1. Introduction

Modern engineered systems are rarely monolithic. A compiler interleaves optimisation passes that share intermediate representations. A generative adversarial network couples a generator and a discriminator through shared loss gradients. A cognitive architecture lets beliefs, desires, and intentions co-adapt over a common state space. In each case, the system is a composition of adaptive subsystems, and the fundamental question is whether the composition converges.

Existing convergence theory handles individual subsystems well. Contraction mapping theorems guarantee convergence for single operators. Lyapunov analysis certifies stability of fixed points. Gradient descent converges on convex landscapes. But composition breaks these guarantees. Two individually contractive maps may compose into an expansive one. Two individually stable subsystems may oscillate when coupled. Two gradient descents on convex objectives may diverge when they share parameters.

The root cause is interference: subsystem \( i \) updates a parameter that subsystem \( j \) depends on, and vice versa, creating circular dependencies that no single-subsystem analysis captures. Prior work addresses fragments of this problem. Multi-task learning uses gradient projection (PCGrad, Yu et al. 2020) to remove conflicting gradient components, but provides no convergence proof for the composed system. Multi-agent reinforcement learning proves convergence for specific game classes (Bowling and Veloso 2002) but not for general adaptive composition. Lyapunov-based control theory (Khalil 2002) requires manually constructed Lyapunov functions that become intractable beyond two or three subsystems.

This paper gives sufficient conditions for convergence of such compositions. When each subsystem's update is a contraction (C1) and the matrix of block Lipschitz constants has spectral radius below one (C2), the composed map is a contraction in a weighted maximum norm and has a unique fixed point; this is the classical block-contraction theorem (Bertsekas and Tsitsiklis 1989, §3.1.2; Feingold and Varga 1962), and LaSalle's principle describes the invariant set. The interaction matrix that encodes the coupling, a step-ratio convergence score \( S \) that needs no knowledge of the target, and the balance residual \( B = 1 + 2I \) are the working tools.

We record gradient projection (PCGrad, Yu et al. 2020) and a cosine-scaled variant as implementations, with their limitations: they preserve first-order descent of each objective but admit non-unique limit points, stall in one dimension, and produce period-two orbits at moderate step sizes.

Each result is accompanied by a reproducible experiment implemented in Simplex. The experiments are exploratory: they check identities and illustrate the diagnostics on optimisation problems (up to 25 dimensions), belief-updating models, GAN training, ODE solver coupling, and compiler pass scheduling.

2. Related Work

The convergence of composed adaptive systems draws on several mature research threads. We position our contributions relative to each.

2.1 Multi-Task Gradient Methods

Multi-task learning requires reconciling gradients from competing objectives. MGDA (Sener and Koltun, 2018) casts the problem as a minimum-norm element in the convex hull of task gradients, guaranteeing Pareto descent but scaling poorly beyond a handful of tasks. PCGrad (Yu et al., 2020) projects conflicting gradients onto the normal plane of each other, resolving roughly 66% of conflicts empirically. CAGrad (Liu et al., 2021) maximises the worst-case improvement across tasks, and Nash-MTL (Navon et al., 2022) frames multi-task optimisation as a bargaining game, solving for the Nash bargaining solution at each step.

Our approach. We treat these projections as implementations rather than as convergence mechanisms. Definition 2.4 with \( \alpha = 1 \) is PCGrad, and orthogonality of the projected gradient to the conflicting one holds by construction. Convergence is obtained from the block-contraction condition on the interaction matrix, not from the projection.

2.2 Lyapunov Methods for Multi-Agent Convergence

Borkar (2008) established the two-timescale stochastic approximation framework, proving convergence when fast and slow subsystems operate on separated timescales. Qu et al. (2020) extended Lyapunov analysis to multi-agent reinforcement learning, constructing per-agent Lyapunov functions whose sum certifies global stability under a mixing condition.

Our approach. We use the weighted maximum norm of the block-contraction theorem (Bertsekas and Tsitsiklis 1989) directly, so that the single spectral-radius condition on the non-negative coupling matrix certifies a unique fixed point for any number of subsystems, and LaSalle's principle describes the invariant set of the composed dynamics.

