The Algebra of the Qubit: Fourteen Theorems on I(V) = γ²(V)

Bharath Srivats

Fourteen theorems on the algebraic structure of binary quantum measurement, built on the identity I(V) = γ²(V) = 1/(1 − V²), where I is the classical Fisher information of a binary measurement with visibility V = 2p − 1 and γ is the Lorentz factor of special relativity. Part A (T1–T5): internal structure. Uniqueness of the Jeffreys prior π_J(V) = γ(V)/π under Diff₊ invariance (T1). Closed-form Thomas–Wigner rotation for sequential binary measurements (T2). SLD saturation of the quantum Cramér–Rao bound (T3). Uniqueness of the Gudermannian bridge between hyperbolic and spherical Fisher–Rao geometries (T4). Closed-form mixed-state retention R(r) interpolating from 1/3 to 1/2 (T5). Part B (T6–T7): observer–manifold theorems. Futility theorem via Lawvere's fixed-point theorem: no embedded observer can surjectively reconstruct the γ² field (T6). Energy-precision uncertainty E·Var(θ̂) ≥ kT ln 2 / γ²(V), saturating Landauer at V = 0 (T7). Part C (T8, held): dimensional selection of 3+1D spacetime from exceptional isomorphisms, with explicit gap acknowledged. Part D (T9–T12): symmetry and classification. Form-invariance of the conformal equivalence under Möbius action (T9). Two-outcome binary qubit effect classification via 4γ²(V) (T10). Modular flow as SL(2,ℂ) boost at imaginary rapidity (T11). SLOCC covariance γ²(V') = γ²(V)·γ²(ξ)·(1+Vξ)² (T12). Part E (T13–T14): quantum gravity connections. Holographic subregion complexity equals Fisher information under boost (T13, proof sketch). γ² Page curve for black hole evaporation (T14, proof sketch). Complete proofs for T1–T7 and T9–T12. T8 presented with scope limitations. T13 and T14 presented as proof sketches. Mathematical companion to: B.G. Srivats, "Fisher Information Is the Squared Lorentz Factor," Zenodo DOI 10.5281/zenodo.19363449 (2026) B.G. Srivats, "From Measurement to Reality," Zenodo DOI 10.5281/zenodo.19486140 (2026) Builds on independently established work by Burns–Greenfield–Dressel (QSMF 13:10, 2026) and Fullwood–Vedral–Guzmán-González (arXiv:2604.07471, 2026). Each theorem is intended for subsequent individual journal publication; this record establishes priority on all fourteen as a collection.

B. Shravan Kumar, S. Balakrishnan

Bharath Srivats

This document collects fourteen theorems extending the Fisher information identity I(V) = γ²(V) = 1/(1−V²), where V is the visibility of a binary quantum measurement and γ is the Lorentz factor. The identity was established in the author's anchor preprint (Zenodo DOI: 10.5281/zenodo.19363449). Part A — Internal Structure (T1–T5): Uniqueness of the Jeffreys prior as a Diff⁺- invariant half-density on the binary Fisher-Rao manifold (T1). Closed-form Thomas-Wigner rotation angle for sequential binary qubit measurements (T2). SLD saturation theorem with explicit mixed-state tilt formula tan β = b(1−r²)/a (T3). Uniqueness of the Gudermannian as the arclength bridge between Bures and Beltrami-Klein metrics (T4). Closed-form isotropic retention ratio R(r) interpolating from 1/3 to 1/2 (T5). Part B — Observer Limits (T6–T7): Lawvere fixed-point obstruction proving no finite observer can reconstruct the full γ² field (T6). Energy-precision uncertainty bound E·Var(θ̂) ≥ kT ln 2 / γ²(V) with distinguished Planckian operating point at V_P = √(1−ln 2) (T7). Part C — Symmetry and Classification (T8–T11): Möbius form-invariance of the conformal equivalence (T8). Classification of unsharp binary qubit effects by visibility (T9). Modular flow as imaginary-time SL(2,ℂ) boost with spinor rapidity ζ = 2itη (T10). SLOCC covariance of single-qubit visibility via Einstein addition (T11). Part D — Quantum Gravity (T12–T13): Scaling correspondence between holographic subregion-CV complexity and Fisher information under boost, with dimension-specific constants κ_d (T12; identity form κ_d = 1 stated as conjecture). γ² Page curve for black hole evaporation (T13; proof sketch). Part E — Partial Result (T14): Partial result deriving 3+1D spacetime signature from SL(2,ℂ) qubit measurement symmetry via exceptional isomorphisms, with explicit gap acknowledged. Complete proofs are provided for T1 through T11. T12 and T13 are proof sketches. T14 is a partial result with an explicit gap acknowledged. Each theorem is intended for subsequent individual journal submission; four companion papers (Bayesian-geometric, Lorentz kinematics, observer thermodynamics, symmetry-holography) are in preparation. This record establishes priority on the theorems as a collection. Related records:Anchor preprint: DOI 10.5281/zenodo.19363449Cross-domain applications: DOI 10.5281/zenodo.19545786

Sergei Kurgalin, Sergei Borzunov

Bharath Srivats

The Fisher information of a binary quantum measurement with visibility V equals the squared Lorentz factor: I(V) = 1/(1 − V²) = γ²(V). This mathematical identity, confirmed through six independent derivations, underlies a conformal equivalence between the Bures and Beltrami-Klein metrics on the qubit Bloch ball, unique to N = 2 via a Weyl tensor obstruction for N ≥ 3. Consequences are traced across quantum metrology, thermodynamics, population genetics, cooperative biology, and pure mathematics. Results are organized by epistemic confidence, with proven theorems separated from conjectural extensions. Falsifiable predictions — including a Thomas-Wigner rotation angle for sequential qubit measurements testable on existing hardware — distinguish this framework from existing alternatives. Three honest tensions constrain its scope. This is the expanded companion preprint to the mathematical proof published as DOI: 10.5281/zenodo.19363449.

Christopher J. Ballance