From Measurement to Reality I(V) = γ²(V) and the Geometry of Binary Distinction

C. Muñoz, A. B. Klimov

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.

Masanari Asano, Masanori Ohya, Yuta Yamamori

Bharath G. Srivats

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.

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.

Allen Stairs