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Operational Identity under Boosts - Classical Transparency and a Reduced-Spin Accessibility Tax

Paul Cooney

Published Jul 13, 2026• 0 citations
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Abstract

This paper investigates whether relativistic motion imposes an intrinsic cost on certifying a system’s identity. Identity refers here to internal physical parameters, such as a spectroscopic splitting, composition marker, or spin orientation, rather than to Poincaré-invariant labels such as mass and total spin. The central result is that a fixed Lorentz boost does not destroy identity information. A fixed, parameter-independent boost acts unitarily on the complete physical state, so the global quantum Fisher information associated with an internal parameter is exactly preserved. Any apparent loss of identity information must therefore arise from restricted measurements, finite detector resources, discarded degrees of freedom, incomplete knowledge of the boost, or averaging over transformations that are not recorded. The paper develops this conclusion through classical spectroscopy, relativistic flyby estimation, photon frequency encoding, and massive spin-momentum systems. Classical spectral identity A Doppler transformation uniformly rescales the line center, line width, and internal splitting of a spectrum. For a spectral doublet, the dimensionless ratio between the splitting and the line center remains unchanged. The per-photon Fisher information about the rest-frame splitting is therefore invariant under an ideal Doppler dilation. Relativistic motion can still change the total information collected because photon number, detector resolution, observing duration, and spectral bandwidth depend on the measurement protocol. The paper compares three detector classes: An ideal spectrometer with unlimited bandwidth A spectrometer with fixed observer-frame resolution A resolution-limited detector with a finite observing band These detectors produce different information-versus-velocity laws. Relativity therefore does not impose a universal identity-collapse exponent independently of the detector and resource convention. Relativistic flyby self-calibration The paper considers a source undergoing a relativistic flyby with unknown velocity, impact parameter, time of closest approach, source amplitude, rest transition frequency, and spectral splitting. The changing line-of-sight velocity generates a Doppler chirp. The same photons can therefore constrain both the trajectory and the source’s internal identity. For a symmetric doublet observed over a window that samples both sides of the encounter, the variance penalty from marginalizing over the unknown trajectory has the form variance inflation = 1 + O[(Δ/ν₀)²], where Δ is the rest-frame splitting and ν₀ is the central rest frequency. The trajectory is determined primarily through the much larger frequency scale ν₀, while the identity parameter is encoded in the smaller separation Δ. For the numerical examples studied, this produces a Fisher-information hierarchy of approximately one million and a trajectory-marginalization penalty below approximately 0.1 percent. The result remains valid when the rest transition frequency is treated as unknown. Two mechanisms make self-calibration possible. First, the ratio of the early and late Doppler asymptotes constrains the source velocity without requiring an external frequency reference. Second, changing the central frequency translates both spectral lines together, while changing the splitting moves them symmetrically apart. For a symmetric doublet, these score directions are approximately orthogonal. The self-calibration result is conditional on adequate signal strength, a resolved spectral structure, a sufficiently long observing window, and a nonsingular nuisance-parameter Fisher matrix. Retarded-time flyby calculation The paper explicitly includes the retarded relation between emission time and observation time. This confirms the small trajectory-marginalization penalty but changes the interpretation of total information accumulation. The blueshifted approach phase is compressed into a short interval of observer time. As a result, an ideal detector can collect less total information at higher velocity, while a resolution-limited detector can benefit when blueshift moves the doublet above its instrumental resolution floor. Whether relativistic motion helps or hinders estimation is therefore protocol-dependent. Quantum frequency encoding For a single photon whose identity parameter is encoded in its frequency wavefunction, a collinear Lorentz boost acts as a unitary frequency dilation. The global quantum Fisher information about the encoded parameter is exactly invariant. This result applies to the complete frequency state. It does not imply that every reduced photonic measurement remains invariant, particularly when momentum and polarization are coupled or when the detector accesses only a restricted measurement algebra. Massive carriers and reduced-spin accessibility For a massive spin-1/2 carrier, a Lorentz boost generates momentum-dependent Wigner rotations. The complete spin-momentum state preserves its global quantum Fisher information. However, a detector that measures spin while ignoring momentum