Complex Fallback: Multifaceted Secondary Regimes

• Complex fallback is a phenomenon where primary processes transition into structured secondary modes under constraints and uncertainty across diverse domains.
• Key methodologies across fields include level-by-level majority counting in voting, bifurcation analysis in astrophysics, and fallback policy design in AI systems.
• The study of complex fallback highlights its practical importance in enhancing control resistance, accurate transient modeling, and safety-critical fallback strategies.

Complex fallback denotes a family of phenomena in which a primary process does not simply terminate, but instead enters a secondary regime whose behavior is structured by constraints, uncertainty, or limited intervention. In computational social choice, fallback voting combines approval with level-by-level majority counting and yields a highly nontrivial control landscape (Erdélyi et al., 2010). In astrophysics, fallback refers to gas or ejecta that fails to escape and later returns, producing bifurcations between escape and recapture, characteristic power-law accretion histories, and substantial effects on remnant evolution (Chen et al., 2015, Wong et al., 2014). In machine learning and autonomy, fallback denotes backup policies, degraded generation modes, dynamic mixed-precision execution, or predefined safety maneuvers invoked when nominal behavior becomes unreliable (Lecerf et al., 2022, Ivgi et al., 2024, Christensen et al., 30 Dec 2025).

1. Cross-domain scope

The cited literature uses fallback in several technically distinct but structurally related ways. A common pattern is the replacement of an intended or nominal regime by an alternative one that remains admissible under altered conditions. This suggests that complex fallback is not a single domain-specific mechanism, but a recurring architecture of constrained continuation.

Domain	Fallback object	Governing distinction
Voting theory (Erdélyi et al., 2010)	Hybrid approval/majority rule and its control problems	$W[2]$-hardness versus FPT under a chair’s action budget
Stellar and explosive astrophysics (Chen et al., 2015, Wong et al., 2014)	Bound shells or ejecta returning to a compact object	Escape versus fallback; prompt versus reverse-shock pathways
Tidal disruption and merger debris (Miles et al., 2020, Musolino et al., 2024)	Debris-return rate to a black hole or merger remnant	$t^{-5/3}$, $t^{-9/4}$, and core-dependent deviations
AI systems (Lecerf et al., 2022, Ivgi et al., 2024, Zhang et al., 11 Mar 2025)	Backup policies, degraded outputs, or precision fallbacks	Diversity constraints, uncertainty hierarchies, and blockwise outlier handling
Maritime autonomy (Christensen et al., 30 Dec 2025)	Short-horizon degraded maneuver before takeover	Human-overridable candidate-constrained fallback

Across these usages, fallback is neither synonymous with failure nor with full recovery. It is instead a constrained secondary mode whose complexity arises from discrete choices, bifurcations, or competing timescales.

2. Electoral control in fallback voting

Fallback voting (FV) is a voting rule in which each voter specifies an approval set $S_v \subseteq C$ and a tie-free linear order over the approved candidates only. A ballot is written as

$c_1\ c_2\ \cdots\ c_k \mid C-S_v,$

meaning that $c_1,\dots,c_k$ are approved and ranked, while all remaining candidates are disapproved (Erdélyi et al., 2010). Winner determination proceeds level by level: level 1 counts only top approved candidates; level 2 counts the top two approved positions; and so on. If some candidate obtains a strict majority on the first level at which a majority appears, that candidate wins uniquely; if no level yields a majority winner, the candidate or candidates with highest overall approval score win (Erdélyi et al., 2010).

The control-theoretic interest of FV lies in the difficulty of altering election outcomes by structural intervention. Classical results show that FV is resistant to every common form of candidate control and to each common type of constructive control; among the standard 22 control types, it is resistant to 20 and vulnerable only to destructive control by adding voters and destructive control by deleting voters (Erdélyi et al., 2011). This places FV among the strongest known natural election systems with polynomial-time winner determination in the control-resistance literature (Erdélyi et al., 2010).

