DOI: [To be assigned]
John Swygert
July 31, 2026
Abstract
A 2026 experiment involving an all–van der Waals WTe₂/Fe₃GaTe₂/hBN heterostructure provides an unusually direct demonstration that the realized expression of electrical input can be redirected by changing a boundary-defined symmetry condition while leaving the principal material stack unchanged. WTe₂ serves as a low-symmetry spin source, Fe₃GaTe₂ as a perpendicular ferromagnet, and optically active spin defects in hBN as an integrated imaging layer. Electrical current applied along the low-symmetry crystallographic axis of WTe₂ permits an out-of-plane spin-polarization component and produces field-free deterministic switching of Fe₃GaTe₂. Current applied along the high-symmetry axis suppresses that route and produces an in-plane or thermally randomized intermediate state followed by stochastic perpendicular remagnetization. Spatial imaging further reveals that charge flow, magnetic nucleation, domain-wall propagation, pinning, and thermal degradation are nonuniform and locally coupled. This paper interprets the published experiment through The Theory of Spatial-Temporal Observational-Expressional Architectural Orientation, TSTOEAO, as a retrospective applied case rather than proof of the theory. It then establishes a prospective experiment in which current orientation is continuously varied relative to the WTe₂ crystallographic axes while spin torque, magnetic switching, current density, temperature, domain structure, latency, and stochasticity are measured simultaneously. The locked prediction is that suppression of one symmetry-permitted route will produce a reproducible redistribution into alternative spin, magnetic, thermal, transport, or unresolved channels rather than an unexplained disappearance of expression. The framework is weakened if withheld current orientations and local magnetic outcomes cannot be predicted from the preregistered symmetry, current-distribution, thermal, and defect architecture. Success would demonstrate the operational value of a second relational map for tracking route selection and cost location; it would not establish TSTOEAO as a unique or complete account of spin-orbit torque physics.
1. Introduction
Electrical energy supplied to a material does not determine one inevitable physical outcome. The result depends upon the pathways through which charge, spin, angular momentum, heat, lattice response, and magnetic order are permitted to interact.
This distinction is central to spin-orbit-torque systems. An applied charge current can generate a nonequilibrium spin polarization. That spin polarization can transfer angular momentum into a neighboring ferromagnet. Depending upon symmetry, interface structure, magnetic anisotropy, current distribution, defects, temperature, and pulse conditions, the final expression may be deterministic magnetic reversal, precession, domain-wall motion, partial switching, heating, stochastic remagnetization, or material degradation.
The conventional theory of spin-orbit torque describes these processes through established condensed-matter physics. Spin-orbit coupling connects charge transport to spin accumulation and spin-current generation. Damping-like and field-like torques act upon magnetization. Magnetic anisotropy establishes preferred orientations and energy barriers. Joule heating changes magnetic parameters and may drive the magnetic material toward or beyond its Curie temperature. Defects and spatial current inhomogeneity determine where domains nucleate and where walls propagate or become pinned. Spin-orbit torques are therefore already understood as transfers of angular momentum mediated by material and interfacial structure.
TSTOEAO does not replace this account. It supplies a second map organized around four questions:
1. What capacity is available?
2. Which routes are permitted by the architecture?
3. Where does the realized expression appear?
4. Where are the costs, exclusions, and unresolved residuals located?
The basic TSTOEAO grammar is:
\[
V=E\times Y,
\]
where:
\(E\) represents available capacity or energetic opportunity;
\(Y\) represents the relational architecture through which that capacity is routed;
\(V\) represents the realized expression.
In the present system, the decisive experimental feature is that the principal material stack remains fixed while the direction of charge current relative to the WTe₂ crystallographic axes changes. The energetic input may be comparable, but the symmetry architecture changes which spin-polarization components are permitted. A change in orientation therefore redirects the expression of the same general input.
This makes the WTe₂/Fe₃GaTe₂/hBN system an unusually clean candidate for testing a boundary-and-route formulation.
2. Epistemic Status
Three levels of claim must remain separate.
2.1 Published observation
The 2026 experiment directly reports electrical transport measurements, hBN-based magnetic imaging, reconstructed current-density distributions, and micromagnetic simulations in WTe₂/Fe₃GaTe₂/hBN heterostructures. It finds deterministic field-free magnetic switching for current applied along the low-symmetry WTe₂ axis, but not for current applied along its high-symmetry axis. It also reports a transition toward stochastic switching when heating becomes sufficiently strong.
2.2 Conventional interpretation
The authors explain the result using crystal-symmetry-controlled spin-orbit torque, out-of-plane and in-plane spin polarization, damping-like and field-like torques, perpendicular magnetic anisotropy, domain-wall dynamics, and Joule heating.
2.3 TSTOEAO interpretation
TSTOEAO classifies the result as symmetry-selected route transfer and localized cost expression. This is a reinterpretive and organizational layer. It is not an independent discovery of the reported effect, and the published result cannot be counted as a prospective confirmation of predictions formulated after the result was known.
The published experiment is therefore:
\[
\text{retrospective compatibility}
\neq
\text{prospective confirmation}.
\]
The prospective value of this paper lies in specifying an angular experiment, observables, route-order predictions, withheld tests, and failure conditions before the proposed measurements are performed.