accesses only the reduced spin state. Tracing over momentum can therefore reduce the spin-only quantum Fisher information. For canonical Wigner spin, a narrow isotropic Gaussian momentum packet, and the spin-angle encoding studied in the paper, the leading fractional accessibility loss is proportional to [2/(1 + cos²θ)] × (σₚ/mc)² × tanh²(ξ/2), where θ is the encoding angle, σₚ is the momentum spread, m is the carrier mass, and ξ is the boost rapidity. The accessibility tax has three important properties. It depends on the spin-encoding direction. It increases with momentum spread. It saturates at high rapidity because tanh²(ξ/2) approaches one. The reduced-spin information loss therefore does not grow without bound as the relative velocity approaches the speed of light. The missing information is recoverable in principle. It has been transferred into momentum dependence and spin-momentum correlations. A joint measurement of spin and momentum restores access to the invariant global information. The effect is therefore a subsystem-accessibility cost, not an irreversible destruction of identity. Localization and action-count scaling For a minimum-uncertainty wave packet, σₓσₚ = ℏ/2. The Compton wavelength is λ_C = ℏ/(mc). Using the localization scale R = σₓ, the rest-frame action-count is Ξ = ER/(ℏc) = mcσₓ/ℏ = σₓ/λ_C. At large rapidity, the reduced-spin accessibility tax scales as Ξ⁻², with an additional factor determined by the spin-encoding angle. A carrier localized over a distance much larger than its Compton wavelength therefore pays a negligible reduced-spin accessibility cost. For packets that do not saturate the uncertainty relation, the same argument gives a bound rather than an equality. Energy and identity information The paper asks whether the kinetic energy required to match the source frame can be interpreted as the price of recovering identity information. Three arguments reject a universal proportionality. First, frame-matching energy grows without bound with the Lorentz factor, while the reduced-spin accessibility loss saturates. Second, frame matching is not necessary. Classical Doppler self-calibration can recover a spectral identity parameter without accelerating the detector into the source frame. In the quantum case, expanding the measured algebra to include momentum can recover the lost spin-only information. Third, kinetic energy invested in frame matching can in principle be recovered by later deceleration, while the acquired measurement record remains. Energy constrains storage capacity, evolution speed, and thermodynamic reset costs, but the paper finds no universal energy price per unit of Fisher information acquired. Main conclusions The paper establishes that: Fixed, parameter-independent Lorentz boosts preserve global quantum Fisher information. Per-photon spectral identity information is invariant under ideal Doppler dilation. Total information remains detector- and protocol-dependent. Relativistic flyby chirps can self-calibrate unknown source kinematics. Trajectory marginalization produces only a small identity-estimation penalty for well-resolved symmetric doublets. Reduced-spin measurements can lose access to information through momentum-dependent Wigner rotations. The reduced-spin accessibility tax depends on encoding angle and saturates with rapidity. Joint spin-momentum measurements restore the invariant global information. For minimum-uncertainty packets, the reduced-spin tax scales as Ξ⁻². No universal proportionality exists between frame-matching energy and identity information. Scope The global invariance theorem applies to fixed Lorentz transformations that do not depend on the identity parameter. The paper does not analyze unknown boosts as joint quantum-estimation parameters or random boost distributions that are averaged without being recorded. The reduced-spin result is derived for a single massive spin-1/2 carrier, canonical Wigner spin, a narrow isotropic Gaussian momentum packet, and a momentum-insensitive spin measurement. Broad or anisotropic packets, composite carriers, alternative relativistic spin observables, generic reduced photonic channels, and sequential non-collinear boosts remain open extensions. The classical flyby conclusions apply to high-signal, sufficiently resolved, symmetric spectral doublets observed over an adequate time window. Relation to the companion paper The companion paper examines the accelerated rather than inertial case: Action-Count, Observer Integrity, and Horizon Saturation: A Unified Framework from Ξ = ER/(ℏc)https://doi.org/10.5281/zenodo.19599288 Together, the two papers distinguish the operational effects of velocity and acceleration. Uniform velocity preserves global identity information but can redistribut

DOI
10.5281/zenodo.21343346

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•Apr 18, 2026•1 citation