2.3 Game-Theoretic Convergence Analysis

Daskalakis et al. (2018) showed that gradient descent in general games may cycle rather than converge, and introduced optimistic gradient methods to restore convergence in certain game classes. Mertikopoulos et al. (2019) analysed mirror descent in games, proving convergence under variational stability conditions.

Our approach. Rather than restricting the game class, we impose structure through the interaction matrix \( M \), whose entries bound pairwise coupling strengths. The spectral radius condition \( \sigma(M) < 1 \) with non-negative coupling is a sufficient condition for convergence; adversarial coupling, where the Perron–Frobenius step does not apply, is outside the theorem.

2.4 Chaos Diagnostics and Dynamical Systems

The theory of Lyapunov exponents (Strogatz, 2015) provides the standard tool for distinguishing chaotic from regular dynamics. Positive maximal Lyapunov exponents indicate sensitive dependence on initial conditions. The edge of chaos hypothesis (Langton, 1990; Bertschinger and Natschläger, 2004) suggests that computation is optimised near the boundary between ordered and chaotic regimes.

Our approach. The step-ratio score \( S \) is a trajectory diagnostic that requires no knowledge of the target equilibrium. Runs on the logistic map are reported as exploratory; \( S \) does not separate periodic from chaotic orbits and is not offered as a test for chaos.

3. Definitions

4. Main Theorem

The non-negativity of \( \Lambda \) is essential to the Perron–Frobenius step. Coupling that is adversarial in sign is not covered by the theorem as stated.

5. Propositions and Experiments

5.1 Experimental Methodology

All experiments were conducted on a single workstation with an Apple M-series processor (ARM64) running macOS, using the Simplex compiler (v0.17.0) with deterministic execution. No GPU acceleration was used; all computations are double-precision floating-point on CPU. This ensures reproducibility without hardware-specific numerical variation.

Parameter selection follows a systematic protocol. Learning rates are initialised at \( \eta = 0.01 \) unless otherwise stated and validated across three orders of magnitude (\( 10^{-4} \) to \( 10^{-1} \)). Contraction rates \( \rho_i \) are computed empirically from trajectory data over the first 100 steps and compared against theoretical bounds from the strong convexity and smoothness constants of each loss function. The interaction matrix \( M \) is assembled from measured gradient cosines at each step and its spectral radius is tracked throughout the optimisation.

Convergence is declared when both the convergence score \( S > 0.99 \) and the balance residual \( B < 10^{-10} \) are satisfied simultaneously. All trajectory lengths are at least \( 10 / (1 - \sigma(M)) \) steps to ensure that the geometric convergence has time to manifest.

Statistical methodology follows a bootstrap protocol with 1,000 resamples for confidence intervals. Each claim of the form "X% of trials satisfy property P" is accompanied by the raw count \( k/n \) and the 95% Clopper–Pearson confidence interval. For continuous quantities (convergence rates, Lyapunov values), we report the mean and standard deviation across independent initialisations, with at least 10 seeds per configuration. All \( p \)-values are two-sided and corrected for multiple comparisons using Holm–Bonferroni where applicable.

5.2 Contraction of Individual Subsystems

Table 1. Empirical contraction rates across 5 subsystems (exp_contraction.sx).

Subsystem	Theoretical \(\rho_i\)	Empirical \(\hat{\rho}_i\)	Steps to \(10^{-6}\)
Quadratic-1	0.80	0.79	132
Quadratic-2	0.85	0.84	178
Rosenbrock	0.92	0.91	287
Rastrigin (local)	0.88	0.87	215
Coupled oscillator	0.75	0.74	108

5.3 Gradient Projection as an Implementation

The projection preserves first-order descent of each objective: \( \tilde{g}_i \cdot g_i = \|\tilde{g}_i\|^2 \geq 0 \). It is not a convergence mechanism. Three limitations are recorded from the experiments (exp_gradient_interference.sx, exp_pcgrad_refinement.sx): the projected dynamics admit non-unique limit points, since any point where the projected aggregate vanishes is stationary; in one dimension every conflict is antiparallel, so the projection removes the whole gradient and the iteration stalls; and at moderate step sizes the alternation between projected and unprojected steps produces period-two orbits. The cosine-scaled variant with \( \alpha = |\cos(g_i, g_j)| \) leaves a residual conflicting component and shares these properties.

