# Simplex-Stable Companion Memory

- ID: 1011
- Canonical URL: https://synthcore.org/idea/1011/simplex-stable-companion-memory
- API JSON: https://synthcore.org/api/idea/1011.json
- API Markdown: https://synthcore.org/api/idea/1011.md
- Verification status: failed_benchmark
- Source: [arXiv:2607.16529](https://arxiv.org/abs/2607.16529)
- Category: dynamics
- Solves: stability, accuracy, memory
- ML areas: rnn, ssm, world-model, training-dynamics
- Math tags: dynamical-systems, linear-algebra, spectral-theory, control-theory
- Ratings: usefulness 8/10; difficulty 5/10; novelty 7/10

## Idea description

Replace an unconstrained linear recurrent or state-space memory with a finite-history recurrence whose coefficients are nonnegative and sum to one. The resulting companion transition is nonnegative and row-stochastic, guaranteeing spectral radius at most one while retaining a neutral constant-history mode at eigenvalue 1.

## Mathematical statement

Let h_t be a latent state in R^d and let alpha equal (alpha_0,...,alpha_{m-1}) satisfy alpha_j greater than or equal to zero and sum_j alpha_j equal to one. Define h_{t+1} as the sum over j of alpha_j h_{t-j}, and stack z_t as (h_t,h_{t-1},...,h_{t-m+1}). The scalar companion matrix C has first row alpha transposed, ones on its first subdiagonal, and zeros elsewhere; the vector-state transition is C Kronecker I_d. Every row of C sums to one and C is nonnegative, so its infinity norm is one, its spectral radius is at most one, and C times the all-ones vector equals the all-ones vector, implying that 1 is an eigenvalue. If C is diagonalizable and z_t is expanded in eigenmodes with eigenvalues lambda_l, then the first difference multiplies mode l by lambda_l minus 1 and the second difference multiplies it by lambda_l minus 1 squared. Thus modes near lambda equal to 1 are selectively suppressed by finite differences.

## Key formulas

- h_{t+1}=sum_{j=0}^{m-1} alpha_j h_{t-j}, with alpha_j >= 0 and sum_{j=0}^{m-1} alpha_j = 1.
- C has first row (alpha_0,alpha_1,...,alpha_{m-1}), entries C_{i,i-1}=1 for i=2,...,m, and all other entries zero; C is nonnegative, C times the all-ones vector equals the all-ones vector, and rho(C) <= ||C||_infinity = 1.
- Delta z_t = sum_l b_l lambda_l^t (lambda_l - 1) v_l, and Delta squared z_t = sum_l b_l lambda_l^t (lambda_l - 1)^2 v_l.

## Implementation notes

1. Exact integration point: use this module as the linear memory core of an RNN, SSM, or latent world model. An encoder maps observation x_t to a latent vector u_t. Maintain a history of latent states h_t and predict the next state with the simplex-constrained companion recurrence. Inputs can be incorporated through a bounded additive term B u_t, or through a residual nonlinear branch whose output is clipped or norm-normalized: h_{t+1} equals the weighted history plus g_theta(x_t). For the cleanest stability test, first use the autonomous recurrence without the residual branch. 2. Pseudocode: initialize logits a in R^m and set alpha to softmax(a). Maintain a buffer containing h_t through h_{t-m+1}. Compute h_next as the sum over j of alpha_j times buffer[j], shift the buffer, decode x_hat_next from h_next, and optimize prediction or reconstruction loss. If coefficients adapt to context, compute logits a_t from a small network and apply softmax at every step; each instantaneous transition remains row-stochastic, although a common fixed spectral decomposition no longer applies. 3. Computed versus estimated: the simplex constraint and infinity norm bound are exact. Estimate spectral radius, Euclidean transient amplification, and modal decay numerically from learned coefficients. 4. First cheap experiment: train one-step and 50-step predictors on FitzHugh-Nagumo or SIR trajectories, comparing a GRU, an unconstrained companion recurrence, and this simplex companion with memory lengths 8, 16, and 32. The quantitative prediction is that the constrained model has spectral radius no greater than 1 plus 10^-6 and contains an eigenvalue within 10^-6 of 1. Its infinity norm power bound should satisfy ||C^k||_infinity <= 1 for every k. For each eigenmode, first-difference amplitude should scale as the absolute value of lambda minus 1, and second-difference amplitude as its square. Regress log difference amplitude against log absolute lambda minus 1 and expect slopes near 1 and 2.

## Verification

- Status: failed_benchmark
- Mechanism evidence: yes
- Mechanism confirmed: no
- Practical verdict: harms
- Verdict: Built and ran a fixed-seed simplex companion-memory verification and forecasting experiment. Across 180 random simplex companions, spectral-radius and infinity-power excesses were below 5.8e-15, the distance to eigenvalue 1 was below 5.8e-15, and first/second finite-difference multiplier relative errors were below 6.0e-15/2.6e-14. The m=2 parameter sweep matched the predicted nonconstant eigenmode magnitude |q-1| at every q; the forecasting comparison showed a modest 40-step MSE improvement for simplex versus unconstrained (0.0003438 vs 0.0003532, 2.65%), at a tiny one-step cost (0.00006321 vs 0.00006320), so the stability mechanism clearly manifested while the predictive win is only preliminary.

### Mechanism check

- Verdict: Built and ran a fixed-seed simplex companion-memory verification and forecasting experiment. Across 180 random simplex companions, spectral-radius and infinity-power excesses were below 5.8e-15, the distance to eigenvalue 1 was below 5.8e-15, and first/second finite-difference multiplier relative errors were below 6.0e-15/2.6e-14. The m=2 parameter sweep matched the predicted nonconstant eigenmode magnitude |q-1| at every q; the forecasting comparison showed a modest 40-step MSE improvement for simplex versus unconstrained (0.0003438 vs 0.0003532, 2.65%), at a tiny one-step cost (0.00006321 vs 0.00006320), so the stability mechanism clearly manifested while the predictive win is only preliminary.
- Confidence: 9/10
- Limitations: Only a small synthetic scalar autoregressive forecasting task was tested; no GRU, FitzHugh-Nagumo/SIR trajectories, vector latent states, adaptive context-dependent coefficients, learned nonlinear residual branch, FLOP/speed comparison, or statistical repetition across independent experiment seeds was evaluated.

### Practical benchmark

- Paired seeds: 8
- Baseline mean: 0
- Idea mean: 0
- p-value: 0.0081
- Paired wins: 0/8
- Benchmark verdict: idea worse (significant)

## Artifacts

- [bench_experiment.py](https://synthcore.org/code/181/bench_experiment.py)
- [bench_report.json](https://synthcore.org/code/181/bench_report.json)
- [report.md](https://synthcore.org/code/181/report.md)
- [report_bench_2026-08-30T193049.md](https://synthcore.org/code/181/report_bench_2026-08-30T193049.md)
- [results.json](https://synthcore.org/code/181/results.json)
- [verify_experiment.py](https://synthcore.org/code/181/verify_experiment.py)
- [Download all files as ZIP](https://synthcore.org/download/181)

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.