Horizon-Maintenance Dark Energy and Horizon Ledger Occupancy: Pre-Euclid Registration, the Minimal w(a) Relation, and Reconstruction Protocols in Finite Distinction Systems

Official website: distinctiontheory.orgPublic portal for the start guide, papers, claim status, failure registry, prior-art boundary, and citation resources. Canonical GitHub repository:https://github.com/yiningwu-research/Distinction-Theory FDS-X1 v1.2 is the integrated canonical version of the X1 dark-energy bridge paper in the Finite Distinction Systems (FDS) / Distinction Theory sequence. It combines the original pre-Euclid prediction registry with the horizon-ledger occupancy model developed in the technical sharpening. FDS-X1 is a conditional physical bridge claim connecting finite observer boundaries to the dark-energy scale. It does not derive general relativity, quantum gravity, the Bekenstein-Hawking coefficient, the exact numerical value of the cosmological constant, perturbations, or a likelihood-ready replacement for ΛCDM. It also does not claim that any departure from ΛCDM automatically supports FDS-X1. The central premise is that if a cosmological horizon is treated as a finite distinguishability boundary, then the relevant dark-energy-like scale should be horizon-like, of order MPl2H02, rather than a Planck-volume vacuum density. Using standard horizon entropy and temperature as physical bridge inputs, a spherical horizon of radius Rh has the horizon-ledger density ρh = ThSh / Vh = 3MPl,red2Rh-2 in natural units. The dark-energy-like horizon-maintenance component is written as ρHM(a) = 3MPl,red2 μh(a) Rh-2(a), where μh(a) is a bounded horizon-ledger occupancy or response factor. It measures how much of the selected horizon ledger appears as the effective dark-energy-like horizon-maintenance sector. If the horizon-maintenance component satisfies an effective continuity equation, then its physical equation of state obeys the minimal relation wHM(a) = -1 - (1/3) d ln μh / d ln a + (2/3) d ln Rh / d ln a. The strict non-phantom version of X1 is therefore equivalent to the ledger-growth condition d ln μh / d ln a ≤ 2 d ln Rh / d ln a. In this interpretation, true physical phantom behavior corresponds to an occupancy response that grows faster than the available horizon-area ledger allowance. Effective phantom behavior in fitted variables may still arise from sector mixing, modified-gravity mapping, curvature, reconstruction choices, systematics, or incomplete modeling of μh(a) and Rh(a). The paper gives an event-horizon specialization. For a future event horizon Re, the equation of state becomes wHM = -1 - (1/3) d ln μh / d ln a + (2/3) (1 - 1/(H Re)). Using the density fraction relation ΩHM = μh / (H2Rh2), the model can be written as a horizon-ledger reconstruction problem rather than as an unconstrained equation-of-state fit. The reconstruction protocol is: given reconstructed H(a), ΩDE(a), and a candidate horizon Rh(a), define μhrec(a) = ΩDE(a) H2(a) Rh2(a). Then compute the response diagnostic σrec(a) = [d ln μhrec / d ln a] / [2 d ln Rh / d ln a], where the denominator is nonzero. The strict X1 expectation is that the correct physical reconstruction should be bounded, regular, horizon-tied, and compatible with the non-phantom ledger-growth condition. FDS-X1 v1.2 preserves the pre-Euclid audit logic. It states in advance how the claim should be updated: a bounded horizon-tied reconstruction with non-phantom physical behavior supports the strict version; constant-density data with no horizon response demote the dynamic version; a true fundamental phantom component falsifies the strict version; and failure to reconstruct any bounded physically justified μh weakens or retires the bridge. The novelty of X1 is not the already-known holographic scaling ρ ∼ MPl2L-2. The novelty is the finite-distinguishability interpretation of the horizon as a boundary ledger, the occupancy factor μh, the minimal w(a) consistency relation, the non-phantom ledger-growth condition, and the pre-registered demotion protocol. The release includes deterministic normal-form demonstrations. They illustrate horizon-ledger area suppression, the non-phantom ledger-growth condition, an event-horizon candidate on a fiducial flat matter-plus-ΛCDM background, minimal response models, and the reconstruction of horizon occupancy. These figures are conceptual normal forms, not empirical fits, not perturbation calculations, and not a likelihood analysis. This release includes the paper PDF, LaTeX source, reproducibility code, generated figures, CSV tables, and model summary outputs.