Primacy Tail: From Black Holes to Learning Bias

• Primacy Tail is a phenomenon where early inputs or initial conditions continue to influence later stages, as seen in both black-hole ringdown and machine learning systems.
• In black-hole perturbation theory, the late-time power-law ringdown tail inherits a significant contribution from the entire inspiral history rather than just the final plunge.
• In reinforcement learning and sequential memory models, primacy bias leads to enduring effects from early data, necessitating strategies like periodic resets to mitigate overfitting.

The literature uses “primacy tail” in more than one sense. Most explicitly, it names a late-time gravitational-wave tail inherited from the full inspiral history of a perturbed Schwarzschild black hole. In other fields, the phrase is used more loosely for persistent effects of early or primary inputs, including early-experience dominance in reinforcement learning, position-dependent recall in sequential models and LLMs, and first-mover asymmetries in cumulative-advantage systems. A common thread is that the final or asymptotic regime is not determined solely by the most recent event: earlier structure remains dynamically active and leaves a measurable tail in the later observable (Amicis et al., 2024).

1. Terminological scope and unifying idea

In black-hole perturbation theory, the term is introduced as the “Primacy Tail” for a late-time ringdown component that is inherited from the entire inspiral history rather than set mainly by the final plunge. The setting is a perturbed Schwarzschild black hole sourced by a point particle on generic orbits, including quasi-circular inspiral, eccentric inspiral, dynamical capture, and radial infall. The resulting late-time waveform is written as an integral over the full inspiral history, and the paper’s central claim is that the dominant asymptotic tail is the one primarily inherited from that history (Amicis et al., 2024).

In several other literatures, the phrase is not introduced as a distinct theorem or canonical framework. In deep reinforcement learning, for example, it is used as a helpful way to describe the long-lasting downstream effect of early overfitting: early interactions shape optimization so strongly that later evidence remains available but becomes difficult to incorporate. In model-based reinforcement learning and diffusion alignment, related work likewise diagnoses persistent early-data or early-neuron effects without treating “primacy tail” as a separate formal object (Nikishin et al., 2022).

This suggests a cross-domain family resemblance rather than a single universal definition. In all cases, a “primary” component—an early trajectory, an initial condition, a first prompt slot, a first-mover group, or a strongest-response subset—continues to structure later behavior after one might naively expect it to have been superseded.

2. Inspiral-inherited ringdown tails in black-hole perturbation theory

The most technical and explicit use of the term appears in the study of inspiral-driven perturbations of a Schwarzschild black hole. There the late-time field is represented as a source integral over the entire particle trajectory, so the waveform does not simply “turn on” at merger. Instead, the ringdown tail retains memory of the prior orbital motion, and the late-time relaxation is captured by an analytic tail expression that exactly reproduces the slow relaxation seen in numerical evolutions performed on a hyperboloidal compactified grid (Amicis et al., 2024).

The asymptotic structure is an infinite power-law expansion,

$$\Psi(t)\sim \sum_{n=0}^{\infty}\frac{a_n}{t^{p+n}}, \qquad \Psi(t)\sim \frac{a_0}{t^p},$$

with the slowest-decaying term given by Price’s law. The key physical point is that the late-time signal is not a single quasinormal remnant. It is a superposition of infinitely many power-law pieces derived from the inspiral history, and every faster-decaying contribution eventually yields to the leading Price-law term (Amicis et al., 2024).

A major result is the several-orders-of-magnitude enhancement of tail terms for non-circular orbits. For eccentric, dynamical-capture, and radial trajectories, the dominant enhancement is activated when the particle is far from the black hole and has small tangential and radial velocities soon before plunge. The source then lingers in the weak-field region, accumulating a larger tail contribution over an extended interval. For large eccentricities, the dominant tail amplitude can therefore be extracted accurately from the last apastron before merger, even though the exact representation is an integral over the full inspiral history (Amicis et al., 2024).

The same work connects the inherited tail to broader issues in black-hole physics. It discusses implications for late-time tail extraction in nonlinear evolutions, possible observational consequences for ringdown interpretation, a scattering scenario, and a connection with the soft graviton theorem. In this usage, “primacy” is literal and asymptotic: the primary late-time contribution is the inspiral-inherited power-law tail rather than the terminal plunge.

