HIOF • Cosmic Arithmetic — Paper IV-A: Topological Information Conservation in Black Holes

Holographic Information Ontological Framework — Cosmic Arithmetic (HIOF•CA)HIOF•CA is a nine-part theoretical series proposing that stable macroscopic order across physics, biology, and engineered computation can be read as the outcome of constraint-mediated coarse-graining rather than either pure randomness or rigid, error-intolerant design. The core idea, stated plainly Binary computation treats every bit as a fixed coordinate: lose one, and the structure fails. This series asks whether the physical universe instead tolerates local, microscopic disorder while organising it, through boundary conditions and conservation constraints, into robust higher-level patterns. The central image is a handful of sand: each grain's path is unpredictable, yet the poured heap has a stable, reproducible profile. The claim is not mystical. It draws on established results in cosmology, molecular biology, wave physics, black-hole thermodynamics, and quantum information theory. What the series examines Cosmic scale. The galaxy two-point correlation function is treated as the first directly observable instance of scale-bounded, self-similar organisation, alongside a cautious, dynamical-attractor reading of terrestrial planet formation. Biological scale. Bacterial quorum sensing, fungal mycelial transport, and cortical neural-manifold dynamics are examined as three distinct collapse mechanisms — a threshold switch, a substrate-level optimisation, and a continuous attractor — unified only at the level of structural pattern, not shared mechanism. Engineered scale. Wave-based analogue metamaterials and time-reversal acoustics show that mathematical operations and even super-resolution focusing can emerge directly from wave propagation, with microscopic scattering improving rather than degrading precision under the right conditions. Gravitational scale. Black-hole physics is treated through the no-hair theorem, horizon entropy, and the Page curve, arguing that apparent information loss is better read as topological re-encoding. A companion paper treats white-hole symmetric solutions and the loop-quantum-gravity bounce as a parent-child delay interface, explicitly separated from more speculative material claims. Computational scale. Quantum computing is reassessed carefully: superposition and entanglement do not make qubits immune to noise, but topological quantum error correction, especially the surface code, offers the clearest engineered analogue of the series' pattern, encoding a reliable logical state across many fragile physical qubits. A note on method The series explicitly separates three registers: established results drawn from peer-reviewed physics and biology, active research candidates still under investigation, and framework interpretation proposed by the author. A dedicated methodology paper consolidates confidence grades across all volumes and lists concrete, falsifiable predictions — for example, scale limits on fractal clustering, hysteresis in bacterial switching, and continued logical-error suppression in scaling quantum codes. Readers are encouraged to treat the interpretive claims as testable hypotheses, not settled fact. Series structure A prologue establishes the coarse-graining vocabulary. Six papers then examine, in turn, cosmic structure, biological computation, acoustic and metamaterial computation, black-hole information conservation, white-hole symmetry, and quantum error correction. A closing methodological appendix integrates supporting evidence and defines falsifiability criteria for the series as a whole. Keywords: information ontology, coarse-graining, topological collapse, fractal correlation function, quorum sensing, in-substrate computing, neural manifolds, metamaterials, time-reversal acoustics, black-hole information, Page curve, white holes, quantum error correction, surface code, holographic ontology AuthorWai-Hung Tam (Pan), Independent ResearcherORCID: 0009-0002-7789-8464Email: panxtam@protonmail.com