5.4 Invariant Set Existence

Table 5. Invariant set validation (exp_invariants.sx).

Experiment	Steps	Escape Violations	\(d(\theta_T, \Omega)\)
3-subsystem quadratic	15,000	0	3.1e-8
5-subsystem coupled	15,000	0	7.2e-7
10-subsystem random	15,000	0	4.5e-5

5.5 Convergence Score Lemma

Table 7. Convergence score validation (exp_convergence_order.sx).

System	Theoretical Rate	Empirical Rate	Final \(S\)
Cooperative K=3	0.80	0.79	0.9997
Mixed K=5	0.91	0.90	0.9974
Adversarial K=5	0.95	0.94	0.9926

6. Diagnostics

6.1 Cosine-Scaled Projection

Standard PCGrad (Yu et al. 2020) projects the gradient of task \( i \) onto the normal plane of task \( j \) whenever the inner product is negative. This binary decision ignores the magnitude of the conflict. The cosine-scaled variant introduces a continuous scaling factor \( |\cos(g_i, g_j)| \), so that near-orthogonal gradients receive minimal correction while antiparallel gradients receive full projection. With \( \alpha = 1 \) it reduces to PCGrad; with \( \alpha = |\cos| \) a residual conflicting component remains. Both are implementations with the limitations recorded in Section 5.3.

6.2 Interaction Matrix

The interaction matrix \( M \) is the central structural object of the framework. Its diagonal entries are individual contraction rates (which are well-studied), and its off-diagonal entries capture pairwise interference through gradient cosines weighted by contraction rates.

For non-negative coupling, the spectral radius \( \sigma(M) \) is the single scalar in the sufficient condition of Theorem 1. This replaces the standard approach of analysing each pair of subsystems individually, which scales as \( O(K^2) \) and does not capture higher-order interactions.

In the experiments (Conjecture 6.8), the eigenvectors of \( M \) reveal the coalition structure of the subsystems: clusters of subsystems that are tightly coupled form blocks in the eigendecomposition.

6.3 Step-Ratio Convergence Score

The convergence score \( S = 1 - \frac{\Delta_{\text{late}}}{\Delta_{\text{early}}} \) requires no knowledge of the target equilibrium. It uses only the trajectory itself, comparing the average step size in the late phase to the early phase. This makes it applicable as a runtime diagnostic in systems where the fixed point is unknown.

The score is a diagnostic of the trajectory's own step sizes. It is bounded below on contracting systems by Proposition 3.7 and is used in the exploratory runs of Section 8 as a monitor; it is not a test that separates periodic from chaotic dynamics.

7. The Balance Residual and B-Flow

7.1 The Balance Identity

Table 8. Numerical check of the identity \( B = 1 + 2I \) across configurations (floating-point residual) (exp_iratio_proof.sx, exp_iratio_proof_statistical.sx).

Test Class	Tests	Pass	Max Error
K=2, analytical	28	28	2.22e-16
K=3, analytical	24	24	4.44e-16
K=5, analytical	20	20	8.88e-16
K=10, random	36	36	1.78e-15
K=100, random	30	30	3.55e-14
Statistical (70 configs)	70	70	1.0e-12

7.2 B-Flow

Gradient descent on \( B \) (B-flow) moves the parameter towards points where the subsystem gradients cancel. Minimising the total loss \( L = \sum_i L_i \) drives each subsystem toward its own minimum; minimising \( B \) drives the aggregate gradient to zero, which is a different objective. A two-phase strategy, loss-flow followed by B-flow once \( B \) is below a threshold, is used in the exploratory runs (exp_balance_residual.sx, exp_equilibrium_mapping.sx). No comparison of final precision between the two flows is claimed; the two objectives have different minimisers.

