Boundary Encoding-Axis Asymmetry

• Boundary encoding-axis asymmetry is a framework where boundary conditions store or reveal directional information that breaks a symmetric exchange between axes.
• It is observed across fields—from quantum dynamics and CFT to directed graph learning—by using controlled measurement protocols that highlight robust aggregate responses.
• Operational diagnostics transform raw boundary signals into integrated topological invariants, offering practical insights into asymmetric transmission and system responses.

“Boundary encoding-axis asymmetry” can be treated as an Editor’s term for a recurrent technical motif in which a boundary, boundary condition, or boundary-local degree of freedom stores, selects, or reveals information that is not symmetric with respect to a distinguished axis, role, or propagation direction. In the works considered here, that motif appears in sharply different forms: boundary symmetry breaking encoded in reduced density matrices of intervals attached to a boundary (Fossati et al., 2024, Kusuki et al., 2024), a frequency–momentum hierarchy in pole-skipping reconstruction where the frequency axis is privileged over spatial momentum (Lu et al., 15 Jun 2025), a boundary-local metrological asymmetry between $x$- and $y$-axis encodings in the open Kitaev chain (Płodzień et al., 1 May 2026), and the transmission of odd boundary shaping into core magnetic-axis asymmetry in tokamak equilibria (Rodrigues et al., 2013). This suggests that the topic is not a single invariant or formalism, but a structural class of asymmetry mechanisms.

1. Structural definition and recurring ingredients

Across these literatures, three ingredients recur. First, the encoding locus is boundary-like: a physical boundary, a contact region, a symmetry-breaking boundary condition, a clamped base, a plasma edge, or a designated role boundary between source and destination tokens. Second, the asymmetry is axis-resolved: time versus space, source versus destination, power versus recovery stroke, left versus right transport, or odd versus even boundary harmonics relative to an axis. Third, the observable of interest is usually not a raw local signal but a robust aggregate or asymptotic quantity, such as a plateau, an algebraic approach to a universal limit, a recursive reconstruction scheme, or a near-axis scaling law.

The surveyed works also show that “axis” need not mean a Euclidean coordinate axis. In the pole-skipping reconstruction of black-hole exteriors, the relevant asymmetry is between the boundary frequency $\omega$ and the spatial momentum invariant $\mu=k^2$, while spatial isotropy is retained because only $k^2$ appears (Lu et al., 15 Jun 2025). In boundary conformal field theory, the axis is a symmetry-sector orientation selected by a boundary condition, so the asymmetry is measured under the group action $U_{A,g}$ rather than by spatial reflection (Fossati et al., 2024). In dynamic graphs, the axis is a semantic role axis—source versus destination—made explicit through separate role-conditioned encodings (Bonnet et al., 26 Feb 2026). In this sense, the phrase designates a hierarchy of non-exchangeable directions or roles whose distinction is encoded at, or revealed by, a boundary.

A further common feature is that the asymmetry is often operational rather than purely descriptive. The topological-QFI work does not merely note that two boundary encoding axes differ; it identifies an exact channel-resolved diagnostic,

$$F_{Q,y}^{(j)}\big|_{\theta_0=0}=|U_{j,k}(t)+V_{j,k}(t)|^2,\qquad F_{Q,x}^{(j)}\big|_{\theta_0=0}=|U_{j,k}(t)-V_{j,k}(t)|^2,$$

and shows that only one of these channels retains an $O(1)$ long-time boundary response in the topological phase (Płodzień et al., 1 May 2026). Likewise, the open-Floquet transport work shows that raw left/right transmission profiles are boundary-sensitive, while the cumulative asymmetry

$$\mathbf A(E_0)=\int_0^{E_0} dE\, \Bigl[ \bar{\mathcal T}^{L}(E)-\bar{\mathcal T}^{R}(E) \Bigr]$$

is boundary-robust and saturates to a topological plateau (Zhang et al., 25 Apr 2026). These examples indicate that boundary encoding-axis asymmetry is typically tied to a measurement protocol or reconstruction rule, not only to a kinematic distinction.

