Default-Direction Asymmetry Overview

• Default-Direction Asymmetry is a phenomenon where nominally symmetric processes show a bias, favoring one direction through intrinsic dynamics, design, or measurement protocols.
• It spans multiple fields—from reinforcement learning and assistant controllability to nonequilibrium transport and astrophysical handedness—each with distinct structural causes.
• Empirical studies reveal that mechanisms like on-policy sampling and biased routing induce measurable differences in metrics such as entropy change, convergence rates, and signal propagation.

Searching arXiv for the provided topic and ids to ground the synthesis in current literature. arxiv_search(query="^^^^^^^^4\4 asymmetry^^^^^^^^4\4 OR ^^^^^^^^4\4 polarity^^^^^^^^4\4 OR ^^^^^^^^4\4 suppression asymmetry^^^^^^^^4\4 OR ^^^^^^^^4\4 computational asymmetry^^^^^^^^4\4 max_results=^^^^4 OR \4\4^^^^) Default-Direction Asymmetry denotes a structurally induced bias by which one direction of update, inference, transport, steering, or observation is easier to trigger, harder to suppress, or more stably realized than its nominal opposite. Across current literature, the term does not refer to a single invariant formalism; rather, it recurs in several technically distinct settings, including reinforcement fine-tuning for LLMs, post-training controllability of assistants, asymmetric routing, causal-direction identification, nonequilibrium first-passage phenomena, engineered nonreciprocity, and preferred-direction signals in astrophysical data. The common pattern is that an apparently symmetric task acquires a privileged orientation through sampling, architecture, optimization, dissipation, or observational geometry (&&&^^^^4\4^^^^&&&, &&&^^^^4 OR \4^^^^&&&, &&&^^^^4 OR \4^^^^&&&, &&&^^^^4 OR \4^^^^&&&).

^^^^4 OR \4^^^^. Conceptual structure

A recurring formulation has three ingredients. First, there is a privileged baseline direction: for example, positive-advantage on-policy reinforcement of likely tokens, anti-underanswering assistant behavior, node-centric routing decisions, the forward causal direction, or a sky axis singled out by anisotropy. Second, there is a mechanism that breaks effective symmetry: sampled-token entropy mechanics, content-budget overshoot plus continuation persistence, representation–decision mismatch, residual dependence, broken detailed balance, or dipole modulation. Third, there is an observable asymmetry: different entropy trajectories, different suppressibility costs, unequal forward/backward hitting times, unequal transmission amplitudes, one-way steering, or direction-dependent parameter estimates (&&&^^^^4\4^^^^&&&, &&&^^^^4 OR \4^^^^&&&, Shin et al., 2020, Downing et al., 2022, Mukherjee et al., 2015).

This suggests a useful cross-domain distinction between mere anisotropy and default-direction asymmetry. The latter is not only a difference between two directions; it is a difference that is preferentially activated by the system’s native dynamics, objective, or measurement protocol. In RLVR, the default direction under positive advantage and on-policy sampling is entropy contraction. In assistant post-training, the default direction is anti-underanswering over-expansion. In asymmetric routing, the default direction is a node-centric scoring bias that under-expresses the currently chosen directed transition. In causal inference from optimization time, the default direction is faster convergence in the true causal orientation (&&&^^^^4\4^^^^&&&, &&&^^^^4 OR \4^^^^&&&, &&&^^^^4 OR \4^^^^&&&, &&&^^^^4 OR \4^^^^&&&).

The literature also makes clear that the sign of the asymmetry need not be universal. In a two-lane lattice, either forward or backward transition can be faster depending on the sign of the nonequilibrium current; in biased run-and-tumble dynamics, first-passage duality can fail at finite distance yet be restored asymptotically when a Gallavotti–Cohen symmetry holds; in driven-dissipative resonators, the preferred transport direction is set by a relative phase and can be reversed by changing that phase (Shin et al., 2020, &&&^^^^4 OR \44^^^^&&&, Downing et al., 2022).