8. Exploratory Applications

The following runs illustrate the diagnostics on systems outside the scope of Theorem 1. They are exploratory and establish nothing beyond what each experiment prints.

8.1 Generative Adversarial Networks

A GAN is a two-subsystem composition with adversarial coupling, which is outside the non-negative case of Theorem 1. The step-ratio score \( S \) applied to GAN training (exp_gan_convergence.sx) drops below 0.5 some hundreds of steps before the discriminator loss diverges in the runs reported; whether this is a usable early warning has not been tested beyond those runs.

8.2 ODE Solvers and Stiff Systems

Splitting methods for stiff ODEs decompose the right-hand side into fast and slow components, each handled by a specialised integrator (exp_ode_solvers.sx). The interaction matrix records the coupling between the integrators; the condition \( \sigma(M) < 1 \) is a sufficient condition for the composed iteration, not a replacement for the stability analysis of the scheme.

8.3 Compiler Pass Scheduling

Treating each compiler optimisation pass as a subsystem and monitoring \( B \) along the pass sequence (exp_compiler_passes.sx) gives a per-program schedule adaptation. The experiment reports the schedules found; no compile-time benchmark is claimed.

9. Conjectures

The following conjectures arise from the experimental programme. Each is stated and assigned a status based on the experiments cited; “Validated” and “Supported” refer to those runs only.

10. Discussion

10.1 Limitations

Scope of the theorem. Theorem 1 is the classical block-contraction theorem applied to sequentially composed updates. It requires non-negative coupling; adversarial coupling (a GAN, a zero-sum game) is outside it. It certifies a unique fixed point of the composition, which in general is not a minimiser of any individual objective and not the centroid of the individual minimisers.

Gradient access. The framework assumes access to the true gradient \( \nabla L_i(\theta) \) of each subsystem. In stochastic settings only noisy estimates are available; the score \( S \) is empirically robust to moderate noise (Appendix B.4) but no stochastic version of the theorem is given.

Computational cost. Assembling the interaction matrix \( M \) at each step requires \( O(K^2 n) \) inner products, where \( K \) is the number of subsystems and \( n \) the parameter dimension.

Status of the experiments. The experiments check identities (Section 7.1), measure contraction rates against their theoretical values (Section 5.2), and illustrate the diagnostics. They are not proofs of anything beyond the identities they evaluate.

10.2 Role of the Assumptions

Without C1 (contraction): two expansive maps on \( [0,1] \) can compose to a map with no fixed point in the interior. Without C2 (bounded interaction): even with individually contractive subsystems, if \( \sigma(\Lambda) > 1 \) the coupling can amplify perturbations and the composition can oscillate indefinitely. Without non-negativity: the Perron–Frobenius weight vector need not exist, and the weighted-norm argument does not go through; this is the adversarial case.

10.3 Future Directions

Stochastic extension. Replacing deterministic gradients with stochastic estimates \( \hat{g}_i = \nabla L_i(\theta) + \xi_i \), where \( \xi_i \) has bounded variance \( \sigma_i^2 \), should yield a stochastic version of the main theorem with convergence rate \( O(\sigma(M)^t + \sigma_{\max}^2 / t) \). The key challenge is controlling the interaction between gradient noise and the projection operator.

Online \( M \) learning. The interaction matrix \( M \) is currently computed from measured gradient cosines. A natural extension is to learn \( M \) itself via a meta-gradient, treating the matrix entries as learnable parameters. This would allow the framework to adapt to non-stationary coupling structures, relevant to continual learning and curriculum design.

Infinite-dimensional extension. For PDE-constrained optimisation and neural operator learning, the parameter space is a function space rather than \( \mathbb{R}^n \). Extending the interaction matrix framework to Hilbert or Banach spaces requires replacing the Euclidean inner products with appropriate function-space inner products and establishing compactness in the weak topology.

11. Robustness

The framework's practical utility depends on its sensitivity to parameter choices. We assess robustness along four axes, validated by exp_sensitivity.sx.

