Closure-Measure Conservation in QMU

Abstract

This paper develops the realization calculus required to derive the four-fifths physical realization law in Quantum Measurement Units (QMU) without treating representation dimensions as sequential physical paths. A preceding temporal--charge analysis of the Aether unit established two distinct representation ranks. The complete exposed realization capacity has rank four, generated by electrostatic polarity crossed with the contemporaneously available left/right temporal-torque routes, while the nonconstant Aether chronogeometry has rank five. This establishes the geometric ratio $4/5$, but rank counting alone does not imply the physical scaling $\alpha_a^{4/5}$. To determine the required realization law, let $R$ be a positive self-adjoint realization operator acting on a finite-dimensional QMU representation space. A scalar realization deficit is required to be continuous, invariant under orthogonal or unitary change of basis, additive under independent direct sums, additive under composition of commuting positive realizations, and antisymmetric under reciprocal inversion. These requirements select, up to normalization, the logarithmic determinant measure $$\mathscr{D}(R)=-\ln\det R.$$ For an isotropic rank-$n$ realization, $$R=xI_n,$$ the realization measure becomes $$\mathscr{D}(xI_n)=-n\ln x.$$ Representation rank therefore enters as an extensive multiplicity rather than as a count of sequential trajectories. The remaining physical input is stated explicitly as the Closure-Measure Conservation Principle: when two complete representations describe redistribution of the same stationary Aether closure asymmetry, with no source or sink of closure content between them, their total logarithmic realization measure is invariant. Applying this principle to the rank-four realization $$R_F=\alpha_a I_4$$ and the rank-five Aether realization $$R_A=\Xi_F I_5$$ gives $$-4\ln\alpha_a=-5\ln\Xi_F,$$ and therefore $$\Xi_F=\alpha_a^{4/5},$$ or equivalently $$\Xi_F^5=\alpha_a^4.$$ The determinant equality is thus a consequence of conservation of an extensive realization measure rather than an independent constitutive assumption. Because QMU defines resonance as $$\mathrm{rson}={F_q}^2,$$ the corresponding resonance realization is quadratic: $$\Xi_R=\Xi_F^2=\alpha_a^{8/5}.$$ The exponent $8/5$ therefore does not arise from eight directed loxodromic paths or from independent channel counting. It follows from the four-to-five closure transfer followed by the quadratic QMU resonance relation. The paper audits this construction against Ledger One, $$A_u\,\mathrm{curl}={F_q}^2{\lambda_C}^2,$$ reciprocal curl--magnetic-flux realization, the six-port Aether decomposition, invariant deformation measures, and the scalar common mode. The common mode is shown not to generate the fractional exponent. Three distinct mathematical structures are kept separate throughout the analysis: the angular closure capacity $16\pi^2$, the logarithmic realization measure $-\ln\det R$, and the later boundary-to-spatial cosmological projection. In particular, the cosmological coefficient $8\pi/3$ is not derived from the logarithmic measure developed here. In the companion cosmology analysis it follows only conditionally from boundary-trace preservation, linearity, and spatial isotropy, while the subsequent Spatial Resonance Response Law remains a separate constitutive premise. The result isolates the new physical assumption required for the four-fifths law, derives all subsequent exponent relations algebraically, and provides the realization foundation used by the current QMU cosmology program while leaving its macroscopic response law logically independent and experimentally falsifiable.