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European Journal of Pure and Applied Mathematics

Three Fisher-Rao Identities on the Binary Statistical Manifold: Jeffreys Uniqueness, the Gudermannian Bridge, and Isotropic Retention

Bharath Srivats

Published Jul 28, 2026• 0 citations
Publisher

Abstract

On the one-dimensional Fisher--Rao manifold of binary probability distributions parameterized by the visibility coordinate $V \in (-1,1)$, the Fisher information equals the squared Lorentz factor: $I(V) = \gamma^2(V) = 1/(1-V^2)$. We derive three consequences of this identity concerning reparameterization-invariant priors, arclength bridges between canonical metrics, and coarse-graining of qubit measurements. First, the Jeffreys prior $\pi_J(V) = \gamma(V)/\pi$ is the unique normalized smooth probability density whose associated half-density coincides with the canonical half-density of the metric, and is therefore the unique prior covariant under every smooth reparameterization. Second, the Gudermannian function is the unique smooth bijection relating the Bures arclength coordinate $s_B = \tfrac{1}{2}\arcsin(V)$ to the Beltrami--Klein arclength coordinate $s_{BK} = \operatorname{arctanh}(V)$ via $2\,s_B = \operatorname{gd}(s_{BK})$; equivalently, $\arcsin(V) = \operatorname{gd}(\operatorname{arctanh}(V))$. Third, for a mixed qubit state of Bloch radius $r$ undergoing transverse rotation, the isotropically averaged ratio of classical to quantum Fisher information has the closed form $R(r) = [2r + (r^2-1)\ln((1+r)/(1-r))]/(4r^3)$, interpolating monotonically between $R(0) = 1/3$ and $R(1) = 1/2$. Each identity is proved from scratch. These three identities are elementary consequences of $I(V) = \gamma^2(V)$; their joint geometric content positions the binary Fisher--Rao manifold at the intersection of Bayesian statistics, hyperbolic geometry, and quantum measurement theory

