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Spokes, not a Ring: Revisiting the Weekday Manifold in Language Models

Diksha Gupta, Rieke Fruengel, Stefan Heimersheim

arXiv:2610.37796v1cs.LGcs.AI

TL;DR

Language models are often interpreted or steered using fitted representation manifolds, but it is unresolved whether points between observed representations carry meaningful concept values. This paper tests that assumption on the weekday ring and finds evidence for seven discrete states rather than a continuous representation.

  • Problem

    Fitted manifolds interpolate between finite observed representations, leaving unresolved whether intermediate points represent meaningful concept values.

    Source: §1 · p. 1

  • Method

    The paper tests weekday-ring continuity using perturbation sensitivity and steering, then examines time-of-day, uncertainty, and natural-text activations for intermediate states.

    Source: §1 · p. 2

  • Results

    The analyses favor seven discrete weekday states: steering is step-like, and time of day, uncertainty, and natural text do not populate the intervening arcs.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The findings motivate explicit tests of continuity in other proposed representational manifolds.

    Source: §1 · p. 2

    The sensitivity pattern is consistent with discrete weekday states but does not establish them conclusively, because its causes remain unclear and low sensitivity may not reliably identify in-distribution regions.

    Source: §5.2 · p. 9

Abstract

from arXiv · show

Recent work has hypothesised that large language models may represent some concepts as multi-dimensional manifolds instead of single directions. Using these manifolds to interpret or steer representations often assumes that moving along them corresponds to a continuous change in the represented concept. In this work, we test this continuous manifold assumption on the prototypical weekday manifold. Despite its striking circular geometry, we find that the weekday representations behave more like seven discrete states arranged in a circle than a continuous ring. First, we show that local sensitivity has seven minima aligned with the weekdays, producing a step-like mapping between representation and behaviour under on-manifold steering. Second, we test whether any concepts are represented along the manifold outside the seven discrete states, and find neither time of day, uncertainty, nor natural text populate the intermediate space. Together, these results challenge the continuous interpretation of the weekday manifold and motivate testing whether movement along other representation manifolds correspond to meaningful changes in model representations and behaviour.

1 INTRODUCTION

The paper tests whether the fitted weekday ring represents meaningful intermediate values or connects seven discrete states. Its analyses favor discrete weekday states: steering changes stepwise, and the intervening arcs lack evidence of meaningful occupancy.

1 INTRODUCTION

  • Manifold geometry is appealing because it may reflect semantic relationships among concept values, but continuity in fitted geometry does not by itself establish meaningful intermediate representations.Source: §1 · pp. 1–2
  • The continuous manifold assumption holds that points throughout a fitted manifold correspond to meaningful values of the represented concept, although the fitted regions between observed representations may only be interpolations.Source: §1 · p. 1
  • The paper tests this assumption on the weekday ring through perturbation sensitivity and steering analyses, then probes whether time of day, uncertainty, or natural text occupies its arcs.Source: §1 · p. 2
  • Weekday steering changes stepwise around seven low-sensitivity states, while time of day, uncertainty, and natural text do not populate the intervening arcs.Source: §1 · p. 2

2 RELATED WORK

Prior work has identified rich geometric structure in model representations and used it to probe or intervene on behavior. This paper focuses on whether continuity in a fitted manifold reflects continuity in the underlying representation.

Geometric structure in representations

  • Mechanistic interpretability has expanded from direction-based concepts to circular, helical, spatial, temporal, and perceptual geometries, which interventions can use to change model behavior.Source: §2 · p. 3

Representation manifolds and continuity

  • Representation manifolds connect activation-space geometry to feature-space structure and have been used in probing, intervention, and feature discovery.Source: §2 · p. 3

Local stability in activation space

  • Prior perturbation studies identify locally stable activation regions separated by boundaries where behavior changes rapidly; this work uses stability to diagnose distinct weekday states.Source: §2 · p. 3

3 METHODS

The methods construct a weekday ring from model activations, then test steering sensitivity and whether time, uncertainty, or natural text occupies interpolated regions. Additional matched-context comparisons assess whether contexts rotate the weekday configuration.