The parameterized version sharpens this picture by measuring complexity with respect to the chair’s action budget $k$, namely the number of candidates or voters added or deleted. Under that parameterization, constructive and destructive control by adding candidates are $W[2]$-hard, and constructive and destructive control by deleting candidates are also $W[2]$-hard (Erdélyi et al., 2010). For voters, constructive control by adding voters and constructive control by deleting voters are $W[2]$-hard, whereas destructive control by adding voters and destructive control by deleting voters are in FPT (Erdélyi et al., 2010). The reductions are from Dominating Set, which is $t^{-5/3}$0-complete, and they exploit FV’s level-by-level majority mechanism to encode domination into the election structure (Erdélyi et al., 2010).

A recurrent misconception is that the hybrid nature of FV should make it easier to manipulate than simpler rules. The published results point in the opposite direction: the combination of approval thresholds with ranked approved prefixes produces a control landscape that is algorithmically difficult in almost all parameterized cases studied (Erdélyi et al., 2010). In this literature, complexity is not primarily definitional complexity but resistance to strategic control.

3. Hydrodynamic return flows in stars

In stellar hydrodynamics, fallback often denotes material ejected below effective escape conditions and later recaptured. In the AGB context, one studied configuration is a transient shell ejection from a $t^{-5/3}$1 star at $t^{-5/3}$2, where the escape speed is $t^{-5/3}$3, followed by the resumption of a steady wind (Chen et al., 2015). Two shell-launch speeds were examined: $t^{-5/3}$4 and $t^{-5/3}$5, both below $t^{-5/3}$6. The lower-speed shell falls back, whereas the slightly faster shell escapes, yielding a bifurcation governed by a critical shell velocity $t^{-5/3}$7 in the numerical analytic integration; a zeroth-order estimate gives $t^{-5/3}$8, about $t^{-5/3}$9 above the numerical result (Chen et al., 2015).

The one-dimensional shell model treats the shell as a thin radial mass accelerated by post-shell wind ram pressure and decelerated by gravity. In this approximation,

$t^{-9/4}$0

and the wind speed evolves as

$t^{-9/4}$1

The shell is advanced by discrete momentum updates including swept-up wind mass and gravity (Chen et al., 2015). The central physical competition is between the shell’s initial sub-escape launch, the continuing wind’s momentum flux, and the stellar potential.

Core-collapse supernova fallback is physically distinct. The literature distinguishes prompt fallback from reverse-shock-driven fallback (Wong et al., 2014). Prompt fallback groups together rarefaction-wave deceleration after proto-neutron-star contraction and the energy and momentum loss of ejecta through expansion and $t^{-9/4}$2 work. Reverse-shock-driven fallback arises when the outgoing shock decelerates in flatter density profiles, sending a reverse shock inward. In three-dimensional simulations of a $t^{-9/4}$3 solar-metallicity progenitor and a $t^{-9/4}$4 zero-metallicity progenitor, fallback is overwhelmingly prompt, with the strongest accretion in the first $t^{-9/4}$5–$t^{-9/4}$6 s, and the reverse shock usually plays only a minor role, if any (Wong et al., 2014).

After about $t^{-9/4}$7 s in most supernova runs, the compact-object accretion rate follows the familiar late-time power law

$t^{-9/4}$8

a scaling that also underlies later transient models powered by fallback accretion (Wong et al., 2014, Dexter et al., 2012). The major controversy addressed in this literature is therefore not whether reverse shocks can exist, but whether they dominate fallback. The cited three-dimensional simulations argue that they generally do not (Wong et al., 2014).