Domain	Default direction	Structural source
RLVR entropy control	Contraction under typical rewarded on-policy updates	Sampled-token term dominates correction term
Assistant controllability	Anti-underanswering over-expansion	Planning overshoot and continuation persistence
Asymmetric routing	Node-centric scoring	Representation–decision mismatch
Causal inference	Faster forward training	Reverse residual dependence and gradient-noise structure
Nonequilibrium transport	Direction selected by current or bias	Broken detailed balance or hidden internal dynamics
Quantum/optical transport	Phase-selected one-way flow	Interference of coherent and dissipative couplings

^^^^4 OR \4^^^^. Learning and control: reinforcement fine-tuning, assistant post-training, and learned defaults

In RLVR for LLMs, default-direction asymmetry is formalized at the token level through a first-order entropy expansion. For a local policy entropy PRESERVED_PLACEHOLDER_^^^^4\4^^^^ and a sampled token PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^ with probability PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^, one gradient-ascent step on PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^ yields

$$\Delta\mathcal{H}_t = -\eta A\,t_1(s_t,y_t) + \eta A\,t_2(s_t) + O(\eta^2),$$

with

$$t_1(s_t,y_t):=p_t(\mathcal{H}_t+\log p_t), \qquad t_2(s_t):=\sum_{v\in\mathcal{V}} p_v^2(\mathcal{H}_t+\log p_v).$$

The paper defines the intrinsic entropy tendency

$\mathcal{T}(s_t,y_t):=-t_1(s_t,y_t)+t_2(s_t),$

and the realized entropy polarity

$$\mathcal{P}(s_t,y_t,A):=A\,\mathcal{T}(s_t,y_t), \qquad \Delta\mathcal{H}_t=\eta\,\mathcal{P}(s_t,y_t,A)+O(\eta^2).$$

The sampled-token term changes sign at the entropy-induced threshold $p_t=\exp(-\mathcal{H}_t)$, while $t_2(s_t)\ge 0$ is always expansive. Because on-policy sampling draws high-probability tokens more often, positive-advantage updates are structurally biased toward contraction; expansion typically requires low-probability samples or a sufficiently strong state-wise correction. This is the paper’s “default direction under PRESERVED_PLACEHOLDER_^^^^4 OR \4\4^^^^ and on-policy sampling.” Empirically, polarity tracks measured entropy change, the polarity magnitude distribution is heavy-tailed, and single-polarity ablations show that the negative branch drives exploitation whereas the positive branch preserves exploration. The proposed Polarity-Aware Policy Optimization reweights advantages by polarity sign and uses an entropy-slope phase signal; on math reasoning it improves over strong baselines, including +^^^^4 OR \4^^^^.^^^^4 OR \4^^^^% on AIME^^^^4 OR \44^^^^ and +^^^^4 OR \4^^^^.^^^^4 OR \4^^^^% on AIME^^^^4 OR \4^^^^5 for Qwen^^^^4 OR \4^^^^.5-^^^^4 OR \44^^^^B, and +^^^^4 OR \4^^^^.4% on Minerva and +^^^^4 OR \4^^^^.^^^^4 OR \4^^^^% on AMC for Qwen^^^^4 OR \4^^^^.5-7B (&&&^^^^4\4^^^^&&&).

In post-trained assistants, the same idea appears as boundary-suppression asymmetry. The default response direction induced by anti-underanswering optimization is more complete, cautious, and proactively helpful, but that direction is harder to suppress when prompts explicitly request narrower answers. The paper operationalizes this through residual expansion under boundary-control prompts, using length-based costs such as

PRESERVED_PLACEHOLDER_^^^^4 OR \4 OR \4^^^^

On the main controlled family, the anti-underanswering policy remains substantially more expansive than baseline under matched controls: for scope_minimal_sufficient, mean lengths are ^^^^4 OR \4 OR \4^^^^.^^^^4 OR \4^^^^6 for baseline, 4^^^^4\4^^^^.^^^^4 OR \4^^^^8 for anti, and ^^^^4 OR \4^^^^7.^^^^4 OR \4^^^^8 for minimal; for avoid_underanswer, ^^^^4 OR \4 OR \4^^^^.44, ^^^^4 OR \4 OR \4^^^^.78, and ^^^^4 OR \4^^^^8.74; and in forced-prefix continuation, ^^^^4 OR \4^^^^.54, 4.^^^^4\4^^^^8, and ^^^^4 OR \4^^^^.5^^^^4 OR \4^^^^, with immediate-stop rates 6^^^^4 OR \4^^^^%, 48%, and 6^^^^4 OR \4^^^^%, respectively. Mechanism probes reject pure EOS failure, pure uncertainty compensation, and pure local continuation bias as complete explanations. The paper instead supports a mixed planning/stopping account in which content-budget overshoot and continuation persistence jointly make the anti-underanswering direction harder to pull back (&&&^^^^4 OR \4^^^^&&&).

