Creative Determinant Research Roadmap¶

Open research directions in the Creative Determinant (CD) framework. Anyone may take one up. Open an issue or PR to coordinate, or work independently and share results.

1. Construct Semiotic Manifolds from Real Systems¶

Motivation: The framework assumes a manifold \(M\) of meanings, but how do you extract \(M\) from a real neural network or cognitive system?

What this involves: - Techniques for dimensionality reduction or manifold learning (e.g., UMAP, diffusion maps, autoencoders) applied to activation spaces. - Riemannian metric inference from local gradients or Hessians. - Validation: does the inferred manifold structure predict behavioral or representational properties?

Technical prerequisites: Python/PyTorch, familiarity with manifold learning, access to model activations.

Starting point: Apply simple manifold learning to a toy 2-layer MLP on MNIST; visualize the 2D latent as a candidate \(M\).

2. Empirically Test the CD Condition on Toy Dynamical Systems¶

Motivation: The CD condition claims coherence observables correlate with Jacobian volume dynamics. Test this on systems where both are computable exactly.

What this involves: - Pick a low-dimensional dynamical system (e.g., 2D or 3D maps, simple ODEs). - Define a coherence observable (e.g., distance to a known attractor, entropy of trajectory distribution). - Compute Jacobian determinants along trajectories. - Measure correlation and variance as in Definition 5.1 of the paper.

Technical prerequisites: Basic dynamical systems theory, Python (NumPy/SciPy).

Starting point: Logistic map, Hénon map, or Lorenz system with hand-chosen coherence metrics. A concrete first setup: iterate the Hénon map \(x_{n+1} = 1 - A x_n^2 + y_n\), \(y_{n+1} = B x_n\), take the coherence observable to be the inverse distance to the attractor or the sign stability of the local Lyapunov exponent, record \(\log|\det J|\) at each step, and measure the correlation along a long trajectory. Conjecture: for a well-chosen observable, the correlation is low in chaotic regimes and higher in periodic or quasi-periodic regimes.

3. Alternative Viability Closures¶

Motivation: The canonical closure \(b = \kappa\gamma - \lambda\mu\) is one choice. Are there others that fit different cognitive or computational contexts?

What this involves: - Propose alternative functional forms for \(b(x)\) (e.g., nonlinear couplings, multiplicative terms). - Analyze how viability thresholds \(\lambda_1^{(b)}\) change under these alternatives. - Numerical experiments: does changing the closure qualitatively alter bifurcation structure?

Technical prerequisites: PDE theory, finite-difference or FEM solvers.

Starting point: Try \(b = \kappa\gamma \cdot e^{-\lambda\mu}\) or \(b = \kappa\gamma - \lambda\mu^2\) in the 1D notebook and observe threshold shifts.

4. Multiplicity of Coherent Configurations¶

Motivation: The existence theorem guarantees at least one solution, but enactivist intuition suggests multiple "interpretive frames" might coexist. Are there regimes with multiple equilibria?

What this involves: - Bifurcation analysis: vary parameters \(\beta\), \(\lambda\) and look for saddle-node, pitchfork, or transcritical bifurcations. - Numerical experiments: initialize Picard iteration from different seeds and find distinct solutions. - Theoretical analysis: prove multiplicity using variational or topological methods.

Technical prerequisites: Nonlinear analysis, bifurcation theory, numerical continuation.

Starting point: Extend the 1D notebook to sweep \(\lambda\) more finely and check for hysteresis.

5. Master Equation with Nonzero Forcing¶

Motivation: The driven case \(f \neq 0\) (Remark 3.17) models systems receiving exogenous input. What does coherence look like under external driving?

What this involves: - Formulate boundary-value or initial-value problems with \(f(x)\) representing external significance or environmental input. - Study how \(f\) perturbs equilibria or creates new stable patterns. - Interpretation: when does external input stabilize vs. destabilize coherence?

Technical prerequisites: Elliptic/parabolic PDE theory, numerical time-stepping.

Starting point: Add a simple Gaussian forcing term to the 1D V1′ model and observe equilibrium shifts.

