Nonequilibrium Phases of Repulsive Self-Attention: Chaos, Attention Condensation, and Emergent Locality

I. INTRODUCTION

• At fixed finite β, circle dynamics can progress from polygon instability to period-two motion, chaos, or clustered states while attention remains diffuse as N grows.The first instability is a flip bifurcation; at smaller β, clustered periodic motion can arise without an intervening chaotic regime.Source: §I · p. 3
• When β scales as N^2, attention localizes on finitely many microscopic neighbors and gap fluctuations drive sparse routing with regular, period-two, or chaotic dynamics.Repulsive updates alter local gaps and can rewire the dominant attention links.Source: §I · p. 3
• In the hard-routing limit, repulsive amplification and partner switches spread local perturbations ballistically, creating a butterfly cone in angular order.Locality emerges from representation geometry rather than input-sequence position.Source: §I · p. 3
• For d=N, Gaussian initial conditions provide evidence of condensation at β=O(1), driven by dynamically generated O(1) overlap gaps.The scaled logits turn these gaps into large attention-weight ratios, while negative feedback allows sparse routing to persistently reroute.Source: §I · p. 3
• High-dimensional diffuse states include simplex-like configurations and consensus flips, while condensed states include active sparse routing and fragmented cluster flips.Intercluster attention can vanish as N grows, allowing fragmentation to persist when the thermodynamic limit precedes the long-time limit.Source: §I · p. 3

II. MINIMAL SINGLE-HEAD SELF-ATTENTION DYNAMICS

• The model isolates recurrent attention feedback with one normalized head: token representations set a directed, state-dependent routing graph, and the same map updates them at each layer.β controls routing sharpness, while γ sets attention feedback strength; the paper focuses on Q=K=I and V=−I.Source: §II · p. 4, A. Single-head attention with general… · p. 4

B. Observables and diagnostics

• Attention condensation is measured by the inverse participation ratio YA, while the Gram matrix tracks pairwise overlaps and cluster geometry.YA=O(N^-1) indicates diffuse routing and YA=O(1) indicates routing concentrated on O(1) targets.Source: B. Observables and diagnostics · p. 5
• A sustained aperiodic trajectory with positive largest Lyapunov exponent is classified as chaotic.The leading transverse exponent removes the neutral common-rotation direction in the isotropic circle model.Source: B. Observables and diagnostics · p. 5

III. REPULSIVE SELF-ATTENTION ON THE CIRCLE

• For fixed finite β, the regular polygon’s first instability is a flip bifurcation, and its post-instability dynamics can include period-two motion, chaos, or clustered periodic states.At smaller β, the polygon can enter cluster-flip or cluster-permutation motion directly, without an intervening chaotic stage.Source: §III · p. 5
• At fixed finite β, bounded attention scores keep routing asymptotically diffuse, whereas β∼N^2 permits localization on finitely many microscopic neighbors and hard routing yields an emergent butterfly cone.The latter regimes arise when the attention-kernel width becomes comparable to neighboring-token spacing.Source: §III · p. 5

A. Model and exact angular map

• The exact angular map shows that negative value feedback reverses the tangential response when the radial component is positive and can produce an angular flip when that component changes sign.The dynamics is equivariant under a common rotation, leaving global orientation as a neutral mode.Source: A. Model and exact angular map · p. 6

B. Uniform polygon state

• The regular polygon is an exact fixed point whenever its pre-normalization radial factor is positive.The fixed-point property alone does not establish stability against nonuniform angular perturbations.Source: B. Uniform polygon state · p. 7

• The linearized message combines a geometric tangential projection with attention rerouting, whose contribution grows as β sharpens attention.The angular response also increases as the radial component rN becomes small.Source: C. Fourier stability and the… · p. 8
• The circulant linearized dynamics separates into independent Fourier modes; the common-rotation mode is neutral rather than a physical deformation.A mode’s wave number gives the number of angular oscillations around the circle.Source: C. Fourier stability and the… · p. 8
• At finite N, the first unstable wave number and its flip threshold follow from the mode-resolved spectrum and can depend on N.Source: C. Fourier stability and the… · p. 9
• In the continuum limit at fixed β, the critical mode reaches multiplier −1 and reverses sign each layer, producing a flip instability.Source: C. Fourier stability and the… · p. 9
• The nonuniform flip occurs while the uniform configuration remains on its fixed-point branch, distinct from a rigid global π flip.Source: C. Fourier stability and the… · p. 9
• Near an isolated first instability, the critical angular deformation reverses each layer, and nonlinear saturation can yield a period-two wave.The linear analysis identifies the critical mode but does not determine which nonlinear attractor is selected.Source: C. Fourier stability and the… · p. 9

