Open Problems in Creative Determinant Theory¶

Theoretical gaps, unresolved questions and conjectures in the Creative Determinant (CD) framework.

1. Relationship Between Weak and Strong Coherence¶

Status: Open; one obstruction is proved. Proposition 4.11 of the paper shows that on a compact manifold with zero boundary data (or a closed manifold) no nonnegative \(C^1\) field satisfies the strong pointwise constraint except \(\Phi \equiv 0\), so the strong ideal is unattainable in that setting and only weak configurations are reachable.

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 decides whether the PDE represents the semiotic and enactivist intuition or only a tractable approximation of it.

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: The zero field always solves the equation, and Theorem 3.16 gives a solution positive in the interior when \(\lambda_1(-\Delta - b; M) < 0\). Under what conditions is the positive solution unique? When do multiple distinct positive equilibria coexist? The Lean result scaling_uniqueness only excludes proportional solutions \(k\Phi\).

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 analysed.

Problem: The driven case \(-\Delta u = a(x)|\nabla u| + b(x)u - c(x)u^p + f(x)\) (Remark 3.17) 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. Applications need the driven case.

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 a toy network would be the first instantiation.

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: Defined (Section 3.7), not analysed.

Problem: Section 3.7 of the paper defines a time-dependent closure: coherence debt \(D(t)\) evolves by the ODE of Definition 3.38 and raises the effective contradiction cost \(\lambda_{\mathrm{eff}}(t)\) (Definition 3.40), which enters the dynamical viability closure \(b(x,t) = \kappa(x,t)\gamma(x,t) - \lambda_{\mathrm{eff}}(t)\mu(x,t)\) (Definition 3.42) and so moves \(\lambda_1(t)\). The well-posedness and dynamics of this loop are not analysed, and the existence theorems concern equilibria (time-independent solutions). Is the closed loop well posed? What are its transient dynamics, its approach to equilibrium, and its oscillatory or 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.

11. Exactness of the Threshold With a Gradient Term¶

Status: Settled: exact for \(a \equiv 0\); sufficient only for \(a > 0\), in the continuum and on finite graphs.

Problem: For \(a \equiv 0\), a positive solution exists if and only if \(\lambda_1(-\Delta - b; M) < 0\) (paper Proposition 3.19). With a gradient term the spectral condition is sufficient (Theorem 3.16) but not necessary: Proposition 3.21 gives a positive solution on an interval with \(\lambda_1 = +1/4\), enclosed between an explicit subsolution and a constant supersolution (a classical barrier construction, not Lean-verified; the solution itself is not given in closed form), and the finite-graph model has its own counterexample (Proposition 3.32, \(\lambda_1 = +1/2\)). What remains open is a characterization of the positive-existence region for \(a > 0\): what is the correct nonlinear spectral problem?

Why it matters: The linear indicator is used throughout the numerics; knowing where it is exact tells us when a zero result is a proof of collapse and when it is only an observation.

Starting point: Bound \(\int a|\nabla\Phi|\phi_1\) against \(\int c\Phi^p\phi_1\), or study the barrier construction of Proposition 3.21 as a function of \(a\) and \(b\).

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. A proof that a problem is harder than expected, or that an approach fails, is a contribution.

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.