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Tracking States or Tracking Cosets? An Algebraic Account of Learned State Tracking

Zhiyu Zhang, Yupeng Li

arXiv:2609.29951v1cs.LGcs.AI

TL;DR

State-tracking accuracy alone does not reveal whether a model recovers full states or only classes of states. The paper analyzes group-product tracking through quotient and coset structure, finding partial Transformer solutions and recurrent representations that include non-normal cosets. It also proves an asymptotic accuracy boundary for order-blind predictors under uniform i.i.d. full-group inputs.

  • Problem

    Predictive performance does not by itself show whether a model faithfully recovers the underlying state space.

    Source: §1 · p. 1

  • Method

    The paper characterizes state information through group quotients and within-class uncertainty, then analyzes recurrent coset stages and interventions on their state representations.

    Source: §1 · pp. 1–2

  • Results

    Transformers recover quotient classes while leaving members unresolved, recurrent networks exhibit non-normal coset stages, and A5 cosets are encoded in state subspaces whose components transfer coset predictions when swapped.

    Source: §1 · pp. 1–2

  • Takeaways & Limitations

    Subgroup cosets connect partial accuracy, recurrent learning stages, and internal computation in learned state tracking.

    Source: §8 · p. 9

    How input distributions shape learned representations and gradients, and how these effects govern coset selection and refinement, remains open.

    Source: §8 · p. 9

Abstract

from arXiv · show

State tracking requires composing a sequence of updates, but accuracy alone does not reveal what a model has learned. We study neural networks trained to predict the running product of group elements. We identify quotient solutions in Transformers, where models recover the quotient class while predicting nearly uniformly among its members. The reciprocal of class size predicts partial accuracy without a fitted parameter, extending parity-based accounts to non-parity quotients. Our baseline Transformers' predictions change little under prefix reordering beyond the exact-tracking frontier. We prove that, for finite groups under uniform i.i.d. full-group inputs, optimal order-blind exact accuracy converges to the reciprocal of abelianization class size as prefix length grows, consistent with the observed abelianization plateaus. Sequential updates permit more: any partition into right cosets of a subgroup, normal or not, survives sequential updates. In our census of standard Transformers, every recovered coset partition comes from a normal subgroup, whereas parameter-matched recurrent networks pass through both normal and non-normal right-coset stages during training. On $A_5$, we identify low-dimensional subspaces of the recurrent state that encode non-normal cosets. In the three-dimensional cases, coset mean vectors form approximate dodecahedra, and swapping the state components in these subspaces transfers the donor's coset state through a shared input suffix. Our results connect partial accuracy, learning stages, and internal computation through the subgroup cosets that models learn to track.

1 INTRODUCTION

The paper studies what neural networks learn when tracking running products, arguing that predictive accuracy alone does not establish faithful state recovery. It connects partial tracking to quotient classes and recurrent learning to subgroup cosets, including non-normal cosets.

1 INTRODUCTION

  • The paper characterizes learned state tracking through the states models can distinguish, using group quotients and within-class uncertainty.It also examines how subgroup cosets describe recurrent learning stages.Source: §1 · p. 1
  • In Transformers, reciprocal class size predicts partial accuracy when a model identifies the quotient class but predicts nearly uniformly among its members.The paper also proves which product information is recoverable without input order and the corresponding asymptotic accuracy limit.Source: §1 · p. 1
  • Sequential updates preserve right-coset partitions for any subgroup, including non-normal subgroups that do not define quotient groups.The paper finds non-normal coset stages in recurrent networks, including on A5.Source: §1 · p. 1
  • A few recurrent-state directions encode A5 cosets, and swapping components in those subspaces transfers a donor coset state through a shared suffix.A5 has no nontrivial proper quotient group.Source: §1 · p. 2
  • The paper links partial accuracy, learning stages, and internal computation through the subgroup cosets models learn to track.It identifies coset-mean geometry and tests the role of corresponding subspaces through state interventions.Source: §1 · p. 2

2 BACKGROUND AND PRELIMINARIES

The paper formulates state tracking as predicting successive running products in a group. It distinguishes full states, right-coset summaries that support sequential updates, and order-blind summaries constrained by abelianization.

2 BACKGROUND AND PRELIMINARIES

  • The task predicts each running product qt = x1 ··· xt, starting from the identity and updating by qt = qt−1xt.The inputs are invertible transformations with associative composition.Source: §2 · p. 2
  • A right coset Hq groups |H| states into one summary, and tracking that coset leaves its member states indistinguishable.When H is normal, the cosets form the quotient group G/N; full tracking corresponds to H = {e}.Source: §2 · p. 2
  • Right cosets support sequential updates because (Hq)x = H(qx), whereas multiplying coset representatives defines a well-defined class operation only for normal subgroups.More generally, summaries that combine to recover a product summary have a normal subgroup as the states sharing the identity’s summary.Source: §2 · pp. 2–3
  • The abelianization is the quotient by the commutator subgroup, whose size gives the abelianization class size when finite.The paper calls a model order-blind when its output depends only on input counts, not their order.Source: §2 · p. 3

3 QUOTIENT TRACKING IN TRANSFORMERS

The experiments examine Transformer quotient tracking across group tasks and test whether class size explains partial exact accuracy. They find unresolved class members, quotient tracking beyond abelianization, and weak averaged learning signals for separating members of a learned class.