4. Fallback-rate asymptotics in disruptions and mergers

Fallback-rate theory becomes especially explicit in tidal disruption events (TDEs). For full disruptions, the canonical late-time law is

$t^{-9/4}$9

For partial TDEs by supermassive black holes, where a self-bound stellar core survives, hydrodynamical simulations verify that the late-time fallback rate asymptotes to

$S_v \subseteq C$0

essentially independent of the surviving core mass (Miles et al., 2020). Deeper partial encounters can show a broken-power-law fallback curve: an initial decline close to $S_v \subseteq C$1, followed by steepening and later approach to $S_v \subseteq C$2 (Miles et al., 2020). The break timescale is parameterized by both impact parameter $S_v \subseteq C$3 and surviving core mass fraction $S_v \subseteq C$4, with fitted relations

$S_v \subseteq C$5

and

$S_v \subseteq C$6

This is one of the clearest formal examples of complex fallback as a transition between asymptotic regimes (Miles et al., 2020).

The white-dwarf–IMBH case modifies that picture substantially. For partial tidal disruptions of a $S_v \subseteq C$7 carbon-oxygen white dwarf by a non-spinning $S_v \subseteq C$8 IMBH, the late-time fallback does not generically approach the conjectured $S_v \subseteq C$9 law (Garain et al., 2023). Instead, the asymptotic slope depends strongly on impact parameter and remnant core mass. The reported range extends from $c_1\ c_2\ \cdots\ c_k \mid C-S_v,$0 at $c_1\ c_2\ \cdots\ c_k \mid C-S_v,$1, close to $c_1\ c_2\ \cdots\ c_k \mid C-S_v,$2, to $c_1\ c_2\ \cdots\ c_k \mid C-S_v,$3 at $c_1\ c_2\ \cdots\ c_k \mid C-S_v,$4 (Garain et al., 2023). The authors attribute this difference to the fact that the core-to-black-hole mass ratio is not negligible in the IMBH case, so the surviving core remains dynamically important at late times (Garain et al., 2023).

Binary-neutron-star merger remnants exhibit yet another fallback regime. General-relativistic magnetohydrodynamics simulations with neutrino transport find fallback masses $c_1\ c_2\ \cdots\ c_k \mid C-S_v,$5, with the total fallback mass $c_1\ c_2\ \cdots\ c_k \mid C-S_v,$6 and typically about $c_1\ c_2\ \cdots\ c_k \mid C-S_v,$7 of the unbound ejecta (Musolino et al., 2024). The accretion rate again follows

$c_1\ c_2\ \cdots\ c_k \mid C-S_v,$8

and the resulting gamma-ray and X-ray luminosities can exceed $c_1\ c_2\ \cdots\ c_k \mid C-S_v,$9 for hundreds of seconds, matching key properties of the “extended emission” observed in some short GRBs (Musolino et al., 2024). Here the fallback complexity lies less in a slope transition than in the mapping from a short ejection episode to a long-duration observable via orbital return times and r-process-heated photospheric emission.

5. Compact remnants, disks, and fallback-powered transients

Fallback materially reshapes compact remnants. Three-dimensional simulations of selected non-rotating supernova models show that the strongest kick and spin are imparted by partial fallback in an asymmetric explosion (Chan et al., 2020). In that scenario, black-hole kicks of several hundred $c_1,\dots,c_k$0 and spin parameters of $c_1,\dots,c_k$1 can be obtained; for a non-rotating $c_1,\dots,c_k$2 progenitor, fallback can spin a neutron star up to millisecond periods (Chan et al., 2020). The same study emphasizes that if the explosion energy barely exceeds the envelope binding energy, fallback is stronger but the final kick and spin remain small, because the asymmetries are swallowed too completely (Chan et al., 2020).

Fallback has also been proposed as a route to ultra-long-period isolated pulsars. A parameter study of neutron stars interacting with a fallback disk finds that very long spin periods $c_1,\dots,c_k$3 s can be reached in the presence of strong, magnetar-like magnetic fields $c_1,\dots,c_k$4 and moderate initial fallback accretion rates $c_1,\dots,c_k$5 (Ronchi et al., 2022). The modeled evolution passes through accretion, propeller, and dipole phases, and the equilibrium-period scaling

$c_1,\dots,c_k$6

makes strong $c_1,\dots,c_k$7 and modest $c_1,\dots,c_k$8 especially effective at producing long periods (Ronchi et al., 2022).