D. Nonlinear dynamics beyond the uniform-state instability

• For sufficiently large finite β, a narrow stable period-two window follows the flip, then recurrence breaks down and chaos precedes a cluster-flip state; at smaller β, cluster orbits appear directly.The direct low-β outcomes include a stable two-cluster exchange orbit or a single-cluster flip.Source: D. Nonlinear dynamics beyond the… · p. 10
• Finite-N Gaussian initial conditions can yield clustered period-two states just above γ = 1/2, with cluster counts and populations depending on initialization.The exact equal-population four-cluster branch has a stability window that narrows as O(β^-1) at large β.Source: D. Nonlinear dynamics beyond the… · p. 10

1. A stable period-two orbit appears above the flip

• Above the flip, the five-wave amplitude saturates into a nonzero period-two orbit whose tested two-layer Floquet multipliers indicate local stability.The tested multipliers, excluding global rotation, have modulus below one.Source: §1 · pp. 11–12

2. Two-layer recurrence is lost farther from the flip threshold

• At δγ = 0.011, loss of two-layer recurrence together with a positive finite-time Lyapunov exponent supports chaos; positive growth alone at δγ = 0.007 is not enough to classify runs as chaotic.Source: §2 · p. 12

3. Evidence for extensive chaos

• At γ = 0.45, the number of finite-time Lyapunov exponents above tested thresholds grows linearly with N, supporting extensive chaos beyond the fivefold-symmetric sector.The scaling is measured in the full relative-angle space and is not an artifact of the fivefold-symmetry restriction.Source: §3 · pp. 12–13
• The measurements characterize irregular motion but do not resolve how it emerges from the locally stable period-two orbit; a parameter-resolved bifurcation analysis is beyond scope.Source: §3 · p. 13

4. Small-β regime

• At sufficiently large β, increasing γ takes the polygon from a period-two orbit to chaos and then to cluster-flip or cluster-permutation attractors.At small β, the chaotic stage is absent and the system enters cluster-flip or cluster-permutation dynamics directly after the polygon loses stability.Source: §4 · p. 14

E. Cluster-permutation attractors at large attention feedback strength

• At fixed β, clustered exchange states remain attention-diffuse as N grows because each macroscopic cluster shares its weight among O(N) targets.Their attention inverse participation ratio scales as YA = O(N^-1), distinguishing them from condensed cluster states.Source: E. Cluster-permutation attractors at… · p. 14
• The consensus state is fixed for γ < 1/2 and flips its common direction each layer for γ > 1/2, while its Gram and attention matrices stay unchanged.The exact thermodynamic stability threshold of the consensus flip is 2/3, but this is not a global phase boundary because other cluster-permutation attractors can coexist above it.Source: E. Cluster-permutation attractors at… · p. 14
• The equal two-cluster exchange orbit combines cluster exchange with global inversion, while its overlap, Gram matrix, and attention weights remain unchanged each layer.The orbit returns labeled centers to their initial positions after two layers, so tokens can move persistently despite a stationary attention pattern.Source: E. Cluster-permutation attractors at… · pp. 14–15
• In the macroscopic exchange geometry, collective perturbations separating the two cluster centers decay, but internal splitting perturbations grow, so each parent cluster can fragment into several descendants.Source: E. Cluster-permutation attractors at… · p. 15
• Regular cluster-flip states with L proportional to N are linearly unstable at fixed finite β on the flipping branch, because every fixed nonzero Fourier mode amplifies density inhomogeneity.This instability occurs even when clusters remain internally coherent; the subsequent nonlinear rearrangement or merging is not determined by the linear analysis.Source: E. Cluster-permutation attractors at… · p. 17