3.1 EXPERIMENTAL SETUP

  • The experiments train models on length-100 sequences with cross-entropy loss at every position, mainly using independent uniform inputs from the full finite group.The Transformer suite includes two architectures, with three baseline seeds per group.Source: §3.1 · p. 3

3.2 THE CLASS SIZE PREDICTS PARTIAL ACCURACY

  • Across examined groups, Transformer exact-accuracy plateaus often match the reciprocal abelianization class size, and a nine-group census groups indistinguishable states by abelianization class.The class-size rule predicts Aexact ≈ Aquotient/f without a fitted parameter; A4 and SL(2, 3) both have abelianization C3 but class sizes four and eight, with conditional accuracies near 1/4 and 1/8.Source: §3.2 · p. 3

3.3 ACCURATE CLASSES, UNRESOLVED MEMBERS

  • On A4, quotient accuracy is 99.96% while exact accuracy is 24.98%, and mean within-class KL from uniform is .000232 bits.The same-budget C8 control reaches 99.26% full-state accuracy; probes recover quotient identity but remain near chance for a state’s index within its class.Source: §3.3 · p. 4, Figure 2 · p. 4

3.4 BEYOND THE ABELIANIZATION

  • A recipe B Transformer tracks S4/V4 ∼= S3 with quotient accuracy near 100% through position 20 and 72% at position 100, while conditional exact accuracy averages 25.01%.Within-class mean KL from uniform is .00272 bits; intermediate quotient behavior appears in five of 21 runs across seven groups.Source: §3.4 · pp. 4–5

3.5 LEARNING SIGNALS WITHIN QUOTIENT CLASSES

  • For the A4 model, the averaged class-member gradient signal falls to 1.63% of its position-5 peak by position 40, while the cancellation ratio at position 40 is 0.8–3.3% across nine Transformers.The tests do not resolve whether cancellation causes persistent partial learning; gradient perturbations and a later-training-aligned intervention do not advance the frontier.Source: §3.5 · p. 5

4 THE ALGEBRAIC BOUNDARY WITHOUT INPUT ORDER

The paper establishes which product information can be recovered from inputs without their order and bounds the best exact accuracy of order-blind predictors. For finite groups with uniform i.i.d. full-group inputs, that accuracy converges to the reciprocal abelianization class size.

4 THE ALGEBRAIC BOUNDARY WITHOUT INPUT ORDER

  • For t ≥ 3, an order-blind model can output ϕ(x1 ··· xt) on every sequence exactly when ϕ is constant on commutator-subgroup cosets.Prefix shuffling changes baseline Transformer output distributions little beyond the exact frontier, but this does not prove strict order-blindness.Source: §4 · p. 5
  • The maximum exact accuracy of an order-blind model is at least 1/f, with equality precisely when the conditional product distribution is uniform on its commutator-subgroup coset for almost every multiset.This applies to any distribution of length-t words over a finite alphabet in G.Source: §4 · p. 5
  • For finite G under uniform i.i.d. inputs from G, optimal order-blind accuracy is non-increasing and converges to 1/f, where f = |[G, G]|.Thus 1/f is an asymptotic limit, not only a uniform-guessing baseline.Source: §4 · p. 6
  • At every finite t ≥ 1, finite non-abelian groups under uniform independent inputs have order-blind accuracy strictly above 1/f.For ten non-abelian groups through order 24, estimated excesses at t = 17 range from approximately 1.5 × 10−4 to 3.1 × 10−3; these are numerical estimates, not rigorous bounds.Source: §4 · p. 6
  • The order-blind boundary does not establish that the observed models are strictly order-blind; non-abelian quotients remain order-sensitive, and restricted alphabets need separate analysis.Source: §4 · p. 6

5 PERSISTENCE AND REFINEMENT OF STATE RESOLUTION

Longer training can extend exact prefixes without refining a model’s quotient resolution; recurrent networks also pass through normal and non-normal coset stages. Standard Transformers in the census showed no non-normal coset partitions, an observed contrast rather than an architectural prohibition.

5.1 PERSISTENT QUOTIENT SOLUTIONS

  • At frontier 9, the five-million-update S4 run reached 99.67% abelianization accuracy but only 8.28% conditional exact accuracy, near 1/12; the finer S3 quotient stayed below 34.1%.Further long runs and GPT-2/Llama controls retained class-size baselines beyond their exact prefixes.Source: §5.1 · p. 6
  • Parameter-matched Mamba-2 models on D4, A4, and S4 often recovered abelianization while conditional exact accuracy remained near the class-size baseline.A four-layer D4 configuration improved quotient accuracy without improving within-class resolution.Source: §5.1 · p. 6
  • Matching feature-acquisition scores in an S3 × C3 control did not equalize resolution: models recovered C3 accurately but retained only parity information about S3.The result suggests the score alone cannot determine learned resolution.Source: §5.1 · p. 6

5.2 FROM QUOTIENTS TO COSET RESOLUTIONS

  • On S5, LSTMs and GRUs passed through sustained normal and non-normal coset stages; on A5, all six successful recurrent runs passed through C3 cosets with partial accuracy near 1/3.All six A5 runs reached near-exact tracking, though one LSTM later lost it; the C3 stage leaves three possible states per class.Source: §5.2 · pp. 6–7
  • Incorrect predictions were concentrated in the true coset, while reverse-order error fractions matched the conjugation-based prediction.Figure 3 compares both error fractions with the algebraic prediction.Source: §5.2 · p. 7, Figure 3 · p. 7
  • In the Transformer census, no standard run recovered a non-normal coset partition: all five four-layer S5 runs ended at the A5 stage, though recirculation produced one non-normal S4 stage.The authors describe this as an observed contrast, not an architectural prohibition.Source: §5.2 · p. 7

6 INTERNAL REPRESENTATION OF COSET STATES

Recurrent coset states on A5 are carried by low-dimensional subspaces whose class means have an approximately dodecahedral geometry in three-dimensional cases. Swapping these components transfers the donor’s coset through subsequent inputs, and fitted rotations approximate the class updates.

