The single-fermion codec must be extended to N-fermion systems via tensor products.
Experiment 7 shows that the boson-residual formalism is compatible with a mass term once a two-band Hamiltonian is assumed, but the Hamiltonian itself was imported rather than derived. A first test of whether the same overlap/exclusion cost functional that produces Pauli exclusion in Experiment 4 forces a band-splitting structure, rather than assuming one, has now been carried out (exclusion_poc.py, exclusion_poc_3level.py; see Supplementary Material).
With two internal bands, Solomonoff/MDL weighting over the band-mixing angle Δ between two fermions’ internal states – using only the exclusion cost ∥Φ∥2, with no Hamiltonian and no imported mass parameter – is maximized at Δ = π∕2: full orthogonality between the two fermions’ band occupations. This reproduces, from pure MDL reasoning, the standard antisymmetric ground state (one fermion per band). With only two bands, however, this is partly a fact of linear algebra rather than new physics: Λ2(ℂ2) is one-dimensional, so the reduced single-particle density matrix is forced to the maximally mixed state diag(1∕2,1∕2), with zero boson residual, for any choice of the two orbitals. A two-band system therefore cannot, even in principle, support a variable residual for a mass term to attach to.
Extending to three internal bands changes this. Because Λ2(ℂ3) is three-dimensional, the occupied
two-fermion subspace can be tilted relative to the fixed band basis, and the reduced single-particle boson
residual ∥B1∥ – evaluated in that fixed basis, not the particle’s own natural-orbital basis, where it is
always exactly zero – becomes nonzero and varies continuously with the tilt angle, up to a maximum of
1∕(2
) ≈ 0.354. This is a genuinely new degree of freedom, absent by construction in the two-band case,
and it is the first structure identified in this framework with the right qualitative shape – a
continuously variable, band-off-diagonal coherence – for a mass-like parameter to be built
from.
Neither script derives an energy scale: both use only counting/overlap (exclusion cost or Slater-determinant coherence), never a dynamical term or an imported constant with units of energy. Converting “Δ is forced to π∕2” or “the residual coherence varies with tilt” into “the two bands differ by a specific rest-mass energy m” remains open, and plausibly requires relating cost-in-bits to cost-in-energy (e.g. via an analogue of kBT or ℏω) rather than a relabelling. Until that step is taken, Result 5 remains at its current evidentiary tier rather than being upgraded to an analytically exact theorem on the footing of Results 1–4.
Paper VI identified the analytic derivation of the Friedmann equation from the relational scale factor R(t). Worldlines in GR are themselves compressible. An elliptic orbit is a single Fourier mode — the minimum-complexity trajectory consistent with the conserved quantities of the system. PoC to show these most probable path through the 2n yield objects orbiting each reproducing Keplerian dynamics without a force law.