An Exact Identity Linking the Cosmological Constant, de Sitter Entropy, and the Unified Coupling

We present an exact identity connecting four quantities drawn from otherwise separate domains of physics: the cosmological constant (large-scale cosmology), the de Sitter horizon entropy S_dS Λ (holography and black-hole thermodynamics), the unified coupling _GUT (high-energy particle α physics), and the dimension of the fundamental representation of the exceptional Lie algebra G2. Within the G2 framework established in prior work, two independently derived results — _GUT = 1/(6×7) = α 1/42 and = 6/S_dS — combine to give × S_dS × _GUT × 7 = 1 exactly. The identity is a theorem, not a Λ Λ α numerical coincidence: it follows algebraically from 6×7 = 42. We interpret it as a holographic constraint in which the G2 symmetry of the vacuum locks together dark energy, information capacity, coupling strength, and compact dimensionality so that they cannot vary independently. A corollary is that if S_dS grows as the universe expands, _GUT must decrease at a rate of order the Hubble rate — a falsifiable, if α currently unmeasurable, prediction of a slowly time-varying unified coupling.