# Certified Fold Map for Recurrent Fixed Points

- ID: 1928
- Canonical URL: https://synthcore.org/idea/1928/certified-fold-map-for-recurrent-fixed-points
- API JSON: https://synthcore.org/api/idea/1928.json
- API Markdown: https://synthcore.org/api/idea/1928.md
- Verification status: mechanism_failed
- Source: [arXiv:2608.07119](https://arxiv.org/abs/2608.07119)
- Category: dynamics
- Solves: stability, accuracy
- ML areas: rnn, ssm, training-dynamics
- Math tags: dynamical-systems, bifurcations, linear-algebra, numerical-analysis
- Ratings: usefulness 8/10; difficulty 6/10; novelty 8/10

## Idea description

Apply interval Krawczyk certification to the augmented equations for a recurrent-network fixed point and a singular state Jacobian. This produces a rigorous local certificate for the gain or feedback value at which two fixed points merge or disappear, allowing training or inference to avoid parameter boxes containing an uncertified fold.

## Mathematical statement

Let a recurrent update be x_(t+1)=Phi(x_t,mu), with state x in R^n and scalar parameter mu. Define F(x,mu)=Phi(x,mu)-x and J(x,mu)=D_x F(x,mu). A saddle-node candidate is a zero of g(z)=[F(x,mu); det(J(x,mu))], where z=(x,mu) is in R^(n+1). For a search box [z], let z_c=midpoint([z]), let C approximate the inverse of Dg(z_c), and let [Dg]([z]) enclose the Jacobian throughout the box. The Krawczyk set is K([z])=z_c-Cg(z_c)+(I-C[Dg]([z]))([z]-z_c). Strict inclusion K([z]) subset int([z]) certifies a unique zero; disjointness K([z]) intersect [z]=emptyset excludes the box. The contraction quantity q=norm(I-C[Dg]([z])) predicts whether certification succeeds.

## Key formulas

- $$F(x,\mu)=\Phi(x,\mu)-x,\qquad J(x,\mu)=D_xF(x,\mu),\qquad g(x,\mu)=\begin{bmatrix}F(x,\mu)\\ \det J(x,\mu)\end{bmatrix}.$$
- $$K([z])=z_c-Cg(z_c)+\left(I-C[Dg]([z])\right)([z]-z_c),\qquad z_c=\operatorname{midpoint}([z]).$$
- $$K([z])\subset\operatorname{int}([z])\Rightarrow\text{one unique saddle-node candidate in }[z],\qquad K([z])\cap[z]=\varnothing\Rightarrow\text{exclude }[z].$$

## Implementation notes

Integration point: use this as an offline certification layer for an RNN, implicit layer, or state-space model, with weights fixed and mu chosen as recurrent gain, residual scale, or feedback strength. For a small tanh RNN, set Phi(x,mu)=tanh(mu W x+b), evaluate F and J with interval arithmetic, and use automatic differentiation at box midpoints. For larger systems replace det(J)=0 with Jv=0 and c^T v-1=0, introducing a null vector v and fixed normalization vector c. Pseudocode: tile a box over x and mu; compute g(z_c), C=inverse(Dg(z_c)), and interval [Dg]([z]); form K; accept strict inclusion; discard disjoint boxes; otherwise bisect the widest coordinate and repeat. The paper contributes the Krawczyk construction, outward-rounded interval evaluation, and inclusion logic. The engineer estimates activation interval bounds and chooses search domains and stopping widths. First experiment: use a two-dimensional tanh RNN, compare certification against dense fixed-point continuation and Newton solves. Prediction: accepted boxes occur only when q=norm(I-C[Dg]) is below one and the Krawczyk image has positive interior margin; the certified mu interval must contain the continuation fold, while q decreases as box diameter shrinks near a regular augmented root.

## Verification

- Status: mechanism_failed
- Mechanism evidence: yes
- Mechanism confirmed: no
- Verdict: Built a 2D tanh-RNN augmented fold solver and Krawczyk sweep. The mechanism appeared numerically: q fell from 1.816 at box width 0.2 to 0.0407 at width 0.00625, strict inclusion began at width 0.1, and narrow boxes offset from the root were excluded. However, the implementation uses finite differences and corner sampling rather than outward-rounded interval arithmetic, so it is not a rigorous certificate; the continuation baseline also failed to localize the fold and reported mu=1.675 versus the augmented root mu=0.7434.

### Mechanism check

- Verdict: Built a 2D tanh-RNN augmented fold solver and Krawczyk sweep. The mechanism appeared numerically: q fell from 1.816 at box width 0.2 to 0.0407 at width 0.00625, strict inclusion began at width 0.1, and narrow boxes offset from the root were excluded. However, the implementation uses finite differences and corner sampling rather than outward-rounded interval arithmetic, so it is not a rigorous certificate; the continuation baseline also failed to localize the fold and reported mu=1.675 versus the augmented root mu=0.7434.
- Confidence: 9/10
- Limitations: No outward-rounded interval library was used, so strict inclusion is only an empirical surrogate. No larger RNN, training/inference integration, certified box tiling/bisection, GPU experiment, or reliable branch-aware continuation comparison was tested.

## Artifacts

- [fold_map_experiment.py](https://synthcore.org/code/711/fold_map_experiment.py)
- [report.md](https://synthcore.org/code/711/report.md)
- [toy_results.json](https://synthcore.org/code/711/toy_results.json)
- [Download all files as ZIP](https://synthcore.org/download/711)

## Disclaimer

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