2. Boundary conditions as symmetry selectors in boundary conformal field theory

In boundary CFT, the most direct realization of the topic is a boundary condition that explicitly breaks a bulk symmetry $G$. For a subsystem $y$0 attached to that boundary, the reduced state $y$1 is generally not invariant under the restricted group action. The relevant symmetrized state is

$y$2

and the entanglement asymmetry is

$y$3

For finite groups, the asymptotic result for a boundary condition fully breaking $y$4 is

$y$5

where $y$6 is the smallest dimension of the symmetry-induced boundary-condition-changing operator; if only a subgroup $y$7 is preserved, $y$8 is replaced by $y$9 (Fossati et al., 2024). The physical content is that the boundary does not merely perturb short-distance entanglement; it selects a symmetry orientation that remains encoded nonlocally in $\omega$0 even for large $\omega$1.

The mechanism is formulated in terms of topological symmetry lines and boundary-condition-changing operators. When a $\omega$2-defect line is dragged into a symmetry-breaking boundary, the local move

$\omega$3

necessarily inserts a BCC operator $\omega$4. For the subsystem string order parameter, the resulting three-point chiral correlator gives

$\omega$5

whereas the neutral replicated charged moments controlling entanglement asymmetry scale as

$\omega$6

The boundary therefore acts as the place where the symmetry defect ceases to be topologically trivial, and the asymmetry exponent is determined by the lightest BCC operator rather than by bulk endpoint data (Fossati et al., 2024).

A closely related BCFT treatment rewrites the same phenomenon directly in defect language. For finite groups, the large-$\omega$7 asymmetry takes the form

$\omega$8

with $\omega$9 the lowest BCC scaling dimension induced by symmetry-related boundary conditions. For compact Lie groups, the leading growth is instead

$\mu=k^2$0

In a global quench from a symmetry-broken boundary state under a symmetry-preserving Hamiltonian, the same framework yields

$\mu=k^2$1

for a finite interval in the $\mu=k^2$2 regime (Kusuki et al., 2024). Here the asymmetry is literally boundary-encoded: it is the failure of a bulk topological symmetry defect to terminate topologically on the boundary.

3. Boundary-local topological asymmetry in quantum dynamics and transport

The open Kitaev chain provides the most explicit use of the phrase itself. A continuous angle $\mu=k^2$3 is encoded by a local spin rotation,

$\mu=k^2$4

and read out from the single-site reduced qubit at the left boundary. The crucial asymmetry is between the physical $\mu=k^2$5- and $\mu=k^2$6-encoding axes. For the optimal operating point $\mu=k^2$7, the local QFI is exactly

$\mu=k^2$8

with

$\mu=k^2$9

At the left boundary and for the real-pairing convention with $k^2$0, the $k^2$1-encoding channel has an $k^2$2 long-time boundary response, while the $k^2$3-encoding channel is exponentially suppressed with system size. The formal zero-mode diagnostic is

$k^2$4

so the $k^2$5 channel couples to the left Majorana envelope $k^2$6 and the $k^2$7 channel to the right Majorana envelope $k^2$8, whose weight at the left edge is exponentially small. The resulting boundary plateau,

$k^2$9

persists up to

$U_{A,g}$0

and is presented as a diagnostic that distinguishes topological boundary memory from a generic localized subgap signal (Płodzień et al., 1 May 2026).

Open Floquet lattices realize a different but closely related asymmetry. A finite driven region is connected to asymptotically free leads, and left- and right-incident transmissions,

$U_{A,g}$1

are strongly reshaped by contact nonadiabaticity. After local energy smoothing and a long-sample limit, however, the cumulative transmission imbalance

$U_{A,g}$2

approaches a plateau set by the bulk Floquet winding number,

$U_{A,g}$3

The mechanism is the deep-bulk branch-population principle: in the long-sample limit, each propagating Floquet–Bloch branch is generically populated with unit weight because true Floquet bound states are nongeneric. The branch-population identity

$U_{A,g}$4

therefore leads generically to $U_{A,g}$5, so the integrated asymmetry counts the net chirality of bulk branches rather than the details of the boundary lineshape (Zhang et al., 25 Apr 2026).

