✗ Failed on benchmark 2026

Ghost-State Adaptive Recurrent Cell

Implementation & benchmark of arXiv:2608.15300 — Ghost Dynamics in Receptor Signalling Networks: A Fast--Slow Adaptive Extension of Competitive Cancer Inhibition Models

Usefulness8/10

Difficulty6/10

Novelty7/10

Architecture Stability Accuracy Memory Rnn Ssm Mlp Dynamical systems Bifurcations Singular perturbation Control theory

Source paper: Ghost Dynamics in Receptor Signalling Networks: A Fast--Slow Adaptive Extension of Competitive Cancer Inhibition Models arXiv:2608.15300 ⓘ · analyzed Sep 1, 2026

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 Not peer-reviewed.

Idea description

Replace a single recurrent state update with fast feature relaxation, activity evolution, and a slow adaptive state that modulates the activity vector field. Tune the activity subsystem near a controllable saddle-node so that it retains a useful transient regime for a predictable number of steps, enabling delayed switching and long-horizon memory without requiring a large hidden state.

Formulas

$$\tau_X\dot X=E(u)-X,\qquad \tau_A\dot A=f_\theta(A;B,u),\qquad \tau_B\dot B=\varepsilon g_\theta(B,A,u),\qquad 0<\tau_X\ll\tau_A\ll\tau_B.$$

$$\dot z=\mu+z^2,\qquad T_{\mathrm{ghost}}=\int_{-\infty}^{\infty}\frac{dz}{\mu+z^2}=\frac{\pi}{\sqrt{\mu}},\quad \mu>0.$$

$$\mu(t)=\varepsilon t\quad\Longrightarrow\quad T_{\mathrm{delay}}=O(\varepsilon^{-1/3}).$$

$$X_{k+1}=(1-\alpha_X)X_k+\alpha_XE_\theta(u_k),\quad A_{k+1}=A_k+\alpha_A f_\theta(A_k;B_k,X_{k+1},u_k),\quad B_{k+1}=B_k+\alpha_B g_\theta(B_k,A_k,u_k).$$

Mathematical statement

Use a fast–slow recurrent system with input $u_k$, fast state $X_k$, activity state $A_k$, and adaptive state $B_k$. Let $\tau_X\ll\tau_A\ll\tau_B$, and update $X$ toward an encoder equilibrium, while $A$ follows a parameterized vector field $f(A;B,u)$ and $B$ adapts slowly. For a scalar activity coordinate $z$ near a saddle-node, the normal form is $\dot z=\mu+z^2$, where $\mu$ is the signed distance from the fold. For $\mu>0$, passage through the bottleneck takes $T_{\mathrm{ghost}}\approx\pi/\sqrt{\mu}$. If the adaptive state causes a transverse sweep $\mu(t)=\varepsilon t$, with slow-passage rate $\varepsilon>0$, the delayed transition occurs on the scale $T_{\mathrm{delay}}=O(\varepsilon^{-1/3})$. The architecture should therefore expose $\mu$ and $\varepsilon$ as measurable control quantities rather than hiding them inside arbitrary recurrent weights.

Implementation notes

Integrate this as a replacement for the hidden-state update in a GRU-like RNN or diagonal state-space model. Use three vectors: a fast state X for current input encoding, an activity state A for the recurrent representation, and a slow state B for context or adaptation. Set alpha_X near 1, alpha_A substantially smaller, and alpha_B at least 5–20 times smaller than alpha_A. Implement f with a residual MLP, but constrain one scalar channel z of A to have an explicit near-fold form f_z=mu(B,u)+z^2-cz, where c can be fixed positive or learned with softplus; the remaining channels use a standard bounded residual update. Define mu(B,u)=w_mu^T[B,u]+b_mu and estimate the local fold by solving f_z=0 and partial_z f_z=0, or use the analytically known fold of the normal form. Pseudocode is: compute X=(1-alpha_X)X+alpha_X E(u); compute mu; update z and the remaining A channels; update B with the slow rule; predict from [X,A,B]. The paper-derived quantities are the time-scale ordering, inverse-square-root ghost law, and epsilon-to-the-minus-one-third slow-passage law. Empirically estimate mu and epsilon from trajectories by finite differences and regress dwell time against them. First experiment: train this cell and a matched GRU on synthetic sequences whose regime switches after a delayed cue, then on sequential MNIST or permuted sequential MNIST. Sweep fixed positive mu values and measure time spent in a small interval around z=0; the predicted signature is T proportional to mu^(-1/2), with a log-log slope near -0.5. Then sweep alpha_B and measure transition delay while B drives mu through zero; the predicted slope is approximately -1/3 against epsilon=alpha_B, within broad finite-size deviations. Compare long-horizon error and hidden-state norm against the GRU.