F. Exponentially slow merging of clusters

• At γ = 0.70 and 0.78, beyond the N = 128 exchange-branch endpoint of approximately 0.668421, coherent clusters form rapidly but their centers move much more slowly.At γ = 0.78, trajectories reach one cluster for β = 4, while multiple clusters persist through 10^7 layers for β = 10.Source: F. Exponentially slow merging of… · pp. 17, 19, Figure 7 · p. 18
• Weak attention between clusters makes their effective center interaction attractive but exponentially small, producing an enormous near-antipodal departure time.At N = 128, γ = 0.70, and β = 10, the predicted e-folding time is approximately 5.4 × 10^11 layers; exact antipodality is preserved by symmetry.Source: F. Exponentially slow merging of… · pp. 18–19, Figure 7 · p. 18

G. Microscopic attention dynamics in the β ∼N 2 scaling

• Attention condensation requires β ~ N^2, which resolves microscopic token spacing and leaves only O(1) relevant targets per token as N → ∞.At fixed finite β, attention remains diffuse in the thermodynamic limit; the β ~ N^2 regime instead yields an effectively sparse, state-dependent attention graph.Source: G. Microscopic attention dynamics in… · pp. 19–20
• In condensed attention, small separation changes redistribute routing among nearby targets, while repulsive updates alter those same separations and can propagate routing changes to neighboring tokens.This local geometry–routing feedback is identified as the physical origin of period-doubling and, at larger γ, routing-chaotic dynamics.Source: G. Microscopic attention dynamics in… · pp. 21–22
• The condensed uniform state first loses stability through a microscopic flip instability, which can saturate into a coherent period-two routing state rather than immediate chaos.Below γ_flip(χ) the state is stable; above it, the unstable mode alternates between layers, and perturbations toward the period-two orbit decay.Source: G. Microscopic attention dynamics in… · pp. 21–22
• Unlike the collective long-wavelength instability at fixed β, the condensed instability begins as a local rearrangement involving only a few neighboring tokens.Its first unstable mode has a wavelength spanning O(1) neighboring tokens, allowing an O(1) response in the thermodynamic limit.Source: G. Microscopic attention dynamics in… · p. 22
• In condensed routing chaos, local perturbations grow exponentially and spread through neighboring tokens with an approximately ballistic butterfly front.At finite χ, each token interacts appreciably with O(1) neighbors, although exponentially small long-range softmax tails make the front an effective—not exact—causal boundary.Source: G. Microscopic attention dynamics in… · p. 22
• Unlike fixed-β collective chaos, condensed routing chaos is governed by a few microscopic neighbors whose switching transmits perturbations through a sparse, rewiring attention graph.Its onset follows the nonlinear loss of stability of the coherent period-two state and can be diagnosed by a positive Lyapunov exponent and ballistic damage spreading; the hard-routing limit reduces the dynamics to a nearest-neighbor toy model.Source: G. Microscopic attention dynamics in… · p. 22
• For γ > 1/2, attention to nearly parallel neighbors reverses a token’s direction after normalization, while sufficiently large γ supports capture into cluster-flip attractors.Active and periodic motion can persist above γ = 1/2; the selected cluster depends on initial conditions, so consensus is not the unique outcome and γ = 1/2 is not a universal clustering transition.Source: G. Microscopic attention dynamics in… · pp. 22–23
• Clustered outcomes are finite-time observations, and the con = 127/190 ≃ 0.668421 line marks local stability against splitting, not the time required for clusters to merge.The gray dotted γ = 1/2 line marks breakdown of the nonflipping local expansion; displayed cell edges are conventions rather than located transitions.Source: G. Microscopic attention dynamics in… · p. 23

c. Chaos: exponential growth and a butterfly cone

• A single-token perturbation spreads outward with an approximately ballistic front in the active hard-routing regime.Figure 9 uses N = 1024 and γ = 0.20; the initial displacement is 10^-8 mean spacings.Source: Figure 9 · p. 24, H. Hard routing: local chaos and the… · p. 24

a. Emergent local dynamics

• The hard-routing map is local and linear for fixed routing choices; it becomes nonlinear when neighboring gaps exchange order and tokens switch partners.Tokens select the adjacent neighbor across the smaller gap and are driven away from it.Source: H. Hard routing: local chaos and the… · pp. 23–24
• The fixed-label lattice description assumes preserved angular ordering; 0 < γ < 1/4 suffices for that technical condition, which is not a phase boundary.Source: H. Hard routing: local chaos and the… · p. 24

