Directional Displacement-Fidelity Response

• Directional displacement-fidelity response is a framework that quantifies both the magnitude of displacement and its direction-specific impact using tailored fidelity metrics.
• It distinguishes scalar displacement measures from directional effects by employing methods such as KL divergence, localization precision, norm ratios, and anisotropic state evaluations.
• This approach guides system design and optimization across various domains, ensuring improved operational reliability and task-specific performance.

Searching arXiv for the exact phrase and closely related papers to ground the article. Across the cited literature, directional displacement-fidelity response denotes a class of analyses in which a displacement, perturbation, or routed response is evaluated jointly with a direction-sensitive criterion and a fidelity measure. In some works the displacement is literal position or motion; in others it is a distributional divergence, a phase-space translation, or a shift of an effective center. The corresponding fidelity is benchmark quality, localization precision, overlap fidelity, one-way transmission purity, or direction-specific reconstruction accuracy. A recurring theme is that displacement magnitude alone is often insufficient: sign, branch, anisotropy, topology, or task dependence must also be resolved (Nikolić et al., 17 Jun 2026, Bag et al., 2019, Kiefer et al., 26 May 2026, Rocklin, 2016).

1. Core formulations and recurring structure

Representative formulations span several technically distinct domains.

Domain	Displacement or perturbation	Fidelity quantity
Quantized LLM deployment	Distributional disagreement with a BF16 reference	Downstream benchmark score
Integrated photonic sensing	Lateral antenna displacement $x$	Localization precision $\sigma_x$
Continuous-variable photonics	Phase-space displacement $\alpha=\epsilon e^{i\phi}$	$F_\psi(\alpha)$ and $R_F(\phi)$
Topological metamaterials	Bulk displacement field under local forcing	$$F=\|\mathbf{u}_{\rm allowed}\|/\|\mathbf{u}_{\rm total}\|$$

In quantized LLM deployment, per-token KL divergence is defined between the next-token distribution $P$ of the high-precision reference and $Q$ of a quantized candidate as

$$D_{\mathrm{KL}}(P\|Q)=\sum_{x\in V}P(x)\log\frac{P(x)}{Q(x)},$$

with mean, median, percentile, and top-$k$ aggregations used as fidelity metrics. In integrated photonic sensing, the normalized directionality signal is

$\sigma_x$0

in the linear regime, and the localization precision is

$\sigma_x$1

In continuous-variable photonics, the displacement fidelity of a pure probe $\sigma_x$2 is

$\sigma_x$3

and the fidelity-threshold radius $\sigma_x$4 is the maximal $\sigma_x$5 along a fixed phase-space ray for which the fidelity remains above a prescribed threshold. In topological Maxwell lattices, directional fidelity is expressed as the norm ratio of allowed to total displacement response (Nikolić et al., 17 Jun 2026, Bag et al., 2019, Kiefer et al., 26 May 2026, Rocklin, 2016).

These definitions show that the term is not a single standardized formalism. Rather, it names a recurring problem class: quantifying how far a system moves or deviates, in which direction the effect propagates, and how faithfully that response meets a task criterion.

2. Quantized LLMs: displacement is not direction

The sharpest recent critique of magnitude-only fidelity proxies appears in quantized LLM deployment. A study of a 28-quant cohort of Qwen3.6-35B-A3B and a 41-quant cohort of Devstral-Small-2-24B found that mean WikiText KLD is strongly correlated with a composite of eight downstream benchmarks over the full cohort, with Spearman $\sigma_x$6 on Qwen and $\sigma_x$7 on Devstral, both with $\sigma_x$8. However, in the near-baseline “silent zone,” the relationship collapses to non-significance: $\sigma_x$9 on Qwen and $\alpha=\epsilon e^{i\phi}$0, $\alpha=\epsilon e^{i\phi}$1, on Devstral. The collapse persists across 14 measurement variants, including different KLD aggregations, perplexity formulations, top-1 agreement, calibration corpora, and context lengths (Nikolić et al., 17 Jun 2026).

The paper formalizes this failure by decomposing benchmark changes into drops $\alpha=\epsilon e^{i\phi}$2, leapfrogs $\alpha=\epsilon e^{i\phi}$3, volume $\alpha=\epsilon e^{i\phi}$4, and a direction factor $\alpha=\epsilon e^{i\phi}$5. In that decomposition, displacement corresponds to the total number of disagreements with the reference, whereas direction captures whether those disagreements are beneficial or harmful. The central empirical result is that KLD primarily measures disagreement volume, not the direction of score change. Inside the silent zone, the composite score correlates strongly with volume through KLD, with $\alpha=\epsilon e^{i\phi}$6 on Qwen and $\alpha=\epsilon e^{i\phi}$7 on Devstral, while the relation to the direction factor is weak and task-conditional.