This paper investigates whether relativistic motion imposes an intrinsic cost on certifying a system’s identity. Identity refers here to internal physical parameters, such as a spectroscopic splitting, composition marker, or spin orientation, rather than to Poincaré-invariant labels such as mass and total spin. The central result is that a fixed Lorentz boost does not destroy identity information. A fixed, parameter-independent boost acts unitarily on the complete physical state, so the global quantum Fisher information associated with an internal parameter is exactly preserved. Any apparent loss of identity information must therefore arise from restricted measurements, finite detector resources, discarded degrees of freedom, incomplete knowledge of the boost, or averaging over transformations that are not recorded. The paper develops this conclusion through classical spectroscopy, relativistic flyby estimation, photon frequency encoding, and massive spin-momentum systems. Classical spectral identity A Doppler transformation uniformly rescales the line center, line width, and internal splitting of a spectrum. For a spectral doublet, the dimensionless ratio between the splitting and the line center remains unchanged. The per-photon Fisher information about the rest-frame splitting is therefore invariant under an ideal Doppler dilation. Relativistic motion can still change the total information collected because photon number, detector resolution, observing duration, and spectral bandwidth depend on the measurement protocol. The paper compares three detector classes: An ideal spectrometer with unlimited bandwidth A spectrometer with fixed observer-frame resolution A resolution-limited detector with a finite observing band These detectors produce different information-versus-velocity laws. Relativity therefore does not impose a universal identity-collapse exponent independently of the detector and resource convention. Relativistic flyby self-calibration The paper considers a source undergoing a relativistic flyby with unknown velocity, impact parameter, time of closest approach, source amplitude, rest transition frequency, and spectral splitting. The changing line-of-sight velocity generates a Doppler chirp. The same photons can therefore constrain both the trajectory and the source’s internal identity. For a symmetric doublet observed over a window that samples both sides of the encounter, the variance penalty from marginalizing over the unknown trajectory has the form variance inflation = 1 + O[(Δ/ν₀)²], where Δ is the rest-frame splitting and ν₀ is the central rest frequency. The trajectory is determined primarily through the much larger frequency scale ν₀, while the identity parameter is encoded in the smaller separation Δ. For the numerical examples studied, this produces a Fisher-information hierarchy of approximately one million and a trajectory-marginalization penalty below approximately 0.1 percent. The result remains valid when the rest transition frequency is treated as unknown. Two mechanisms make self-calibration possible. First, the ratio of the early and late Doppler asymptotes constrains the source velocity without requiring an external frequency reference. Second, changing the central frequency translates both spectral lines together, while changing the splitting moves them symmetrically apart. For a symmetric doublet, these score directions are approximately orthogonal. The self-calibration result is conditional on adequate signal strength, a resolved spectral structure, a sufficiently long observing window, and a nonsingular nuisance-parameter Fisher matrix. Retarded-time flyby calculation The paper explicitly includes the retarded relation between emission time and observation time. This confirms the small trajectory-marginalization penalty but changes the interpretation of total information accumulation. The blueshifted approach phase is compressed into a short interval of observer time. As a result, an ideal detector can collect less total information at higher velocity, while a resolution-limited detector can benefit when blueshift moves the doublet above its instrumental resolution floor. Whether relativistic motion helps or hinders estimation is therefore protocol-dependent. Quantum frequency encoding For a single photon whose identity parameter is encoded in its frequency wavefunction, a collinear Lorentz boost acts as a unitary frequency dilation. The global quantum Fisher information about the encoded parameter is exactly invariant. This result applies to the complete frequency state. It does not imply that every reduced photonic measurement remains invariant, particularly when momentum and polarization are coupled or when the detector accesses only a restricted measurement algebra. Massive carriers and reduced-spin accessibility For a massive spin-1/2 carrier, a Lorentz boost generates momentum-dependent Wigner rotations. The complete spin-momentum state preserves its global quantum Fisher information. However, a detector that measures spin while ignoring momentum accesses only the reduced spin state. Tracing over momentum can therefore reduce the spin-only quantum Fisher information. For canonical Wigner spin, a narrow isotropic Gaussian momentum packet, and the spin-angle encoding studied in the paper, the leading fractional accessibility loss is proportional to [2/(1 + cos²θ)] × (σₚ/mc)² × tanh²(ξ/2), where θ is the encoding angle, σₚ is the momentum spread, m is the carrier mass, and ξ is the boost rapidity. The accessibility tax has three important properties. It depends on the spin-encoding direction. It increases with momentum spread. It saturates at high rapidity because tanh²(ξ/2) approaches one. The reduced-spin information loss therefore does