Spectral Structure of Finite-Domain Vacuum Energy: The Channel-Counting Principle

This paper is part of a research program proposing that the cosmological constant arises not from total vacuum energy but from the finite-domain residual to which gravity actually couples. This paper develops the sixth component of a research program examining how the particle content of quantum field theory determines the magnitude of the finite-domain vacuum energy residual that acts as the cosmological constant. Paper of this series derived the cosmological constant prefactor α = N/(16π²) by physical argument, identifying three formal gaps: the derivation of UV cancellation and mass-term suppression in the finite-domain residual; the channel-counting principle that each degree of freedom contributes one independent geometric fluctuation channel; and the formal basis for the unsigned count N = NB + NF. The present paper supplies the missing formal structure. The central result is this: once the finite-domain residual is defined as the spectral difference ρres = ρ[L] − ρ[∞] and normalized by the standard QFT phase-space measure in 3+1 dimensions, the coefficient α = N/(16π²) follows as a natural consequence of the channel-counting framework — not a fitted parameter. Four propositions support this result. Proposition formalizes the residual as a spectral density difference. Proposition shows that boundary-insensitive contributions are suppressed or absent in this difference. Proposition shows channel additivity via path-integral factorization. Proposition shows that the coefficient 1/(16π²) is fixed by the geometry of 3+1-dimensional momentum space. The unsigned sum N = NB + NF = is supported by the positivity of boundary-sensitive fluctuation intensity. Together, these propositions convert Paper 5's channel-counting rule from a physically motivated hypothesis into a structural consequence of the finite-domain subtraction. Series Context This paper forms the sixth component of the Finite-Domain Vacuum Energy research program addressing the cosmological constant problem. Papers 1–4 establish the conceptual framework, gravitational consistency conditions, and cosmological self-consistency relations underlying finite-domain vacuum energy. Paper introduces the channel-counting principle, proposing that the prefactor governing the residual vacuum energy density is determined by the number of independent quantum field degrees of freedom. The present paper provides the spectral justification for this principle by showing that when the finite-domain residual vacuum energy is defined as the spectral differenceρ_res(L) = ρ[L] − ρ[∞]and evaluated using the standard quantum field theory phase-space measure in + dimensions, the coefficient α = N/(16π²) emerges as a structural consequence of the subtraction. The result links the magnitude of the cosmological constant to the total number of independent quantum field degrees of freedom whose fluctuations survive the gravitational admissibility bound.