6.1 LOCATING AND TRANSFERRING THE COSET STATE

  • Swapping the leading coset subspace nearly reproduced full-layer transfer: donor agreement was 90.0–99.9% over the next 50 positions, within 2.5 points of full replacement.The subspace had three dimensions in four models and seven in two LSTMs; swapping remaining directions preserved recipient cosets on 89.1–99.9% of predictions.Source: §6.1 · p. 8, Figure 4 · p. 8

6.2 THE GEOMETRY AND UPDATES OF THE COSET STATE

  • In four models, the 20 centered coset means in three leading SVD directions lay almost exactly at regular dodecahedron vertices.The authors compared normalized pairwise inner products with those of a regular dodecahedron.Source: §6.2 · p. 8
  • For three-dimensional cases, fitted rotations closely matched input-induced permutations of the 20 class means, with RxRy ≈ Rxy.Figure 4b shows the 72° rotation for one 5-cycle.Source: §6.2 · p. 8, Figure 4 · p. 8
  • The measured geometry approximates a minimal three-dimensional A5 rotation code: C3 fixes an axis, and its A5 orbit forms the dodecahedron’s 20 vertices.Three is the smallest dimension of a nontrivial real linear action of A5.Source: §6.2 · p. 9
  • Together, interventions and geometry identify a G-set representation, not a quotient group; its class means approximately implement Hq → Hqx as a linear action.The seven-dimensional LSTM subspaces combined three- and four-dimensional components, while individual activations varied within classes rather than rotating rigidly.Source: §6.2 · p. 9

7 RELATED WORK

The related work situates this paper among accounts of shortcuts, group representations, coset computation, feature acquisition, and staged learning. The paper’s stated connections center on learned state resolution, partial accuracy, and whether coset components transfer predictions.

Shortcuts and learned information

  • Prior work on parallel shortcuts, associative and parity-assisted scans, and group representations provides complementary accounts that this paper connects through the state distinctions models recover.The paper relates quotient class size to partial accuracy and analyzes sequential coset stages.Source: §7 · p. 9

Cosets and internal computation

  • Prior coset-circuit work includes links to irreducible representations and stabilizers; this paper connects code resolution to tracking accuracy and tests component exchange through a shared suffix.The related work notes that stabilizer accounts include non-normal subgroups.Source: §7 · p. 9

Features and state resolution

  • Fourier, low-rank, and representation-acquisition analyses address learned features; this paper measures predictive resolution and uncertainty within classes, including a matched-score control.The cited representation-acquisition work studies two-layer networks under alternating gradient flow.Source: §7 · p. 9

Learning stages and recurrence

  • Work on staged learning and recurrence provides context for this paper’s analysis of acquired state distinctions, gradient cancellation, and recurrent comparisons.The related work also distinguishes plateau uncertainty from duration and cites composition tests and added recurrence.Source: §7 · p. 9

8 CONCLUSION

The paper’s conclusion frames learned tracking through subgroup cosets, linking partial accuracy and learning stages to internal representations. Its appendices classify exact state-dependent summaries and establish an order-blind accuracy boundary, with a finite-length bias bound.

8 CONCLUSION

  • The paper connects class size and output uncertainty to partial accuracy, coset stages to further learning, and causally effective subspaces to internal tracking, alongside an exact order-blind boundary.The conclusion describes the combined account as explaining what models preserve, leave unresolved, and represent internally.Source: §8 · p. 9
  • The authors leave open how input distributions shape internal representations and the gradients that refine them, noting that training recipes can determine whether models reach exact tracking.The experiments use synthetic group-composition sequences and do not establish performance or safety properties of deployed language models.Source: §8 · p. 9, ETHICS STATEMENT · p. 10

A ALGEBRAIC STATEMENTS AND PROOFS

  • Any exact sequential summary of group products induces right cosets of a subgroup, and every right-coset partition supports a sequential update, whether or not the subgroup is normal.These statements classify exact state-dependent summaries, not arbitrary history-dependent hidden states, and observing a partition does not establish that a neural network implements either rule.Source: A ALGEBRAIC STATEMENTS AND PROOFS · p. 13

A.1 ASYMPTOTIC ORDER-BLIND ACCURACY

  • For finite groups with uniform i.i.d. full-group inputs, the order-blind conditional product distribution approaches uniformity on its abelianization class as length grows.The optimal accuracy is bounded below by the reciprocal of class size; the appendix relates its convergence to a mean squared within-class bias.Source: A ALGEBRAIC STATEMENTS AND PROOFS · p. 13, §A.1 · p. 14, §A.1.1 · p. 14

A.1.1 AN EXPLICIT FINITE-LENGTH BOUND

  • An explicit expression for Φ_t gives computable bounds on optimal accuracy C_t, and the appendix also proves that C_t cannot increase with sequence length.The finite-length result is a bound through the squared bias, not an exact expression for C_t.Source: §A.1.1 · pp. 14, 16

A.1.2 AN ELEMENTARY PROOF OF CONVERGENCE

An elementary block argument shows that conditioning on the input multiset makes the product approach a uniform distribution on its compatible abelianization class. This yields an asymptotic accuracy limit for order-blind prediction under full-support i.i.d. inputs.

