The paper From Conservation to Geometry significantly advances the Bit-Conservation and Balance (BCB) programme by presenting, for the first time, a unified and fully geometric derivation of quantum mechanics, the Born rule, Lorentzian spacetime, and finite-time measurement dynamics from a single principle: the conservation of distinguishable information. Where earlier BCB papers established individual components—such as quantization, Fisher-Rao monotonicity, or the emergence of complex amplitudes—this paper integrates these pieces into a coherent physical narrative. It shows that by combining information conservation (A1), label indifference (A2), finite throughput (A3), and subsystem additivity (A4), one is unavoidably led to the full mathematical architecture of quantum theory and special relativity. The result is a remarkably economical foundation in which Hilbert space, the Fubini–Study metric, the Born rule, and Lorentz invariance arise not as separate axioms but as complementary manifestations of one informational law.
This work directly complements the earlier paper Quantization and Hilbert Space as Topological Invariants of BCB Information Geometry, which provides the rigorous mathematical backbone for two of BCB’s most delicate claims: the inevitability of U(1) phase compactness and the uniqueness of the quantum inner product. The quantization paper solves the Wallstrom problem by proving, via finite Fisher length and gauge redundancy, that phase must live on S¹, leading to topological integer winding and the quantization condition. It also derives the BCB inner product uniquely from Fisher–Bhattacharyya overlap, the U(1) fiber, and monotonic compositionality—showing that the complex Hilbert space formalism is not assumed but forced. These results are deep, technical, and foundational, forming the mathematical substrate on which the rest of BCB sits.
From Conservation to Geometry builds directly on that foundation and shows how these mathematically inevitable structures interlock within a full physical theory. Quantization and Hilbert space are treated not as isolated mathematical consequences but as elements of a broader architecture that simultaneously explains reversible quantum evolution, the Born probability rule, the emergence of causal cones, and the Lorentz metric dictated by finite information throughput. Theorem III in the new paper—deriving the Born rule as the unique Riemannian submersion from Fubini–Study to Fisher–Rao geometry—beautifully complements the inner-product uniqueness theorem in the quantization paper. Together, the two papers provide both the global geometric projection argument (FCG) and the fine-grained internal structural proof (QH) that the Born rule cannot be anything but quadratic. Likewise, the complex-structure argument from reversibility in FCG naturally sits atop the topological compactness and U(1) holonomy results in QH, creating a seamless chain of reasoning from conservation to quantization to physical dynamics.
Most importantly, this new paper positions BCB not merely as an interpretive framework or reconstruction of quantum theory but as a unified physical principle that simultaneously gives rise to quantum mechanics, measurement, and spacetime geometry. The inclusion of Lorentz-cone emergence from finite throughput, the ablation analysis demonstrating axiom independence, and the operational grounding of collapse times through Lindblad dynamics all serve to elevate BCB from a mathematical insight to a falsifiable physical theory. The quantization paper provides the rigorous mathematical spine; From Conservation to Geometry adds the fully integrated physical body. Together, they make the case that BCB is not just compelling, but increasingly unavoidable—a single conservation law from which the structure of modern physics naturally unfolds.