3. The All–van der Waals System
The device consists principally of three functional layers.
3.1 WTe₂: symmetry-selective spin source
WTe₂ possesses reduced in-plane crystal symmetry. It has mirror symmetry with respect to its \(bc\)-plane but not its \(ac\)-plane. Consequently, charge current along different crystallographic directions does not generate identical spin responses.
When current flows along the low-symmetry \(a\)-axis, the reduced symmetry permits a spin-current component with out-of-plane polarization. When current flows along the high-symmetry \(b\)-axis, the preserved mirror symmetry suppresses that unconventional out-of-plane component, and the conventional mutually orthogonal charge, spin-current, and spin-polarization arrangement dominates. Earlier WTe₂ work had already established that crystal symmetry can control the form of spin-orbit torque.
3.2 Fe₃GaTe₂: perpendicular magnetic receiver
Fe₃GaTe₂ is an intrinsic van der Waals ferromagnet with strong perpendicular magnetic anisotropy and magnetic order capable of persisting above room temperature in suitable samples. The 2022 foundational report described Curie temperatures of approximately 350–380 K and substantial perpendicular anisotropy.
The switching and wide-field imaging measurements central to the 2026 study were nevertheless conducted primarily at approximately 200 K and 260 K. The material’s above-room-temperature capability must not be confused with the temperature of every measurement in the reported experiment.
3.3 hBN: protective layer and embedded observer
The hBN layer protects the underlying structure and contains negatively charged boron-vacancy spin defects. Their optically detected magnetic resonance allows spatial mapping of local magnetic fields. Related hBN platforms have previously imaged van der Waals ferromagnets and mapped current and thermal behavior in two-dimensional devices.
In the present device, hBN is not merely a passive cap. It becomes an embedded observational layer capable of mapping:
\[
\text{magnetic stray field}
\rightarrow
\text{domain configuration},
\]
and:
\[
\text{current-generated Oersted field}
\rightarrow
\text{current-density distribution}.
\]
This is particularly valuable because it allows the experiment to inspect the hidden middle of the route rather than infer all internal behavior from terminal electrical measurements.
4. The Published Boundary-Control Result
The published experiment applies electrical current pulses along two principal WTe₂ crystallographic axes.
4.1 Low-symmetry current orientation
For current along the \(a\)-axis:
\[
J_c^{a}
\rightarrow
S_z+S_{xy}
\rightarrow
\tau_{\mathrm{SOT}}
\rightarrow
\Delta M_z.
\]
The out-of-plane spin component permits field-free deterministic reversal of perpendicular Fe₃GaTe₂ magnetization. Positive and negative current polarities produce corresponding terminal magnetic states. The unconventional spin-orbit-torque efficiency was estimated as:
\[
\xi_{\mathrm{SOT}}=0.180\pm0.045.
\]
At the plateau of the anomalous Hall loop, approximately 70% of the imaged magnetic area had reversed without an external magnetic field. Switching began locally where the energy barrier was lowest and then expanded through domain nucleation and domain-wall propagation.
4.2 High-symmetry current orientation
For current along the \(b\)-axis:
\[
J_c^{b}
\rightarrow
S_{xy}
+
Q_{\mathrm{thermal}}
\rightarrow
M_{xy}\ \text{or demagnetization}
\rightarrow
\text{stochastic }M_z.
\]
The deterministic out-of-plane switching signature vanished. The authors describe an intermediate condition that may be predominantly in-plane magnetized or thermally randomized during the pulse. When the pulse ends, spontaneous perpendicular remagnetization produces randomly oriented domains and approximately zero net out-of-plane magnetization.
4.3 Increasing-current transition
Current amplitude introduces another route transition.
At an operating current that remains within the deterministic regime, field-like and damping-like torques assist directed switching. As current becomes larger, local Joule heating increasingly affects magnetic order. The published images show that apparently similar anomalous Hall signals can conceal growing microscopic stochasticity. At sufficiently strong pulses, some regions may be heated above the local Curie temperature, followed by random demagnetization and remagnetization.
Thus:
\[
J\uparrow
\not\Rightarrow
D_{\mathrm{switch}}\uparrow
\]
without limit.
Instead:
\[
J\uparrow
\rightarrow
\begin{cases}
\text{threshold crossing and directed switching},\\
\text{thermal competition and partial stochasticity},\\
\text{loss of deterministic control}.
\end{cases}
\]
4.4 Spatial current architecture
The hBN sensor also reconstructs current flow from measured Oersted fields. The reported current distribution is nonuniform and includes filament-like conducting paths. Magnetic reversal occurs preferentially in current-active regions and expands as additional local areas cross their switching thresholds. The ultimate switched fraction is affected by current density, material inhomogeneity, defects, anisotropy, pinning, edges, and geometry.
The experiment therefore does not show a spatially uniform slab responding homogeneously to one scalar current. It shows a route network.
5. The Controlled Boundary Condition
The most important feature is not simply that WTe₂ is anisotropic. It is that orientation serves as a controlled boundary variable within the same material.
Let:
\[
\theta
\]
represent the angle between the applied in-plane charge current and the WTe₂ low-symmetry \(a\)-axis.