3 METHODS

  • The authors study Llama-3.1-8B, Mistral-7B-v0.1, and GPT-2 XL, constructing the weekday ring from seven prompt-averaged weekday centroids fit with a periodic cubic spline.Source: §3 · p. 3
  • They steer at the weekday token from layer 2 toward a late readout, using third-person templates and comparing against arithmetic prompts that strongly constrain weekday outputs.Source: §3 · p. 3
  • They measure downstream changes under equal-step ring steering and calculate the Frobenius norm of the layer-28 residual Jacobian across a polar grid of the ring and surrounding space.Source: §3 · pp. 3–4
  • Clock-time prompts and weekday-pair uncertainty prompts test whether these concepts move activations into interpolated ring positions or otherwise alter weekday representations.Source: §3 · p. 4
  • Natural-text occupancy is tested by projecting activations from 16-token FineWeb excerpts ending in weekdays onto a weekday subspace fitted from a subset of excerpts.Source: §3 · p. 4
  • Six matched variants replace each sequence’s final weekday, allowing comparison of weekday configurations across contexts after removing within-context mean shifts.Source: §3 · p. 4

4 RESULTS

Across the weekday ring, sensitivity is lowest near the seven weekdays and higher between them. Tests of clock time, uncertainty, and natural text do not show these inputs occupying meaningful intermediate positions along the ring.

4.1 PERTURBATION SENSITIVITY HAS SEVEN WEEKDAY-CENTRED MINIMA

  • Equal ring steps produce slow downstream changes near weekdays and rapid changes between them; Jacobian sensitivity has seven weekday-centred minima extending radially into spokes.The sevenfold contrast in residual-stream response grows across later layers.Source: §4.1 · pp. 5–6
  • The sevenfold sensitivity pattern appears across perturbation directions in Llama-3.1-8B and Mistral-7B-v0.1, but is most pronounced for tangential perturbations in GPT-2 XL.It is also robust across prompt templates and readout positions, though its magnitude varies.Source: §4.1 · p. 6

4.2.1 TIME-OF-DAY VARIATION ALONG THE WEEKDAY RING IS SMALL AND NOT CAUSAL

  • 99.8% of hour-centroid variation lies outside the weekday plane, and ring steering changes predicted hours by less than 0.02 h, unlike off-ring late–early steering.The off-ring interventions shift predicted hours by −1.8h toward “early” and +1.1h toward “late”; small in-plane shifts are comparable to non-temporal controls.Source: Figure 2 · p. 5, §4.2.1 · pp. 6–7

4.2.2 UNCERTAINTY VARIES PRIMARILY ALONG WEEKDAY SPOKES

  • With greater stated uncertainty, activations move inward along candidate weekday spokes rather than through the intervening arc, then move outward along the favoured spoke as confidence rises.At an even split, mean radius is 21–27% smaller than for fully committed prompts, while angular offset from the nearest spoke is unchanged.Source: Figure 3 · p. 7, §4.2.2 · p. 8

4.3 NATURAL TEXT DOES NOT SYSTEMATICALLY OCCUPY INTERPOLATED REGIONS

  • After mean-centering matched natural-text variants, weekday activations form seven weekday-aligned clusters, with within-weekday scatter falling to 0.23 ring radii.Preceding context primarily translates the weekday representations together while preserving their relative arrangement and orientation.Source: Figure 4 · p. 8, §4.3 · p. 9

5 DISCUSSION

The discussion argues that the weekday ring’s smooth geometry does not establish meaningful continuity between weekdays. It qualifies prior evidence, notes limits on interpreting the sensitivity pattern, and confines the study’s conclusions to the weekday manifold.

