This roadmap summarizes the next research steps after Dynamical Genesis of Complex Structure on Graphs: Neimark–Sacker Bifurcation and Non-Abelian Holonomy (v2).
It is organized in phases and can be revisited and refined as results accumulate.
Contents
Phase 0 – Finalize v2 and Make It Reproducible
- 0.1 Publish & archive
- Upload Version 2 PDF to Zenodo and rtgtheory.org.
- Write a short blog post summarizing the main results (K₃ NS, K₄ double NS, SU(2) lift idea).
- 0.2 Public code repository
- Create a GitHub (or similar) repository for the project.
- Include:
- Core simulation scripts:
rtg_core.py,rtg_ns_scout,rtg_k4_quat_scout_v5.py,rtg_post_diag.py. - Example configs / CLI calls to reproduce:
- K₃ single NS sweep + post-NS diagnostics.
- K₄ double-NS sweep, post-NS run, and extra observables.
README.mdwith installation instructions and “how to reproduce figures”.
- Core simulation scripts:
Phase 1 – Close the K₄ Story Analytically (Highest Priority)
1A. Codimension-Two Neimark–Sacker Normal Form on K₄
Goal: Derive and validate the coupled 4D normal form that explains why both NS modes on K₄ survive with comparable amplitudes and generate a 4D invariant torus.
- 1A.1 Locate the true double-NS point
- In the parameter space \((K, w_{\text{diag}}, \sigma_2, \varepsilon_{\text{asym}}, \dots)\), solve numerically for a point where two complex pairs satisfy \(|\lambda_1| = |\lambda_2| = 1\) and all other eigenvalues are inside the unit circle.
- Verify transversality: \(d|\lambda_1|/dK \neq d|\lambda_2|/dK\) at this codimension-two point.
- 1A.2 Build the 4D center manifold basis
- Compute the Jacobian at the double-NS point.
- Extract the four critical eigenvectors (two complex pairs), form a real 4D basis, and define a projection from the full 8D state space to complex coordinates \((z_1, z_2)\).
- 1A.3 Fit the cubic normal form coefficients
- Simulate the full K₄ system near the fixed point for many short runs.
- Project the dynamics onto \((z_1,z_2)\) and fit the coupled NS normal form:
z₁' = e^{μ₁ + iω₁} z₁ − β₁₁|z₁|²z₁ − β₁₂|z₂|²z₁ + …
z₂' = e^{μ₂ + iω₂} z₂ − β₂₁|z₁|²z₂ − β₂₂|z₂|²z₂ + …
- Check qualitative signs: \(\Re β_{11},\Re β_{22} < 0\) (self-damping), \(\Re β_{12},\Re β_{21} > 0\) (mutual saturation).
- 1A.4 Compare with long-run torus data
- Use the normal form to predict steady amplitudes \(A_1^*,A_2^*\) and compare with measured modal amplitudes on the v2 4D torus.
- Check if the normal form reproduces the observed winding ratio \(\rho \approx 0.9\) and confirms absence of low-order resonances (supports KAM/Diophantine condition in the paper).
- 1A.5 Draft “Part II” paper
- Working title: “Coupled Neimark–Sacker Modes and a 4D Invariant Torus on a Frustrated K₄ Graph”.
- Target journals: Chaos, Nonlinearity, or SIAM J. Applied Dynamical Systems.
1B. Explicit SU(2) Lift Algorithm for Small Graphs
Goal: Turn the SU(2) lift and trace–angle relation from a definition + conjecture into a constructive algorithm.
- 1B.1 Single-triangle solver
- Given an abelian flux \(F_\triangle\), construct an SU(2) holonomy
\(\mathbf{H}_\triangle = \exp\big(\frac{F_\triangle}{2}\,\mathbf{n}\cdot\sigma\big)\). - Solve for edge transports \(U_{ij} = \exp(\mathbf{A}_{ij})\) that minimize
\(\sum \|\mathbf{A}_{ij}\|^2\) subject to \(U_{ij}U_{jk}U_{ki} = \mathbf{H}_\triangle\). - This gives a minimal-norm SU(2) lift, unique up to global conjugation.
- Given an abelian flux \(F_\triangle\), construct an SU(2) holonomy
- 1B.2 Multi-triangle compatibility on K₄
- Extend the solver to the four triangular faces of K₄, enforcing shared edges.
- Study when a consistent global SU(2) connection exists and how “large” the resulting \(\mathbf{A}_{ij}\) must be as frustration parameters vary.
- 1B.3 Holonomy gap observable
- Along K₃/K₄ trajectories, compute both:
- Abelian flux \(F_\triangle(t)\),
- SU(2) trace \(T_\triangle(t) = \frac{1}{2}\mathrm{Tr}\,\mathbf{H}_\triangle(t)\).
