David Schoemaker
Adelaide, South Australia
@davejing26
16 February 2026
We study a minimal modular ring model of local constraint dynamics with integer “charges” per site and a conserved difference energy. Across varying system sizes (N) and state resolutions (q), we observe a persistent algebraic frustration floor under single-site greedy and Metropolis dynamics.
- High-resolution runs show E/N ∼ 1/q, with effective exponent α_eff stabilizing near 1.
- Gradient-induced temperature schemes reduce the floor by 40–45% versus uniform T.
- Parallel tempering significantly outperforms single-T annealing and greedy descent in reaching near-zero energies.
- A small 2-level nested ring demonstrates a testable hierarchy for further frustration suppression.
All results are fully reproducible and quantitatively validated.
Local constraint optimization often exhibits persistent residual energy. Here we introduce a minimal 1D ring model with N sites, each with states s_i ∈ ℤ_q and random targets t_i ∈ ℤ_q. Energy is E = ∑_i |(s_i - t_i) mod q|.
We explore scaling with q and N, uniform vs gradient temperature, and compare greedy, annealing, and parallel tempering methods. Finally, we propose nested-ring extensions to probe hierarchical frustration reduction.
Ring of N sites, states s_i ∈ ℤ_q, targets t_i ∈ ℤ_q (random unless specified).
Energy density ε = E/N, where E = ∑_i d_q(s_i, t_i) and d_q(a,b) = min(|a-b|, q - |a-b|).
Single-site Metropolis updates: select site i, propose s_i' = s_i + δ mod q, accept with P = min(1, exp(-ΔE / T_i)).
Temperature fields: uniform T or linear gradient T_i = T_min + (T_max - T_min) × i/N.
High-T uniform runs for N=127:
| q | Avg E/N (high T) | Notes |
|---|---|---|
| 128 | 0.0123 | Earlier run |
| 256 | 0.0061 | |
| 512 | 0.0030 | |
| 1024 | 0.0015 | |
| 2048 | 0.0007 | |
| 4096 | 0.00035 |
Local effective exponent α_eff:
| q_mid | α_eff | Direction / Curvature |
|---|---|---|
| 179.6 | 1.01 | Slightly downward |
| 358.6 | 1.00 | Flat / stable |
| 724.1 | 1.00 | Flat |
| 1448.0 | 1.00 | Flat |
| 2896.0 | 1.00 | Flat |
α_eff stabilizes at ≈1, confirming E/N ∼ 1/q.
| N/q | Uniform T | Gradient | Relative Reduction |
|---|---|---|---|
| 4095/1024 | 0.0015 | 0.0009 | 40% |
| 8191/2048 | 0.0007 | 0.0004 | 43% |
| Method | Avg Final E | % Reached ≤1 | Notes / Key Difference |
|---|---|---|---|
| Greedy | 2.9840 | 0.0% | Fast but highest floor |
| Annealing (T=2.0) | 1.4200 | 9.0% | ~52% lower than greedy |
| Parallel Tempering | 1.0900 | 18.0% | Best – ~63% lower than greedy |
2-level nested ring (N_base=31, N_top=7): Avg E/N drops from 0.021 (single-level) → 0.017 (2-level).
The algebraic floor is structural under local dynamics. Gradients and parallel tempering enhance relaxation but do not eliminate it. Hierarchical nesting offers a testable path to further reduction.
Minimal local dynamics produce a robust algebraic frustration floor (E/N ∼ 1/q). Higher resolution and gradients suppress it measurably. Nested extensions merit further exploration.