DOI
10.29020/nybg.ejpam.v19i2.7819

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On the one-dimensional Fisher--Rao manifold of binary probability distributions parameterized by the visibility coordinate V∈(−1,1)V \in (-1,1)V∈(−1,1), the Fisher information equals the squared Lorentz factor: I(V)=γ2(V)=1/(1−V2)I(V) = \gamma^2(V) = 1/(1-V^2)I(V)=γ2(V)=1/(1−V2). We derive three consequences of this identity concerning reparameterization-invariant priors, arclength bridges between canonical metrics, and coarse-graining of qubit measurements. First, the Jeffreys prior πJ(V)=γ(V)/π\pi_J(V) = \gamma(V)/\piπJ​(V)=γ(V)/π is the unique normalized smooth probability density whose associated half-density coincides with the canonical half-density of the metric, and is therefore the unique prior covariant under every smooth reparameterization. Second, the Gudermannian function is the unique smooth bijection relating the Bures arclength coordinate sB=12arcsin⁡(V)s_B = \tfrac{1}{2}\arcsin(V)sB​=21​arcsin(V) to the Beltrami--Klein arclength coordinate sBK=arctanh⁡(V)s_{BK} = \operatorname{arctanh}(V)sBK​=arctanh(V) via 2 sB=gd⁡(sBK)2\,s_B = \operatorname{gd}(s_{BK})2sB​=gd(sBK​); equivalently, arcsin⁡(V)=gd⁡(arctanh⁡(V))\arcsin(V) = \operatorname{gd}(\operatorname{arctanh}(V))arcsin(V)=gd(arctanh(V)). Third, for a mixed qubit state of Bloch radius rrr undergoing transverse rotation, the isotropically averaged ratio of classical to quantum Fisher information has the closed form R(r)=[2r+(r2−1)ln⁡((1+r)/(1−r))]/(4r3)R(r) = [2r + (r^2-1)\ln((1+r)/(1-r))]/(4r^3)R(r)=[2r+(r2−1)ln((1+r)/(1−r))]/(4r3), interpolating monotonically between R(0)=1/3R(0) = 1/3R(0)=1/3 and R(1)=1/2R(1) = 1/2R(1)=1/2. Each identity is proved from scratch. These three identities are elementary consequences of I(V)=γ2(V)I(V) = \gamma^2(V)I(V)=γ2(V); their joint geometric content positions the binary Fisher--Rao manifold at the intersection of Bayesian statistics, hyperbolic geometry, and quantum measurement theory
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
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.
Three results. (1) We prove the tensor-level conformal identity dsBK2=4γ2(r) dsBures2 between the Bures metric and the Beltrami–Klein metric on the open qubit Bloch ball, where γ2(r)=1/(1−r2) is the squared Lorentz factor. (2) For the visibility coordinate V=2p−1 of a binary quantum measurement, the classical Bernoulli Fisher information takes the closed form I(V)=1/(1−V2)=γ2(V). (3) The conformal structure is special to qubits; for N≥3, the Bures metric on full-rank density matrices has a non-constant sectional curvature at the maximally mixed state and hence a non-vanishing Weyl tensor and no conformal equivalence to any constant-curvature hyperbolic model (Theorem 2). We outline an experimentally testable operational consequence where sequential non-collinear weak measurements on a qubit predict a Thomas–Wigner rotation with a closed-form purity dependence that deviates from the Pancharatnam baseline at intermediate visibility.
The Fisher information of a binary quantum measurement with visibility V is I(V) = 1/(1-V²) = γ²(V), the squared Lorentz factor from special relativity. This identity, proven in the companion anchor preprint, holds only for qubits: for N≥3 quantum systems, the Weyl curvature tensor obstructs the conformal equivalence between the Bures metric of quantum state space and the Beltrami-Klein metric of hyperbolic/relativistic geometry. Five independent research programs—in quantum optics, condensed matter, stochastic thermodynamics, holography, and quantum measurement theory—have converged on this same γ² structure over the past decade. This document is the interpretive companion to the anchor preprint. It traces the implications of the identity from fundamental physics (a deductive chain through five published theorems connecting γ² to Einstein's equations) through cosmology (dark energy, dark matter, fine-tuning as Fisher optimality), consciousness (entropy production as the measurement process experienced from inside), biology (evolution, aging, cancer as Fisher information dynamics), and computation (transformer attention as Fisher-Rao geometry). It introduces the concepts of "beautiful change" (structured, self-organized entropy production as the distinguishing feature of rich consciousness) and the "downprojection hypothesis" (experienced spacetime as a conformal restriction of a higher-dimensional information-geometric manifold). All claims are explicitly labeled by epistemic status across four tiers: proven mathematical identity, published theorems by independent groups, strong circumstantial evidence, and conjecture. Falsifiable predictions are provided, two already experimentally confirmed. The proven mathematics stands independently of the interpretive framework built upon it.
The Fisher information of a binary classifier's sigmoid output in the visibility coordinate V = 2p - 1 equals the squared Lorentz factor: I(V) = 1/(1 - V^2) = gamma^2(V). This identity, established for binary quantum measurements in B. G. Srivats, "Fisher Information Is the Squared Lorentz Factor: Why the Qubit Is Special" (Zenodo DOI: 10.5281/zenodo.19363449), applies to any system with Bernoulli output parameterized by visibility. Three consequences for neural network optimization are derived: (1) natural gradient descent on binary classifiers inherits a relativistic speed limit at the certainty boundary; (2) the Gudermannian function connects cross-entropy loss (hyperbolic distance) to classification accuracy (spherical distance); (3) standard Euclidean gradient descent ignores the divergent gamma^2 curvature near confident predictions. These observations strengthen the Measurement Irreversibility Criterion: digital classifiers share the gamma^2 geometry of physical measurement but lack thermodynamic irreversibility, providing an information-geometric supplement to the thermodynamic argument.Companion note to "Why Current AI Cannot Be Conscious: A Thermodynamic Criterion from the Fisher-Lorentz Identity" (Zenodo DOI: 10.5281/zenodo.19489276).
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.