These two cases share a precise structural analogy. In the Kitaev chain, the boundary readout exposes a topological separation of Majorana quadratures along an encoding axis. In the Floquet lattice, the contacts expose a bulk directional imbalance along the transport axis. In both settings, a raw boundary-sensitive signal exists, but the robust diagnostic is an integrated or asymptotic quantity tied to a bulk topological structure.

4. Asymmetric encoding axes in reconstruction, quantum algorithms, and directed graph learning

In pole-skipping reconstruction of static planar-symmetric black holes, the encoding asymmetry is between frequency and momentum. The bulk metric is written as

$U_{A,g}$6

and a probe scalar after Fourier decomposition depends on

$U_{A,g}$7

Pole-skipping points occur at discrete frequencies

$U_{A,g}$8

while the geometry-dependent data at each level are the roots $U_{A,g}$9 of a degree-$$F_{Q,y}^{(j)}\big|_{\theta_0=0}=|U_{j,k}(t)+V_{j,k}(t)|^2,\qquad F_{Q,x}^{(j)}\big|_{\theta_0=0}=|U_{j,k}(t)-V_{j,k}(t)|^2,$$0 polynomial in $$F_{Q,y}^{(j)}\big|_{\theta_0=0}=|U_{j,k}(t)+V_{j,k}(t)|^2,\qquad F_{Q,x}^{(j)}\big|_{\theta_0=0}=|U_{j,k}(t)-V_{j,k}(t)|^2,$$1. The reconstruction is therefore organized as a frequency tower with momentum data attached to each level. The paper is explicit that the encoding is isotropic in spatial directions but anisotropic between temporal and spatial axes: frequency is discrete and universal, while momentum carries the nontrivial root data. At $$F_{Q,y}^{(j)}\big|_{\theta_0=0}=|U_{j,k}(t)+V_{j,k}(t)|^2,\qquad F_{Q,x}^{(j)}\big|_{\theta_0=0}=|U_{j,k}(t)-V_{j,k}(t)|^2,$$2,

$$F_{Q,y}^{(j)}\big|_{\theta_0=0}=|U_{j,k}(t)+V_{j,k}(t)|^2,\qquad F_{Q,x}^{(j)}\big|_{\theta_0=0}=|U_{j,k}(t)-V_{j,k}(t)|^2,$$3

so $$F_{Q,y}^{(j)}\big|_{\theta_0=0}=|U_{j,k}(t)+V_{j,k}(t)|^2,\qquad F_{Q,x}^{(j)}\big|_{\theta_0=0}=|U_{j,k}(t)-V_{j,k}(t)|^2,$$4 sets the thermal scale while $$F_{Q,y}^{(j)}\big|_{\theta_0=0}=|U_{j,k}(t)+V_{j,k}(t)|^2,\qquad F_{Q,x}^{(j)}\big|_{\theta_0=0}=|U_{j,k}(t)-V_{j,k}(t)|^2,$$5 supplies the nontrivial boundary datum. For $$F_{Q,y}^{(j)}\big|_{\theta_0=0}=|U_{j,k}(t)+V_{j,k}(t)|^2,\qquad F_{Q,x}^{(j)}\big|_{\theta_0=0}=|U_{j,k}(t)-V_{j,k}(t)|^2,$$6, only two new horizon coefficients appear at each level, while the remaining momentum-root data are constrained by universal homogeneous polynomial identities, such as

$$F_{Q,y}^{(j)}\big|_{\theta_0=0}=|U_{j,k}(t)+V_{j,k}(t)|^2,\qquad F_{Q,x}^{(j)}\big|_{\theta_0=0}=|U_{j,k}(t)-V_{j,k}(t)|^2,$$7

This makes the boundary encoding both asymmetric and redundant (Lu et al., 15 Jun 2025).