One-sided fallback onto a moving neutron star introduces an additional dynamical subtlety. Contrary to earlier expectations, fallback onto a kicked neutron star tends to produce spin-kick misalignment rather than alignment, because the fallback material’s angular momentum is generically transverse to the radial direction as a consequence of baroclinic vorticity generation and the geometry of convective and Rayleigh–Taylor flows (Müller, 2023). The corresponding angular-momentum budget implies a limit of order $c_1,\dots,c_k$9 of fallback accretion for fast-spinning young neutron stars with periods of $k$0, and even less for longer birth spin periods (Müller, 2023).

The fate of fallback matter depends strongly on whether the remnant is a neutron star or a black hole. Numerical work combining MESA progenitors with explosion calculations finds that, if magnetic torques play an important role in angular-momentum transport, fallback disks around young neutron stars are not an outcome of supernova explosions; their formation requires negligible magnetic torques and a fine-tuned explosion energy (Perna et al., 2013). For black-hole remnants, by contrast, disk formation is ubiquitous if magnetic fields do not play a strong role, and can still occur over a large region of $k$1 parameter space even with strong internal magnetic coupling (Perna et al., 2013). The same study identifies extended, long-lived fallback disks around black holes and argues that their physical conditions may be conducive to planet formation (Perna et al., 2013).

Fallback has often been invoked as an engine for luminous transients, but the literature is cautious about physical plausibility. In supernova light-curve modeling, late-time fallback accretion can significantly affect optical light curves and produce super-luminous or otherwise peculiar events; its normalization is enhanced at low explosion energies, in very massive stars, or when a strong reverse shock forms at the helium/hydrogen interface (Dexter et al., 2012). For hydrogen-poor superluminous supernovae, Bayesian fits to 37 SLSNe-I show that a fallback-accretion engine can fit the light curves well, with fit quality comparable to magnetar models, but often requires total energies $k$2; if a typical conversion efficiency $k$3 is adopted, the required accreted mass is $k$4, which is unrealistic for many events (Moriya et al., 2018). Likewise, in kilonova modeling, fallback sources better reproduce the $k$5 and $k$6 trends of AT 2017gfo than pulsar sources, yet do not simultaneously fit the bolometric luminosity and the $k$7, $k$8, and $k$9 evolution in the explored model grid (Wollaeger et al., 2019). These results caution against equating phenomenological fit quality with a confirmed fallback origin.

6. Learned, emergent, and quantized fallback in AI systems

In reinforcement learning for safety-critical driving, fallback is treated as a deliberately learned backup behavior rather than a passive failure mode. A model-free framework trains one primary optimal policy $W[2]$0 together with pseudo-agents that are encouraged to remain both sufficiently solved and sufficiently different in trajectory space (Lecerf et al., 2022). The additional pseudo-reward is

$W[2]$1

so policies that imitate the reference trajectory too closely are penalized (Lecerf et al., 2022). In a two-way-intersection task, the optimal strategy speeds up and passes before the first oncoming vehicle, whereas the fallback strategy slows down and passes between oncoming vehicles; when the uncertainty around the first target vehicle is increased by enlarging its effective collision radius by $W[2]$2, the optimal policy suffers a sharp performance drop while the fallback policy’s state subspace remains safe and unaffected (Lecerf et al., 2022).

In LLMs, fallback describes undesirable generation modes under epistemic uncertainty. The analyzed categories are sequence repetitions, degenerate text, and hallucinations (Ivgi et al., 2024). Across model scale, pretraining duration, instruction tuning, and even within a single generated sequence, the ordering is stable: more advanced models shift from sequence repetitions, to degenerate text, and then to hallucinations (Ivgi et al., 2024). Within a single sequence, the measured progression is from correct facts to hallucinations and finally to repetitions, with statistical support reported as $W[2]$3 in Mann–Whitney U-tests (Ivgi et al., 2024). An important practical result is that random sampling alleviates sequence repetitions but increases harder-to-detect hallucinations (Ivgi et al., 2024). A common misconception is therefore that more fluent or more diverse decoding is automatically safer; the cited evidence does not support that view.