11.1 Learning Rate Sensitivity

Contraction of a gradient-descent subsystem (Proposition 3.1) requires \( \eta_i \leq \frac{2}{\mu_i + L_i} \). Empirically, the framework tolerates learning rates up to 3 orders of magnitude above and below the optimal value. The convergence score \( S \) remains above 0.9 for \( \eta \in [10^{-4}, 10^{-1}] \) in all tested configurations. At \( \eta > 0.5 \), oscillations appear but the system still converges (though more slowly).

11.2 Component Scaling

The normalised Lyapunov function ensures that the analysis is invariant to rescaling of individual subsystem losses. Multiplying \( L_i \) by a constant \( c_i > 0 \) rescales \( g_i \) but does not change the cosine similarities, so the off-diagonal entries of \( M \) are invariant. The diagonal entries scale by \( c_i \), which can change \( \sigma(M) \), but the framework remains valid as long as the contraction rates after scaling satisfy \( \rho_i < 1 \).

11.3 Dimensionality

The contraction modulus \( \sigma(\Lambda) \) is independent of the parameter dimension \( n \). The interaction matrix is \( K \times K \), where \( K \) is the number of subsystems, not the number of parameters. This makes the framework applicable to high-dimensional problems without additional cost. Experiments ran with \( n \) from 2 to 1,000 dimensions (Appendix B.3).

11.4 Convergence Speed

Table 11. Convergence speed across configurations.

K	\(\sigma(M)\)	Steps to \(S > 0.99\)	Steps to \(B < 10^{-10}\)
2	0.72	45	230
3	0.84	110	580
5	0.91	280	1,400
10	0.96	720	3,600

Convergence speed scales logarithmically with \( 1/\sigma(M) \), as expected from the geometric rate bound. The practical implication is that systems with \( \sigma(M) > 0.95 \) may require thousands of iterations, while systems with \( \sigma(M) < 0.8 \) converge in under 100 steps.

12. Reproducibility

12.1 Experiment Index

Table 12. Experiment index. All experiments implemented in Simplex.

File	Domain	Relates to	What it prints
exp_contraction.sx	Core	Prop 3.1	Empirical vs theoretical contraction rates, 5 subsystems
exp_gradient_interference.sx	Core	Prop 3.2	Projection orthogonality check; projected-dynamics behaviour
exp_lyapunov.sx	Core	Section 4 remark	Normalised distance function along trajectories
exp_interaction_matrix.sx	Core	Theorem 1	Spectral radius of the measured interaction matrix
exp_convergence_order.sx	Core	Prop 3.7	Convergence score \( S \) against the contraction bound
exp_invariants.sx	Core	Prop 3.5	Escape checks over 15K steps
exp_timescale.sx	Core	Exploratory	Behaviour under separated learning rates
exp_anima_deep.sx	Cognitive	Conj. 6.5	Calibration error with and without a prior term
exp_anima_correlated.sx	Cognitive	Exploratory	Correlated beliefs with a misaligned prior
exp_skeptical_annealing.sx	Cognitive	Conj. 6.3, 6.5	Constant vs annealed prior weight across horizons
exp_belief_cascade.sx	Cognitive	Conj. 6.4	Chain topology recovered from \( M \)
exp_memory_dynamics.sx	Cognitive	Conj. 6.6–6.10	Forgetting rate, transfer threshold, self-reference
exp_chaos_boundary.sx	Dynamics	Exploratory	\( S \) along the logistic-map parameter
exp_s_vs_lyapunov.sx	Dynamics	Exploratory	\( S \) alongside the Lyapunov exponent
exp_nash_equilibrium.sx	Games	Exploratory	Iterated Prisoner's Dilemma with an added prior term
exp_iratio_proof.sx	Core	Section 7.1	Floating-point residual of \( B = 1 + 2I \), \( K = 2 \ldots 100 \)
exp_iratio_proof_statistical.sx	Core	Section 7.1	Same identity on 70 random configurations
exp_balance_residual.sx	Core	Section 7.2	Final \( B \) under loss-flow, B-flow and two-phase
exp_iratio_applications.sx	Cross-domain	Section 7.1	\( B \) evaluated on five example systems
exp_equilibrium_mapping.sx	Optimisation	Section 7.2	B-flow trajectories
exp_symmetry_breaking.sx	Core	Conj. 6.2, 6.8	Group structure from the eigenvectors of \( M \)
exp_sensitivity.sx	Robustness	Section 11	Learning-rate and dimension sweeps
exp_code_gates.sx	Code	Exploratory	\( S \) on a gated code-structure model
exp_compiler_passes.sx	Compiler	Section 8.3	Per-program pass schedules
exp_structure_discovery.sx	Topology	Exploratory	Constraint graph from gradient topology
exp_gan_convergence.sx	GANs	Section 8.1	\( S \) during GAN training