A related but distinct asymmetry appears in KL-regularized RL with a learned default policy. The objective

PRESERVED_PLACEHOLDER_^^^^4 OR \4 OR \4^^^^

combines an information asymmetry, because PRESERVED_PLACEHOLDER_^^^^4 OR \4 OR \4^^^^ is restricted to PRESERVED_PLACEHOLDER_^^^^4 OR \44^^^^, with a divergence-direction asymmetry, because the forward KL pulls the full policy toward the default. The default learns broad, reusable behaviors precisely because it sees less information, while the main policy deviates only when reward justifies it. The paper links this to an information-bottleneck view and reports faster learning and improved performance in tasks with repeated structure (&&&^^^^4 OR \4^^^^8&&&).

^^^^4 OR \4^^^^. Inference and optimization: routing decoders, convergence-time causality, and topological residual geometry

In neural asymmetric routing, default-direction asymmetry is formulated as a decoder-side failure to expose the decision-critical directed transition. For ATSP, the Bellman-style action value is

PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^5

yet the RADAR-style baseline decoder scores candidates primarily through context–node compatibility,

PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^6

The paper identifies a representation–decision mismatch: pairwise directed information may be encoded upstream, but the final score remains node-centric. Its edge-aware decoder adds explicit local edge, closure, and static lookahead terms through a candidate descriptor PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^7 and a learned bias PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^8, yielding PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^9. On a controlled SVD/Sinkhorn asymmetric backbone, this reduces the ATSP-^^^^4 OR \4\4\4\4^^^^ gap from 4.^^^^4 OR \4 OR \4^^^^% to ^^^^4 OR \4^^^^.7^^^^4 OR \4^^^^%; removing the local-edge term degrades the ATSP-^^^^4 OR \4\4\4\4^^^^ gap to 5.^^^^4\44^^^^%, worse than the RADAR reference, which sharpens the claim that explicit exposure of the current directed edge is the decisive corrective signal (&&&^^^^4 OR \4^^^^&&&).

In bivariate causal discovery, the paper on Causal Computational Asymmetry posits an optimization-time default direction. Under the additive-noise model PRESERVED_PLACEHOLDER_^^^^4 OR \4\4^^^^ with PRESERVED_PLACEHOLDER_^^^^4 OR \4 OR \4^^^^ and PRESERVED_PLACEHOLDER_^^^^4 OR \4 OR \4^^^^ nonlinear and injective, one trains parity-matched networks in both directions and compares the hitting times

PRESERVED_PLACEHOLDER_^^^^4 OR \4 OR \4^^^^

The CCA score is

PRESERVED_PLACEHOLDER_^^^^4 OR \44^^^^

The theoretical asymmetry arises because forward residuals converge to independent noise, whereas reverse residuals remain statistically dependent on the input, which induces a higher irreducible loss floor and non-separable gradient noise. Under local PL conditions, the paper states that

PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^5

Z-scoring of both variables is mandatory, since otherwise scale alone can invert the convergence ordering. Empirically, the method achieves ^^^^4 OR \4^^^^6/^^^^4 OR \4\4^^^^ correct synthetic causal identifications across six architectures, including ^^^^4 OR \4\4^^^^/^^^^4 OR \4\4^^^^ on sine and exponential data-generating processes (&&&^^^^4 OR \4^^^^&&&).