The per-prompt analysis sharpens the point. On LiveCodeBench, failed prompts have geometric-mean per-prompt KLD values only $\alpha=\epsilon e^{i\phi}$8 to $\alpha=\epsilon e^{i\phi}$9 higher than passed prompts across five silent-zone Qwen quants. As a cross-model router on disagreement prompts, choosing the lower per-prompt KLD model succeeds only $F_\psi(\alpha)$0 to $F_\psi(\alpha)$1 of the time, worse than chance. The study therefore distinguishes a lossy zone, where scalar fidelity metrics remain useful for filtering out grossly degraded quants, from a silent zone, where smaller divergence no longer implies better downstream quality. A common misconception in deployment practice is that lower KLD or perplexity automatically yields better near-baseline model selection; the reported results explicitly reject that inference.

3. Sensing and measurement systems

In integrated photonic displacement sensing, directional response is produced by Huygens-dipole emission in a six-way photonic crystal waveguide crossing. A tightly focused, radially polarized beam at $F_\psi(\alpha)$2 and $F_\psi(\alpha)$3 excites a Si nanoparticle so that off-axis motion drives both a longitudinal electric dipole and a transverse magnetic dipole. Under the Kerker or Huygens condition, their interference becomes unidirectional and routes light into a preferred waveguide arm. For a small lateral displacement $F_\psi(\alpha)$4, the horizontal-arm powers obey $F_\psi(\alpha)$5 and $F_\psi(\alpha)$6, so the normalized directionality is linear in $F_\psi(\alpha)$7. Experimentally, the relevant sensitivity is $F_\psi(\alpha)$8, the measured noise floor is $F_\psi(\alpha)$9, and the achieved localization precision is $R_F(\phi)$0; the prototype also demonstrated a standard deviation of the position accuracy below $R_F(\phi)$1 at room temperature and ambient conditions (Bag et al., 2019).

Full-range LVDT modeling addresses a different failure mode: linear-region characterization does not suffice once the response becomes non-linear or multivalued. A unified analytic expression parameterized by $R_F(\phi)$2 reproduces the LVDT differential output across the entire measured stroke and provides closed-form first and second derivatives. The first derivative is interpreted as instantaneous sensitivity, the second as curvature or non-linearity. In the central region $R_F(\phi)$3, the gain is nearly constant; at larger $R_F(\phi)$4, geometric non-linearity appears; and near extrema, the sign of $R_F(\phi)$5 distinguishes rising from falling branches when $R_F(\phi)$6. Over the tested interval $R_F(\phi)$7 to $R_F(\phi)$8, the relative deviation between fit and data remains below $R_F(\phi)$9, with central slope $$F=\|\mathbf{u}_{\rm allowed}\|/\|\mathbf{u}_{\rm total}\|$$0 and quadratic curvature coefficient $$F=\|\mathbf{u}_{\rm allowed}\|/\|\mathbf{u}_{\rm total}\|$$1. The framework is proposed for unambiguous reconstruction over large quasi-static excursions, overload recovery, and directional reversals (Kukkadapu, 13 Jun 2026).

Vision-based structural displacement measurement introduces direction-specific fidelity metrics explicitly. VFM-SDM combines VFM-inferred camera parameter estimation, point tracking, stereo triangulation, metric scale recovery, and structural geometry refinement to reconstruct vertical and lateral displacements without task-specific training, marker installation, or manual camera calibration. The reported metrics are range-normalized RMSE, Pearson correlation coefficient, and relative peak-to-peak amplitude error, computed separately for vertical and lateral directions. On a representative field sequence, the framework achieved $$F=\|\mathbf{u}_{\rm allowed}\|/\|\mathbf{u}_{\rm total}\|$$2, correlation coefficient $$F=\|\mathbf{u}_{\rm allowed}\|/\|\mathbf{u}_{\rm total}\|$$3, and $$F=\|\mathbf{u}_{\rm allowed}\|/\|\mathbf{u}_{\rm total}\|$$4 for vertical and lateral displacements. The summary over all sequences indicates stronger performance vertically and more variable performance laterally, with structural geometry refinement providing its greatest benefit in the smaller-amplitude lateral channel (Xian et al., 10 May 2026).

