We summarise what has been established, what is strongly suggested, and what remains open.
Established (exact, verified).
- Pauli exclusion as zero compression residual (Paper VII).
- Boson amplitude ∥B(𝜃)∥ = sin(2𝜃)∕
, independent of system size (Paper VII).
- Born rule as interference term of the density matrix (Paper VII).
- Ricci/Weyl decomposition from the diagonal/off-diagonal split of metric density matrices
(Paper VIII).
- Two graviton polarisations as degenerate eigenmodes (Paper VIII).
- Graviton amplitude ∥W(𝜖)∥ = sin(2𝜖)∕
(Paper VIII).
- Newtonian 1∕r potential from graviton flux conservation, with G = 1∕(8π) in Planck units
(Paper VIII).
- Keplerian orbits as minimum-complexity closed worldlines (Paper VIII).
Strongly suggested (structural agreement, not yet derived).
- The aspect ratio 𝒜(n) = 2n∕(nlnn) matching inflationary expansion in Planck units
(Papers IV–V).
- The ΛCDM expansion profile recovered from the lognormal emergence law without a
cosmological constant (Paper V).
- The Friedmann equation under the identification of matter as bound bit structures (Paper V).
- The proportionality ∥W∥∝M needed to make Theorem 4 of Paper VIII fully self-contained.
- The correct topology: why the boundary is a 3-sphere rather than some other manifold?
The minimum-complexity compact boundaryless 2-boundary in 3D Euclidean space is S2,
with surface area 4πr2. Under Solomonoff induction, all other topologies are exponentially
suppressed. The factor 4π is therefore not a free parameter but a topological selection result.
Open.
- Solving the saddle-point equation (6) and recovering n∗≈ 184.
- Deriving the decomposition (5) from equation (1) without additional assumptions.
- The continuum limit: recovering the full Einstein field equations from the discrete toy model
of Paper VIII.
- The emergence of energy: how E ∝ℏω arises from the spectral cost Cψ in physical units.
- Massive bosons (W, Z, Higgs) and the gauge structure of the Standard Model.