Then:
\[
\theta=0^\circ
\]
corresponds to current along the low-symmetry axis, while:
\[
\theta=90^\circ
\]
corresponds to current along the high-symmetry axis.
The stack need not be chemically changed. Fe₃GaTe₂ need not be replaced. The hBN sensor need not be altered. The decisive change is:
\[
Y_{\mathrm{symmetry}}(\theta).
\]
The orientation of the current changes which conversion routes are symmetry-permitted.
This is a particularly strong boundary-control case because the experimental distinction is not merely:
\[
\text{material A versus material B}.
\]
It is:
\[
\text{one material architecture viewed and driven through different directional relations}.
\]
The controlled variable is relational.
6. TSTOEAO Translation
The basic scalar expression:
\[
V=E\times Y
\]
is insufficient by itself for a multichannel physical system. The operational form must be treated as a mapping:
\[
\mathbf V
=
\mathcal Y_{\theta,\mathcal I,\mathcal G,\mathcal D,T}
[\mathbf E],
\]
where:
\(\mathbf E\) is the input vector, including charge-current density, voltage, pulse duration, and initial thermal condition;
\(\theta\) is current orientation relative to crystal axes;
\(\mathcal I\) is the WTe₂/Fe₃GaTe₂ interfacial architecture;
\(\mathcal G\) is the device and electrode geometry;
\(\mathcal D\) is the spatial defect and pinning architecture;
\(T\) is the thermal field;
\(\mathbf V\) is the measured response vector.
The system can be represented recursively:
\[
V_{\mathrm{charge}}
\rightarrow
Y_{\mathrm{spin}}
\rightarrow
V_{\mathrm{spin}}
\rightarrow
Y_{\mathrm{interface}}
\rightarrow
V_{\mathrm{torque}}
\rightarrow
Y_{\mathrm{magnetic}}
\rightarrow
V_{\mathrm{domain}}.
\]
Each realized expression becomes part of the architecture confronting the next stage:
\[
V^{(n)}
\rightarrow
Y^{(n+1)}.
\]
The compact route is:
\[
\boxed{
\text{charge flow}
\rightarrow
\text{symmetry-selected spin orientation}
\rightarrow
\text{interfacial angular-momentum transfer}
\rightarrow
\text{magnetic-barrier crossing}
\rightarrow
\text{domain reconfiguration}
}
\]
But the complete route must include alternative expressions:
\[
\boxed{
\text{charge flow}
\rightarrow
\begin{cases}
\text{out-of-plane spin torque},\\
\text{in-plane spin torque},\\
\text{Joule heating},\\
\text{lattice and interfacial dissipation},\\
\text{transport remaining in the charge channel}.
\end{cases}
}
\]
TSTOEAO predicts that changes in \(Y\) redistribute the realized response among these accessible channels.
7. Route Selection Is Not Energy Creation
The low-symmetry orientation does not create additional energy. It permits a conversion route unavailable or strongly suppressed under the high-symmetry orientation.
The distinction is:
\[
E_{\mathrm{available}}
\neq
V_{\mathrm{selected}}.
\]
At fixed nominal input, the current orientation can change:
the polarization of generated spin current;
the magnitude and orientation of torque;
the probability of crossing the magnetic anisotropy barrier;
the spatial sites at which reversal begins;
the degree of deterministic domain growth;
the amount and localization of thermal dissipation;
the final domain state.
The appropriate TSTOEAO statement is therefore:
> Symmetry determines which routes of expression are available to the input; it does not supply the input independently.
This prevents the relational account from being mistaken for a violation of conservation laws.
8. Cost Location
A route is not costless merely because the desired outcome occurs.
The present system identifies at least five physically distinguishable cost locations.
8.1 Joule heating
The local volumetric heating rate is:
\[
q_J(\mathbf r,t)
=
\mathbf J(\mathbf r,t)\cdot
\mathbf E(\mathbf r,t).
\]
Because current density is nonuniform, heating is also nonuniform. High-current filaments and electrode-defined constrictions may accumulate disproportionate thermal cost.
8.2 Magnetic anisotropy barrier
Fe₃GaTe₂ possesses perpendicular magnetic anisotropy. Reversal requires the magnetic state to cross or circumvent an energy barrier. Spin torque provides a route through this barrier, while temperature, defects, and local fields modify its height.
Let:
\[
U_A(\mathbf r,T)
\]
represent the local anisotropy barrier.
The barrier need not be uniform over the flake.
8.3 Domain nucleation and pinning
Reversal begins preferentially where the effective barrier is lowest. Defects may assist nucleation while also pinning domain walls after nucleation.
Let:
\[
U_P(\mathbf r)
\]
represent the local pinning potential.
A defect may therefore reduce the cost of initiating one transition while increasing the cost of propagating the resulting boundary.
8.4 Interfacial transfer
Angular momentum must cross the WTe₂/Fe₃GaTe₂ interface. Imperfect transmission, spin dephasing, interfacial disorder, and charge-current shunting can reduce the fraction of input that reaches the magnetic channel.
8.5 Stochastic domain competition
When thermal fluctuations and competing domains dominate, energy continues to be dissipated even though a stable deterministic terminal state is not obtained.
