Open Questions
The Quantum Family Tree conjecture is not a finished theory. It is a research program. These are the open problems — stated honestly, with proposed directions. If you can solve any of them, get in touch.
1. Derive α from first principles.
The mean MI decay exponent α ≈ 1.2 is empirically stable across depths 4–12 with R² > 0.999 but has no theoretical derivation. This is the central open problem. It may be accessible via random matrix theory applied to the Haar unitary ensemble, or via p-adic holography where the Bruhat–Tits tree structure may predict the decay exponent.
Proposed direction: Random matrix theory analysis of the branching channel. p-adic holographic prediction for α in the p=2 (binary tree) case.
2. Derive the σ/μ scaling exponent.
The far-field fluctuation growth rate (~1.3–1.4 per two distance steps) is empirically observed but not derived. A second universal exponent distinct from α would strongly support the two-law framework.
Proposed direction: Compute the scaling exponent across depths 4–12 to establish universality. Relate to the statistics of random products of unitary matrices.
3. Establish the CMB correspondence quantitatively.
We conjecture that the σ/μ fluctuation spectrum corresponds to the CMB angular power spectrum. The near-field suppression matches the observed low quadrupole anomaly. Establishing this requires a conversion constant K connecting MI in nats to CMB temperature fluctuations in μK².
Proposed direction: Match the shape of the σ/μ vs. dG curve to the CMB power spectrum l(l+1)Cl/2π vs. multipole l. Shape matching alone, without K, would be significant.
4. Tighten the first law result.
The first law δ⟨H⟩/δS = 1.0068 ± 0.015 currently comes from depth-4 tests only. Robustness across depth 6 and 8, multiple subsystem choices, and multiple perturbation classes would significantly strengthen this result — potentially making it the paper's headline claim.
Proposed direction: Run first law tests at depths 6 and 8. Test multiple subsystem sizes and shapes. Verify that the modular Hamiltonian proxy H = −log(ρA) is appropriate for this discrete setting.
5. Reconstruct the metric tensor.
The scalar distance d = (1/α)·ln(1/MI) must be extended to a full metric tensor gμν. This requires embedding the leaf-entanglement matrix into a continuous coordinate system and computing Ricci curvature.
Proposed direction: MDS or manifold learning on the N×N MI matrix at depth 12 (4,096 leaves). Test whether Ricci curvature satisfies linearized Einstein equations δGμν = 8πG · δTμν.
6. Establish the continuum limit.
The Gromov–Hausdorff convergence of the tree metric to a smooth manifold must be demonstrated formally. This is required to connect the discrete simulation to the Faulkner–Van Raamsdonk derivation of Einstein's equations.
Proposed direction: p-adic analysis. The binary tree (p=2) corresponds to the Q2 Bruhat–Tits case. The continuum limit may be the p-adic numbers rather than the reals — a natural habitat for ultrametric geometry.
7. Extend to de Sitter space.
The current framework is most naturally AdS-like (negative curvature). The observable universe is de Sitter (positive cosmological constant, accelerating expansion). This extension is a limitation shared with the entire holographic program.
Proposed direction: Modify the branching rate to accelerate with depth. Test whether this produces de Sitter-like geometry in the MDS embedding.