6. CD Condition on Small Neural Networks¶

Motivation: Can the CD condition be measured on a small trained neural network?

What this involves: - Train or fine-tune a small transformer or RNN. - Define a coherence observable (e.g., intrinsic dimensionality of neural representations, attention entropy, inverse perplexity). - Approximate Jacobian determinants using low-rank methods or trace estimators. - Compute correlations across a dataset or along generated trajectories.

Technical prerequisites: ML frameworks (PyTorch/JAX), Jacobian estimation techniques, interpretability tools.

Starting point: 2-layer transformer on a toy language task (for example next-token prediction over a small vocabulary); compute correlations layer-by-layer and compare coherent, low-perplexity inputs against noise.

7. Care-Weighted Metrics and Behavioral Experiments¶

Motivation: Proposition 4.9 states that if every homotopy contracting a loop must meet the zero-care set \(\{\kappa = 0\}\), then no admissible homotopy exists: the care-weighted energy is defined only on homotopies that stay in \(\{\kappa > 0\}\), so care acts by excluding paths from the admissible class, not by making them expensive (Remark 4.10). Can this obstruction be tested behaviorally?

What this involves: - Design a cognitive task where "care" (attention, importance weighting) is experimentally manipulated. - Measure whether transitions that must pass through zero-care (unattended) conceptual regions fail or are avoided even when geometrically short. Whether low-care paths are merely slower or more error-prone is a separate empirical question that Proposition 4.9 does not decide. - Map results onto a simple manifold model.

Technical prerequisites: Experimental design, behavioral data analysis, basic topology.

Starting point: Simple lexical decision, semantic priming or category learning task with attention manipulation; compare reaction times and error rates for transitions through high-care versus low-care regions and map the results onto a simple 2D manifold model.

8. Falsification Experiments¶

Motivation: Definition 5.3 lists five falsifiability criteria. Actively trying to break the framework is a first-class contribution.

What this involves: - Design experiments targeting F1–F5 (e.g., look for coherence without coupling, Hamiltonian systems with CD signatures, contradiction resolution without care). - Document every result, including null results and violations.

Technical prerequisites: Depends on the criterion chosen.

Starting point: F3 (Hamiltonian coherence) --- construct a volume-preserving map and test for CD coupling.

9. Extensions to Multi-Modal or Multi-Agent Settings¶

Motivation: Real cognition involves multiple modalities (vision, language, affect) and social interaction. Can CD scale to these?

What this involves: - Vector-valued presence fields \(\mathbf{\Phi}(x)\) with different components for modalities or agents. - Coupled PDEs on shared or overlapping manifolds. - Interpretation: multi-modal coherence as alignment across components; social meaning-making as coupled dynamics.

Technical prerequisites: Systems of PDEs, multi-agent dynamical systems.

Starting point: Two coupled scalar fields on the same domain, representing two agents or two modalities.

10. Computational Pathology Models¶

Motivation: Can CD formalize cognitive/psychiatric pathologies as breakdown regimes of coherence?

Status: Heuristic. The mappings below are interpretive proposals, not theorems or derived predictions.

What this involves: - Model depression as \(\kappa \to 0\) (where \(\kappa\) is the care parameter; zero care everywhere). - Model schizophrenia as high \(\mu\) (contradiction measure) without resolution (contradictions persist). - Model OCD as failure of saturation (presence fields don't stabilize).

Technical prerequisites: Conceptual modeling, clinical intuition, dynamical systems.

Starting point: Modify notebook parameters to simulate these regimes; visualize equilibrium behavior.

How to claim a direction¶

1. Open an issue tagged roadmap describing which direction you want to pursue (or propose a new one).
2. Or just start working and open a PR when you have something to share.
3. Collaboration is encouraged --- tag others if you want co-investigators.

Adding new directions¶

If you see a direction not listed here, propose it! Open an issue tagged roadmap-proposal with: - One-sentence motivation. - What the direction involves. - Suggested starting point.

We'll add it to this document if it aligns with the framework.