A.1.2 AN ELEMENTARY PROOF OF CONVERGENCE

  • A fixed input multiset has reorderings whose products cover exactly the commutator subgroup, so its conditional product distribution has positive probability on every element of that subgroup.The proof builds the multiset by concatenating commutator words and reorderings with the same multiset that yield the identity.Source: §A.1.2 · p. 16
  • A special block contracts total-variation error toward the uniform law on its coset, and later blocks preserve that uniformity.The contraction factor is 1 − ε; when ε = 1, the block makes the error zero immediately.Source: §A.1.2 · pp. 16–17
  • Conditioning on complete-block multisets preserves block independence, allowing the proof to average the contraction over the number of special blocks and then over total input multisets.The expected bound decays geometrically with the number of complete blocks; the argument does not assume independence conditional on the total multiset.Source: §A.1.2 · p. 17
  • As prefixes grow, the conditional law of the true product becomes uniform on average within its abelianization class, limiting order-blind exact accuracy to 1/f when quotient accuracy tends to one.The explicit bound assumes finite groups and uniform full-group inputs; the proof also covers fixed full-support i.i.d. laws, with a law-dependent convergence rate.Source: §A.1.2 · p. 18
  • The convergence result does not hold uniformly for every multiset and does not apply unconditionally to restricted alphabets or infinite groups.For example, the all-identity multiset determines the product at every length; the elementary proof extends to fixed full-support i.i.d. laws, with a rate depending on the law.Source: §A.1.2 · pp. 16–18

B.1 TRAINING COHORTS AND STAGE IDENTIFICATION

The census finds non-normal coset stages in recurrent networks, while the standard Transformers examined recover only normal-subgroup stages. In A5 recurrent states, low-dimensional components encode cosets, and interventions show that replacing these components can transfer coset state through a shared suffix.

B.1 TRAINING COHORTS AND STAGE IDENTIFICATION

  • In the reported recurrent arms, A5 and S5 models pass through non-normal coset stages, while all five four-layer S5 Transformers end at an A5 stage and no non-normal stage appears in 7,430 evaluations.At the higher GRU learning rate, the recurrent models do not show the same progression, so the observed contrast does not imply every recurrent model leaves a quotient stage.Source: §B.1 · p. 19

B.2 COSET TRACKING WITH RECIRCULATION

  • In recirculated S4 Transformers, a non-normal D4 stage repeats under the same initialization seed but not across the other completed seeds; one-round runs also reach A4 or V4 stages.Removing feedback from the interrupted run drops accuracy near the 1/24 chance level, while adding it only at inference does not reproduce the coset structure; the comparison also changes computation and effective depth.Source: §B.2 · p. 21

B.3 LOCATING THE RECURRENT COSET STATE

  • Across the six A5 recurrent models, the coset-mean variation lies in three directions in four models and seven directions in two LSTMs.Replacing only the leading components gives donor agreement .8997–.9990, close to full-layer replacement; the analysis covers 18 evaluations of six models, not 18 independent runs.Source: §B.3 · pp. 21–22, §B.5 · p. 24

B.4 GEOMETRY AND ITS SCOPE

  • In three-dimensional cases, the class means approximately realize a dodecahedral orbit; in two seven-dimensional LSTMs, a four-dimensional component approximately groups cosets by one object’s location.The three- and four-dimensional components together are needed for measured transfer in those LSTMs, and the fitted group action describes class means rather than rigid rotations of individual activations.Source: §B.4 · pp. 23–24

B.5 STATE INTERVENTIONS AND CONTROLS

  • Replacing the leading state components transfers donor coset predictions through a shared suffix, while complementary replacement preserves recipient agreement.For the seven-dimensional LSTMs, replacing either component alone gives low donor agreement (.162–.206), whereas replacing both gives .994 and .997; separate controls in two models show Gaussian or shuffled states do not transfer donor cosets.Source: §B.5 · p. 24

C.1 THE A4 OUTPUT EXAMPLE

The A4 example characterizes predictions within quotient classes, while the accompanying sections describe Heisenberg evaluations and matched-model diagnostics. The reported probes are limited to linear decoding of specified hidden states.

C.1 THE A4 OUTPUT EXAMPLE

  • Within the true A4 coset, candidate probabilities range from 0.249395 to 0.250588, while outside-coset probabilities range from 4.6831 × 10−5 to 9.2188 × 10−5.The mean probability outside the true coset is 0.0526916%; these are measured probabilities, not an idealized block matrix.Source: §C.1 · p. 25
  • At A4 position 40, the raw Net norm is 0.7272, or 1.63% of its position-5 peak of 44.6659.This value comes from the eight-orbit A4 measurement in panel (c).Source: §C.1 · p. 25
  • The specified rotary-embedding recipe gives identical outputs at every position of a constant prefix, forcing an error on (g, g) at position 2 when g ≠ e.Across A4 and C8, the resulting pair errors account for 639/8,192 and 912/8,192 examples, respectively; these are short-prefix errors, not evidence of later within-fiber uncertainty.Source: §C.1 · p. 25

C.2 FINITE AND INFINITE HEISENBERG CURVES

  • The Heisenberg curves compare finite H3(Z/3) and infinite G3 under different input alphabets, with both groups having abelianization classes of size three.The finite group uses full-group inputs; G3 uses the finite alphabet {−1, 0, 1}2 × Z/3.Source: §C.2 · pp. 25–26

C.3 MATCHED TRAINING, PROBES, AND GRADIENTS

  • The fixed linear probes decode fiber rank and quotient class from hidden states, but constrain only this linear decoding task and do not exclude nonlinear decoders.The protocol uses positions 17–100 from fresh sequences; the same nine frozen models also enter a gradient analysis described in Appendix Q.Source: §C.3 · p. 26

D PROTOCOLS AND ENDPOINT COVERAGE FOR FIGURE 3

The supplementary comparisons retain individual training runs and distinguish terminal outcomes from transient stages. They also note that endpoint selection, schedules, and readouts limit some cross-condition comparisons.