3. Sequential memory, state-space dynamics, and end-of-sequence behavior

In sequential-memory studies, primacy-tail language is tied to serial-position effects. A structured state-space model, specifically S4, was shown to exhibit a primacy effect in a serial probe recognition task, despite the theoretical expectation that exponentially decaying memory should mainly favor recent inputs. Accuracy was highest for items at the beginning of the list, while the most recent study items were not always easiest to retrieve immediately after presentation; the latest items could instead show relatively low immediate accuracy, indicating a lag between encoding and usable retrieval (Morita, 19 Feb 2025).

The S4 architecture is described through the continuous-time linear state-space dynamics

$$\frac{d}{dt}\mathbf{h}(t)=A\mathbf{h}(t)+Bx(t),\qquad \mathbf{y}(t)=C\mathbf{h}(t),$$

and the discretized update

$$\mathbf{h}(t_j)=\bar{A}\mathbf{h}(t_{j-1})+\bar{B}x(t_j),$$

with a learnable step size $\Delta t$ controlling the effective memory timescale. The paper argues that many learned $\Delta t$ values fall below roughly $\le 0.03$, favoring long-range retention. This does not fully resolve why early items become more stably represented than recent ones, but it establishes that the learned discrete implementation can behave differently from the ideal continuous-time decay picture (Morita, 19 Feb 2025).

A mechanistic account of analogous behavior in Mamba identifies two distinct memory regimes inside the selective state-space block. Primacy is supported by a sparse subset of channels that preserve early input tokens, whereas recency is produced by delta-modulated recurrence:

$$\mathbf{h}_t^{(i)}=\sum_{j=1}^{t}\left(\prod_{k=j+1}^{t}\mathbf{A}_k^{(i)}\right)\mathbf{B}_j^{(i)}x_j^{(i)}.$$

Older inputs are attenuated by repeated recurrent factors, so late tokens enjoy an end-of-sequence advantage because they have undergone fewer forgetting steps. This recency benefit is fragile: inserting distractor tokens sharply degrades recent-item recall, showing that the tail end of the sequence is a limited-capacity short-term store rather than a permanent memory bank (Airlangga et al., 18 Jun 2025).

Related work on “lost in the middle” frames the U-shaped serial-position curve as an emergent consequence of mixed retrieval demands. Uniform long-term demand induces primacy when combined with autoregressive structure and attention sinks, whereas end-weighted short-term demand induces recency. When both are present, performance is high at the beginning and end and weak in the middle. Here the “tail” is the end-of-context edge of the U-shape, but the important point is that primacy and recency arise from different mechanisms rather than a single monotone decay law (Salvatore et al., 11 Oct 2025).

4. Primacy bias and long-lived optimization tails in reinforcement learning and diffusion alignment

In deep reinforcement learning, primacy bias is defined as “a tendency to overfit initial experiences that damages the rest of the learning process.” The compounding mechanism is specific to RL: agents collect the data they train on, so early overfitting can produce a worse policy, which produces worse future data, which makes recovery still harder. Experience replay, high replay ratio, function approximation, bootstrapped TD learning, and larger $n$-step targets all exacerbate this effect. In this setting, a primacy tail is the enduring influence of early data long after it was collected (Nikishin et al., 2022).

A practical mitigation is periodic parameter resetting while preserving the replay buffer. The implementation is algorithm-specific: for SPR on Atari 100k, only the final linear layer of the 5-layer Q-network is reset every $2\times 10^4$ steps; for SAC on DeepMind Control Suite, the agent’s networks are reset entirely every $2\times 10^5$ steps; and for DrQ, the last 3 out of 7 layers of the policy and value networks are reset every $$\frac{d}{dt}\mathbf{h}(t)=A\mathbf{h}(t)+Bx(t),\qquad \mathbf{y}(t)=C\mathbf{h}(t),$$0 steps. The buffer remains intact, which is crucial to rapid recovery. On Atari 100k, SPR improves from IQM 0.380 to 0.478 with resets; on DeepMind Control Suite, SAC rises from 501 to 656 and DrQ from 569 to 762 (Nikishin et al., 2022).

