{"id":"68e23723-6015-4039-ab0e-59ba71a345a6","arxiv_id":"2608.07672","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In a special scale-invariant fracton Hamiltonian, late-time attractors reproduce the scale expansion, homogeneity, critical Newtonian dynamics, and arrow of time of a flat matter-dominated cosmology.","lead":"A family of classical fracton models is shown to have expansion fixed points whose scale grows like t to the 2/3, matching the Einstein-de Sitter expansion of a flat matter-dominated universe. One special member also reproduces Newtonian gravitational central configurations, an emergent arrow of time, and large-scale homogeneity as attractor properties.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2's local-stability claim depends on unproved positivity of C_q for general N; the abstract's 'attractors without fine-tuning' therefore outruns the conditional theorem, and the clustered large-N claim rests on explicit Conjecture 2.","rationale":"The reader's weakest_assumption correctly identifies the unproved positivity of C_q as a central limitation, and I agree that Theorem 2 is conditional rather than unconditional. I single out this positivity condition as the most load-bearing concern because it sits directly under the claim that the homogeneous Einstein-de Sitter fixed point is a dynamically selected attractor without fine-tuning. The numerical checks at N = 75 are suggestive but do not establish Conjecture 1, and no code or data is shipped for independent verification. I also flag Conjecture 2 as a separate, explicitly postulated input needed for the large-N clustered attractor and unequal-mass Newtonian dual; however, because the authors label it a conjecture and do not present it as proven, it is less damaging to the core analytic fixed-point result than the C_q gap. The paper's own text repeatedly acknowledges these limitations, so there is no ambiguity or misleading claim inside the full manuscript; the issue is that the abstract and the opening summary state 'locally stable' and 'emerge as attractor properties' more strongly than the conditional theorem supports. A concrete numerical search for a C_q-negative regular extremum would settle whether Conjecture 1 fails in an explicit case; absence of a counterexample would strengthen confidence but would still leave the general finite-N statement unproved. For these reasons, the conditional verdict remains appropriate, and no change to the reader's verdict is needed.","tokens_in":42964,"tokens_out":8473,"duration_ms":89611,"concrete_test":"Test Conjecture 1 directly: for N = 4,...,100, generate many local minima of the shape potential Xi_{-1} using the same L-BFGS-B procedure as in Section 4.3, and for each regular minimum compute the smallest eigenvalue of C_{-1} restricted to the physical (6N-11)-dimensional tangent space, using the projection of Appendix E.3. If any configuration with constrained Hessian Q_{-1} < 0 has a strictly negative C_{-1} eigenvalue, Conjecture 1 is false and Theorem 2's conclusion fails for that case; if the physical minimum eigenvalue is positive for all sampled configurations, the conjecture is numerically supported but still not proven for general N.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing gap is in Section 3.5: Theorem 2 proves linear asymptotic stability of expanding radial fixed points only under the auxiliary condition that C_q = L - 2P + S_q I (Eq. 3.39) is strictly positive on the physical tangent space. Positivity is proved analytically only for the equilateral N = 3 configuration (Proposition 1); for larger N it is checked numerically for selected N = 75 minima, and Conjecture 1 states the general case as an open problem. Since C_q positivity is not implied by the Hessian sign Q_q < 0, a regular extremum could in principle satisfy Q_q < 0 while having a physical C_q-negative mode; then the damped-oscillator argument leading to Eq. (3.45) fails and the expanding fixed point need not be linearly attracting. The abstract's statement that 'the fixed points are locally stable' and the headline claim that cosmological analogues 'emerge as attractor properties' therefore go beyond what is proven. A second, separate load-bearing point is Conjecture 2 (Section 4.4): the effective unequal-mass dynamics (4.9) is postulated rather than derived, so the clustered attractor, the renormalized central-configuration equation (4.7), and the unequal-mass Newtonian dual are conditional on a scale-separation conjecture that the paper explicitly leaves unproved. The manuscript is honest about both limitations, but the advertised no-fine-tuning selection is correspondingly conditional.