- Define a measure like
\(\Delta_{\text{Tr}} = \mathrm{std}_t\big(T_\triangle(t) – \cos(F_\triangle(t)/2)\big)\). - Report small \(\Delta_{\text{Tr}}\) as quantitative support for the trace–angle relation in dynamics.
- Along K₃/K₄ trajectories, compute both:
Phase 2 – Genericity and Robustness of the K₄ Double-NS
2A. Parameter-Space Survey Around the Tuned K₄ Point
Goal: Determine whether the K₄ double-NS event is an isolated curiosity or part of a codimension-one structure in parameter space.
- 2A.1 Choose scan parameters
- Keep \((dt,\gamma,\phi_\triangle)\) fixed.
- Scan a 2D grid over, e.g.:
- \(w_{\mathrm{diag}} \in [2.55, 2.69]\),
- \(\sigma_2 \in [-0.095, -0.088]\).
- 2A.2 Detect NS crossings
- For each grid point, run a K-scan in a small interval around 0.24.
- Record:
- Number of NS crossings (0, 1, or 2).
- For double-NS cases: the separation \(\Delta K = |K_2^* – K_1^*|\) and the angles \(\omega_1,\omega_2\).
- Define a threshold, e.g. “true double NS” if \(\Delta K < 10^{-3}\) and both angles nonzero.
- 2A.3 Quick nonlinear tests
- For promising double-NS points, run short post-NS simulations.
- Check:
- PSD of \(r(t)\) (1 vs 2 dominant frequency peaks).
- Largest two Lyapunov exponents (are both near zero?).
- 2A.4 Visualization and interpretation
- Produce heatmaps over \((w_{\mathrm{diag}},\sigma_2)\) showing:
- Number of NS crossings.
- Value of \(\Delta K\) where double NS exists.
- Discuss whether double NS appears along a 1D manifold, in small clusters, or only at a single point.
- Produce heatmaps over \((w_{\mathrm{diag}},\sigma_2)\) showing:
Phase 3 – Beyond Minimal Motifs (Exploratory)
3A. Embedding K₄ Modules in Larger Graphs
Goal: Test whether the K₄ 4D torus can act as a local “module” inside larger networks.
- Construct systems with two (or more) K₄ blocks, each tuned near the double-NS point.
- Introduce weak couplings between blocks with strength ε.
- Study:
- ε → 0: two independent 4D tori (8D product attractor?).
- Small ε: do the blocks phase-lock or remain quasi-independent?
- Larger ε: does the system collapse to a lower-dimensional attractor or become chaotic?
- Use Lyapunov spectra and PSD analysis to characterize the resulting attractors.
3B. Octonion / k=7 Experiments
Goal: Test the conjecture that higher rungs (k=7, octonionic) may lead to torus breakdown and chaos.
- Generate random moderately sized frustrated graphs (e.g. 8–9 nodes, mixed positive/negative couplings with multiple frustrated triads).
- Use an automated NS scout to search for points where three complex pairs approach the unit circle.
- For candidates, run nonlinear simulations and check:
- PSD of \(r(t)\) (3 peaks vs broad spectrum).
- Lyapunov spectrum (3 near-zero vs one or more positive exponents).
- Record both positive and negative results:
- Finding no stable 6D/8D tori supports the idea that non-associative structures naturally produce chaos.
- Finding even one stable higher-dimensional torus would be a major result.
Phase 4 – RTG-Facing Applications (After Phase 1–2)
Once the K₄ normal form and robustness analysis are in place, it will be safe to connect these dynamical results back to Relational Time Geometry (RTG).
- Write a conceptual paper or long-form note:
- Map K₃ (1 NS mode → 2D torus) and K₄ (2 NS modes → 4D torus) to RTG’s emergent dimension picture.
- Relate the normal-form coefficients and SU(2) lift to RTG’s resonant couplings and geometric data.
- Discuss how chains or lattices of K₃/K₄ motifs might model higher-dimensional RTG structures.
Summary of Priorities
- Phase 1:
- Derive and validate the coupled NS normal form on K₄ (1A).
- Set up and document a public code repository.
- Begin parameter sweep around the tuned K₄ point (2A).
- Phase 2:
- Complete SU(2) lift algorithm and holonomy gap analysis (1B).
- Interpret parameter-space results and finalize a “Part II” paper.
- Phase 3:
- Explore embedding of K₄ modules in larger graphs (3A).
- Run exploratory octonion / k=7 searches (3B).
- Phase 4:
- Write an RTG-oriented application paper building on the rigorous dynamical results.