A second causal-direction method, Topological Residual Asymmetry, makes the asymmetry geometric rather than optimization-dynamic. After cross-fitted regression in both directions and rank-based copula standardization, the correct-direction regressor–residual cloud is approximately two-dimensional, whereas the wrong-direction cloud concentrates near a one-dimensional tube, especially in a small-noise regime. TRA measures this bulk–tube contrast with a ^^^^4\4^^^^D persistent-homology functional computed from Euclidean MST edge-length profiles. The signed score

PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^6

is positive in the forward direction under the paper’s consistency regime, while TRA-s extends the method to fixed noise by binning reverse residuals, and TRA-C adds a confounding-aware abstention rule calibrated by a Gaussian-copula plug-in bootstrap (&&&^^^^4 OR \4 OR \4^^^^&&&).

4. Nonequilibrium transport and first-passage asymmetry

In stochastic transport, default-direction asymmetry is often literal: forward and backward transition or first-passage times cease to be interchangeable once hidden structure or nonequilibrium currents are present. In a two-lane lattice random walk with rates PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^7, microscopic reversibility no longer forces equal forward and backward transition times, because parallel pathways and inter-lane switching create state-space cycles with thermodynamic affinity

PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^8

The mean forward and backward transition times, PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^9 and PRESERVED_PLACEHOLDER_^^^^4 OR \4\4^^^^, coincide only at equilibrium,

PRESERVED_PLACEHOLDER_^^^^4 OR \4 OR \4^^^^

For the illustrative parameter set PRESERVED_PLACEHOLDER_^^^^4 OR \4 OR \4^^^^, PRESERVED_PLACEHOLDER_^^^^4 OR \4 OR \4^^^^, PRESERVED_PLACEHOLDER_^^^^4 OR \44^^^^, PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^5, varying PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^6 changes the sign of the asymmetry: PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^7 gives PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^8, PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^9 restores symmetry, and $$\Delta\mathcal{H}_t = -\eta A\,t_1(s_t,y_t) + \eta A\,t_2(s_t) + O(\eta^2),$$^^^^4\4^^^^ gives $$\Delta\mathcal{H}_t = -\eta A\,t_1(s_t,y_t) + \eta A\,t_2(s_t) + O(\eta^2),$$^^^^4 OR \4^^^^. The asymmetry thus functions as an experimentally accessible signature of broken detailed balance and hidden mechanistic complexity (Shin et al., 2020).

Biased run-and-tumble dynamics generalize this to full first-passage-time distributions. With absorbing boundaries at $$\Delta\mathcal{H}_t = -\eta A\,t_1(s_t,y_t) + \eta A\,t_2(s_t) + O(\eta^2),$$^^^^4 OR \4^^^^, the conditional first-passage distributions $$\Delta\mathcal{H}_t = -\eta A\,t_1(s_t,y_t) + \eta A\,t_2(s_t) + O(\eta^2),$$^^^^4 OR \4^^^^ obey first-passage duality only if

$$\Delta\mathcal{H}_t = -\eta A\,t_1(s_t,y_t) + \eta A\,t_2(s_t) + O(\eta^2),$$4

The paper shows that this duality generically fails in biased run-and-tumble processes, so default-direction asymmetry appears as different means, variances, and shapes for the two conditional distributions. It quantifies the violation with KL divergences and a signal-to-noise proxy,

$$\Delta\mathcal{H}_t = -\eta A\,t_1(s_t,y_t) + \eta A\,t_2(s_t) + O(\eta^2),$$5

A key distinction is between visible tumbles and hidden tumbles. For ballistic visible tumbles with $$\Delta\mathcal{H}_t = -\eta A\,t_1(s_t,y_t) + \eta A\,t_2(s_t) + O(\eta^2),$$6, the scaled cumulant generating function obeys a Gallavotti–Cohen symmetry and asymptotic duality is restored as $$\Delta\mathcal{H}_t = -\eta A\,t_1(s_t,y_t) + \eta A\,t_2(s_t) + O(\eta^2),$$7. For hidden tumbles with $$\Delta\mathcal{H}_t = -\eta A\,t_1(s_t,y_t) + \eta A\,t_2(s_t) + O(\eta^2),$$8, the GC symmetry generically fails, and a robust asymptotic second peak in the asymmetry survives (&&&^^^^4 OR \44^^^^&&&).