not grow without bound as the relative velocity approaches the speed of light. The missing information is recoverable in principle. It has been transferred into momentum dependence and spin-momentum correlations. A joint measurement of spin and momentum restores access to the invariant global information. The effect is therefore a subsystem-accessibility cost, not an irreversible destruction of identity. Localization and action-count scaling For a minimum-uncertainty wave packet, σₓσₚ = ℏ/2. The Compton wavelength is λ_C = ℏ/(mc). Using the localization scale R = σₓ, the rest-frame action-count is Ξ = ER/(ℏc) = mcσₓ/ℏ = σₓ/λ_C. At large rapidity, the reduced-spin accessibility tax scales as Ξ⁻², with an additional factor determined by the spin-encoding angle. A carrier localized over a distance much larger than its Compton wavelength therefore pays a negligible reduced-spin accessibility cost. For packets that do not saturate the uncertainty relation, the same argument gives a bound rather than an equality. Energy and identity information The paper asks whether the kinetic energy required to match the source frame can be interpreted as the price of recovering identity information. Three arguments reject a universal proportionality. First, frame-matching energy grows without bound with the Lorentz factor, while the reduced-spin accessibility loss saturates. Second, frame matching is not necessary. Classical Doppler self-calibration can recover a spectral identity parameter without accelerating the detector into the source frame. In the quantum case, expanding the measured algebra to include momentum can recover the lost spin-only information. Third, kinetic energy invested in frame matching can in principle be recovered by later deceleration, while the acquired measurement record remains. Energy constrains storage capacity, evolution speed, and thermodynamic reset costs, but the paper finds no universal energy price per unit of Fisher information acquired. Main conclusions The paper establishes that: Fixed, parameter-independent Lorentz boosts preserve global quantum Fisher information. Per-photon spectral identity information is invariant under ideal Doppler dilation. Total information remains detector- and protocol-dependent. Relativistic flyby chirps can self-calibrate unknown source kinematics. Trajectory marginalization produces only a small identity-estimation penalty for well-resolved symmetric doublets. Reduced-spin measurements can lose access to information through momentum-dependent Wigner rotations. The reduced-spin accessibility tax depends on encoding angle and saturates with rapidity. Joint spin-momentum measurements restore the invariant global information. For minimum-uncertainty packets, the reduced-spin tax scales as Ξ⁻². No universal proportionality exists between frame-matching energy and identity information. Scope The global invariance theorem applies to fixed Lorentz transformations that do not depend on the identity parameter. The paper does not analyze unknown boosts as joint quantum-estimation parameters or random boost distributions that are averaged without being recorded. The reduced-spin result is derived for a single massive spin-1/2 carrier, canonical Wigner spin, a narrow isotropic Gaussian momentum packet, and a momentum-insensitive spin measurement. Broad or anisotropic packets, composite carriers, alternative relativistic spin observables, generic reduced photonic channels, and sequential non-collinear boosts remain open extensions. The classical flyby conclusions apply to high-signal, sufficiently resolved, symmetric spectral doublets observed over an adequate time window. Relation to the companion paper The companion paper examines the accelerated rather than inertial case: Action-Count, Observer Integrity, and Horizon Saturation: A Unified Framework from Ξ = ER/(ℏc)https://doi.org/10.5281/zenodo.19599288 Together, the two papers distinguish the operational effects of velocity and acceleration. Uniform velocity preserves global identity information but can redistribut
We ask whether Kochen-Specker contextuality is Lorentz invariant. On the full Hilbert space, it is, since boosts act unitarily. Restricted to spin, it is not: momentum acts as an environment applying momentum-dependent Wigner rotations, leaving spin observables unsharp and breaking the operator identities behind state-independent contextuality. For spin 1/21/21/2 one scalar governs the loss, the packet average ΔΔΔ of sin⁡2(θW/2)\sin^2(θ_W/2)sin2(θW​/2) (where θWθ_WθW​ is the Wigner angle); for any spin, a ray degrades at a rate set by its spin variance transverse to the boost. In the spin-only account, the Free Will Theorem's SPIN axiom fails for every Δ>0Δ>0Δ>0; the Yu-Oh set loses contextuality between Δ≈0.008Δ\approx0.008Δ≈0.008 and 0.0160.0160.016, with state independence going first; the Peres-Mermin square loses contextuality at Δ≈0.059Δ\approx0.059Δ≈0.059 or 0.0740.0740.074 by orientation; singlet CHSH loses nonlocality for Δ≥0.159Δ\geq0.159Δ≥0.159; and a GHZ-state OR gate loses Raussendorf's contextuality certificate at Δ≈0.206Δ\approx0.206Δ≈0.206. A qqq-plate implements the channel exactly on photon polarisation, with a retardation in place of the rapidity, so the nonlocality, GHZ and aimed-source results are testable without motion.