5.1 RELATION TO PRIOR EVIDENCE FOR CONTINUITY

  • Smooth steering appears only at late layers near the logits; temporal-modifier shifts are weekday-small, dusty-like, and do not causally affect time-of-day representations.These findings qualify earlier interpretations of smooth steering and temporal-modifier shifts as evidence of continuity.Source: §5.1 · p. 9

5.2 SCOPE AND LIMITATIONS

  • The authors do not reject circular weekday geometry, but note that higher sensitivity between weekday centroids could depend on their adopted periodic-spline construction.They use the construction from Wurgaft et al. (2026), rather than introducing a new interpolation procedure.Source: §5.2 · p. 9
  • The sevenfold sensitivity pattern is consistent with discrete states but not conclusive, and the study leaves continuously varying concepts such as colour for future work.Weekday is one particularly well-studied manifold, and the authors say further work must clarify what causes activation plateaus and whether low sensitivity reliably marks in-distribution regions.Source: §5.2 · pp. 9–10

5.3 CONCLUDING REMARKS

  • The analyses favor seven discrete weekday spokes over meaningful intermediate states, despite the representations’ circular arrangement.Smooth fitted geometry does not necessarily imply continuity in the underlying representation.Source: §5.3 · p. 10

AI USE STATEMENT

The authors disclose their use of generative AI for implementation and manuscript-related tasks, describe ethical considerations, and provide model, method, and code information for reproducibility.

AI USE STATEMENT

  • Generative AI assisted with analysis and plotting code, figures, manuscript editing, references, and literature search; authors tested code and checked figures and references.They report no AI use for hypothesis development, experiment design, or result interpretation and take responsibility for the final work.Source: AI USE STATEMENT · p. 10

ETHICS STATEMENT

  • The study uses open-weight models and public text without human subjects or private data; the authors note dual-use potential and possible bias in FineWeb.They reproduce four text excerpts and welcome scrutiny of unforeseen implications.Source: ETHICS STATEMENT · p. 10

REPRODUCIBILITY STATEMENT

  • The authors identify three open-weight models, specify experimental procedures in appendices, and provide code for reproducing experiments and figures.The models are Llama-3.1-8B, Mistral-7B-v0.1, and GPT-2 XL.Source: REPRODUCIBILITY STATEMENT · p. 10

A METHODS

The methods construct a weekday spline manifold, steer activations along it, and measure behavioral and representational sensitivity. Additional prompt designs test whether time of day and uncertainty correspond to positions between weekdays.

A.1 SETUP

  • The study uses Llama-3.1-8B, Mistral-7B-v0.1, and GPT-2 XL, steering at layer 2 and reading near the output.Readout layers are 28 for Llama and Mistral and 42 for GPT-2 XL.Source: §A.1 · p. 12

A.2 PROMPTS

  • Prompt families include 140 day-neutral weekday sentences, 49 arithmetic queries, clock-time and modifier prompts, uncertainty phrasing variants, and FineWeb excerpts.Time-of-day prompts vary clock hours and include controls; uncertainty prompts vary phrasing and weekday pairs.Source: §A.2 · pp. 12–13

A.3 THE ACTIVATION MANIFOLD

  • The activation manifold is a periodic cubic spline through seven weekday centroids, parameterized by an intrinsic coordinate with knots at the weekday representations.Centroids are per-day means, and the spline interpolates them exactly.Source: §A.3 · p. 13

A.4 ACTIVATION STEERING

  • Steering replaces the recorded activation with its original value plus the spline target minus the centroid for the prompt’s stated day.Polar targets extend this construction from the ring to its centre and beyond; Figure 5 schematizes the construction and sensitivity analysis.Source: §A.4 · p. 13, Figure 5 · p. 13

A.5 BEHAVIOURAL READOUT

  • Behavioral readout compares next-token distributions over weekday tokens and an “other” class in Hellinger space, with path lengths formed from successive distances.The distributions are mapped into square-root coordinates, where Hellinger distance is Euclidean.Source: §A.5 · p. 13