The same logic extends to turbulence, where the asymmetry concerns short-time two-particle dispersion forward and backward in time. The paper proves Lagrangian identities equating the antisymmetric cubic coefficient of short-time dispersion growth to the local energy defect. For strong limits of ^^^^4 OR \4^^^^D Navier–Stokes solutions, the measure equals $$\Delta\mathcal{H}_t = -\eta A\,t_1(s_t,y_t) + \eta A\,t_2(s_t) + O(\eta^2),$$9, the viscous dissipation anomaly, implying that particles initially disperse faster backward-in-time than forward-in-time. In a ^^^^4 OR \4^^^^D inverse-cascade regime with increasingly high-wavenumber forcing, the same measure equals $$t_1(s_t,y_t):=p_t(\mathcal{H}_t+\log p_t), \qquad t_2(s_t):=\sum_{v\in\mathcal{V}} p_v^2(\mathcal{H}_t+\log p_v).$$^^^^4\4^^^^, the anomalous input, implying that particles typically disperse faster forward-in-time than backward-in-time. Here the default direction is set by cascade direction: downscale in $$t_1(s_t,y_t):=p_t(\mathcal{H}_t+\log p_t), \qquad t_2(s_t):=\sum_{v\in\mathcal{V}} p_v^2(\mathcal{H}_t+\log p_v).$$^^^^4 OR \4^^^^, upscale in $$t_1(s_t,y_t):=p_t(\mathcal{H}_t+\log p_t), \qquad t_2(s_t):=\sum_{v\in\mathcal{V}} p_v^2(\mathcal{H}_t+\log p_v).$$^^^^4 OR \4^^^^ (&&&^^^^4 OR \44^^^^&&&).

5. Engineered directionality, steering, and operational asymmetry

In open quantum systems, default-direction asymmetry can be programmed. For a pair of driven-dissipative resonators, coherent hopping with phase $$t_1(s_t,y_t):=p_t(\mathcal{H}_t+\log p_t), \qquad t_2(s_t):=\sum_{v\in\mathcal{V}} p_v^2(\mathcal{H}_t+\log p_v).$$^^^^4 OR \4^^^^ and dissipative coupling with phase $$t_1(s_t,y_t):=p_t(\mathcal{H}_t+\log p_t), \qquad t_2(s_t):=\sum_{v\in\mathcal{V}} p_v^2(\mathcal{H}_t+\log p_v).$$4 combine into generalized off-diagonal couplings

$$t_1(s_t,y_t):=p_t(\mathcal{H}_t+\log p_t), \qquad t_2(s_t):=\sum_{v\in\mathcal{V}} p_v^2(\mathcal{H}_t+\log p_v).$$5

with relative phase $$t_1(s_t,y_t):=p_t(\mathcal{H}_t+\log p_t), \qquad t_2(s_t):=\sum_{v\in\mathcal{V}} p_v^2(\mathcal{H}_t+\log p_v).$$6. The two directions are unequal whenever both $$t_1(s_t,y_t):=p_t(\mathcal{H}_t+\log p_t), \qquad t_2(s_t):=\sum_{v\in\mathcal{V}} p_v^2(\mathcal{H}_t+\log p_v).$$7 and $$t_1(s_t,y_t):=p_t(\mathcal{H}_t+\log p_t), \qquad t_2(s_t):=\sum_{v\in\mathcal{V}} p_v^2(\mathcal{H}_t+\log p_v).$$8 are nonzero and $$t_1(s_t,y_t):=p_t(\mathcal{H}_t+\log p_t), \qquad t_2(s_t):=\sum_{v\in\mathcal{V}} p_v^2(\mathcal{H}_t+\log p_v).$$9. The transmission amplitudes satisfy

$\mathcal{T}(s_t,y_t):=-t_1(s_t,y_t)+t_2(s_t),$^^^^4\4^^^^

so isolation is set by $\mathcal{T}(s_t,y_t):=-t_1(s_t,y_t)+t_2(s_t),$^^^^4 OR \4^^^^. Perfect one-way transport occurs at $\mathcal{T}(s_t,y_t):=-t_1(s_t,y_t)+t_2(s_t),$^^^^4 OR \4^^^^ with $\mathcal{T}(s_t,y_t):=-t_1(s_t,y_t)+t_2(s_t),$^^^^4 OR \4^^^^ for rightward transport and $\mathcal{T}(s_t,y_t):=-t_1(s_t,y_t)+t_2(s_t),$4 for leftward transport. The paper interprets this as dissipation-induced quantum directionality without magnetic bias (Downing et al., 2022).

