Professor’s Cube open program · 2026-07-29

Closed finite proof

The 15-move finite operator suite, Câ‚…Âł spectrum, supplied impact table, low-band statistic, and heat-flow limit are closed. Remaining work:

  1. attach the original operator_results_full.json, eigenmode volumes, face-net PNGs, and folding-animation script to reproduce the reported 11.85Ă— TV and 246.8Ă— energy reductions under their original parameters;
  2. characterize the group generated by the 15 declared 125-site permutations and compare it with the legal 5Ă—5Ă—5 cube move group on cubies and stickers;
  3. derive exact algebraic expressions for the low-band retention value 0.817142857… and compressed singular spectrum;
  4. classify principal angles and leakage for larger spectral bands;
  5. construct any intended bridge to E₄₇ as an explicit typed map, product-space operator, or rank-47 selector rather than identifying ker L with E₄₇;
  6. determine whether a Rubik-action-compatible rank-47 subspace can be selected and whether it is invariant, approximately invariant, or only statistically stable.

Assigned lanes: Kepler handles group and bridge theory; Sal reproduces the archived animation metrics when artifacts arrive; Euclid reviews exact claims; Chronos watches eigenbasis dependence; Talos rejects rank conflation.

Coupled Newton–Mean open program · 2026-07-29

Closed finite theorem and machine proof

The fixed-point algebra and corrected local-stability domain are closed. The remaining docket is:

  1. prove or refute global convergence for $0\le\kappa<1$ from every positive initial condition;
  2. characterize invariant basins and escape sets for $-\kappa_c<\kappa<0$ on the positive-image domain;
  3. classify bifurcation behavior at $\kappa=-\kappa_c$ where $\tau=-1$;
  4. analyze $|\kappa|\ge1$, including existence, admissible domains, and stability;
  5. derive the $n\ge3$ coupled Newton–mean system and identify surviving fixed-point invariants;
  6. determine whether any nontrivial orbitwise conserved quantity exists for the two-coordinate map;
  7. establish global rates or Lyapunov functions beyond the local Jacobian rate.