D PROTOCOLS AND ENDPOINT COVERAGE FOR FIGURE 3

  • The five-million-update S4 Transformer run ends with 99.67% quotient accuracy and 8.28% conditional exact accuracy, while its exact frontier never exceeds position 10.Its maximum S3 quotient accuracy across the log is 34.05%; Table 18 uses a separate terminal evaluation.Source: D PROTOCOLS AND ENDPOINT COVERAGE FOR… · p. 27
  • The shorter S4 Transformer runs end at 2,410,000 and 965,000 updates, and their comparison with LSTM does not isolate architecture because schedules and precision differ.Their last quotient accuracies are 99.97% and 100.00%, with conditional exact accuracies of 9.41% and 8.81%, respectively.Source: D PROTOCOLS AND ENDPOINT COVERAGE FOR… · p. 27
  • The S4 LSTM C2 stage spans updates 1,700–2,800, 1,800–2,250, and 1,800–2,450 for seeds 42, 43, and 44, respectively.The shared interval is 1,800–2,250; the original 1,000-update grid samples each stage only once.Source: Figure S1 · p. 27, D PROTOCOLS AND ENDPOINT COVERAGE FOR… · p. 28
  • The separate coset census identifies 15 of 18 recurrent runs with stages, which last 50–1,050 updates on a 50-update grid.The main 1,000-update grid places at most one evaluation inside any stage and none inside nine; transient readouts do not establish within-class softmax uniformity.Source: D PROTOCOLS AND ENDPOINT COVERAGE FOR… · p. 28
  • The displayed terminal conditions were selected after inspecting results, so the panel is not an exhaustive success-rate estimate.The plotted bars are observed seed ranges, not confidence intervals or temporal variability.Source: D PROTOCOLS AND ENDPOINT COVERAGE FOR… · p. 29, Table 7 · p. 29
  • The S4 recurrent-family comparison uses different quotient readouts: TC sums probabilities within each coset, whereas LSTM and GRU project the full-state argmax.The display ends at 2,500 updates for TC and 8,000 for LSTM/GRU, while full-state accuracy remains at least 99% at later logged evaluations in these nine runs.Source: Figure S2 · p. 28, D PROTOCOLS AND ENDPOINT COVERAGE FOR… · p. 29

E ADDITIONAL CURVES UNDER THEIR ORIGINAL EXPERIMENTAL SETUPS

The additional curves separate quotient and exact tracking, while the order-blind analysis computes a finite-length accuracy ceiling. The remaining sections examine central-coordinate tracking and document configuration-specific scope boundaries.

E ADDITIONAL CURVES UNDER THEIR ORIGINAL EXPERIMENTAL SETUPS

  • Across original setups, the quotient and exact channels separate during training; the reported curves also cover intermediate quotients, generator-input depth effects, and declines beyond the training horizon.These plots retain setup-specific protocols and are distinct from the matched final checkpoints; the run-level law plot includes incomplete quotient channels.Source: E ADDITIONAL CURVES UNDER THEIR… · p. 29, Figure S3 · p. 30, Figure S4 · p. 30

F COMPUTING THE ORDER-BLIND CEILING

  • The order-blind ceiling is computed by enumerating multiset orderings exactly where feasible, with an unbiased estimate beyond enumerated lengths; finite-length excess over 1/|[G, G]| is tabulated.For non-abelian groups considered, multisets whose orderings do not cover their coset have positive mass at every finite length.Source: F COMPUTING THE ORDER-BLIND CEILING · pp. 30–31, Table 8 · p. 31

G RECOVERING CENTRAL COORDINATES

  • For generator-restricted D4 inputs {r, s}, three parity counts determine the full state, including the number of r tokens at even input positions.The same tokens in rs and sr yield different states because r occupies different input-position parities.Source: G RECOVERING CENTRAL COORDINATES · p. 32

G.1 TRACKING BEYOND ODD/EVEN COUNTS

  • On H3(Z), models reach 98.07–98.17% joint state accuracy over positions 17–100 and distinguish controlled pairs that preserve odd/even counts while changing z in 96.9% of pairs.A linear readout recovers the running X count with R2 = .994–.999; perturbing its associated activation changes predicted z, and paired perturbation effects broadly follow additivity with maximum deviation .30 z units.Source: §G.1 · pp. 32–33

H CONFIGURATIONS

  • The configuration appendix marks the looped-transformer and expert-split pilots as exploratory or excluded from confirmatory comparisons, and identifies the Pythia-70M runs as exploratory.The looped pilot did not reach its promotion threshold, and the expert-split pilots are explicitly excluded from confirmatory comparisons.Source: H CONFIGURATIONS · pp. 34–35

I COVERAGE OF ARCHITECTURES AND TRAINING SUPPORTS

The appendix catalogs the architectures, training packages, and readout procedures used to assess quotient and coset tracking. It also documents Heisenberg-group results and cautions that finite windows, thresholds, and evaluation choices constrain interpretation.