A more geometric treatment uses the Fisher Information Matrix to characterize a two-phase trajectory: an early memorization phase with sharp growth in $$\frac{d}{dt}\mathbf{h}(t)=A\mathbf{h}(t)+Bx(t),\qquad \mathbf{y}(t)=C\mathbf{h}(t),$$1, followed by a reorganization phase in which sensitivity declines even if reward continues to improve. The proposed Fisher-Guided Selective Forgetting (FGSF) perturbs parameters using Fisher-structured noise,

$$\frac{d}{dt}\mathbf{h}(t)=A\mathbf{h}(t)+Bx(t),\qquad \mathbf{y}(t)=C\mathbf{h}(t),$$2

to prevent early experiences from dominating the learned geometry. The paper reports strong gains in complex DeepMind Control tasks, including Humanoid at 136.645 \pm 14.360 versus 68.503 \pm 21.938 for baseline SAC, and Quadruped at 873.473 \pm 21.287 versus 582.909 \pm 37.262 (Falzari et al., 2 Feb 2025).

In model-based reinforcement learning, the dominant source of primacy bias shifts from the agent to the world model. Resetting the agent can hurt MBPO, whereas resetting the world model improves performance because the model overfits early policy-induced dynamics and then fails to track the evolving data distribution. The recommended procedure is periodic reinitialization of the most adaptation-sensitive part of the world model—for MBPO, all ensemble models but only the last 2 hidden layers every $$\frac{d}{dt}\mathbf{h}(t)=A\mathbf{h}(t)+Bx(t),\qquad \mathbf{y}(t)=C\mathbf{h}(t),$$3 environment steps; for DreamerV2, the transition predictor’s hidden layer every $$\frac{d}{dt}\mathbf{h}(t)=A\mathbf{h}(t)+Bx(t),\qquad \mathbf{y}(t)=C\mathbf{h}(t),$$4 steps (Qiao et al., 2023).

In diffusion-model alignment, the relevant diagnosis combines temporal inductive bias with primacy bias in the critic. Existing methods optimize only a final-image reward $$\frac{d}{dt}\mathbf{h}(t)=A\mathbf{h}(t)+Bx(t),\qquad \mathbf{y}(t)=C\mathbf{h}(t),$$5, thereby overemphasizing the end of the denoising trajectory. TDPO-R instead distributes credit across intermediate timesteps and periodically resets active critic neurons every $$\frac{d}{dt}\mathbf{h}(t)=A\mathbf{h}(t)+Bx(t),\qquad \mathbf{y}(t)=C\mathbf{h}(t),$$6 epochs, with dormant threshold $$\frac{d}{dt}\mathbf{h}(t)=A\mathbf{h}(t)+Bx(t),\qquad \mathbf{y}(t)=C\mathbf{h}(t),$$7. The paper’s core empirical claim is that dormant neurons regularize against overoptimization, whereas active neurons reflect primacy bias; resetting active neurons improves cross-reward generalization and reduces reward overoptimization (Zhang et al., 2024).

5. LLMs, multiple-choice ordering, and multimodal retrieval

Order effects in LLMs provide a more behavioral use of primacy-tail language. In a direct adaptation of Asch’s impression-formation experiment, ChatGPT in a joint forced-choice setup preferred the candidate with positive adjectives listed first 65.5% of the time, whereas Gemini split evenly between positive-first and negative-first at 47.5% each, and Claude refused in 100% of trials. In a separate-rating setup, most outputs became equal ratings, but when a distinction was made the preference often flipped toward negative-first, which the paper interprets as a possible recency effect rather than classic primacy (Hämäläinen, 29 Apr 2025).

In multiple-choice question answering, the positional effect is operationalized directly. Fine-tuned LLMs show stronger primacy bias than pre-trained ones: accuracy is higher when the correct option appears near the beginning of the option list. A training-free exploitation strategy sorts options by descending semantic similarity to the query, placing semantically most relevant answers first. On CLINC, this raises Llama-3-8B fine-tuned accuracy from 0.37 to 0.49, and Mistral-7B pre-trained accuracy from 0.41 to 0.53. The paper therefore treats primacy not only as a nuisance bias but as an inference-time signal that can be harnessed (Raimondi et al., 18 Jul 2025).

A different pattern appears in multimodal retrieval-augmented question answering. In KB-VQA with frozen 7B/8B VLM readers, the classic text-LLM U-shape does not transfer. Instead, the shape becomes strongly primacy-dominant: at $$\frac{d}{dt}\mathbf{h}(t)=A\mathbf{h}(t)+Bx(t),\qquad \mathbf{y}(t)=C\mathbf{h}(t),$$8, gold-at-first beats gold-at-last by 16–26 percentage points in every reader-by-benchmark cell, a phenomenon termed “Lost at the End.” The effect is roughly

$$\frac{d}{dt}\mathbf{h}(t)=A\mathbf{h}(t)+Bx(t),\qquad \mathbf{y}(t)=C\mathbf{h}(t),$$9

Text-only control shows that multimodal KB-VQA amplifies an already present text-mode primacy effect by roughly 2.2× to 4.5×, and image-position and distractor-shuffle ablations localize the bias to prompt slot 0 of the instruction-tuned reader (Liu et al., 15 Jun 2026).