The cost is not simply “wasted energy.” It appears as:
increased domain entropy;
repeated nucleation;
domain-wall competition;
temporal fluctuation;
incomplete net switching;
increased error probability.
These different quantities must not be prematurely compressed into one arbitrary scalar.
The initial cost representation should instead be a field of observables:
\[
\mathbf C(\mathbf r,t)
=
\left[
q_J,\,
\Delta T,\,
U_A,\,
U_P,\,
H_D,\,
P_{\mathrm{error}}
\right].
\]
Here \(H_D\) is a domain-disorder measure and \(P_{\mathrm{error}}\) is the probability that the terminal state differs from the current-selected target.
9. Dimensional Discipline
A conceptual route equation such as:
\[
R_{z}
+
R_{xy}
+
C_{\mathrm{heat}}
+
C_{\mathrm{domain}}
=
\text{total response}
\]
cannot be interpreted as a physical conservation equation when the terms use different dimensions.
Spin-torque efficiency, magnetic area, temperature, entropy, current, and energy cannot be added directly.
The first-stage experiment must therefore preserve a response vector:
\[
\mathbf O(\theta,J,\tau,T_0)
=
\begin{bmatrix}
\xi_z\\
\xi_{xy}\\
R_{xy}\\
m_z\\
f_{\mathrm{sw}}\\
H_D\\
\ell_D\\
\tau_{\mathrm{sw}}\\
\Delta T\\
q_J\\
P_{\mathrm{error}}\\
R
\end{bmatrix},
\]
where:
\(\xi_z\) is the out-of-plane spin-torque measure;
\(\xi_{xy}\) represents in-plane torque-sensitive response;
\(R_{xy}\) is anomalous Hall resistance;
\(m_z\) is normalized net perpendicular magnetization;
\(f_{\mathrm{sw}}\) is switched magnetic-area fraction;
\(H_D\) is domain entropy or disorder;
\(\ell_D\) is domain correlation length;
\(\tau_{\mathrm{sw}}\) is switching latency;
\(\Delta T\) is temperature rise;
\(q_J\) is local or integrated Joule heating;
\(P_{\mathrm{error}}\) is terminal-state error probability;
\(R\) is an explicitly unresolved residual.
Separate physical budgets may later be constructed.
9.1 Electrical-energy budget
\[
P_{\mathrm{electrical}}
=
P_J
+
P_{\mathrm{magnetic}}
+
P_{\mathrm{lattice/interface}}
+
P_{\mathrm{electromagnetic}}
+
\frac{dU}{dt}
+
P_{\mathrm{unresolved}}.
\]
9.2 Angular-momentum budget
\[
\mathbf J_s^{\mathrm{in}}
=
\frac{d\mathbf L_M}{dt}
+
\mathbf J_s^{\mathrm{out}}
+
\boldsymbol{\tau}_{\mathrm{lattice}}
+
\boldsymbol{\tau}_{\mathrm{diss}}
+
\boldsymbol{\tau}_{\mathrm{unresolved}}.
\]
The electrical-energy and angular-momentum budgets are related, but they are not interchangeable.
A quantitative closure claim must wait until each included term has been converted into compatible units with calibrated uncertainty.
10. The Prospective Experiment
The central prospective experiment continuously varies \(\theta\), rather than comparing only the two principal crystallographic axes.
The primary question is:
> At fixed material composition, thickness, pulse duration, initial magnetic condition, and nominal current density, how does the complete response vector change as current orientation rotates relative to WTe₂ crystal symmetry?
10.1 Device architecture
Two complementary device families should be used.
Family A: multi-terminal same-flake device
A geometrically symmetric device should contain multiple electrode pairs allowing current to be driven through the same WTe₂/Fe₃GaTe₂ region at different in-plane angles.
Advantages include:
constant material stack;
constant interface;
constant defect landscape;
constant magnetic target;
direct within-device angular comparison.
Its primary risk is that different electrode pairs may generate different current-density distributions.
That risk must be measured rather than assumed away.
Family B: replicated fixed-channel devices
Nominally identical devices should be fabricated with current channels aligned at different angles relative to the measured WTe₂ crystal axes.
Advantages include:
comparable channel geometry;
simpler current flow;
independent replication.
Its primary risk is device-to-device material variation.
The two families therefore address complementary confounders.
10.2 Crystal-axis determination
WTe₂ crystallographic orientation must be determined before switching measurements using rotational-anisotropy second-harmonic generation or another validated orientation-sensitive method.
The axis labels and angular coordinate system must be locked before outcome data are inspected.
10.3 Angular sampling
The exploratory phase should measure:
\[
0^\circ\leq\theta<180^\circ
\]
at intervals no larger than \(15^\circ\).
The response under current polarity reversal supplies the corresponding directional information across the full \(360^\circ\) cycle.
A confirmatory phase should withhold a preregistered subset of intermediate angles. These angles are predicted only after the model has been fitted to the exploratory set.
10.4 Current regimes
Three current regimes must be established separately for each device:
1. Subthreshold regime
Torque insufficient for substantial reversal.
2. Deterministic switching regime
Directed magnetic reversal occurs with limited thermal degradation.
3. Thermally degraded regime
Increased heating produces partial stochasticity, demagnetization-remagnetization, or chaotic multidomain competition.