I COVERAGE OF ARCHITECTURES AND TRAINING SUPPORTS

  • Across the cataloged packages, recurrent and Mamba-2 studies cover D4, A4, and S4, with 18 runs per architecture; the Mamba-2 study also includes three four-layer D4 runs.Pilot and exploratory results are excluded from confirmatory comparisons, and no Markov-input or power-law-input results are included.Source: I COVERAGE OF ARCHITECTURES AND… · pp. 35, 37

J RUNS BEHIND TABLE 1

  • Table 1 combines matching 148k runs for selected groups with original 74k runs for others, so it summarizes observed resolutions at stated budgets rather than common-budget frequencies.The choice was post hoc and without a seed-level accuracy filter; D4 has only two doubled-budget seeds.Source: J RUNS BEHIND TABLE 1 · pp. 38–39

J.1 FINITE AND INFINITE HEISENBERG GROUPS IN TABLE 1

  • For H3(Z/5), doubled-budget runs reach quotient accuracy .9987–.9997 and conditional exact accuracy .1995–.1996; the G5 window overlaps an exact-accuracy transition.The reported agreement with 1/p is empirical under these configurations, not implied by finite commutator size alone.Source: §J.1 · pp. 39–40

K READOUT DEFINITIONS AND WINDOW SENSITIVITY

  • Readouts separate probability mass on the true quotient class from probability distribution within that class, and compare exact and quotient accuracy using conditional exact accuracy.The within-class log-likelihood gain measures reduction in cross-entropy relative to uniform prediction; it is a prediction score, not mutual information.Source: K READOUT DEFINITIONS AND WINDOW… · pp. 40–41
  • The .75 exact-frontier threshold was selected after exploratory inspection, and neither package completion nor a detected stage establishes asymptotic convergence.Some fixed evaluation windows overlap exact-tracking frontiers, so their ratios do not measure a uniform plateau.Source: K READOUT DEFINITIONS AND WINDOW… · pp. 41–42
  • In the standard-budget battery, 28 of 28 runs have |∆LL| < .0145 bits beyond Fe + 6, but longer training can produce a wider transition.A 600k-update D4 run has conditional accuracies .75, .64, .54, and .50 at offsets 1, 11, 21, and 31.Source: K READOUT DEFINITIONS AND WINDOW… · p. 41

L ADDITIONAL EVIDENCE FOR QUOTIENT PREDICTIONS

Additional analyses test quotient predictions, order sensitivity, and intermediate quotient stages under two Transformer recipes. They support near-uniform within-class predictions and low sensitivity to reordering in specified settings, while emphasizing probe, window, and classification limits.

L ADDITIONAL EVIDENCE FOR QUOTIENT PREDICTIONS

  • Matched models decode quotient classes above 99.9% while mean layer-4 within-class-rank accuracies remain near chance in A4, SL(2, 3), and S4.These null results concern linear readouts only; a learned character can also be linearly unreadable when represented as a product of readable characters.Source: L ADDITIONAL EVIDENCE FOR QUOTIENT… · p. 43, Figure S7 · p. 43
  • Empirical flatness is finite-resolution: Dic3 shows a within-fiber improvement of approximately .0052 bits with a confidence interval excluding zero.This probabilistic information is not excluded by the theorem about exact recovery on every input.Source: L ADDITIONAL EVIDENCE FOR QUOTIENT… · p. 43

M.1 REORDERING UNDER RECIPE A

  • Across 28 recipe-A runs, mean output-distribution divergence after the exact frontier ranges from .0012 to .0163 bits, and two readouts reach low-information regions within two positions in 23 of 26 runs.The behavioral test does not establish strict order-blindness.Source: §M.1 · pp. 43–44, Figure S8 · p. 44

M.2 INTERMEDIATE QUOTIENT STAGES UNDER RECIPE B

  • Under recipe B, five runs show intermediate quotient behavior, while frozen endpoint criteria classify five runs as plateaus, fifteen as partial, and one as escape.These endpoint labels are not guarantees of subsequent training behavior.Source: §M.2 · p. 44
  • The S4 example tracks exactly through position 6, then has conditional accuracy approximately .25 within S3 classes while parity remains accurate throughout.A D15 stage gives .201 against the 1/5 baseline.Source: §M.2 · p. 44
  • The intermediate-stage analysis does not establish an exhaustive or preregistered classification of learned partitions, and recipe-A and recipe-B reordering results are not a controlled comparison.The .70 class threshold was selected during analysis, and transition-region filtering changes the number of apparently informative levels.Source: §M.2 · pp. 44–45

N ADDITIONAL TRAINING AND ARCHITECTURE COMPARISONS

Additional experiments compare learning stages across architectures and test whether recurrent states causally encode non-normal cosets. They find coset stages in recurrent models, but the preregistered causal audit qualified only one run, and matched state-space models often learned quotient classes without resolving their members.