b. Unstable ordered solutions

• The uniform polygon and alternating dimer cycle are reference trajectories, not attractors; small imbalances destabilize them and generic dynamics enters active routing.A perturbation of the dimer orbit grows by (1−2γ)^−1 > 1 per layer.Source: H. Hard routing: local chaos and the… · p. 24

c. Chaos: exponential growth and a butterfly cone

• Alternating perturbations grow exponentially at every layer, giving the hard map a positive maximal Lyapunov exponent.At γ = 0.20, the reported exponent is λmax = log(5/3) ≃ 0.511.Source: H. Hard routing: local chaos and the… · p. 24

IV. THE HIGH-DIMENSIONAL LIMIT

• At fixed finite d, oscillations, chaos, and clustered periodic or flipping states can occur even as attention remains diffuse for N → ∞.The bounded-logit argument gives YA = O(N^-1) at fixed d and β.Source: §IV · p. 25
• In d = N, dynamically generated O(1) overlap gaps create O(N) logit separations and can support attention condensation at β = O(1).Source: §IV · p. 25
• Gaussian initial conditions yield four principal outcomes: diffuse simplex-like states, active condensed routing, consensus flips, and fragmented cluster flips.The proposed phase boundary is schematic, and the finite-size scans sample representative regions rather than establishing the full diagram.Source: §IV · p. 25, Figure 10 · p. 25

A. High-dimensional model and diagnostics

• The diagnostics distinguish diffuse from condensed attention, continued motion from fixed points or two-cycles, and geometric clustering from alignment.YA measures attention concentration, D2 tracks two-layer displacement, and Kq and mmax characterize geometric clusters.Source: A. High-dimensional model and… · pp. 25–26

B. The diffuse simplex and its loss of stability

• For every β and 0 < γ < 1, the simplex is an exact fixed point; its existence should be distinguished from its stability, rather than treated as evidence of stable behavior.The loss of stability of a specially prepared simplex is parametrically distinct from condensation starting from a Gaussian cloud.Source: B. The diffuse simplex and its loss of… · p. 26
• The scans show a stable diffuse simplex, then a diffuse dynamical phase, then condensed dynamics; the observed condensation onset lies near the dipolar threshold.The shape threshold is the simplex spinodal, while the later dipolar crossing is not a second simplex spinodal.Source: B. The diffuse simplex and its loss of… · p. 26, Figure 11 · p. 27
• Chaos appears before condensation: at c = 3, YA remains O(N^-1) while finite-time Lyapunov estimates are positive.The reported estimates range from 0.14 to 0.27 per layer.Source: B. The diffuse simplex and its loss of… · p. 26

C. Phase selection from Gaussian initial conditions

• For γ < 2/3, increasing β selects condensed active routing; for γ > 2/3, it selects condensed fragmented flips from diffuse consensus flips.The condensation line separates YA = O(N^-1) from YA = O(1).Source: C. Phase selection from Gaussian… · pp. 27–28
• From Gaussian initial conditions, finite overlap gaps can yield condensation at order-one β, distinct from the O(N) stability scale of a prepared simplex.The initial overlaps vanish as N^-1/2, but dynamically generated O(1) gaps produce O(N) logit differences.Source: B. The diffuse simplex and its loss of… · p. 26, C. Phase selection from Gaussian… · p. 27
• At γ = 0.3, condensed attention accompanies sustained, potentially chaotic motion without geometric clustering, as negative feedback continually changes routing partners.The active state remains essentially unclustered while its attention network rewires.Source: C. Phase selection from Gaussian… · p. 28

2. Consensus and fragmented flip phases for γ > 2/3

• In high dimension, weakly communicating clusters can flip independently; stable clusters of bounded size require γ > 2/3.The minimum stable cluster size grows as γ approaches 2/3 from above.Source: §2 · pp. 29–30
• At γ = 0.8, consensus flips have uniform attention, whereas increasing β can produce condensed fragmented flips, so alignment and attention concentration are distinct.The scans show the largest geometric cluster shrinking from the consensus state to O(10) tokens at β = 5.Source: §2 · pp. 29–30, Figure 15 · p. 30