Current density should be expressed both absolutely and as a device-normalized value:
\[
j^*
=
\frac{J}{J_{\mathrm{critical},a}},
\]
where \(J_{\mathrm{critical},a}\) is the deterministic switching threshold measured along the low-symmetry axis under the locked pulse protocol.
10.5 Pulse conditions
The confirmatory angular comparison must hold constant:
pulse duration;
pulse rise time;
duty cycle;
starting magnetic state;
base temperature;
time between write and read operations;
external magnetic field;
sensing current;
device history within the randomized run order.
Additional pulse-duration studies should be treated as a separate experimental dimension rather than mixed into the primary angular test.
11. Required Measurements
No one observable is sufficient.
11.1 Anomalous Hall response
Anomalous Hall measurements provide a global electrical estimate of net perpendicular magnetization and hysteretic switching.
They are necessary but not sufficient because a near-zero Hall value can represent:
no magnetic response;
uniform in-plane magnetization;
thermal demagnetization;
equal populations of oppositely oriented perpendicular domains;
dynamically fluctuating magnetic order.
11.2 Full spatial magnetic-domain maps
The hBN sensor should record the magnetic stray-field map after each pulse.
The images should be converted into:
local \(M_z\) state;
switched-area fraction;
domain count;
domain-size distribution;
domain correlation length;
wall position;
nucleation coordinates;
terminal-state reproducibility.
11.3 Current-density maps
Local current flow must be reconstructed for each electrode configuration and relevant current amplitude.
The experiment must not assume:
\[
J_{\mathrm{nominal}}
=
J(\mathbf r).
\]
The spatially resolved value is required because torque and heating depend upon local current density.
11.4 Temperature maps
Local transient and residual temperature changes should be measured using a calibrated thermometry method compatible with the heterostructure.
The desired output is:
\[
\Delta T(\mathbf r,t;\theta,J).
\]
This allows stochastic magnetic changes to be separated from symmetry-controlled spin-torque effects.
11.5 In-plane torque-sensitive observables
The high-symmetry orientation is expected to suppress the unconventional out-of-plane component, not necessarily all spin conversion.
A torque-sensitive measurement capable of detecting in-plane spin excitation is therefore essential. Without it, the experiment may observe the loss of deterministic perpendicular switching but fail to identify where the spin response moved.
11.6 Switching latency
Terminal imaging alone does not show whether the state was reached through:
immediate coherent or quasicoherent reversal;
nucleation followed by wall propagation;
thermal demagnetization;
delayed self-remagnetization.
Time-resolved or stroboscopic measurements should therefore estimate:
\[
\tau_{\mathrm{nucleation}},
\qquad
\tau_{\mathrm{propagation}},
\qquad
\tau_{\mathrm{terminal}}.
\]
11.7 Structural and defect mapping
Before the confirmatory switching trials, the device should be characterized for:
flake boundaries;
thickness variation;
bubbles and folds;
electrode edges;
interface contamination;
strain;
visible defects;
repeated pinning locations.
These maps are predictors, not explanations added only after the outcome.
12. Operational Metrics
12.1 Net perpendicular state
\[
m_z
=
\frac{A_\uparrow-A_\downarrow}
{A_\uparrow+A_\downarrow}.
\]
A fully upward state approaches \(+1\), a fully downward state approaches \(-1\), and an equal multidomain population approaches zero.
12.2 Switched fraction
\[
f_{\mathrm{sw}}
=
\frac{A_{\mathrm{changed}}}
{A_{\mathrm{imaged}}}.
\]
This distinguishes substantial local reconfiguration from a small net Hall change.
12.3 Domain entropy
For segmented magnetic classes \(k\):
\[
H_D
=
-\sum_k p_k\ln p_k.
\]
This simple entropy measure should be accompanied by spatial correlation length because two patterns with equal up/down fractions may have radically different domain geometry.
12.4 Trial-to-trial determinism
For repeated applications of the same pulse condition:
\[
D
=
1-
\frac{1}{N}
\sum_{n=1}^{N}
\frac{A_{\mathrm{mismatch},n}}
{A_{\mathrm{imaged}}}.
\]
The mismatch is measured relative to the preregistered target-state map or target magnetic polarity.
12.5 Spatial nucleation predictability
Let:
\[
P_N(\mathbf r)
\]
represent the predicted probability of domain nucleation based upon current density, temperature, geometry, defect maps, and prior pinning measurements.
Prediction quality should be tested on pulses and devices not used to fit the model.
12.6 Residual
For each observable:
\[
R_i
=
O_{i,\mathrm{observed}}
–
O_{i,\mathrm{predicted}}.
\]
The residual must remain visible. It must not automatically be relabeled as an unknown route that supposedly confirms the theory.
13. Locked Prospective Predictions
13.1 Symmetry null
At the high-symmetry \(b\)-axis:
\[
\xi_z(90^\circ)
\approx0
\]
within the calibrated sensitivity and experimental uncertainty.
Near the low-symmetry \(a\)-axis:
\[
|\xi_z(0^\circ)|
>
|\xi_z(90^\circ)|.
\]
The detailed angular function should be derived from WTe₂ point-group symmetry and locked before the confirmatory angles are revealed.