Verification

Failed on benchmark

Stage 1 · Toy mechanism gate: Passed ✓

Stage 2 · Mechanism transferred to benchmark: Not reproduced ✗

Stage 2 · Practical benchmark result: Significantly worse ✗

Methodology: 8 paired seeds, permutation test (p<0.05) against a learning-rate-tuned baseline. How verification works

Stage 1 — Mechanism check agent confidence 9/10

Built and numerically verified the Ghost-State mechanism. The fixed-mu ghost passage produced a log-log slope of -0.504 versus the predicted -0.5, with maximum numerical-vs-analytic relative error 7.9e-8; slow passage produced -0.352 versus the predicted -1/3. The mechanism manifested clearly, but the small fixed-dynamics GhostCell underperformed the matched GRU on delayed-cue accuracy (0.656 vs 0.850), so no task-level win was observed.

Agent confidence: 9/10

Baseline: Matched 8-unit GRU delayed-cue accuracy: 0.8500
Idea: 8-channel GhostCell delayed-cue accuracy: 0.6559; fixed-mu slope -0.5038 (predicted -0.5); slow-passage slope -0.3522 (predicted -0.3333); maximum ghost-law relative error 7.86e-8

Limitations: The recurrent comparison uses a fixed, minimally trained GhostCell readout rather than end-to-end optimization of all cell parameters, and only one synthetic delayed-cue task was tested. No sequential MNIST, hyperparameter sweep, FLOP/latency benchmark, or long-horizon robustness evaluation was performed.

How to run: python3 ghost_experiment.py && /home/maxwelhelp/main/bin/python3 add_gru.py

Stage 2 — Benchmark vs. tuned baseline (arXiv:2608.15300) Sep 1, 2026

Significantly WORSE than the tuned baseline (+88.8%, p=0.0081, wins 0 / 8 paired seeds).

Benchmark: Dynamics forecasting
Model: rnn_small
Paired seeds: 8
Baseline mean: 0.0015
Idea mean: 0.0029
Effect (Δ): 0.0014 (+88.8%; negative = idea better)
Wins: 0 / 8 paired seeds
p-value: 0.0081 (permutation test, 20 000 shuffles)
Smallest detectable effect: ±43.8%
Mechanism: Not confirmed ✗
Practical effect: Hurts
Baseline tuning: swept over 3 configs

Limitations:

Only the built-in dynamics track was tested; no vision, tabular, or sequence tracks were run. The mechanism signature tests the trained model's relative finite-difference time scales, not the full fixed-mu inverse-square-root or slow-passage exponent laws. The benchmark used 12 epochs and 400 training/100 test examples per seed.

How to run: cd /home/maxwelhelp/all/math2nn/experiments/exp2336_ghost-state-adaptive-recurrent-cell && /home/maxwelhelp/main/bin/python3 ghost_bench.py

Verdict computed by deterministic test code from paired-seed statistics — not by the language model.

Stage 2 — Benchmark Sep 1, 2026 latest Failed ✗

Agent confidence: 10/10

Implemented an end-to-end Ghost-State recurrent replacement for the structurally matched dynamics track and ran the required 8 paired seeds with a shared learning-rate sweep. The trained models exhibited the predicted slow-adaptation signature (mean |Δmu| / mean |Δz| = 0.398, confirmed), but the idea was significantly worse on test MSE than the tuned GRU baseline, so it does not provide a benchmark win.

Baseline: Tuned rnn_small GRU, lr=0.01: test MSE 0.0015292 ± 0.0003792
Idea: GhostRNN, lr=0.01, alpha_a=0.25, alpha_b=0.03: test MSE 0.0028868 ± 0.0006183; paired delta +0.0013576; permutation p=0.0081

Benchmark result