A. General value maps and effective low-dimensional dynamics

• A D-dimensional eigenspace with a degenerate negative eigenvalue supports the repulsive model after parameter rescaling; D = 2 recovers the circle dynamics.The reduction is exact within the invariant subspace, but generic nearby configurations require transverse perturbations to decay.Source: A. General value maps and effective… · pp. 30–31
• Orthogonal rotation blocks can yield effective circle dynamics, but cluster selection favors the plane with the largest amplification and does not always select repulsive behavior.For an aligned cluster, the selected plane depends on the rotation spectrum; a +1 eigenspace can instead dominate.Source: A. General value maps and effective… · p. 31
• A general real value map reduces exactly on an invariant plane to dynamics governed by its in-plane matrix, which can introduce anisotropic amplification and shear.Whether generic configurations approach that plane depends on transverse perturbation decay, not the spectrum of V alone.Source: A. General value maps and effective… · p. 32

B. Connections to transformer dynamics and computation

• The model’s phases are controlled examples, not a literal phase diagram for trained transformers; in untied networks they describe finite-depth evolution.The most direct connection is to architectures that repeatedly apply a shared block.Source: B. Connections to transformer dynamics… · p. 32
• Attention concentration, representation clustering, and routing activity are distinct; in untied networks, changing attention maps also reflect changing layer parameters.The model’s active condensed regime suggests refreshed sparse routing, but concentration alone does not guarantee an efficient sparse approximation.Source: B. Connections to transformer dynamics… · pp. 32–33

VI. CONCLUSION

• A computational benefit of active routing is not established by geometric or dynamical observables alone; it requires task-conditioned evaluation and controlled interventions.The conclusions also distinguish persistent rerouting and perturbation spreading from representation collapse.Source: B. Connections to transformer dynamics… · p. 33, §VI · p. 33
• For d = 2 and fixed β, attention remains diffuse as N grows despite active dynamics; condensation instead appears at β ∼ N^2, while hard routing yields a ballistic butterfly cone in representation space.In high dimensions, simulations from Gaussian initial conditions support condensation at β = O(1), with overlap gaps organizing four principal regimes.Source: §VI · p. 33

Appendix A: Derivation of the Fourier stability spectrum

• The appendix derives the linearized angular dynamics, the variation of the tangential attention message, and the exact continuum eigenvalue.Its derivation proceeds from the angular map and softmax-weight changes to the Fourier response.Source: Appendix A · p. 34

1. Linearization of the angular map

• The angular map is linearized around the unperturbed polygon, whose radial coefficient is rN = 1 − γ − γaN > 0.The derivative of atan2 with respect to the radial variable vanishes at the unperturbed state.Source: §1 · pp. 34–35

2. Variation of the softmax weights

• A token’s angular perturbation changes its relative angle and attention logit, and the softmax variation includes a term for normalization.The appendix applies this first-order weight variation to the uniform polygon.Source: §2 · p. 35

3. Variation of the tangential message

• The tangential message’s first linear contribution is its direct geometric change; symmetry cancels the first term and a further sum involving changing weights.Combining these contributions yields the linearized tangential message.Source: §3 · p. 35

4. Finite-N Fourier spectrum

• The finite-N Fourier spectrum follows by substituting Fourier modes into the linearized equation; the q = 0 eigenvalue is exactly one.The continuum calculation then gives a positive Fourier response μq for every q ≥ 1 when β > 0.Source: §4 · pp. 35–36, §5 · p. 36

5. Continuum evaluation

• For β > 0, the continuum Fourier response satisfies μq > 0 for every q ≥ 1.The appendix obtains this sign from the continuum expression and a modified-Bessel recurrence relation.Source: §5 · p. 36

6. Critical line and flip threshold

• At the flip threshold, the critical mode reaches −1; beyond it, the deformation alternates and grows, while nonlinear terms may stabilize period-two motion.The linear calculation does not determine whether the nonlinear branch is supercritical or subcritical.Source: §6 · p. 36

1. Exact consensus dynamics

• At consensus, masked attention is uniform; the state is fixed for γ below 1/2 and reverses as a rigid period-two orbit above 1/2.At γ = 1/2 the normalized map is singular, so that value is excluded from stability analysis.Source: §1 · p. 37