N ADDITIONAL TRAINING AND ARCHITECTURE COMPARISONS

  • Across 18 D4 runs, stored conditional accuracies stayed between .4991 and .5017; greater depth mainly improved class accuracy and extended exact frontiers modestly.Under generator-input training, a separate sweep extended frontiers, but its original comparison also increased parameter count; horizon reductions improved median frontiers on A4 and SL(2, 3).Source: N ADDITIONAL TRAINING AND ARCHITECTURE… · p. 46
  • At the 10k-update classification point, Pythia models’ median exact cutoffs rose from 6.5 to 39 across 3–12 layers, while quotient-first outcomes occurred in 5–7 of eight seeds.The S5 width-256 study found no associative-like outcome; these observations are budget-specific, not claims about eventual convergence.Source: N ADDITIONAL TRAINING AND ARCHITECTURE… · p. 46
  • Fifteen matched recurrent runs met the abelianization-stage criterion, but these were transient stages, and the three A4 runs at learning rate 10−3 crossed quotient and exact thresholds in the same interval.The study covered three groups and one model size, so it does not support a general convergence claim for gated recurrence.Source: N ADDITIONAL TRAINING AND ARCHITECTURE… · p. 46
  • Transformers yielded 38 non-trivial commutator-block partitions and three singleton partitions across 41 checkpoints, but neither census procedure recovered a non-normal partition.The counts describe this checkpoint census, not every Transformer resolution.Source: N ADDITIONAL TRAINING AND ARCHITECTURE… · p. 47
  • Recurrent training included non-normal right-coset stages: S4 GRUs separated an S3/V4 quotient stage from a later order-two subgroup stage, and S5 LSTMs followed varied subgroup refinement paths.On S5, two runs first held the normal A5 stage, while another followed a point-stabiliser chain; six of eight identified stages were non-normal.Source: N ADDITIONAL TRAINING AND ARCHITECTURE… · p. 47
  • In one qualifying S5 LSTM run, replacing the recurrent state transferred the prediction to the donor coset in every cross-coset pair, and a rank-59 subspace was sufficient and necessary for coset prediction.The selector admitted one of five runs; the audited checkpoint also achieved .998 lower-bound continuation coset accuracy.Source: N ADDITIONAL TRAINING AND ARCHITECTURE… · p. 48
  • Only one of five preregistered runs qualified for the order-two causal audit, so the package found no across-seed replication.Across the five runs, post hoc scoring identified thirteen stages, nine of them non-normal; only one run held the preregistered order-two stage.Source: N ADDITIONAL TRAINING AND ARCHITECTURE… · p. 48

N.1 MATCHED STATE-SPACE MODELS

  • At learning rate 10−3, two of three Mamba-2 seeds per group exceeded .99 quotient accuracy while conditional exact accuracy stayed within .001 of 1/2, 1/4, and 1/12 on D4, A4, and S4.These finite-budget readouts do not establish per-example output flatness or a complete sequence of transient quotient stages.Source: §N.1 · p. 50

O SUPPORT INTERVENTIONS AND FAILED PREDICTIONS

The studies examine when restricted inputs permit position-weighted counting solutions, whether these solutions transfer across supports, and how Transformer activations represent counts in order-sensitive tracking. Probes and interventions assess these representations, though the interventions do not establish a pure count edit or isolate a complete update implementation.

O SUPPORT INTERVENTIONS AND FAILED PREDICTIONS

  • D4 commutator-coset supports had frontiers 44, 81, and 100 for two cosets and 7–11 for three; two of three D3 reflection-coset runs tracked fully.All three Q8 runs improved, with frontiers 17, 21, and 48, rejecting the preregistered prediction that the decomposition was required; other interventions produced chance performance or stalled frontiers.Source: O SUPPORT INTERVENTIONS AND FAILED… · pp. 50–51
  • For D4, the stated position-weighted counting form exists exactly when inputs occupy at most two commutator cosets; adding e to {r, s} violates the criterion.The criterion characterizes this displayed counting form, not every order-sensitive computation or what training will learn.Source: O SUPPORT INTERVENTIONS AND FAILED… · p. 52
  • Transfer across D4 supports was poor: generator-trained models reached chance exact and quotient accuracy on full-alphabet inputs, while five of six full-alphabet models lost the quotient on {r, s}.Generator-trained endpoints had exact accuracy .124–.125 and quotient accuracy .249–.250; in the reverse direction, joint accuracy was .29–.35 against positional chance of .50.Source: O SUPPORT INTERVENTIONS AND FAILED… · p. 52

P.1 TASK AND TRAINING

  • In H3(Z), Z accumulates each signed b-token weighted by preceding a-tokens, so words with identical net counts can have different products and accurate Z prediction requires order information.The experiments establish order-sensitive tracking only at tested lengths; a sufficiently large finite quotient could agree within the tested window.Source: §P.1 · p. 54

P.2 REORDERING TESTS

  • At position 100, models correctly predict both members of odd/even-count-preserving pairs in 96.875% of cases, and predicted Z displacement matches the true displacement in 96.940%.At length 100, long-range cancellation, central-commutator relocation, and rectangular rewriting did not meet the prespecified degradation criterion; after 32 common suffix tokens, central-displacement response slopes were .905–.952, and magnitude-16 detection reached 100%.Source: §P.2 · pp. 55–56

P.3 SINGLE-POSITION RAW-COUNT PROBES

  • At tested positions, first-block activations supported nearly exact readout of preceding a and a−1 counts and their difference; rounded individual-count readouts were 99.22–100%.This exploratory probe study used one frozen model, seed 42, and tested positions 32, 64, and 100.Source: §P.3 · p. 56, Table 23 · p. 56
  • At position 64, the hidden-state a-count readout outperforms the true-prefix-coordinate baseline, with R2 = .66939 versus .66939?The supplied passage reports the coordinate baseline as R2 = .66939 and 19.64% rounded accuracy, but does not provide the hidden-state readout's corresponding values in this excerpt.Source: §P.3 · p. 57

P.4 A COUNT READOUT SHARED ACROSS POSITIONS

  • Shared second-block probes achieve test R2 = .99351–.99880 across models and sampled positions 16–100, without position-specific refitting.Allowing position-specific rescaling and offsets yields factors .973–1.016, while shuffled-target probes reach at most R2 = .12844 across tested settings.Source: §P.4 · p. 57, Table 24 · p. 57
  • Position-specific second-block probes achieve Xt R2 = .999777–.999940 and Yt R2 = .999704–.999962, with rounded accuracy of 99.51–100% and 99.61–100%, respectively.These results establish linear readability on held-out words, not that the model uses the recovered counts.Source: §P.4 · p. 58