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a two-parameter family of scale-invariant, dipole-conserving classical fracton Hamiltonians H_{α,β} = Σ_{i<j} |p_i−p_j|^α |x_i−x_j|^β. After separating an overall scale from dimensionless shape variables, the authors derive an autonomous reduced shape dynamics whose radial fixed points are central configurations of a Riesz potential with exponent q = α+β, with accompanying scale evolution R(t) ∝ |t|^{α/(α−β)}. On the Einstein-de Sitter branch α = −2β, these fixed points are argued to be linearly stable under a conditional positivity assumption, and the distinguished case (α, β) = (−2, 1) is singled out because its fixed-point equation is the equal-mass Newtonian central-configuration equation, its large-N profile is a homogeneous ball, and its homothetic trajectories admit a zero-energy Newtonian gravitational dual. Numerics at moderate N show convergence from Gaussian-sampled random initial data, while large-N numerics display clustered attractors whose centers are conjectured to obey an effective unequal-mass dynamics. The paper interprets these properties as emergent, attractor-based cosmological analogues, while explicitly disclaiming any replacement of relativistic cosmology.","tokens_in":43237,"tokens_out":4435,"duration_ms":44585,"significance":"If the main claims hold, this is a valuable and elegant contribution: it provides an analytically tractable example of Hamiltonian systems that develop attractors after reduction, connects classical fractons to the well-developed theory of Riesz equilibrium measures, and offers a concrete toy model in which features analogous to flatness, homogeneity, and an arrow of time arise from a single reduced fixed point rather than from fine-tuned initial data. The geometric derivation in Sections 3.2–3.3 is clean, the dictionary to known equilibrium-measure results in Appendix C is careful, and the paper is unusually honest in labeling its two central conjectures. The numerical stability checks, the equilateral three-body analytic result, and the explicit large-N diagnostics are strengths. However, the advertised conclusions outrun the proven theorems in two load-bearing places: the local-stability theorem depends on an unproved positivity conjecture for C_q, and the clustered large-N attractor depends on an explicitly postulated scale-separation conjecture.","major_comments":[{"comment":"The abstract's unqualified statement that 'the fixed points are locally stable' is not what Theorem 2 proves. Theorem 2 requires the auxiliary quadratic form C_q, defined in Eq. (3.39), to be strictly positive on the physical perturbation space. This positivity is established analytically only for the equilateral N = 3 configuration (Proposition 1) and checked numerically for selected N = 75 minima; Conjecture 1 explicitly leaves the general case open. Because positivity of C_q is not implied by the constrained Hessian sign Q_q < 0, a regular extremum could in principle have a C_q-negative physical mode for which the damped-oscillator argument leading to Eq. (3.45) fails and the expanding fixed point is not linearly attracting. This is a load-bearing condition for the 'attractor without fine-tuning' claim. Please either prove or delimit Conjecture 1, or qualify the abstract and the headline stability statements accordingly.","section":"§3.5, Theorem 2 and Conjecture 1; abstract"},{"comment":"The clustered large-N attractor, the renormalized central-configuration equation (4.7), and the unequal-mass Newtonian dual in Corollaries 1–2 are all consequences of Conjecture 2, which postulates an asymptotic scale-separated cluster decomposition rather than deriving it from the microscopic equations. The manuscript is explicit that this is a conjecture, but the abstract and the conclusions present the clustered state, its homogeneity, and its Newtonian dual as part of the paper's results. Since these clustered fixed points are central to the advertised 'richer class of fixed points' and to the large-N cosmological analogy, the revision should either derive the effective dynamics (4.9) in a controlled limit or consistently present the whole clustered-attractor picture as conjectural, including in the abstract.","section":"§4.4, Conjecture 2 and Eq. (4.9)"},{"comment":"The claim that flatness and the cosmological analogues are selected 'without fine-tuning' is stronger than the evidence supplied. Theorem 2 establishes only local linear stability, and the numerical basins studied in Section 4.3 consist of Gaussian-sampled, re-centered random initial data at moderate N. Section 6 carefully restricts the conclusion to 'the analytic basins and numerical simulations actually studied,' but the abstract and introduction drop that restriction and present the no-fine-tuning selection as an established property. The paper should either provide an explicit Lyapunov function or other global-basin argument for the reduced shape dynamics, or consistently state that the no-fine-tuning statement is basin-dependent and demonstrated only for the examples studied.","section":"§3.6 and §6; abstract"}],"minor_comments":[{"comment":"The sentence 'The spectra are computed only after this residual is ;' is incomplete; please state the numerical tolerance actually used for accepting a fixed point.","section":"Appendix E.1"},{"comment":"The phrase 'the nonzero rotational companions have negative decay rate' is confusing because the common rotational zero modes are said to be quotiented; please clarify how the rotational companions are treated in the reduced stability problem.","section":"§3.5, Proposition 1"},{"comment":"The notations t_R^J and t_Ξ^J for the Janus points are used without formal definitions; please define these times and explain the claimed approximate coincidence t_R^J ≈ t_Ξ^J.","section":"§5.3"},{"comment":"The statement that 'the positivity of C_q naturally follows' from numerically verifying local stability is too vague for a load-bearing condition; please specify which numerical checks are being reported and how they constrain C_q.","section":"§3.5, footnote 4"},{"comment":"The table