BCB and the Uniqueness of Quantum Mechanics: A Multi-Path Derivation

Position Within the BCB Program

This paper is part of a broader research program that develops the Bit Conservation and Balance (BCB) principle as a unifying foundation for quantum theory. Earlier works established three key pillars:

1. From Conservation to Geometry
   This paper showed that once one treats distinguishability as a conserved current, Čencov–Petz–style information geometry forces the state space to carry a unique monotone metric. In the classical case this is the Fisher metric; in the quantum case, it becomes the Fubini–Study (FS) metric on projective state space. In other words, BCB ⇒ Fisher/FS geometry, rather than geometry being postulated.
2. Quantization and Hilbert Space
   A second line of work demonstrated that demanding reversible, information-preserving dynamics on this Fisher/FS background recovers the Hilbert-space formalism: linear superposition, complex phases, and the Kähler structure required for unitary evolution. There, BCB was shown to be sufficient to promote a purely probabilistic information manifold into a quantum-like phase space.
3. Hilbert Space Uniqueness and Information Metrics
   A third set of results clarified that the Fisher/FS metric is not one choice among many: under BCB-style monotonicity and symmetry assumptions, it is the only Riemannian structure compatible with data-processing inequalities and reversible flow. This ruled out a large class of ad hoc metric choices and showed that standard quantum geometry is tightly constrained by information conservation.

What was still missing in that sequence was a fully explicit derivation of:

• the Born rule itself (the |cᵢ|² law, and only that law), and
• a systematic exclusion of near-by “quantum-like” theories (real/quaternionic Hilbert spaces, ε-deformed probabilities, nonlinear dynamics) within the same BCB framework.

This paper fills exactly that gap.

• Geometrically, we show that once BCB is imposed on the operational framework, complex projective Hilbert space ℂℙⁿ⁻¹ with Fubini–Study metric is not just natural, but forced: real and quaternionic alternatives fail local tomography, additivity, or locality in ways that directly violate BCB.
• Probabilistically, we prove that any deformation P(i) ∝ |cᵢ|^(2+ε) inevitably breaks distinguishability monotonicity, using explicit Fisher-information calculations. This upgrades the Born rule from an axiom to the unique BCB-compatible probability law.
• Dynamically and compositionally, we show that BCB then leaves only unitary Schrödinger evolution and tensor-product composition as consistent options.

Taken together with the previous BCB papers, this moves BCB from a “promising interpretive idea” to a highly constrained, over-determined framework. In physics, a theory is said to be highly constrained or over-determined when:

Multiple different derivation routes produce the same conclusion, so the theory isn’t relying on a single fragile assumption or one clever trick.

In the BCB program:

• The geometric route (monotone metrics)
• The probabilistic route (ε-deformation & Fisher info)
• The operational route (local tomography & distinguishability)
• The dynamical route (Wigner + Stone from metric isometries)
• The compositional route (tensor-product additivity)

all independently force the same mathematical structure:

• Complex Hilbert space
• Fubini–Study metric
• Born rule |c|²
• Unitary evolution
• Tensor products
• No-signaling

No single line of reasoning carries the whole weight — they all mutually reinforce one another.

Quantum mechanics emerges repeatedly, from multiple independent constraints.

• Each major ingredient of standard quantum mechanics (Hilbert space, FS geometry, Born rule, unitarity, tensor products, no-signaling) is derived from BCB in more than one way.
• Competing nearby theories (alternative metrics, non-complex fields, modified Born rules, nonlinear dynamics) are not merely disfavoured but shown to conflict directly with BCB’s continuity and monotonicity requirements.

In this sense, the present work substantially strengthens the evidential status of BCB. If one accepts the basic physical premise that distinguishability is conserved and cannot be increased by physical processes, then the convergence of all these derivations makes the standard complex-Hilbert-space, Born-rule quantum formalism appear not as a contingent choice, but as the only theory left standing within that information-theoretic envelope.