We have shown that a photon-like boson emerges from a purely fermionic system compressed by a complex-valued spectral codec, without postulating interactions, coupling constants, or particle species. The central result is:
where P = sin2π is the Born-rule hop probability. The boson amplitude is the interference term of the Born rule. Pauli exclusion follows from the analytical result B = 0 for single-site states. The virtual propagator lifecycle β born, peaked, vanished β follows from the π-dependence of β₯Bβ₯.
These results are analytically exact, encoding-independent, and hold for all system sizes and hop distances. They suggest that quantum mechanics is an emergent description: the story an observer constructs to explain fermions that are sampled from a minimum-description-length codec operating on a finite universe of n bits.
We additionally reported an exploratory extension (ExperimentΒ 7) in which the same boson-residual formalism, applied to a two-band lattice Hamiltonian, reproduces the relativistic dispersion relation E2 = (pc)2 + (m0c2)2 as a continuum limit. This result is numerically exact given its assumptions, but those assumptions β a two-band internal structure and a specific band-splitting term β are not yet derived from the spectral complexity functional Cs. We report it as a genuinely open problem, in keeping with the three-tier accounting standard of distinguishing exact results, strongly suggested correspondences, and open problems, rather than as a closed derivation of relativistic mass from first principles.