2. Angular perturbations and mode decomposition

• Linearization separates a neutral global-rotation mode from N −1 internal splitting modes, and attention weights have no first-order correction, making the consensus threshold independent of β.The diagonal mask contributes a finite-size displacement to the average of the other tokens.Source: §2 · p. 37

3. Stability of the stationary consensus branch

• The stationary consensus fixed point is unstable to every nontrivial internal splitting because negative feedback reinforces displacement and normalization amplifies angular deviations.A displaced token receives an average shifted in the opposite direction from the cluster center.Source: §3 · p. 38

4. Stability of the consensus flip orbit

• For the consensus flip orbit, splitting perturbations alternate; they decay above its stability threshold and grow below it.The stability test uses the two-layer Floquet multiplier, with the common-rotation mode remaining neutral.Source: §4 · p. 38

5. Thermodynamic limit and physical interpretation

• In the thermodynamic limit, the consensus flip is locally stable for γ > 2/3, but other cluster-permutation attractors may remain stable at the same parameters.The boundary describes return from sufficiently nearby initial conditions, not generic convergence.Source: §5 · p. 39

1. Exchange orbit and finite-size attention weights

• The equal two-cluster exchange orbit has a two-cycle of labeled centers while its overlap and attention weights remain constant.The finite-size setup divides an even number of tokens into two equally populated clusters and retains the diagonal-mask effect.Source: §1 · p. 39, Appendix C · p. 39

2. Collective separation mode

• The exchange orbit is locally stable only when both its collective separation and internal splitting perturbations decay.A negative separation multiplier can still be stable when its magnitude is below unity.Source: §2 · p. 40

3. Internal cluster-splitting modes

• Zero-sum constraints remove coherent center shifts, leaving N −2 internal perturbations that are degenerate and do not couple across clusters at linear order.The perturbations describe splitting within the two clusters while their macroscopic centers remain fixed to linear order.Source: §3 · pp. 40–41

4. Floquet spectrum and stability criteria

• If separation is restored but internal splitting is unstable, each cluster fragments and can generate higher-cluster descendants; the exchange orbit attracts only when both multipliers lie inside the unit circle.The two conditions distinguish instability of the center separation from fragmentation within clusters.Source: §4 · p. 41

Appendix D: Derivation of the local map in the attention-condensed regime

• The condensed-regime local map updates each rescaled gap through its attention-weighted signed separation of nearby targets, under a small-angle expansion.The expansion applies for γ < 1/2; as γ approaches 1/2 from below, the radial component vanishes and the expansion breaks down.Source: Appendix D · p. 42

1. An exact four-cluster orbit

• The symmetric four-cluster construction yields labeled period-four motion with nontrivial period-two Gram motion.The antipodal pairs turn in opposite directions, and the local environments reappear up to exchange of labels.Source: §1 · p. 43, Figure 16 · p. 43

2. Stability and the shrinking window

• At large β, the stability window narrows as β^-1, while cluster separation at fixed κ(γ −1/2) scales as β^-1/2.The upper endpoint is set by a fold; weak antipodal channels are needed to locate the lower edge.Source: §2 · p. 43
• Direct evolution reaches relative period-two states over a broader region than the equal-population orbit’s local stability interval, often with unequal populations or different cluster counts.The finite-time sampled region is not the analytic local-stability boundary, and additional periodic pockets occur outside the stability strip.Source: §2 · pp. 43–44, Figure 17 · p. 44

1. Periodic orbits and their stability

• All 39 physical multipliers lie inside the unit circle for the tested N = 40 upper P2 and lower uniform states, but a larger-N trajectory shows slow shape drift rather than a certified exact P2.At N = 128, the RMS angular displacement reaches 2.62×10^-5 radians over 20,000 layers; eventual convergence is not excluded.Source: §1 · p. 45

2. Why the P2 reached above the flip depends on β

• At β = 10, a five-wave perturbation reaches a small P2 above the polygon flip, while at β = 0.5 a one-wave perturbation reaches an equal-population two-cluster exchange P2.The β = 0.5 example shows no observed intermediate chaotic stage at the tested offset.Source: §2 · p. 45
• The low-β flip is locally subcritical and the high-β flip supercritical; the low-side exchange P2 persists when reached by switching an already formed orbit, showing initial-state dependence at the tested point.The cubic expansion does not predict the distant exchange orbit's amplitude or establish a global branch connection.Source: §2 · p. 45

