Forced Spectral Complements of Biological Systems: Phase-Shifted Field Patterns from Conservation on a Discrete D=3 Lattice

Any spatially coherent, nontrivial, balanced pattern on a discrete Z3 lattice with period-8 conservation necessarily generates a spectral complement in the surrounding field whose DFT-8 is the pointwise negation of the source [1]. We argue that biological organisms satisfy all hypotheses of this theorem: nontriviality (metabolic activity produces nonzero recognition signals), neutrality (homeostasis maintains 8-tick balance), spatial localization (organisms occupy bounded connected regions), and signal dominance (the organism's metabolic signal exceeds the ambient background in its local neighborhood). The resulting complement is phase-shifted by π at each of the three conjugate mode pairs (1,7),(2,6),(3,5), carries equal spectral energy at every mode, and satisfies the conservation constraint independently of the source — making it admissible for autonomous evolution after the organism ceases. We connect these predictions to existing measurements: biophoton emission spectra, EEG/MEG field topography, and biofield detection experiments. The complement framework provides a mathematical account of why extended field patterns around living systems have been repeatedly observed but lacked a first-principles derivation.