13.2 Determinism tracks the out-of-plane route below the thermal threshold
Within the nonthermal operating window:
\[
|\xi_z(\theta)|\uparrow
\Rightarrow
D(\theta)\uparrow,
\]
\[
|m_z(\theta)|\uparrow,
\]
and:
\[
H_D(\theta)\downarrow.
\]
The strongest deterministic switching is expected near orientations where the symmetry-allowed out-of-plane torque is greatest.
13.3 High-symmetry stochasticity
Near the high-symmetry axis:
\[
|\xi_z|\downarrow
\Rightarrow
|m_z|\downarrow,
\]
\[
D\downarrow,
\]
\[
H_D\uparrow,
\]
with increased in-plane excitation, thermal randomization, multidomain formation, or some measured combination of those channels.
A zero Hall signal accompanied by changing microscopic domain maps will be classified as active stochastic reconfiguration, not as no response.
13.4 Polarity reversal
Near the low-symmetry axis and within the deterministic operating window:
\[
J\rightarrow-J
\]
should reverse the preferred terminal perpendicular state.
The magnetic result should therefore retain a reproducible relationship to current polarity.
13.5 Thermal crossover
At fixed favorable orientation:
\[
J>J_{\mathrm{thermal}}
\]
should produce:
\[
\Delta T\uparrow,
\qquad
H_D\uparrow,
\qquad
D\downarrow.
\]
The loss of deterministic behavior should begin in spatial regions where the combination of current density and thermal vulnerability is greatest.
13.6 Spatial route prediction
The probability of initial nucleation should increase with:
local current density;
local temperature rise;
reduced anisotropy barrier;
preidentified defects or strain;
proximity to reproducible nucleation sites.
After training on exploratory pulses, the model should predict held-out nucleation regions significantly better than area-weighted chance.
13.7 Current-path migration
When electrode geometry redirects current concentration, the spatial switching region and thermal-cost region should migrate correspondingly.
The prediction is not that every hotspot must switch. It is that regions lacking sufficient current density should have a substantially lower switching probability, all else being equal.
13.8 Structured route transfer
As \(\theta\) moves from the low-symmetry toward the high-symmetry axis, the complete response should show a reproducible redistribution:
\[
\Delta\mathbf O
=
\left[
-\Delta\xi_z,\,
+\Delta\xi_{xy},\,
+\Delta H_D,\,
+\Delta P_{\mathrm{error}},\,
\Delta T,\,
\Delta R_{\mathrm{transport}}
\right],
\]
with the signs of the thermal and transport terms determined by measured anisotropic resistance and local current architecture.
The central prediction is not that every alternative channel must increase monotonically. It is that a preregistered multichannel model based on symmetry, current distribution, temperature, and magnetic barriers should predict the changing response better than a model using nominal current magnitude alone.
14. Route-Transfer Function
The desired experimental product is a symmetry-governed angular response map:
\[
\mathcal R(\theta,J,T)
:
\mathbf E
\rightarrow
\mathbf O.
\]
For visualization, each measured response may be standardized relative to its independently calibrated range:
\[
z_i(\theta)
=
\frac{O_i(\theta)-\mu_i}
{\sigma_i}.
\]
This permits comparison of angular patterns without pretending that the underlying physical quantities share dimensions.
A provisional dimensionless route vector may then be defined:
\[
\mathbf r(\theta)
=
\left[
r_z,\,
r_{xy},\,
r_{\mathrm{thermal}},\,
r_{\mathrm{domain}},\,
r_{\mathrm{transport}},\,
r_{\mathrm{unresolved}}
\right],
\]
but only through a preregistered calibration model.
The route vector is an explanatory allocation, not a fundamental conservation equation.
The stronger long-term objective is physical closure in common units. Until that is achieved:
\[
\sum_i r_i=1
\]
would describe the model’s allocation of explained response, not literal conservation of energy or angular momentum.
15. Conventional Model and TSTOEAO Second Map
The conventional physical model may be represented through magnetization dynamics incorporating effective magnetic fields, damping-like torque, field-like torque, temperature dependence, exchange, anisotropy, and interfacial interactions.
The 2026 study used the Landau–Lifshitz–Gilbert–Slonczewski framework in its simulations, with the charge-to-spin conversion efficiency and torque terms acting upon the Fe₃GaTe₂ magnetization.
TSTOEAO does not propose a substitute differential equation.
Its proposed contribution is to require that the experiment track the full architecture:
\[
\theta
\rightarrow
\boldsymbol{\sigma}
\rightarrow
\boldsymbol{\tau}
\rightarrow
\text{barrier crossing}
\rightarrow
\text{domain trajectory}
\rightarrow
\text{terminal state},
\]
together with:
\[
J(\mathbf r)
\rightarrow
q_J(\mathbf r)
\rightarrow
\Delta T(\mathbf r)
\rightarrow
\text{thermal competition}.
\]
The conventional model explains the physical mechanisms.
The TSTOEAO second map asks whether the route architecture can be represented prospectively as a constrained transfer among measurable expressions and localized costs.
The two maps are complementary when the second map adds:
better experimental organization;
more complete simultaneous measurement;
explicit cost accounting;
withheld prediction;
residual preservation;
cross-regime comparison.