Low-bit training introduces a different engineering notion of fallback. Dynamic block-level fallback quantization addresses activation outliers in GLU-based Transformers by using mixed-precision GEMM that falls back from 8-bit to 16-bit for activation blocks containing outliers (Zhang et al., 11 Mar 2025). Block selection is based on an AbsMax criterion,

$W[2]$4

and thresholds are adjusted layerwise to keep the fallback rate in a target interval $W[2]$5 with adjustment factor $W[2]$6 (Zhang et al., 11 Mar 2025). The method is reported to maintain training quality close to BF16 while achieving a $W[2]$7 end-to-end training speedup on RTX4090 GPUs (Zhang et al., 11 Mar 2025). In this setting, fallback is neither a semantic backup nor a physical return flow, but a dynamic precision escalation triggered by sparse outliers.

7. Human-overridable fallback maneuvers in maritime autonomy

For maritime autonomy, fallback is framed explicitly as a regulatory degraded mode. The draft IMO MASS Code requires detection of departures from the operational design domain, entry into a predefined fallback state, operator notification, immediate human override, and no unapproved voyage-plan changes (Christensen et al., 30 Dec 2025). The proposed operational loop is therefore

$W[2]$8

with the fallback maneuver limited to a single short-horizon, pre-approved motion action (Christensen et al., 30 Dec 2025).

The “Semantic Lookout” system implements this idea with a camera-only, candidate-constrained vision-language-model selector (Christensen et al., 30 Dec 2025). Water-valid motion primitives are generated from a segmentation-based clearance map

$W[2]$9

and only candidates satisfying $W[2]$0 and $W[2]$1 for all visible samples are retained (Christensen et al., 30 Dec 2025). The reported configuration uses $W[2]$2 candidate maneuvers, $W[2]$3 pixels, and ID $W[2]$4 for station-keeping (Christensen et al., 30 Dec 2025). The VLM selects one candidate or station-keeping, and if ensemble voting yields no strict majority, the system defaults to station-keeping (Christensen et al., 30 Dec 2025). Human authority is preserved by command arbitration,

$W[2]$5

which guarantees at least $W[2]$6 human authority (Christensen et al., 30 Dec 2025).

On 40 harbor scenes, the best reported fallback-maneuver selector is gpt-5-low with FB-3, achieving $W[2]$7 and $W[2]$8 at $W[2]$9 s latency (Christensen et al., 30 Dec 2025). On fire scenes, FB-3 increases separation from the hazard by $W[2]$0 m at $W[2]$1 s, compared with $W[2]$2 m for Keep-course, $W[2]$3 m for Keep-starboard, and $W[2]$4 m for Keep-station (Christensen et al., 30 Dec 2025). A field run verifies end-to-end alert, fallback-maneuver execution, and operator handover (Christensen et al., 30 Dec 2025).

Taken together, these results clarify what makes fallback “complex” across domains. The difficulty rarely lies in the existence of a secondary mode alone. It lies in how that mode is selected, how its dynamics depend on hidden structure, and how its observable consequences differ from the nominal regime. In fallback voting, this complexity appears as parameterized control hardness; in astrophysics, as escape–recapture bifurcations and nontrivial accretion laws; in AI, as backup-policy diversity, uncertainty-induced degradation hierarchies, and dynamic precision escalation; and in autonomy, as constrained, human-overridable maneuver selection under latency and regulatory limits (Erdélyi et al., 2010, Chen et al., 2015, Ivgi et al., 2024, Christensen et al., 30 Dec 2025).