Creative Determinant Research Roadmap¶

This document outlines open research directions stemming from the Creative Determinant (CD) framework. These are not private plans --- they are invitations. If you pick up any of these, you are doing canonical CD work. Open an issue or PR to coordinate, or just run with it 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, Henon map, or Lorenz system with hand-chosen coherence metrics.

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 we actually measure the CD condition on a real (if small) neural network?

What this involves: - Train or fine-tune a small transformer or RNN. - Define a coherence observable (e.g., intrinsic dimension of activations, attention entropy). - 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, compute correlations layer-by-layer.

7. Care-Weighted Metrics and Behavioral Experiments¶

Motivation: The care-weighted homotopy energy principle (Proposition 4.9) suggests care acts as a topological obstruction. Can this be tested behaviorally?

What this involves: - Design a cognitive task where "care" (attention, importance weighting) is experimentally manipulated. - Measure whether paths through "low-care" conceptual regions are indeed costly (slower, error-prone) even when geometrically short. - Map results onto a simple manifold model.

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

Starting point: Simple lexical decision or category learning task with attention manipulation.

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 results honestly: null results or violations are valuable.

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?

What this involves: - Model depression as \(\kappa \to 0\) (zero care everywhere). - Model schizophrenia as high \(\mu\) 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.

These directions are intentionally open. There is no central authority. If you make progress on any of these, you are extending the CD research program.

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.