A formally different but operationally similar asymmetry appears in block encodings of finite-difference Laplacians. In $$F_{Q,y}^{(j)}\big|_{\theta_0=0}=|U_{j,k}(t)+V_{j,k}(t)|^2,\qquad F_{Q,x}^{(j)}\big|_{\theta_0=0}=|U_{j,k}(t)-V_{j,k}(t)|^2,$$8 dimensions, the discrete operator is assembled as a Kronecker sum,

$$F_{Q,y}^{(j)}\big|_{\theta_0=0}=|U_{j,k}(t)+V_{j,k}(t)|^2,\qquad F_{Q,x}^{(j)}\big|_{\theta_0=0}=|U_{j,k}(t)-V_{j,k}(t)|^2,$$9

with axis-specific grid sizes $O(1)$0, spacings $O(1)$1, and boundary conditions $O(1)$2. After scaling,

$O(1)$3

where

$O(1)$4

Here the “axis asymmetry” is literal: each coordinate axis is independently configurable, and mixed-boundary, anisotropic discretizations are implemented by a selector-controlled sum of modular 1D block encoders. The resulting circuit is an exact

$O(1)$5

block encoding of the scaled multidimensional Laplacian (Boutot et al., 12 Mar 2026).

Directed dynamic graphs provide a machine-learning analogue. DyGnROLE begins from the premise that source and destination nodes in a directed dynamic graph should not be encoded in a single role-blind latent space. The model uses separate role-semantic positional embeddings,

$O(1)$6

prepends role-specific pooling tokens,

$O(1)$7

and learns separate global representations

$O(1)$8

The self-supervised pretraining objective,

$O(1)$9

aligns the disentangled spaces using unlabeled interaction history. In the asymmetry-score analysis,

$$\mathbf A(E_0)=\int_0^{E_0} dE\, \Bigl[ \bar{\mathcal T}^{L}(E)-\bar{\mathcal T}^{R}(E) \Bigr]$$0

the baselines score $$\mathbf A(E_0)=\int_0^{E_0} dE\, \Bigl[ \bar{\mathcal T}^{L}(E)-\bar{\mathcal T}^{R}(E) \Bigr]$$1 on all datasets, whereas DyGnROLE has nonzero scores on all datasets, and it achieves the best Macro F1 on 7 of 8 datasets and ties on the 8th (Bonnet et al., 26 Feb 2026). This is a role-boundary version of encoding-axis asymmetry: source and destination are separated first, then coupled through a learned cross-boundary geometry.

5. Boundary-to-axis transmission in continuum geometry and mechanics

The tokamak equilibrium analysis is perhaps the most literal boundary-to-axis encoding problem in the corpus. Near the magnetic axis, the poloidal flux is

$$\mathbf A(E_0)=\int_0^{E_0} dE\, \Bigl[ \bar{\mathcal T}^{L}(E)-\bar{\mathcal T}^{R}(E) \Bigr]$$2

where $$\mathbf A(E_0)=\int_0^{E_0} dE\, \Bigl[ \bar{\mathcal T}^{L}(E)-\bar{\mathcal T}^{R}(E) \Bigr]$$3 is the odd shaping coefficient and therefore the direct local carrier of up-down asymmetry. The key transmission law is

$$\mathbf A(E_0)=\int_0^{E_0} dE\, \Bigl[ \bar{\mathcal T}^{L}(E)-\bar{\mathcal T}^{R}(E) \Bigr]$$4

so the odd $$\mathbf A(E_0)=\int_0^{E_0} dE\, \Bigl[ \bar{\mathcal T}^{L}(E)-\bar{\mathcal T}^{R}(E) \Bigr]$$5 boundary harmonic is propagated inward linearly with penetration factor $$\mathbf A(E_0)=\int_0^{E_0} dE\, \Bigl[ \bar{\mathcal T}^{L}(E)-\bar{\mathcal T}^{R}(E) \Bigr]$$6, while the even part receives an internal source correction. The geometric asymmetry measure satisfies

$$\mathbf A(E_0)=\int_0^{E_0} dE\, \Bigl[ \bar{\mathcal T}^{L}(E)-\bar{\mathcal T}^{R}(E) \Bigr]$$7

showing that core axis asymmetry is the ratio of penetrating odd boundary shaping to on-axis toroidal current density. Hollow current-density profiles and reverse magnetic shear are therefore favorable because they reduce the screening effect of $$\mathbf A(E_0)=\int_0^{E_0} dE\, \Bigl[ \bar{\mathcal T}^{L}(E)-\bar{\mathcal T}^{R}(E) \Bigr]$$8 and can even produce an inward asymmetry build-up (Rodrigues et al., 2013).

