3 Evidence from Papers VII and VIII

The strongest evidence for the programme comes from two recent exact results, which we summarise here. Both concern the same mathematical object: the off-diagonal density matrix B = ρ − diag(ρ) of a compressed configuration.

3.1 Bosons as Compression Residuals (Paper VII)

Paper VII [meskanen2026vii] encodes two-frame fermion configurations as complex wavefunctions ψ = f₀ + if₁ and computes the density matrix ρ = |ψ⟩⟨ψ|. The off-diagonal part B captures everything the codec records when a fermion moves between frames.

Four results are established analytically and verified numerically:

1. Pauli exclusion. For any stationary single fermion, B = 0 exactly. Double occupancy produces no compression residual; it is informationally invisible. Pauli exclusion is not a postulate but the statement that double occupancy has nothing for the codec to encode.
2. Universal boson. Every single fermion hop produces ∥B∥ = 1∕, independent of chain length, hop distance, and encoding. The residual carries a universal −π∕2 phase, the defining signature of a propagator.

3. Born rule identity. For a fermion in superposition ψ(𝜃) = cos𝜃e_i + isin𝜃e_j,

   (2)

   where P = sin²𝜃 is the Born-rule hop probability. The boson amplitude is the interference term of the Born rule, expressed as a matrix norm.

4. Virtual propagator lifecycle. ∥B∥ = 0 when the fermion is stationary, rises to 1∕ at equal superposition, and returns to 0 when the hop is complete — the lifecycle of a virtual particle, derived without field theory.

These results follow from the single definition B = ρ− diag(ρ) and the normalisation of ψ. No interaction terms, coupling constants, or particle species were introduced.

3.2 Gravitational Structure as Compression Residual (Paper VIII)

Paper VIII [meskanen2026viii] applies the identical decomposition to metric configurations: a two-frame metric pair (g⁽⁰⁾,g⁽¹⁾) is encoded as ψ(x) = g⁽⁰⁾(x) + ig⁽¹⁾(x). The same ρdiag(ρ) + B split then yields:

1. Curvature decomposition. ρ decomposes exactly as ρ_Ricci + ρ_tidal + ρ_graviton, with Tr(ρ_Weyl) = 0 in every case. The diagonal is identified with local Ricci curvature (the Einstein source term); the symmetric off-diagonal with tidal (Coulomb Weyl) forces; the antisymmetric off-diagonal with gravitational waves.
2. Two polarisations. The two degenerate graviton polarisation modes emerge as eigenvectors of ρ_tidal with eigenvalue −1∕4, protected by the reflection symmetry of the metric profiles. No spin-2 field is postulated.

3. Graviton amplitude theorem. For orthogonal metric profiles at mixing angle 𝜖,

   (3)

   the exact gravitational analogue of equation (2). The formula is independent of metric profiles, system size, and overall scale.

4. Newtonian potential. Graviton flux conservation over spherical shells gives A(r) ∝ 1∕r exactly, and the overlap integral yields

   (4)

   The factor 8π = 2 × 4π arises from spherical geometry alone — the same factor that recovers Bekenstein–Hawking entropy in Paper IV [meskanen2026iv].

5. Minimum-complexity orbits. The minimum spectral-complexity closed worldline is ψ(t) = Ae^iωt — a single Fourier mode, tracing an ellipse. Under Solomonoff induction, all other closed trajectories are exponentially suppressed. Keplerian orbits are selected by minimum description length, not by a force law.

3.3 The Shared Structure

Equations (2) and (3) are identical in form. This is not a coincidence of presentation. Both follow from the density matrix decomposition applied to a normalised superposition of two orthogonal configurations. The formula sin(2α)∕ is the unique consequence of this structure, regardless of whether α parametrises a fermion in quantum superposition or a metric in gravitational superposition.

The conjecture is that this shared structure reflects a shared origin: quantum field theory and general relativity are two projections of the same compression principle onto different physical degrees of freedom.