A.6 SENSITIVITY MEASUREMENTS

  • Sensitivity is measured by residual change per input step and by Jacobian norms across a polar grid, with tangential, radial, and orthogonal directional gains also computed.The Jacobian norm is direction-free; finite-difference measurements use the full-dimensional readout residual and patched input step.Source: §A.6 · p. 14

A.7 TIME OF DAY

  • Time-of-day analysis projects weekday-token activations into a PCA-defined weekday plane and measures angular shifts, with separate steering tests of predicted-hour changes.The plane spans components predominantly capturing weekday identity; steering compares along-ring, off-ring, and random directions across layers.Source: §A.7 · pp. 14–15

A.8 UNCERTAINTY BETWEEN TWO WEEKDAYS

  • Uncertainty prompts vary confidence between weekday pairs; activations are represented by angular position and normalized radius, then compared by a relative coordinate between the named days.The relative coordinate assigns 0 to day A’s spoke, 1 to day B’s, and 0.5 to the intervening arc’s midpoint.Source: §A.8 · p. 15

A.9 NATURAL TEXT IN THE WEEKDAY SUBSPACE

The natural-text analysis tests whether apparent positions between weekdays reflect movement around the ring or context-dependent translation. It samples weekday-ending excerpts and compares matched weekday variants within the same contexts.

A.9 NATURAL TEXT IN THE WEEKDAY SUBSPACE

  • The analysis asks whether natural-text activations between weekdays reflect movement around the ring or context-dependent translation.Weekday-ending excerpts are compared with excerpts ending in other capitalized words as controls.Source: §A.9 · p. 15
  • After deduplication, the sample contains 49,678 weekday excerpts from 574,217 documents, split into 16,879 fitting and 32,799 held-out examples.The model reads the final 16 tokens of each excerpt, preceded by a beginning-of-sequence token.Source: §A.9 · p. 16
  • Matched-context variants replace each excerpt’s final token with each weekday, then subtract the seven-variant mean to remove common translation without rotating the representations.The held-out clean weekday contexts are used for Figure 4B–C.Source: §A.9 · p. 16
  • Context-specific weekday arrangements are compared with a reference configuration defined by the mean-centred weekday arrangement from the fitting subset.Four examples in panel B were selected for large translations within predefined context categories.Source: §A.9 · p. 16

B REPRODUCING SENSITIVITY RESULTS WITH OTHER MODELS

Across three models, Jacobian sensitivity along the weekday-ring tangent varies most in Llama and Mistral, while GPT-2 XL shows a smaller tangent contrast and near-unity results in other directions.

B REPRODUCING SENSITIVITY RESULTS WITH OTHER MODELS

  • Tangent sensitivity varies 3.71× in Llama, 6.68× in Mistral, and 1.34× in GPT-2 XL.Measurements use day-neutral, third-person templates.Source: B REPRODUCING SENSITIVITY RESULTS WITH… · p. 17
  • In Llama and Mistral, the sevenfold pattern also appears in norm, radius, and orthogonal-direction Jacobians, with ratios of 1.12×/1.30×, 1.14×/1.39×, and 1.09×/1.25×, respectively.GPT-2 XL's corresponding ratios are 1.01×, 0.98×, and 1.00×.Source: B REPRODUCING SENSITIVITY RESULTS WITH… · p. 17
  • Figure 7 contrasts weekday-token and answer-slot steering, behavior gain along the ring, and layer-28 readout Jacobians on arithmetic prompts.The Jacobian panels separate overall norm from tangent, radial, and orthogonal-direction gains.Source: Figure 7 · p. 18

C NON-UNIFORM SENSITIVITY WITH WEEKDAY ARITHMETIC

On arithmetic prompts, steering effects depend on both the intervention site and ring position: weekday-token behavior changes stepwise, whereas answer-slot behavior changes smoothly.