Twist-free axisymmetric vacuum spacetimes show a more geometric version of the same theme. After Geroch reduction and conformal rescaling, the norm of the axial Killing field is written as

$$\mathbf A(E_0)=\int_0^{E_0} dE\, \Bigl[ \bar{\mathcal T}^{L}(E)-\bar{\mathcal T}^{R}(E) \Bigr]$$9

and the axis is the boundary where $G$0. Regularity is not imposed by arbitrary boundary data but by elementary flatness,

$G$1

which in the reduced $G$2-dimensional formulation is equivalent to

$G$3

Regular axisymmetric scalars admit expansions in powers of $G$4, and the $G$5-adapted triad makes the axis prescription explicit through parity, near-axis series, and vanishing geodesic-obstruction coefficients. The axis is thus encoded geometrically as a boundary of the reduced problem rather than as an independently prescribed surface (Brink et al., 2013).

In soft robotic artificial cilia, the boundary is mechanical rather than geometric or field-theoretic. The propulsors are clamped to a base substrate, and asymmetry is passively encoded by a mismatch between the cast rest shape, the magnetization direction set during poling, and the geometry of the translating magnetic field. The convex and concave propulsors are identical curved elements rotated $G$6 about the vertical axis, so the sign of curvature relative to the power-stroke direction becomes the relevant encoding axis. Spatial and temporal asymmetry are quantified by

$G$7

The convex configuration aligns against the cast curvature during the power stroke and produces the strongest pumping; specifically, it generates $G$8 to $G$9 times the horizontal cycle-averaged momentum of the concave configuration at similar $y$00 (Peterman et al., 2024). This is a boundary-anchored actuation-axis asymmetry: the same driving waveform yields different nonreciprocal beating because the encoded shape–magnetization mismatch is referenced to a clamped base and a signed stroke axis.

6. Intrinsic axis asymmetry, related notions, and scope

Not every preferred-axis problem is a boundary-encoding problem. In planar gauges, asymmetry is intrinsic to the unit disk relative to the distinguished interior point $y$01. For smooth gauges, the outer asymmetry function

$y$02

compares support-direction data at opposite boundary points $y$03 and $y$04. For strictly convex gauges, the inner asymmetry function

$y$05

compares the two support points selected by a direction $y$06. Both vanish exactly for norms, are isometry-invariant, and are continuous in Hausdorff distance. The normalized duality relation

$y$07

shows that boundary-point $y$08 support-direction asymmetry and direction-axis $y$09 support-point asymmetry are exact dual encodings of the same nonsymmetry (Balestro et al., 2019). This is closely related in spirit, but the asymmetry is intrinsic to convex geometry rather than induced by an external boundary condition.

Plane convex bodies exhibit a bulk geometric notion of axis asymmetry through axiality,

$y$10

Every planar convex body has axiality at least

$y$11

while a family of convex quadrilaterals approaches

$y$12

This is a reflection-overlap measure of line symmetry rather than a boundary-encoding mechanism, but it clarifies that “axis asymmetry” can be quantified globally without any boundary-local data structure (Goenka et al., 2023).

Cosmological preferred-axis anomalies provide a further contrast. In CMB parity asymmetry, directional estimators built from rotationally variant power surrogates yield a stable preferred axis near

$y$13

in Galactic coordinates. That axis remains nearly unchanged across WMAP ILC7 and Planck Commander/NILC/SMICA, across several statistics, and under masking; it aligns within less than $y$14 of the CMB kinematic dipole and also aligns with the quadrupole and octopole axes, with the mutual alignment of these CMB axes claimed at more than $y$15 (Zhao et al., 2017). Here the axis is a preferred direction extracted from data rather than an encoding at a physical boundary.

These contrasts sharpen the scope of boundary encoding-axis asymmetry. The phrase is most precise when a boundary or boundary-like partition does one of three things: it selects a symmetry sector, transmits odd or directional data into a core or bulk observable, or enforces a non-exchangeable role structure that becomes visible in a robust measurement or reconstruction. Where the asymmetry is purely intrinsic, bulk statistical, or reflection-geometric, the topic becomes a related but distinct theory of axis asymmetry rather than boundary encoding in the strict sense.