It adds little if it merely renames already-known mechanisms after the result.
16. Controls and Confounders
16.1 Current-density anisotropy
WTe₂ may exhibit direction-dependent conductivity. The same nominal current can therefore produce different voltage, local power density, and spatial flow.
This must be measured directly.
16.2 Heating-only explanation
A heating-only model should predict the magnetic outcome using:
\(J(\mathbf r)\);
resistance;
pulse duration;
\(\Delta T(\mathbf r,t)\);
temperature-dependent magnetic parameters.
The symmetry-and-spin model must outperform this control model on withheld angles.
16.3 Geometry-only explanation
Electrode placement may alter current concentration independently of crystallographic symmetry.
The replicated fixed-channel devices and measured current maps are required to separate geometry from orientation.
16.4 Defect-only explanation
Repeated nucleation at defects does not by itself demonstrate symmetry-selected route transfer.
A valid model must explain both:
why certain sites are locally susceptible;
why the same sites respond differently when the spin route changes.
16.5 Magnetic-history effects
Switching probabilities may depend upon the preceding domain state.
Trial order must be randomized, and each confirmatory pulse must begin from a verified initial condition.
16.6 Sensor perturbation
The hBN sensor, optical excitation, microwave drive, and thermal environment must be tested for possible effects upon the device.
16.7 Device degradation
Electromigration, irreversible contact change, oxidation, strain relaxation, or interface degradation could imitate evolving stochasticity.
Pre-run and post-run transport and structural checks are required.
16.8 Temperature qualification
Measurements at 200 K, 260 K, and room temperature must be treated as separate regimes. A result at one temperature should not be silently generalized to another.
17. Exploratory and Confirmatory Phases
Phase I: reproduction
Reproduce the principal low-symmetry and high-symmetry results in multiple devices.
Phase II: exploratory angular mapping
Measure the full response vector over the majority of angular conditions.
Use this phase to:
estimate sensitivity;
derive the symmetry-compatible angular model;
identify thermal thresholds;
calibrate imaging segmentation;
determine replicate requirements;
select model parameters.
Phase III: preregistration
Lock:
axis definitions;
excluded runs;
measurement thresholds;
angular model;
error model;
response-order predictions;
held-out angles;
held-out devices;
success and failure criteria.
Phase IV: confirmatory withheld-angle test
Predict the full response vector at withheld angular orientations.
No parameter refitting is permitted before the primary test is scored.
Phase V: spatial prediction
Use measured current, temperature, defect, and geometry fields to predict nucleation and propagation in held-out pulses.
Phase VI: replication
Repeat the protocol using independently fabricated devices and, preferably, an independent laboratory.
Failed devices and failed predictions must remain in the report.
18. Success Criteria
The framework receives prospective support only if all primary conditions are met.
18.1 Angular prediction
The preregistered model predicts the direction and magnitude range of the principal response changes at withheld angles.
18.2 Route discrimination
The full multichannel model outperforms:
nominal-current-only models;
heating-only models;
geometry-only models;
unconstrained interpolation baselines.
18.3 Spatial prediction
Current, thermal, and defect maps predict nucleation and switching regions better than chance on held-out trials.
18.4 Residual containment
Unexplained residuals remain within the preregistered uncertainty limits and do not systematically cluster by angle, device, or temperature.
18.5 Replication
The symmetry-governed route-order pattern recurs across multiple devices.
Even complete success would establish only that the TSTOEAO route-and-cost map is operationally useful in this system.
It would not demonstrate that:
\(V=E\times Y\) is a complete physical law;
TSTOEAO uniquely explains spin-orbit torque;
conventional spintronics is incomplete;
all omitted channels have been measured;
the same route architecture applies unchanged to unrelated systems.
19. Failure Criteria
The proposed interpretation is weakened if any of the following occurs.
19.1 No withheld predictive power
The angular model fits known orientations but fails at withheld orientations.
19.2 Heating accounts for the complete effect
Measured thermal variables explain the apparent symmetry dependence without requiring a distinct out-of-plane spin-transfer route.
19.3 Current geometry accounts for the complete effect
Once local current-density distributions are included, crystallographic orientation contributes no reproducible additional predictive information.
19.4 Missing expression remains missing
Suppression of deterministic perpendicular switching produces no measured increase in in-plane excitation, stochastic domain activity, heating, altered charge transport, reflected spin flow, lattice transfer, or a bounded residual.
19.5 Post hoc route creation
A new route is introduced after every failed prediction, making the framework impossible to reject.
19.6 Arbitrary normalization
The result depends upon adjustable weights used to add physically incompatible quantities.
19.7 No spatial predictability
Nucleation and propagation maps cannot be predicted better than chance from the supposed route and cost architecture.
19.8 Replication failure
The reported relationships depend upon one device or one laboratory-specific configuration.
19.9 Residual growth
The residual:
\[
R
=
V_{\mathrm{observed}}
–
V_{\mathrm{predicted}}
\]
grows systematically under precisely the boundary conditions the model claims to organize.
Failure under these conditions must be reported as failure, not reframed automatically as evidence of deeper hidden complexity.