Gaussian quantum steering provides a related but intrinsically asymmetric example. In the nondegenerate three-level cascade laser studied in the paper, the Gaussian steerability from mode $\mathcal{T}(s_t,y_t):=-t_1(s_t,y_t)+t_2(s_t),$5 to mode $\mathcal{T}(s_t,y_t):=-t_1(s_t,y_t)+t_2(s_t),$6 is always at least as strong as the reverse, and one-way steering occurs only from $\mathcal{T}(s_t,y_t):=-t_1(s_t,y_t)+t_2(s_t),$7. The underlying reason is the parameter-independent positivity of the mode-intensity difference

$\mathcal{T}(s_t,y_t):=-t_1(s_t,y_t)+t_2(s_t),$8

which enforces $\mathcal{T}(s_t,y_t):=-t_1(s_t,y_t)+t_2(s_t),$9 in the covariance matrix and therefore $$\mathcal{P}(s_t,y_t,A):=A\,\mathcal{T}(s_t,y_t), \qquad \Delta\mathcal{H}_t=\eta\,\mathcal{P}(s_t,y_t,A)+O(\eta^2).$$^^^^4\4^^^^. The steering asymmetry is bounded by $$\mathcal{P}(s_t,y_t,A):=A\,\mathcal{T}(s_t,y_t), \qquad \Delta\mathcal{H}_t=\eta\,\mathcal{P}(s_t,y_t,A)+O(\eta^2).$$^^^^4 OR \4^^^^, so the state never reaches an extremal asymmetry state (&&&^^^^4 OR \4^^^^6&&&).

Operational time reversal yields a sharper logical asymmetry. For bipartite devices with conditional probabilities $$\mathcal{P}(s_t,y_t,A):=A\,\mathcal{T}(s_t,y_t), \qquad \Delta\mathcal{H}_t=\eta\,\mathcal{P}(s_t,y_t,A)+O(\eta^2).$$^^^^4 OR \4^^^^, Bayes inversion with prior $$\mathcal{P}(s_t,y_t,A):=A\,\mathcal{T}(s_t,y_t), \qquad \Delta\mathcal{H}_t=\eta\,\mathcal{P}(s_t,y_t,A)+O(\eta^2).$$^^^^4 OR \4^^^^ defines a reversed device

$$\mathcal{P}(s_t,y_t,A):=A\,\mathcal{T}(s_t,y_t), \qquad \Delta\mathcal{H}_t=\eta\,\mathcal{P}(s_t,y_t,A)+O(\eta^2).$$4

The paper shows that there exist devices with a local hidden-variable representation whose time-reverses allow perfect signaling. A concrete example is

$$\mathcal{P}(s_t,y_t,A):=A\,\mathcal{T}(s_t,y_t), \qquad \Delta\mathcal{H}_t=\eta\,\mathcal{P}(s_t,y_t,A)+O(\eta^2).$$5

which is forward no-signaling and LHV, but under uniform prior reverses to the deterministic channel $$\mathcal{P}(s_t,y_t,A):=A\,\mathcal{T}(s_t,y_t), \qquad \Delta\mathcal{H}_t=\eta\,\mathcal{P}(s_t,y_t,A)+O(\eta^2).$$6, i.e. a perfect ^^^^4 OR \4^^^^-bit channel from Alice to Bob when $$\mathcal{P}(s_t,y_t,A):=A\,\mathcal{T}(s_t,y_t), \qquad \Delta\mathcal{H}_t=\eta\,\mathcal{P}(s_t,y_t,A)+O(\eta^2).$$7 is fixed. For PR boxes, time reversal also enables signaling, though never as a perfect channel. This is a strong form of directional asymmetry: “perfect channel in one time direction” becomes “non-channel in the other direction” (&&&^^^^4 OR \4^^^^7&&&).