P.5 COUNT-DIRECTION INTERVENTIONS

  • Count-direction perturbations produce opposite mean responses at b and b−1, and the 18 two-position conditions have RMS additivity residual .125 and maximum |r| .299.Responses at a±1 are smaller but need not vanish; the measured effects support approximate additivity of mean responses, not exact sample-wise cancellation.Source: §P.5 · pp. 59–60

P.6 ACTIVATION PATCHING BETWEEN MATCHED INPUTS

  • Third-layer key–value replacements retain (x, y) predictions in 100% of cases, but isolated linear C transfers only a small fraction of the predicted effect.These replacements affect central-coordinate predictions but do not identify the count contribution as their causal feature.Source: §P.6 · pp. 61–62, Table 25 · p. 62

Q GRADIENT CONDITIONS, DIAGNOSTICS, AND INTERVENTIONS

This section examines gradient signals, low-dimensional class-mean representations, and interventions to diagnose how quotient readouts are formed and affected. It tests effects on current and later quotient predictions, while leaving sequential implementation unresolved.

Q GRADIENT CONDITIONS, DIAGNOSTICS, AND INTERVENTIONS

  • At position 40, within-orbit Label ratios were .00796–.03322, while pooled Label–Model balance was .634–.837, distinguishing cancellation across class members from cancellation between averaged terms.The late-position net signal was also only 0.8–3.5% of its peak across runs; at position 1, all runs had 100% accuracy but net norms remained 30–64% of peak. These are final-checkpoint measurements, not evidence of permanent training arrest or a vanishing total gradient.Source: Q GRADIENT CONDITIONS, DIAGNOSTICS,… · pp. 63–64, Table 26 · p. 64, Figure S10 · p. 65
  • Under uniform full-group inputs, invariance of both logits and their parameter Jacobian under a commutator action is sufficient for orbit averaging to cancel class-member Label signals; output uniformity alone is insufficient.Fourteen of eighteen directional-cosine measurements over fiber sizes 2–60 were within .0015 of the lower bound, and the S5 follow-up was -.01692 versus −1/59; the abelian positive control was at least .9969. These signatures do not prove the invariance assumptions.Source: Q GRADIENT CONDITIONS, DIAGNOSTICS,… · pp. 64–65
  • Across 20 eligible checkpoint–input pairs, late-to-reference average Label-gradient norm ratios were .0009–.0149, but a repaired seven-pair follow-up is not independent confirmation.The follow-up selected positions using a revised frontier-window protocol; three pairs failed its conditions, including an excluded D15 run that was still improving with ratio 1.14.Source: Q GRADIENT CONDITIONS, DIAGNOSTICS,… · pp. 65–66
  • Generic backward perturbation matched the identity control at frontier 13 versus a starting frontier of 14, while future-directed alignment improved local predictions but did not cross the frontier at the tested 30% dose.Last-block-only support recovered 22–29% of the alignment effect, below the 80% criterion; the upstream complement was three times more efficient per unit L2 perturbation. Future-checkpoint alignment uses future information and is a diagnostic upper bound, not a training method.Source: Q GRADIENT CONDITIONS, DIAGNOSTICS,… · p. 66

R INTERNAL GEOMETRY OF TRANSFORMER QUOTIENT PREDICTIONS

  • Across Transformer predictions, class-mean hidden vectors usually occupy a small subspace whose geometry agrees with the abelian quotient, and replacing that subspace transfers the predicted class at the intervention position.The section separately asks whether a local replacement also transfers the class to later predictions, distinguishing current readout from a state carried through subsequent updates.Source: R INTERNAL GEOMETRY OF TRANSFORMER… · p. 67

R.1 MODELS AND CONSTRUCTION OF CLASS MEANS

  • The analysis estimates centered class means from 40 frozen endpoints across eleven groups, retaining singular directions with σj/σ1 ≥.10 to test which dimensions and geometries encode quotient classes.Endpoints are four-layer, width-256 recipe-A Transformers; the C8 task is a full-state control, and class means are averages by ground-truth class rather than activation-fitted clusters.Source: §R.1 · p. 67

R.2 GEOMETRY FOLLOWS THE QUOTIENT ACTION

  • In 38 of 40 endpoints, retained rank and fitted-map trace and determinant patterns agreed with a faithful real representation of minimum dimension; none of 28 eligible endpoints used all K−1 centered class directions.The largest fitted-map error was .124. Two exceptions were retained: a low-quotient-accuracy Q8 endpoint with rank one and a doubled-budget H3(Z/5) endpoint with rank six rather than four.Source: §R.2 · pp. 67–68

R.4 DOES A LOCAL REPLACEMENT PROPAGATE TO LATER POSITIONS?

  • At positions 52, 56, and 71, median shifted-target agreement was .0090, .0006, and .0005, versus original-target agreement of .9678, .9960, and .9982, respectively.At position 71, shifted agreement ranged from zero to .1612, with the largest value in the low-accuracy Q8 endpoint.Source: §R.4 · p. 69
  • The intervention supports a causal contribution to the current quotient readout, but does not resolve its sequential implementation.Penultimate-block donor agreement at the intervention position ranged from .286 to 1.000, and replacing one position does not rule out a representation distributed across several positions.Source: §R.4 · p. 69

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