A Lorentz boost acts on the canonical spin of a massive particle through a momentum-dependent Wigner rotation. We show that, for one fixed observer boost, reducing over an uncertain momentum can strictly contract every pairwise spin-state trace distance without producing a channel that is degradable from the less contracted one. For spin 1/21/21/2, we first characterize the exact inversion-symmetric channel cone generated by a fixed Wigner angle and transverse momentum directions. Inside this cone lies the Pauli family Mα=diag⁡(1−α,1−α,1−2α)M_α=\operatorname{diag}(1-α,1-α,1-2α)Mα​=diag(1−α,1−α,1−2α), 0≤α<1/20\leqα<1/20≤α<1/2. For 0<α<β<1/20<α<β<1/20<α<β<1/2, all trace distances between distinct spin states are strictly smaller after MβM_βMβ​ than after MαM_αMα​, yet the unique linear post-processing factor has a negative normalized Choi eigenvalue. We solve the optimization over all physical converters exactly: 12inf⁡Λ∈CPTP∥Φβ−Λ∘Φα∥⋄=α(β−α)2−3α\frac{1}{2}\inf_{Λ\in\mathrm{CPTP}}\|Φ_β-Λ\circΦ_α\|_\diamond=\frac{α(β-α)}{2-3α}21​infΛ∈CPTP​∥Φβ​−Λ∘Φα​∥⋄​=2−3αα(β−α)​, whereas the reverse deficiency is β−αβ-αβ−α. Thus the identity dominates the family, while all positive-noise members are pairwise incomparable under CPTP post-processing. The ideal construction is realized as the narrow-packet limit of pure, normalizable five-component momentum states, and explicit perturbation and finite-shot tomography bounds certify an open set of examples. Separately, every nonidentity member fails embedding in a time-homogeneous Pauli-diagonal Lindblad semigroup. Hence ordering all unassisted spin distinguishabilities does not determine the quantum statistical post-processing order.
Abstract We investigate the limits of reconstructing relativistic particle-mass information from finite causal-source records, with particular emphasis on stochastic event histories, spectral identifiability, and classical–quantum jump dynamics. Izanagi Core Ver.3.0.0 distinguishes information determined by a finite Markov source from additional structures required to define quantum time evolution, local field interactions, stable particle excitations, and physical mass ratios. Three interconnected problems are examined: the nonuniqueness of source-compatible quantum spectral realizations, the preservation of classical event histories under quantum operations, and the existence of stationary states in source-driven bosonic field models. For finite irreducible reversible continuous-time Markov sources, we analyze stationary occupation measures, relative event clocks, typed transition histories, and spectral representations. A specified Markov generator determines its own relaxation spectrum, but that spectrum does not uniquely determine a physically admissible quantum Hamiltonian or relativistic particle masses. Explicit constructions demonstrate that compatible quantum representations can possess different internal spectral-gap ratios while reproducing the prescribed classical source information. The canonical Hilbert-space representation of a stationary Markov record further distinguishes stochastic time translation from positive-energy quantum dynamics: the nontrivial spectrum of the record-shift generator is symmetric about zero. This obstruction applies to the specified representation and does not exclude alternative quantum reconstructions. We formulate classical–quantum jump instruments in which transitions of a classical source induce completely positive operations on an independently specified quantum system. Exact preservation of the complete typed source-history law, including transition identities and waiting-time distributions, is characterized by the scalar-effect condition J_ji*(I) = q_ij I on every directed transition, when preservation is required independently of the quantum state. For source-conditioned transition intensities whose total discrepancies are uniformly bounded by δ, a coupling construction yields the finite-time total-variation estimate TV ≤ 1 − exp(−δT). The estimate is sharp in an explicit comparison model, while control over finite observation intervals does not imply uniform agreement over arbitrarily long times. To examine locality and quantum measurement structure, we construct conditional interactions with externally supplied bosonic free fields using future-supported Weyl displacement operators. The analysis distinguishes relativistic locality, overlap of localized operations, commutation of quantum state channels, and phase-sensitive coherent control. Weyl displacement channels can commute even when their implementing unitary operators differ under an exchange of order by a nontrivial central phase. Such phases can be detected only when an additional coherent ordering interface is available. These constructions clarify the distinction between preserving a classical event record and reconstructing the full operational content of an interacting quantum field. Two stationary-state results identify further limitations of one-way classical–quantum coupling. First, a product state consisting of the stationary source distribution and a designated pure free-field vacuum is stationary precisely when every positive-rate jump operation preserves the corresponding vacuum projector, J_ji(P) = q_ij P, under the stated autonomous-source dynamics. Nontrivial Weyl interactions can violate this condition without changing the classical source-history statistics. Second, for a stationary source driving a bosonic Fock field through equally