Scaled Projection
A PCGrad-type projection \( g_i' = g_i - \alpha|\cos(g_i,g_j)|\,\frac{g_i\cdot g_j}{\|g_j\|^2}g_j \) preserves first-order descent of each objective. Limit points are not unique; the scheme can stall in one dimension and admits period-two orbits at moderate step sizes.
Statement and limitations →Normalised Lyapunov
\( V = \sum_i L_i/L_i(\theta_0) \) is non-negative, equals \( K \) at the start, and is non-increasing whenever every \( L_i \) is non-increasing along the update. Scale-free, no weights.
Statement →Interaction Matrix
For a coupling matrix with block rates \( \rho_i \) and off-diagonal coupling \( c \), Gershgorin gives spectral radius \( \le \rho_{\max}(1 + c(K-1)) \). The balanced matrix has eigenvalues \( \rho(1 + c(K-1)) \) and \( \rho(1-c) \).
Statement →Higher-Order Convergence
For a contraction with rate \( \sigma < 1 \), the convergence score over a window of length \( T \) satisfies \( S \ge 1 - \dfrac{2\sigma^{T/2}}{(1-\sigma)^2} \).
Composed System Convergence
Block contraction with a non-negative coupling matrix of spectral radius below one gives convergence of the sequential composition; LaSalle's invariance principle locates the limit in the largest invariant set on which the Lyapunov function is stationary.
I-Ratio Identity
With \( D = \sum_i \|g_i\|^2 \) and \( I = \sum_{i
B-Flow
The balance residual \( B = \|\sum_i g_i\|^2 / D = 1 + 2I \). At a point \( \theta^* \) where the gradients sum to zero, \( \nabla^2 B(\theta^*) = 2J^{\mathsf T}J/D \) with \( J \) the Jacobian of \( \sum_i g_i \). \( B \) is a diagnostic.
Statement →Exploratory experiments
Runs that motivate the framework but do not establish a theorem. Each page describes what was run and reports the numbers as data.
Belief Update Experiments
Bayesian belief agents composed with the interaction matrix; correlated evidence streams (rain/wet ground) with and without a learned interaction.
Experiment → · Correlated streams →Desire as Regulariser
Belief agents with a desire prior that partially contradicts the evidence stream, compared with aligned and absent desire at several observation horizons.
Experiment note →Code Gate Weights
Neural gate weights in a code-generation loop tracked with the convergence score \( S \) over training steps.
Compiler Pass Interaction
Optimisation passes composed as sequential updates, with an interaction matrix fitted per program.
Convergence Score on the Logistic Map
\( S \) computed along a sweep of the logistic-map parameter \( r \), alongside the Lyapunov exponent.
Experiment note →Multi-Agent Games
Belief agents with partially contradicting desires in repeated Prisoner's Dilemma, a zero-sum game and a three-player coordination game.
Experiment →PID-S Controller
Learning rate driven by the convergence score and its dual-number derivatives: \( \text{lr}(t) = \text{base}\times\text{clamp}(1 + w_1 S + w_2 S' + w_3 S'') \), gains fitted on shifting landscapes.
Perturbation Stability Margin
A stability margin \( M(\theta,\varepsilon) \) measuring perturbation amplification, carried as a dual number so that \( \partial M/\partial t \) and \( \partial^2 M/\partial t^2 \) are available along a run.
Dual Agent Architecture
Two control agents on a viscous Burgers simulation: one injects perturbation, the other adds artificial viscosity. Leverage ratios and the balance point are reported on the experiment page.
Experiment →