12.2 How to Run

All experiments require the Simplex compiler. Build from source:

git clone https://github.com/senuamedia/lab.git
cd lab
./build.sh
./run_all.sh            # Core theorem experiments
./run_math_tests.sh     # 188 compiler math tests

Individual experiments can be compiled and run with:

./build/sxc exp_iratio_proof.sx -o exp_iratio_proof
./exp_iratio_proof

12.3 Citation

@article{higgins2026unified,
  title   = {Unified Adaptation Theorem: Convergence of Composed
             Adaptive Systems via Interaction Matrices and
             Higher-Order Convergence Diagnostics},
  author  = {Higgins, Rod},
  year    = {2026},
  url     = {https://lab.senuamedia.com/papers/unified-adaptation-theorem.html},
  doi     = {10.5281/zenodo.22714409},
  note    = {v3}
}

References

Appendix A: Derivations

A.1 The Identity \( B = 1 + 2I \)

We record the identity behind the balance residual. Let \( g_1, \ldots, g_K \in \mathbb{R}^n \) be the gradient vectors of \( K \) subsystems. The squared norm of the aggregate gradient is

\[ \left\|\sum_{i=1}^K g_i\right\|^2 = \sum_{i=1}^K \|g_i\|^2 + 2\sum_{i < j} g_i \cdot g_j. \]

Define \( D = \sum_{i=1}^K \|g_i\|^2 \) (the total self-energy) and \( C = \sum_{i \[ \left\|\sum_{i=1}^K g_i\right\|^2 = D + 2C = D(1 + 2I). \]

Dividing by \( D > 0 \) gives \( B = 1 + 2I \). Because \( B \geq 0 \), \( I \geq -\tfrac12 \) always, and \( B = 0 \) exactly when the aggregate gradient vanishes. The identity is dimension-free and holds for any \( K \geq 2 \); it says nothing about whether a given dynamics reaches such a point.

A.2 Local Hessian of the Balance Residual

Write \( N(\theta) = \|\sum_i g_i(\theta)\|^2 \) and \( D(\theta) = \sum_i \|g_i(\theta)\|^2 \), so \( B = N/D \). At a point \( \theta^* \) with \( \sum_i g_i(\theta^*) = 0 \) and \( D(\theta^*) > 0 \) we have \( N(\theta^*) = 0 \) and \( \nabla N(\theta^*) = 2 J^{\mathsf T} \sum_i g_i(\theta^*) = 0 \), where \( J = \sum_i \nabla g_i(\theta^*) = \sum_i \nabla^2 L_i(\theta^*) \). The quotient rule for second derivatives then leaves only the term in \( \nabla^2 N \):

\[ \nabla^2 B(\theta^*) = \frac{\nabla^2 N(\theta^*)}{D(\theta^*)} = \frac{2\, J^{\mathsf T} J}{D(\theta^*)} . \]

This matrix is positive semidefinite, and positive definite exactly when \( \ker J = \{0\} \). In that case \( B \) is locally strongly convex with constant \( \mu_B = 2\sigma_{\min}(J)^2 / D(\theta^*) \), and gradient descent on \( B \) with step \( \eta \) converges linearly at rate \( 1 - \eta \mu_B \) in a neighbourhood of \( \theta^* \).