4. Mechanical, topological, and microfluidic response

In topological Maxwell lattices, directional displacement-fidelity response is a bulk property enforced by topology. Small displacements $$F=\|\mathbf{u}_{\rm allowed}\|/\|\mathbf{u}_{\rm total}\|$$5, bond extensions $$F=\|\mathbf{u}_{\rm allowed}\|/\|\mathbf{u}_{\rm total}\|$$6, and forces $$F=\|\mathbf{u}_{\rm allowed}\|/\|\mathbf{u}_{\rm total}\|$$7 are related through the rigidity matrix $$F=\|\mathbf{u}_{\rm allowed}\|/\|\mathbf{u}_{\rm total}\|$$8, with dynamical matrix $$F=\|\mathbf{u}_{\rm allowed}\|/\|\mathbf{u}_{\rm total}\|$$9. The topological polarization vector $P$0, obtained from the winding of $P$1, determines the half-space into which response can propagate. For a localized force, the real-space Green’s function satisfies $P$2 if $P$3, so the response has strictly one-sided support in a fully polarized lattice. Fidelity is then quantified as

$P$4

In ideal fully polarized systems $P$5, while finite-size effects, Guest modes, and edge-mode leakage reduce it. The generalized kagome example yields fidelities $P$6 for lattices of order $P$7, and the deviation from perfect directionality scales as $P$8 (Rocklin, 2016).

Deterministic lateral displacement microfluidics expresses directionality through locking steps rather than topological half-spaces. For spherical particles moving through a square obstacle array, the average migration angle $P$9 locks to rational directions $Q$0 satisfying $Q$1. The allowed forcing angles obey

$Q$2

which generates the Devil’s-staircase structure of $Q$3. Here $Q$4 is the critical offset, absorbed into an effective-radius point-particle model. In binary fractionation, the separation fidelity is the angular separation of the mean migration directions, and the maximum possible resolution among simple staircases is $Q$5, attained when one species remains in $Q$6 and the other locks to $Q$7 (Risbud et al., 2014).

These two systems embody different mechanisms—topological Green’s-function support versus collision-induced directional locking—but both make directionality a discrete, structured property rather than a smooth function of displacement magnitude alone.

5. Thermodynamic, quantum-optical, and photonic-information realizations

In stochastic nanoscale transport, directional fidelity is a thermodynamic quantity. For an isothermal nanoscale motor or particle, the universal equality

$Q$8

relates the maximum achievable directional fidelity $Q$9 to the free-energy input $$D_{\mathrm{KL}}(P\|Q)=\sum_{x\in V}P(x)\log\frac{P(x)}{Q(x)},$$0, with inverse form

$$D_{\mathrm{KL}}(P\|Q)=\sum_{x\in V}P(x)\log\frac{P(x)}{Q(x)},$$1

The bound is derived from cycle-flux and entropy-production arguments and is supported by translational and rotational experiments, including kinesin and force-induced F$$D_{\mathrm{KL}}(P\|Q)=\sum_{x\in V}P(x)\log\frac{P(x)}{Q(x)},$$2-ATPase motion (Wang et al., 2013).

For F$$D_{\mathrm{KL}}(P\|Q)=\sum_{x\in V}P(x)\log\frac{P(x)}{Q(x)},$$3-ATPase itself, the load-dependent maximal fidelity becomes

$$D_{\mathrm{KL}}(P\|Q)=\sum_{x\in V}P(x)\log\frac{P(x)}{Q(x)},$$4

with stall at $$D_{\mathrm{KL}}(P\|Q)=\sum_{x\in V}P(x)\log\frac{P(x)}{Q(x)},$$5. The small-load sensitivity is

$$D_{\mathrm{KL}}(P\|Q)=\sum_{x\in V}P(x)\log\frac{P(x)}{Q(x)},$$6

The experimentally measured stepping ratio satisfies $$D_{\mathrm{KL}}(P\|Q)=\sum_{x\in V}P(x)\log\frac{P(x)}{Q(x)},$$7, and the reported data imply tight chemomechanical coupling up to stalemate, with directionality approaching its thermodynamic limit (Hou et al., 2013).