20. Broader Implications
The importance of the system extends beyond one magnetic-switching device.
It demonstrates that orientation inside an anisotropic material can function as an engineering control variable capable of selecting among distinct expression routes.
This suggests a broader design principle:
> Do not ask only how much energy is supplied. Ask which orientation, interface, symmetry, phase, and geometry allow that energy to enter the desired route.
The same general architecture may apply to systems involving:
charge-to-spin conversion;
spin-to-orbital conversion;
phonon routing;
polarized light;
anisotropic heat flow;
catalytic surface orientation;
piezoelectric response;
phase-selective transport;
topological boundary states;
multiferroic coupling.
The system also demonstrates the importance of embedded observation. A terminal electrical signal can conceal radically different microscopic states.
Two conditions may produce similar Hall values while containing:
a stable uniform state;
balanced opposing domains;
active thermal disorder;
repeated stochastic reconfiguration.
The observer architecture therefore changes which expression can be distinguished:
\[
V_{\mathrm{measured}}
=
E_{\mathrm{physical\ event}}
\times
Y_{\mathrm{observation}}.
\]
The hBN sensing layer makes local pathways visible without becoming identical to the pathways being observed.
21. TSTOEAO Interpretation
The deepest TSTOEAO result is not simply that symmetry affects magnetism.
It is:
> Symmetry determines which forms of conversion are available, while interfaces, current paths, thermal conditions, barriers, and defects determine where those permitted conversions become physically expressed.
The low-symmetry orientation permits a route:
\[
\text{charge}
\rightarrow
S_z
\rightarrow
\text{directed torque}
\rightarrow
\text{deterministic perpendicular state}.
\]
The high-symmetry orientation suppresses that route:
\[
\text{charge}
\rightarrow
S_{xy}
+
\text{heat}
\rightarrow
\text{in-plane or disordered intermediate state}
\rightarrow
\text{stochastic perpendicular domains}.
\]
Increasing current beyond the stable operating window redirects additional input toward thermal expression:
\[
\text{additional electrical input}
\rightarrow
\text{additional heat}
\rightarrow
\text{loss of deterministic magnetic control}.
\]
The system therefore contains three separate but connected forms of route selection:
1. Crystallographic selection
Determines which spin orientations are permitted.
2. Interfacial selection
Determines how spin angular momentum reaches the ferromagnet.
3. Spatial magnetic selection
Determines where domains nucleate, propagate, pin, compete, or fail.
The realized magnetic result is not located in any one component alone.
It emerges from their relation:
\[
V_{\mathrm{magnetic}}
=
E_{\mathrm{charge}}
\times
Y_{\mathrm{symmetry}}
\times
Y_{\mathrm{interface}}
\times
Y_{\mathrm{current}}
\times
Y_{\mathrm{thermal}}
\times
Y_{\mathrm{domain}}.
\]
This extended multiplication is conceptual rather than a literal product of independently measured scalars. Its scientific value depends upon translating every factor into observables, models, uncertainties, and prospective tests.
Conclusion
The WTe₂/Fe₃GaTe₂/hBN heterostructure offers an unusually direct view of symmetry-selected physical routing.
The material stack can remain substantially unchanged while current orientation alters the permitted spin polarization. The resulting change propagates through the van der Waals interface, into magnetic torque, across the anisotropy barrier, through domain nucleation and propagation, and finally into a deterministic or stochastic magnetic state.
The published experiment already establishes that:
low-symmetry current orientation permits an out-of-plane spin route;
the out-of-plane route supports field-free deterministic perpendicular switching;
the high-symmetry orientation suppresses that deterministic route;
local current density helps determine where switching occurs;
Joule heating can degrade deterministic control;
terminal transport signals can conceal complex microscopic domain behavior.
These observations are conventionally explained by spin-orbit coupling, crystal symmetry, magnetic anisotropy, domain dynamics, and thermal physics.
TSTOEAO adds a second organization:
\[
\boxed{
\text{input capacity}
\rightarrow
\text{symmetry-permitted route}
\rightarrow
\text{interfacial transfer}
\rightarrow
\text{localized barrier crossing}
\rightarrow
\text{realized magnetic expression}
\rightarrow
\text{located cost}
}
\]
The decisive future test is not another retrospective statement that boundaries matter.
It is a preregistered continuous-angular experiment capable of predicting:
which spin route dominates;
how deterministic switching changes;
where magnetic domains nucleate;
where current and heat concentrate;
when deterministic control collapses;
where the suppressed expression reappears;
and how much response remains unresolved.
The strongest possible result would be a reproducible map:
\[
\theta
\rightarrow
\text{spin orientation}
\rightarrow
\text{local torque}
\rightarrow
\text{domain trajectory}
\rightarrow
\text{terminal state}
\rightarrow
\text{cost location}.
\]
The theory is strengthened only if that map predicts withheld conditions.
It is weakened if the missing expression cannot be located, if ordinary heating or geometry explains the entire result, if prediction fails outside the fitted orientations, or if unexplained residuals are hidden behind newly invented routes.
The scientific opportunity is therefore precise:
> Change one relational boundary condition, observe the redistribution of the available expression, locate the cost, preserve the residual, and test whether the route map predicts what happens next.
References
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