A common misconception is that retrieval quality alone should determine answer quality once the gold passage appears in the top-$$\mathbf{h}(t_j)=\bar{A}\mathbf{h}(t_{j-1})+\bar{B}x(t_j),$$0 set. The multimodal results contradict this. On a frozen reader, MMR, oracle reranking, and rank-based reordering do not yield a separable improvement that closes the first-versus-last gap, which is why the paper argues that recall@k is the wrong metric for deployed KB-VQA and that the necessary intervention is reader-side rather than retrieval-side (Liu et al., 15 Jun 2026).

6. First-mover asymmetry, biological coding, and tidal-tail morphology

Outside machine learning, primacy-tail ideas arise in cumulative advantage, sensory coding, and stellar dynamics. In Simon’s rich-get-richer model, the first group is not just another sample from the asymptotic distribution. Its expected size exceeds the bulk Zipf law by a factor of order

$$\mathbf{h}(t_j)=\bar{A}\mathbf{h}(t_{j-1})+\bar{B}x(t_j),$$1

where $$\mathbf{h}(t_j)=\bar{A}\mathbf{h}(t_{j-1})+\bar{B}x(t_j),$$2 is the innovation probability. The paper argues that this first-mover excess is not a transient and that the expected waiting time to first replication for a newly born group is infinite. In this usage, a primacy tail is the permanent elevation of the first group above the heavy-tailed bulk (Dodds et al., 2016).

In olfaction, primacy coding identifies odorant identity with the top $$\mathbf{h}(t_j)=\bar{A}\mathbf{h}(t_{j-1})+\bar{B}x(t_j),$$3 responding receptor types rather than absolute response magnitudes. The primacy set $$\mathbf{h}(t_j)=\bar{A}\mathbf{h}(t_{j-1})+\bar{B}x(t_j),$$4 is defined by inequalities of the form

$$\mathbf{h}(t_j)=\bar{A}\mathbf{h}(t_{j-1})+\bar{B}x(t_j),$$5

for receptors inside and outside the set. Because the code depends on response ordering, it is invariant to overall concentration changes and to any monotone nonlinearity in receptor output. The remaining weaker responders form what the paper effectively treats as a primacy tail: a large portion of the response pattern can vary without changing identity, because only the strongest responders determine the code (Kepple et al., 2016).

In stellar-cluster dynamics, primordial mass segregation leaves a strong but fading signature on tidal tails. Direct $$\mathbf{h}(t_j)=\bar{A}\mathbf{h}(t_{j-1})+\bar{B}x(t_j),$$6-body simulations with NBODY7, SSE/BSE, and Plummer initial conditions show that segregated clusters develop earlier, denser, smoother, and longer tails, with bottom-heavy early tail mass functions. Initially mass-segregated models can develop tails up to 1.4 times longer than non-segregated models at fixed mass loss, but the difference in mean-mass gradients and morphology shrinks over time as relaxation, tidal stripping, and stellar-evolution mass loss erase the memory of the initial state (Ghasemi et al., 1 Jul 2026).

A more specific asymmetry appears in open clusters. For Hyades, Praesepe, Coma Berenices, COIN-Gaia 13, and NGC 752, the leading tidal tail is reported to be more populated than the trailing tail within about $$\mathbf{h}(t_j)=\bar{A}\mathbf{h}(t_{j-1})+\bar{B}x(t_j),$$7 pc, and Milgromian simulations predict such asymmetry naturally through the external field effect, whereas Newtonian models remain near-symmetrical. Here the primacy-like feature is not temporal but geometric: the leading arm becomes the preferential escape channel (Kroupa et al., 2022).

Across these literatures, the term “primacy tail” does not designate a single invariant formalism. Its most precise meaning is the inspiral-inherited late-time tail of black-hole ringdown. More broadly, the phrase denotes persistent privilege of an early, primary, strongest, or first-position component—whether an inspiral history, an early replay buffer, a first prompt slot, a first mover, or a strongest receptor subset—whose influence remains visible deep into the later regime.