A.3 Convergence Score Lower Bound

We derive Proposition 3.7. Let the composed system contract with rate \( \sigma < 1 \) towards \( \theta^* \), and put \( D_0 = \|\theta_0 - \theta^*\| \). Then \( \|\theta_t - \theta^*\| \leq \sigma^t D_0 \), so the step sizes decay geometrically:

\[ \|\theta_{t+1} - \theta_t\| \leq \|\theta_{t+1} - \theta^*\| + \|\theta_t - \theta^*\| \leq (1 + \sigma)\sigma^t D_0 \leq 2\sigma^t D_0 . \]

The early-window mean step size over \( t \in [0, T/2) \) is bounded below by its first term, and the first step satisfies \( \|\theta_1 - \theta_0\| \geq \|\theta_0 - \theta^*\| - \|\theta_1 - \theta^*\| \geq (1 - \sigma) D_0 \):

\[ \Delta_{\text{early}} = \frac{2}{T}\sum_{t=0}^{T/2-1} \|\theta_{t+1} - \theta_t\| \geq \frac{2(1-\sigma)D_0}{T} . \]

The late-window mean over \( t \in [T/2, T) \) is bounded above by the geometric tail:

\[ \Delta_{\text{late}} = \frac{2}{T}\sum_{t=T/2}^{T-1} \|\theta_{t+1} - \theta_t\| \leq \frac{2}{T}\sum_{t=T/2}^{T-1} 2\sigma^t D_0 \leq \frac{4D_0}{T} \cdot \frac{\sigma^{T/2}}{1-\sigma} . \]

Therefore

\[ S = 1 - \frac{\Delta_{\text{late}}}{\Delta_{\text{early}}} \geq 1 - \frac{2\sigma^{T/2}}{(1-\sigma)^2}, \]

and \( S \to 1 \) as \( T \to \infty \) for any \( \sigma < 1 \). The constant \( 2/(1-\sigma)^2 \) is what the argument gives; it is not sharp for small \( T \), and the bound is vacuous until \( \sigma^{T/2} < (1-\sigma)^2/2 \).

Appendix B: Additional Experimental Results

B.1 Learning Rate Sensitivity

Table B1. Convergence metrics across learning rates (K=5, n=25, exp_sensitivity.sx).

Learning Rate \(\eta\)	Final \(V\)	Final \(S\)	Final \(B\)	Steps to \(S > 0.99\)
1e-4	8.2e-7	0.9998	3.1e-14	4,200
1e-3	3.1e-7	0.9999	1.7e-15	1,400
1e-2	1.1e-6	0.9997	8.8e-16	280
1e-1	4.5e-5	0.9941	2.3e-11	520
5e-1	1.2e-3	0.9102	4.7e-6	1,800

B.2 Scaling with K

Table B2. Convergence metrics as a function of subsystem count K (n=25, exp_interaction_matrix.sx).

K	\(\sigma(M)\)	Final \(V\)	Final \(S\)	Steps to \(B < 10^{-10}\)
2	0.72	1.8e-8	0.9999	230
5	0.91	1.1e-6	0.9974	1,400
10	0.96	8.3e-5	0.9926	3,600
20	0.982	2.1e-3	0.9847	8,900
50	0.993	1.4e-2	0.9712	22,000

B.3 High-Dimensional Validation

Table B3. Convergence metrics across parameter dimensions (K=5, exp_sensitivity.sx).

Dimension \(n\)	\(\sigma(M)\)	Final \(V\)	Final \(S\)	Lyapunov Violations
2	0.91	9.7e-7	0.9981	0
10	0.91	1.1e-6	0.9974	0
25	0.91	1.3e-6	0.9971	0
50	0.91	1.5e-6	0.9968	0
100	0.91	1.8e-6	0.9963	0

B.4 Stochastic Gradient Experiments

Table B4. Effect of gradient noise on convergence (K=5, n=25, 10 seeds per row).

Noise \(\sigma_\xi\)	Mean Final \(V\)	Std Final \(V\)	Mean \(S\)	Convergence Rate (%)
0 (exact)	1.1e-6	0.0	0.9974	100
0.01	3.8e-5	1.2e-5	0.9951	100
0.1	2.7e-3	8.4e-4	0.9832	100
0.5	4.1e-2	1.5e-2	0.9214	90
1.0	1.8e-1	6.3e-2	0.8147	70