In continuous-variable photonics, the central object is the fidelity-threshold displacement radius $$D_{\mathrm{KL}}(P\|Q)=\sum_{x\in V}P(x)\log\frac{P(x)}{Q(x)},$$8, defined from the overlap $$D_{\mathrm{KL}}(P\|Q)=\sum_{x\in V}P(x)\log\frac{P(x)}{Q(x)},$$9. Photon-conditioned squeezed states, Fock states, and cat states are compared at matched mean photon number $k$0. Fock states are isotropic, cat states show moderate anisotropy, and squeezed single-photon or two-photon-subtracted states exhibit the expected $k$1 anisotropy between conjugate quadratures. The two-photon-subtracted squeezed state shows favorable displacement-fidelity radii over selected quadrature directions at matched $k$2, and the advantageous directions form a finite angular sector around the squeezed axis. The same anisotropy is proposed both for homodyne-aligned displacement-noise mitigation and for weak-displacement sensing along the conjugate axis (Kiefer et al., 26 May 2026).

Engineered conical intersections in trapped Rydberg ions provide a control-theoretic version of the same theme. In the full spinor model, localized nonadiabatic coupling near the conical intersection breaks left-right symmetry and yields directed motion: the wave packet passes the coupling region twice early in the protocol and then proceeds predominantly toward the target. In the Born–Oppenheimer limit, by contrast, the optimized field produces symmetric multi-cycle oscillations. Both cases reach similar final fidelities at $k$3, $k$4 with the conical intersection and $k$5 without it, but the trajectories are qualitatively different (Belfakir et al., 14 Sep 2025).

Device-level photonic implementations translate geometric or directional control into operational fidelity. Composite segmented directional couplers reduce CNOT-gate sensitivity to fabrication variation, lowering the measured mean error probability from $k$6 to $k$7, while interferometric directional Josephson devices achieve $k$8 qubit readout fidelity, isolation up to $k$9, and in-situ enhancement of $\sigma_x$00 and $\sigma_x$01 by two orders of magnitude (Piasetzky et al., 29 Sep 2025, Abdo et al., 2020).

6. Computational design, optimization, and cross-domain principles

Several recent methods make direction-specific fidelity an explicit optimization target. Phase-Center-Constrained Beamforming minimizes phase-center displacement while preserving directional gain through a constrained nonconvex optimization problem with PCO, energy-compactness, and beampattern-fidelity terms. In a simulated $\sigma_x$02 GNSS array, the method reduces the phase-center norm from about $\sigma_x$03 under conventional beamforming to about $\sigma_x$04, a fivefold reduction, while keeping the maximum sidelobe gain increase below $\sigma_x$05 and the main-lobe loss below $\sigma_x$06. The reported stability analysis over 50 random initializations shows that repeated optimization further reduces the PCO norm (Steckel et al., 26 Mar 2025).

DS-HGNN addresses directional displacement fidelity in structural surrogate modeling by preserving separate longitudinal and transverse streams throughout message passing. Edge states are initialized from edge type, sinusoidal positional encoding, and boundary kinematics; geometry and loading enter through FiLM-conditioned spectral convolutions; and the final displacement field is reconstructed by a spectral-bypass low-rank readout. The model achieves the lowest stress and displacement RMSE among six benchmark heterogeneous GNNs and reaches comparable accuracy to the strongest benchmarks using $\sigma_x$07–$\sigma_x$08 fewer training samples. The direction-specific metrics are $\sigma_x$09 and $\sigma_x$10, corresponding to normalized errors of approximately $\sigma_x$11 and $\sigma_x$12 of the peak in-plane displacement (Cai et al., 18 Jun 2026).

Taken together, these results suggest three recurring principles. First, scalar displacement proxies often capture how much a system departs from a reference but not whether that departure is beneficial, harmful, or even on the correct branch; the silent-zone failure of KLD and the multivalued regions of LVDTs are direct examples. Second, high directional fidelity typically requires explicit structural information—topological polarization, branch derivatives, constrained geometry, separate directional channels, or anisotropic state design—rather than post hoc ranking by a scalar error. Third, many systems exhibit a dual-use regime in which the same directional asymmetry can either suppress unwanted perturbations or enhance sensitivity to desired ones, as seen in squeezed-state displacement response, Josephson isolation, and phase-center-constrained beamforming.

A common misconception is that fidelity is always monotone in displacement magnitude. The cited literature repeatedly shows otherwise. Lower KLD need not imply higher downstream quality near baseline; larger Wigner negativity need not imply larger directional displacement radii; linear-region calibration need not characterize full-stroke sensor response; and preserving directional gain alone need not control phase-center displacement. Directional displacement-fidelity response is therefore best understood not as a single metric, but as a family of formalisms for separating displacement magnitude from directional consequence and for designing systems in which that consequence remains operationally faithful.