C NON-UNIFORM SENSITIVITY WITH WEEKDAY ARITHMETIC

  • Weekday-token steering succeeds at roughly 0.7 through layer 14 and falls to zero by layer 22, while answer-slot steering stays at zero until layer 18 and reaches 0.83 by layer 31.Answer-slot day recovery is 0.122 at layer 2, below the 1/7 chance level, and does not exceed 0.80 until layer 18.Source: C NON-UNIFORM SENSITIVITY WITH WEEKDAY… · pp. 18–19
  • At the weekday token, behavior distance advances in steps across the ring, but the same sweep at the answer slot is smooth.The weekday-token behavior gain has minima at the seven weekday centroids; the answer-slot profile is much flatter.Source: C NON-UNIFORM SENSITIVITY WITH WEEKDAY… · p. 19
  • The weekday-token Jacobian profile has ratios of 1.22× for Frobenius norm, 3.56× along the ring tangent, 1.18× radially, and 1.15× orthogonal to the ring span.The comparison is at the layer-28 readout.Source: C NON-UNIFORM SENSITIVITY WITH WEEKDAY… · p. 19
  • Masking weekday-token logits reduces overall output-distribution change but leaves the sevenfold pattern, which is strongest among non-weekday tokens.Individual weekday logits peak near their own day rather than showing sevenfold periodicity.Source: Figure 8 · p. 20

D WEEKDAY-RELATED LOGITS DO NOT EXPLAIN THE SEVENFOLD

The sevenfold sensitivity pattern persists without weekday logits, and graded uncertainty across five prompt families produces candidate-aligned clusters rather than a continuum between weekdays.

D WEEKDAY-RELATED LOGITS DO NOT EXPLAIN THE SEVENFOLD

  • Masking the 14 weekday logits reduces the overall signal but leaves sevenfold modulation; weekday logits peak near their own day, while periodicity is strongest in non-weekday tokens.The pattern also appears across the vocabulary and randomly sampled token subsets.Source: D WEEKDAY-RELATED LOGITS DO NOT… · p. 20

E UNCERTAINTY BETWEEN WEEKDAYS ACROSS PROMPT FAMILIES

  • Across five uncertainty prompt families, pairs two or three days apart form clusters aligned with the candidate weekday directions rather than a continuum through the arc.As stated uncertainty increases, clusters move inward; numerical wordings sit closer to the ring centre than lexical wordings while remaining spoke-aligned.Source: E UNCERTAINTY BETWEEN WEEKDAYS ACROSS… · p. 22
  • For adjacent weekday pairs, the two spokes are too close for the clusters to separate cleanly.The tested families include either/or wording, certainty and frequency words, stated odds, and stated percentages, with both day orders and pairs one to three days apart.Source: E UNCERTAINTY BETWEEN WEEKDAYS ACROSS… · p. 22

F THE WEEKDAY RING ACROSS READ POSITIONS AND LAYERS WITH NATURAL TEXT

In FineWeb excerpts, weekday identity is decodable at the weekday token but largely disappears at later read positions, where projected weekday clusters contract toward the ring centre.

F THE WEEKDAY RING ACROSS READ POSITIONS AND LAYERS WITH NATURAL TEXT

  • Figure 10 tracks decoding across layers and read positions and shows projected activations both on a shared scale and zoomed to each position's ring.Its panels compare position-specific and transferred weekday subspaces, plus projected layer-28 activations.Source: Figure 10 · p. 23
  • At the weekday token, day identity has near-perfect held-out decoding accuracy from the earliest layers; at later positions, decodability drops sharply and stays near chance except for sporadic layers.This holds whether the weekday subspace is fitted at that position or transferred from the weekday token.Source: F THE WEEKDAY RING ACROSS READ… · p. 23, Figure 10 · p. 23
  • At later read positions, layer-28 activations projected into the weekday-token ring plane collapse toward the centre, with ring radius shrinking to roughly one tenth of its weekday-token value.Figure 10 gives the held-out rate of projecting closest to the activation's own weekday centroid beneath each panel.Source: F THE WEEKDAY RING ACROSS READ… · p. 23, Figure 10 · p. 23

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