Active matter supplies a hydrodynamic analogue. In polar flocks, an antisymmetric fore–aft exchange coupling $$\mathcal{P}(s_t,y_t,A):=A\,\mathcal{T}(s_t,y_t), \qquad \Delta\mathcal{H}_t=\eta\,\mathcal{P}(s_t,y_t,A)+O(\eta^2).$$8 makes the response to a neighbor ahead differ from the response to a neighbor behind. In the continuum equations, $$\mathcal{P}(s_t,y_t,A):=A\,\mathcal{T}(s_t,y_t), \qquad \Delta\mathcal{H}_t=\eta\,\mathcal{P}(s_t,y_t,A)+O(\eta^2).$$9 enters the spin dynamics through the term

$p_t=\exp(-\mathcal{H}_t)$^^^^4\4^^^^

and in the overdamped limit it modifies the advective coefficient in a Toner–Tu–type equation. The same asymmetry creates a difference between the speed of information advection and the speed of the flock itself, and the turning instability threshold is

$p_t=\exp(-\mathcal{H}_t)$^^^^4 OR \4^^^^

Thus default-direction asymmetry here is a polarity-selected fore–aft bias that both transports information asymmetrically and, if strong enough, destabilizes the flock (&&&^^^^4 OR \4^^^^8&&&).

6. Preferred directions in astronomy and cosmology

In observational cosmology and extragalactic astronomy, default-direction asymmetry appears as a preferred axis or handedness rather than as a controllability or transport bias. In JWST deep fields overlapping the HST Ultra Deep Field, one study reports ^^^^4 OR \44^^^^ clockwise and ^^^^4 OR \4\4^^^^ counterclockwise spiral galaxies among ^^^^4 OR \44^^^^ galaxies with determinable winding direction, with one-tailed binomial probability $p_t=\exp(-\mathcal{H}_t)$^^^^4 OR \4^^^^. The paper emphasizes that this result is suggestive rather than definitive because of the small sample, but places it in continuity with prior survey-based analyses that reported a clockwise excess in that sky region and a magnitude of asymmetry that increases with redshift (&&&^^^^4 OR \4^^^^9&&&).

A larger HST/SDSS comparison studies a dipole-like handedness pattern rather than a single field excess. In the HST CANDELS data, the best-fit dipole axis is reported at $p_t=\exp(-\mathcal{H}_t)$^^^^4 OR \4^^^^ with significance $p_t=\exp(-\mathcal{H}_t)$4, while in an SDSS sample with $p_t=\exp(-\mathcal{H}_t)$5 the best-fit axis is $p_t=\exp(-\mathcal{H}_t)$6 with significance $p_t=\exp(-\mathcal{H}_t)$7. The HST and SDSS axes agree within the HST $p_t=\exp(-\mathcal{H}_t)$8 range, and COSMOS-region comparisons across HST, SDSS, and Pan-STARRS all show a clockwise excess, though with different strengths. In this usage, default-direction asymmetry denotes a large-scale directional preference in galaxy spin handedness (&&&^^^^4 OR \4\4^^^^&&&).

Cosmic hemispherical asymmetry in the CMB is a still more formal preferred-direction model. The modulated temperature field is written as

$p_t=\exp(-\mathcal{H}_t)$9

where $t_2(s_t)\ge 0$^^^^4\4^^^^ is the preferred axis. In the paper’s simulations, this asymmetry primarily affects the scalar amplitude $t_2(s_t)\ge 0$^^^^4 OR \4^^^^ and, more weakly, the scalar tilt $t_2(s_t)\ge 0$^^^^4 OR \4^^^^. For a Planck-like scale-dependent modulation profile, the induced directional effect is modest: only $t_2(s_t)\ge 0$^^^^4 OR \4^^^^ deviation in $t_2(s_t)\ge 0$4 relative to the isotropic estimate. Here the asymmetry is not a default update direction but a directional modulation of the statistical field itself (Mukherjee et al., 2015).

Taken together, these observational cases suggest a broad semantic extension of the term. In learning systems and engineered transport, default-direction asymmetry is typically mechanistic and actionable; in astronomy and cosmology, it is inferential and tied to anisotropy, handedness, or preferred axes. The unifying idea is nonetheless stable: symmetry expected at the problem statement is broken in a directionally structured manner by the dynamics, the architecture, the measurement protocol, or the large-scale geometry of the phenomenon under study.