Open Problems in Creative Determinant Theory¶

This document lists explicit theoretical gaps, unresolved questions, and conjectures in the Creative Determinant (CD) framework. These are honest admissions of what is not yet known, not weaknesses to hide. Contributions addressing any of these are highly valuable.

1. Relationship Between Weak and Strong Coherence¶

Status: Open.

Problem: The paper distinguishes "weak coherent configurations" (solutions to the PDE) from "strong coherent configurations" (fields satisfying the pointwise CD constraint \(\Phi(x) = \Phi(\kappa(x), \gamma(x), \mu(x))\)). Under what conditions are these equivalent? When does strong coherence imply weak, and vice versa?

Why it matters: This gap determines whether the PDE is a faithful representation of the underlying semiotic/enactivist intuition or merely an analytically tractable approximation.

Starting point: Assume a specific functional form for the pointwise constraint and derive the resulting PDE. Check consistency with the V1 model.

2. Justification of the Canonical Viability Closure¶

Status: Modeling choice, not derived.

Problem: The canonical closure \(b(x) = \kappa(x)\gamma(x) - \lambda\mu(x)\) is motivated heuristically but not derived from first principles. Are there contexts where other closures (e.g., multiplicative, nonlinear in \(\mu\)) are more natural?

Why it matters: The choice of closure affects viability thresholds and bifurcation structure. A principled way to select closures would strengthen applications.

Starting point: Propose alternative closures and compare their spectral and equilibrium properties numerically.

3. Multiplicity and Uniqueness of Equilibria¶

Status: Partial results.

Problem: Theorem 3.12 guarantees existence of at least one solution. Under what conditions are solutions unique? When do multiple distinct equilibria coexist?

Why it matters: Multiplicity would correspond to "multi-stable interpretive frames" in cognitive systems --- a key enactivist prediction. Uniqueness might suggest strong convergence to a single interpretation.

Starting point: Numerical bifurcation analysis varying \(\beta\), \(\lambda\). Theoretical tools: Morse index, degree theory, variational methods.

4. Master Equation with Exogenous Forcing¶

Status: Defined (Remark 3.17), not analyzed.

Problem: The driven case \(-\Delta u + a(x)|\nabla u| + b(x)u - c(x)u^p = f(x)\) encodes external input or environmental significance. How does nonzero \(f\) affect existence, stability, and multiplicity of equilibria?

Why it matters: Real cognitive systems are open, not closed. Understanding the driven case is essential for applications.

Starting point: Analyze perturbative effects of small \(f\) on known equilibria; numerical experiments with localized forcing.

5. Regularity and Smoothness of Solutions¶

Status: \(C^{2,\alpha}\) regularity established; higher regularity unclear.

Problem: Can solutions be shown to be \(C^\infty\) under stronger assumptions on coefficients? Are there contexts where solutions have singularities or shocks?

Why it matters: Smoothness properties affect numerical approximation and physical interpretation (e.g., are "meaning gradients" differentiable?).

Starting point: Apply elliptic bootstrapping; investigate boundary regularity near \(\partial M\).

6. Geometric Interpretation of the Gradient Term¶

Status: Interpretive.

Problem: The term \(a(x)|\nabla u|\) introduces nonlinearity and breaks symmetry. What is its precise geometric or dynamical meaning? Is there a variational principle from which it naturally arises?

Why it matters: If the PDE can be derived from a Lagrangian or energy functional, it would connect to physics and provide additional analytic tools.

Starting point: Investigate whether the V1 equation is the Euler--Lagrange equation of some action functional.

7. Bridging to Neural Dynamics¶

Status: Operational proposal, not yet instantiated.

Problem: How do you map a trained neural network (discrete layers, finite weights) onto a continuous semiotic manifold \((M, g)\)? How do you extract \(\kappa\), \(\gamma\), \(\mu\) from activations or gradients?

Why it matters: Without a concrete mapping, the framework remains abstract. A worked example on even a toy network would be a major advance.

Starting point: Apply manifold learning to a 2-layer MLP; hand-label regions as high/low coherence and fit simple \(\gamma\) fields.

8. CD Condition: Sufficient or Necessary?¶

Status: Proposed as sufficient, not proven.

Problem: The CD condition (Definition 5.1) claims that high correlation between coherence and Jacobian volume indicates a CD regime. Is this correlation necessary for coherence, or only sufficient? Could there be coherent systems that don't exhibit the CD coupling?

Why it matters: If the condition is only sufficient, CD describes a subclass of coherent systems, not all of them.

Starting point: Look for counterexamples --- systems with high coherence but no CD coupling.

9. Falsifiability Criteria: Operationalization¶

Status: Defined conceptually (Definition 5.3), not yet tested empirically.

Problem: The five falsifiability criteria (F1--F5) are stated qualitatively. What are precise, quantitative thresholds? How do you pre-register experiments to test them without post-hoc fitting?

Why it matters: Falsifiability is only meaningful if the criteria are concrete enough to test unambiguously.

Starting point: Design one experiment targeting F1 (coherence without coupling) with explicit statistical thresholds.

10. Time-Dependent Dynamics¶

Status: Not addressed.

Problem: The current framework studies equilibria (time-independent solutions). What about transient dynamics, approach to equilibrium, or oscillatory/chaotic regimes?

Why it matters: Real cognitive processes are temporal. A dynamical (not just static) CD theory would be more realistic.

Starting point: Formulate the parabolic PDE \(\partial_t u = -\Delta u + a(x)|\nabla u| + b(x)u - c(x)u^p\) and study stability of equilibria.

How to contribute¶

If you make progress on any of these, please:

1. Open an issue tagged open-problem referencing the problem number.
2. Share your results (even partial, even negative).
3. If you resolve a problem, open a PR adding your result to the appropriate section of the repo or a linked document.

Negative results count. If you show a problem is harder than expected, or a proposed approach doesn't work, that is valuable information.

Proposing new open problems¶

If you identify a gap or unresolved question not listed here, open an issue tagged open-problem-proposal with: - A clear statement of the problem. - Why it matters. - Suggested starting point (if any).

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