weighted positive and negative Weyl displacements, we obtain an exact energy-balance relation, EH_f − EH_f = Γ_Q t. Here Γ_Q is the nonnegative transition-rate-weighted displacement energy. When Γ_Q > 0, the specified dynamics admit no normal joint stationary state with finite expected free-field energy. This heating obstruction depends on unbiased displacement kicks and nondissipative free-field evolution between transitions. It is not a universal consequence of classical-source autonomy: dissipative dynamics can balance energy injection and permit finite-energy stationary states. The results are placed in the context of reversible Markov processes, quantum spectral theory, GNS representations, canonical commutation relations, generalized free fields, completely positive quantum instruments, quantum dynamical semigroups, and classical–quantum hybrid dynamics. Particular attention is given to the distinction between established mathematical structures and their conditional application to source-to-matter reconstruction. The stationary-state and heating criteria describe a specified class of one-way quantum jump models; they do not constitute a general decoherence–diffusion theorem or a universal prohibition on stationary classical–quantum coupling. The principal unresolved issue is the physical generation of matter from causal-source dynamics. In the constructions studied here, the Markov generator, spacetime geometry, quantum carrier, interaction operators, and measurement interfaces are supplied as independent inputs rather than derived from the Minimal Realization Source Law (MRSL). Accordingly, the results do not establish a dynamically generated Lorentz-covariant matter sector, persistent physical identity between source events and particle excitations, stable isolated particle poles in the coupled relativistic theory, or a uniquely identifiable physical particle-mass ratio. Izanagi Core Ver.3.0.0 therefore provides a mathematically explicit account of the distinction between source-history preservation, quantum dynamical consistency, and physical mass identification, while leaving the source-generated origin and quantitative determination of relativistic particle masses open.
At eleven parameters a frame is a body with a momentum. The quantum step of the theory makes that momentum an operator, and a frame body in a superposition of momenta is a quantum reference frame. This letter asks what using such a frame does to it, in a model with one assumption. An ideal meter comoving with the frame body reads the canonical spins of test bodies, and nothing else. Relative to the frame each spin is turned by the Wigner rotation of the composition law, by an angle that depends on the frame's own rapidity. So every spin the frame serves is a weak measurement of the frame's velocity. The quantum Fisher information per spin is the square of the derivative of the Wigner angle, in closed form. For a frame at rest it is tanh 2 (θt/2), θt the test body's rapidity, and it is bounded by 1/ cosh 2 θF . With polarised spins and the records kept, n spins sharpen the frame's rapidity, to the van Trees bound 1/(σ -2 + n F ) within one percent. With the records discarded, or with unpolarised spins, the frame's momentum coherence decays by the overlap of the two rotated spin states per spin. For a Gaussian frame its purity is 1/ 1 + N F σ 2 , equal to the purity of the spins relative to the frame. The origin of the frame is the conjugate reference. Each discarded spin kicks the frame's position by ±ℏφ ′ /2M , a random walk, and the kept records widen it by the Heisenberg bound. The velocity reference is kept or sharpened while the origin degrades, the two being the conjugate blocks of the eleven. The frame of an entangled pair with a sharp total momentum has nothing to learn and no origin to lose. The sixteen of a body being the operators of two qubits, a boost is a coupling that is not a gate, and its cost in a simulation is stated. The statements are checked numerically.
Spectral constraints can alter the observable manifestation of quantum geometry without changing the local eigenspace. Relativistic electron vortices provide a sharp example: the spin orbit interaction redistributes spin and orbital angular momentum, yet its magnetic contribution differs between localized three-dimensional wave packets and stationary monochromatic beams. We resolve this discrepancy by showing that fixing the energy replaces ambient momentum derivatives in position-weighted current moments by shell-tangent derivatives and derive a Hermitian selection rule for first-order observables in momentum-dependent internal subspaces. Its extension-independent form applies beyond Dirac systems, including a broad class of antisymmetric first moments of scalar-charge velocity currents. For the free positive-energy Dirac doublet, the magnetic correction is an exact non-Abelian Berry-curvature contraction with a pointwise saturable quantum-metric bound. Changing the relative phase between the two σzσ_zσz​ spinor components switches this correction from zero in a conserved transverse-helicity sector to a finite value in a radial coherent superposition. At fixed finite Lorentz factor, the radial response scales as O(θ/γ)O(θ/γ)O(θ/γ), compared with O(θ2)O(θ^2)O(θ2) for longitudinal polarization. Spectral reduction and internal coherence thus provide distinct controls of the observable imprint of quantum geometry.
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