Gravity from the Ledger of the Euler Universe: Dark energy and dark matter as the two faces of de Sitter horizon entropy

Read as a local identity rather than a global sum, the zero-energy ledger of the Euler Cosmos series — the Hamiltonian constraint QE = QG holding at every point — becomes a theory of the whole dark sector. Taking the single principle that only the imbalance of the ledger gravitates, and joining it to the two-part entropy of the de Sitter cosmological horizon, this paper derives a modified law of gravity, g_obs^2 = g_N^2 + g_N a0, in which Newton's law, the cosmological constant, dark matter, and the galactic acceleration scale a0 = cH emerge as four readings of a single entropy: dark energy is the horizon's uniform area-law value, and dark matter is the displacement of its thermal volume-law term by matter. A finite- entropy screening returns the modification to Newton inside the solar system, and the aether supplied by the ledger's preferred foliation gives a relativistic completion (Aether-Scalar-Tensor gravity) that reaches the microwave background and cluster lensing. The reinterpretation of dark energy from a breathing substance to an entropy floor resolves the sound-speed liability the series has carried since the Single Breath paper. The construction rests on a proved result — that matter removes de Sitter entropy, the Schwarzschild-de Sitter horizon deficit dS = -2 pi M L — and on one named open question: whether that deficit's dynamical back-reaction yields the observed force. It is offered as a hypothesis subject to six falsification tests, the primary one — the evolving acceleration scale a0(z) = a0(0) H(z)/H(0) — buildable from existing data and independent of how the open question is resolved. Claude (Anthropic) was used as an analytical and writing tool and is not credited as an author.