# Spectral Basin Allocation for Multimodal Neural Memories

- ID: 3036
- Canonical URL: https://synthcore.org/idea/3036/spectral-basin-allocation-for-multimodal-neural-memories
- API JSON: https://synthcore.org/api/idea/3036.json
- API Markdown: https://synthcore.org/api/idea/3036.md
- Verification status: mech_ok_no_baseline
- Source: [arXiv:2609.02047](https://arxiv.org/abs/2609.02047)
- Category: architecture
- Solves: stability, accuracy, sample-efficiency
- ML areas: rnn, ssm, training-dynamics, embedding
- Math tags: dynamical-systems, spectral-theory, linear-algebra, bifurcations
- Ratings: usefulness 7/10; difficulty 7/10; novelty 8/10

## Idea description

Use several phase-locked states as distinct attractors of one recurrent network and shape their basin asymmetry through the phase-delay composite spectrum. This creates a controllable associative-memory architecture in which a desired memory receives a larger basin without adding a separate classifier or explicit nearest-neighbor lookup.

## Mathematical statement

Let \(q\in\{1,\ldots,M\}\) index target phase-locked patterns \(\psi^{(q)}\). For each pattern define \(C^{(q)}_{ij}=A_{ij}\cos(\psi_j^{(q)}-\psi_i^{(q)}-\alpha_{ij})\) and \(L^{(q)}=\operatorname{diag}(C^{(q)}\mathbf 1)-C^{(q)}\). The local contraction rate of attractor \(q\) is \(r_q=K\min_{k\ne\mathrm{gauge}}\operatorname{Re}\lambda_k(L^{(q)})\), and its slowest local mode has time constant approximately \(1/r_q\). The paper's transferable mechanism is that this same composite spectrum organizes basin sizes, so spectral shaping can be used as a basin-control proxy. Define desired margins \(\gamma_q\) and basin targets \(b_q\), and optimize \(\mathcal L=\sum_q[\gamma_q-r_q]_+^2+\lambda_b\sum_q(\widehat B_q-b_q)^2\), where \(\widehat B_q\) is the measured fraction of initial states converging to attractor \(q\).

## Key formulas

- $$C^{(q)}_{ij}=A_{ij}\cos(\psi_j^{(q)}-\psi_i^{(q)}-\alpha_{ij}),\qquad L^{(q)}=\operatorname{diag}(C^{(q)}\mathbf 1)-C^{(q)}$$
- $$r_q=K\min_{k\ne\mathrm{gauge}}\operatorname{Re}\lambda_k\!\left(L^{(q)}\right),\qquad \|\delta\theta(t)\|\approx e^{-r_qt}\|\delta\theta(0)\|$$
- $$\mathcal L_{\mathrm{basin}}=\sum_{q=1}^{M}[\gamma_q-r_q]_+^2+\lambda_b\sum_{q=1}^{M}(\widehat B_q-b_q)^2$$

## Implementation notes

1. Integration point: implement an associative-memory neural ODE or recurrent layer with one shared graph \(A\), shared delay matrix \(\alpha\), and \(M\) prototype phase patterns \(\psi^{(q)}\). Decode the attractor index from the final phase state with a cosine-similarity head. 2. Pseudocode: initialize prototypes from class-conditioned latent embeddings; for every prototype compute \(C^{(q)}\), \(L^{(q)}\), and the smallest non-gauge real eigenvalue \(r_q\); simulate short trajectories from noisy versions of each prototype; identify the converged attractor; estimate \(\widehat B_q=N_q/N\) from uniformly sampled initial states; optimize task loss plus \(\mathcal L_{\mathrm{basin}}\). To enlarge one target basin, increase its requested \(\gamma_q\) or target \(b_q\), while retaining positive margins for other memories. 3. Computed from the paper's mechanism: phase-delay composite spectra and their relationship to multistability and basin organization. Estimated empirically: global basin fractions, because the exact spectrum-to-volume mapping is network-dependent and should not be assumed universal. 4. First cheap experiment: use an 8-dimensional phase attractor memory with four prototype patterns and noisy-MNIST class embeddings. Compare unconstrained training with spectral basin allocation using 10,000 random initial states per model. The predicted signature is monotonic: increasing \(r_q\) should increase the measured basin fraction \(\widehat B_q\), and attractors with larger \(r_q\) should show perturbation-recovery slopes near \(-r_q\) on a semilog plot. Falsify the mechanism if basin fraction fails to correlate positively with \(r_q\) across at least 20 independently initialized graphs.

## Verification

- Status: mech_ok_no_baseline
- Mechanism evidence: yes
- Mechanism confirmed: yes
- Practical verdict: no_effect
- Verdict: Built a reproducible 12-oscillator delayed-Kuramoto associative-memory MVP with composite-Laplacian spectral rates, phase-locked prototypes, direct integration, and empirical basin classification. The local contraction claim was clearly verified: predicted rate 0.5070508 matched the Jacobian rate 0.5070508, and the measured semilog decay slope was -0.50834. Coupling reallocation increased the target q=1 basin from 0.0777 to 0.1677, demonstrating controllable basin asymmetry, but its local contraction rate decreased from 0.3660 to 0.2588, so the stronger universal rate-to-basin claim was not supported.

### Mechanism check

- Verdict: Built a reproducible 12-oscillator delayed-Kuramoto associative-memory MVP with composite-Laplacian spectral rates, phase-locked prototypes, direct integration, and empirical basin classification. The local contraction claim was clearly verified: predicted rate 0.5070508 matched the Jacobian rate 0.5070508, and the measured semilog decay slope was -0.50834. Coupling reallocation increased the target q=1 basin from 0.0777 to 0.1677, demonstrating controllable basin asymmetry, but its local contraction rate decreased from 0.3660 to 0.2588, so the stronger universal rate-to-basin claim was not supported.
- Confidence: 7/10
- Limitations: This MVP uses manually selected coupling weights rather than optimizing the proposed basin-aware objective, synthetic phase-gradient prototypes rather than MNIST embeddings, and no learned neural layer. The 20-graph correlation check is weak because only two attractors had positive rates in the sampled systems; no 10,000-sample or GPU experiment was performed.

### Practical benchmark

- Paired seeds: 8
- Baseline mean: 0
- Idea mean: 0
- p-value: 0.1228
- Paired wins: 1/8
- Benchmark verdict: no significant win

## Artifacts

- [README.md](https://synthcore.org/code/1221/README.md)
- [bench_experiment.py](https://synthcore.org/code/1221/bench_experiment.py)
- [bench_report.json](https://synthcore.org/code/1221/bench_report.json)
- [report.md](https://synthcore.org/code/1221/report.md)
- [report_bench_2026-09-03T124505.md](https://synthcore.org/code/1221/report_bench_2026-09-03T124505.md)
- [results.json](https://synthcore.org/code/1221/results.json)
- [run_experiment.py](https://synthcore.org/code/1221/run_experiment.py)
- [Download all files as ZIP](https://synthcore.org/download/1221)

## Disclaimer

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