3. Long trajectories and Lyapunov estimates

• At δγ = 0.011, finite-time Lyapunov estimates remain positive in both halves of all twelve records, while two N = 512 records at 0.007 are not classified as chaotic from that estimate alone.The estimates are finite-time, and the full-space calculation—not the fivefold-restricted one—counts exponents above the thresholds.Source: §3 · p. 47

4. Temporary amplification on a stable orbit

• A stable P2 with maximum Floquet multiplier modulus 0.998734 initially amplifies a five-wave perturbation by 1.059, before the finite-time estimate becomes negative and approaches its long-time value.Temporary amplification is consistent with the orbit's long-time stability.Source: §4 · p. 47

5. Numerical tests in the condensed scaling

• At χ = 0.05 and γ = 0.35, polygon-started cycles at N = 128, 256, and 512 have spectral radii 0.994499, 0.997902, and 0.999481; a near-half example has ρ2 ≃ 2.26 × 10^9 despite nearly returning.A small return error alone therefore does not certify stability.Source: §5 · p. 48

1. Long-time observations

• All completed β = 4 runs reach one cluster by layer 590,000, whereas at β = 10 mergers continue much later; final multicluster counts are robust to gap-threshold changes in the 18 saved configurations.The different observation times and two incomplete runs prevent interpreting the table as infinite-time basin probabilities.Source: §1 · p. 48

2. Exact center equation and weak coupling

• The center equation predicts observed three-cluster drift through 10^7 and 4 × 10^6 layers with RMS relative-angle errors below 9 × 10^-9 radians, but checks center motion rather than internal cluster stability.The compared records have different initial center angles and population fractions, so they do not isolate a size effect.Source: §2 · p. 49

3. Finite-time overview

• For display, edges are halfway to adjacent sampled parameters; they therefore do not locate transitions, and neither a small return distance nor a long-lived cluster count replaces a Floquet test.Source: §3 · p. 50

4. Residual weight and merger at fixed cluster geometry

• For the same three-cluster geometry, the first merger occurs later as γ increases: after 26.50, 27.68, and 32.81 million layers at γ = 0.78, 0.80, and 0.90, respectively.This comparison isolates center motion with cluster centers and populations held fixed.Source: §4 · p. 50, Figure 20 · p. 51

1. Tangent modes of the regular simplex

• Simplex shape modes destabilize first below γ = 1/2, while dipolar modes destabilize first above it; for fixed β > 0 and γ < 2/3, the simplex is locally attracting at sufficiently large finite N, modulo rotations.These local results do not establish selection of the simplex or consensus from arbitrary initial conditions.Source: §1 · pp. 51–52

2. Numerical protocols and tangent growth

• The scans use independent samples, matched windows N ≤ t < 4N, and a tangent alignment interval of N layers; reported bands are SEM across samples, not across layers.The Lyapunov estimate is finite-time, includes neutral global rotations, and is distinct from an exact cycle multiplier or an established infinite-time exponent.Source: §2 · p. 53

3. Cluster identification and internal stability

• An exact attention-connected triad can have YA = 1/2 and D2 = 0 while its tokens do not all reach their antipodes; existence does not establish attraction from Gaussian starts or nonzero thermodynamic-limit width.A strict geometric cutoff can split that triad into three clusters.Source: §3 · p. 54
• Geometric clusters join tokens by overlap, whereas attention clusters join them by directed attention weight; in the fragmented snapshot, these criteria yield Kq = 33 and KA = 25.Four finite-width triads are each split into individual tokens by the geometric criterion despite remaining attention-connected.Source: §3 · pp. 53–54
• At (β, γ) = (5, 0.78), the internal perturbation test predicts and measures growth for size-three clusters and contraction for size-four clusters.The fitted width multipliers agree with ρ3 = 61/56 > 1 and ρ4 = 6/7 < 1; the test does not determine which clusters form from Gaussian initial conditions.Source: §3 · p. 54