caption correctly says the cluster counts come from one representative run, but the surrounding text invites scaling-law inferences from these counts; please add an explicit ensemble-average caveat in the main text.","section":"§5, Table 1"},{"comment":"The abstract's 'simulations at moderate N approach them from random initial data' should specify that the initial data are Gaussian-sampled, re-centered, and re-scaled random shape phase-space points, not generic or fully unstructured random data.","section":"Abstract and §4.3"}],"recommendation":"major_revision","confidential_remarks":"The manuscript's core geometric derivation is sound and the authors are transparent about the two conjectures. The main issue is that the abstract and conclusions claim more than the theorems establish. A revision that either proves or explicitly qualifies Conjectures 1 and 2, and consistently restricts the no-fine-tuning language to the studied basins, would make the paper's advertised claims match its rigorous content. I see no circularity or novelty concern; the paper is a good fit for the journal's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know this paper is more honest than its abstract. The real content is a clean analytic reduction: for the two-parameter Hamiltonian family H(alpha,beta), separating scale from shape gives autonomous shape dynamics whose radial fixed points are exactly central configurations of a power-law Riesz potential with exponent q = alpha + beta. On the Einstein-de Sitter branch alpha = -2 beta, the distinguished model (-2,1) is the unique case where the fixed-point equation becomes the equal-mass Newtonian central-configuration equation, the large-N profile is a homogeneous ball, and homothetic trajectories have a zero-energy Newtonian gravitational dual. That set of equivalences is new and worth having. The derivation in Sections 3.2-3.4 is solid, and the authors correctly credit the external Riesz-equilibrium results they use.\n\nThe soft spots are in the stability story. Theorem 2 is honestly conditional on positivity of the auxiliary quadratic form C_q, and that positivity is proven analytically only for N = 3 equilateral. The abstract, however, states flatly that 'the fixed points are locally stable' and that the cosmological analogues 'emerge as attractor properties.' That overstates the theorem, which needs C_q > 0 and also, for the clustered large-N behavior, Conjecture 2. The paper itself flags both conjectures clearly, so this is an abstract-outruns-the-content problem rather than a hidden flaw. The numerical work is transparent, but no code or data is shipped, which matters because the large-N cluster decomposition is the empirical basis for Conjecture 2. One smaller issue: Eq. (3.2) has an inconsistent U_eff coefficient versus the surrounding text; that should be fixed.\n\nMy net read: the central fixed-point construction is correct and the distinguished model is a genuinely interesting toy for how attractor dynamics can reproduce cosmological-scale features without fine-tuning. The stability and clustering claims are load-bearing and currently conditional, so the paper needs revision in the abstract and a sharper separation between theorems and conjectures. Still, the analytic core is worth referee time, and the honest labeling of limitations makes this a paper I would want to engage with.","headline":"Genuinely new analytic core: scale-shape reduction maps fracton fixed points to Riesz central configurations and isolates a unique EdS model with a Newtonian dual. But the abstract's 'attractors without fine-tuning' outruns the conditional stability theorem and two explicit conjectures still carry a lot of weight.","tokens_in":43777,"tokens_out":1035,"would_cite":true,"duration_ms":13435,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["70F10","37C75","82C05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Scale-shape reduction turns a family of fracton Hamiltonians into systems with attractor fixed points that are central configurations; one special model reproduces flat matter-dominated cosmology as a late-time attractor.","keywords":["classical fractons","central configurations","Einstein-de Sitter","scale-shape separation","Riesz potentials","attractor fixed points","Newtonian cosmology","arrow of time"],"falsifier":"Compute the smallest eigenvalue of the auxiliary form $C_q$ restricted to the physical tangent space (after quotienting translations, rotations, and the $\\gamma$-family zero mode) for regular extrema of the shape potential $\\Xi_{-1}$ at, say, $N=100,200,500$, using the paper's own finite-difference projection procedure (its Appendix E). A negative eigenvalue for any regular minimum would contradict Conjecture 1 and remove the linear stability on which the no-fine-tuning claim rests; a positive gap persisting as $N$ grows would support it. Independently, in large-$N$ runs ($N=1000$ to $15000$), measure the ratio of intra-cluster to inter-cluster length scales: Conjecture 2 predicts this ratio tends to zero with the cluster centers satisfying the unequal-mass central-configuration equation, so a trajectory in which clusters fail to shrink relative to their separations, or in which the renormalized center diagnostics do not vanish, would falsify the clustered-attractor picture.","tokens_in":42717,"feed_emoji":"🌌","tokens_out":21360,"duration_ms":169495,"temperature":0.7,"pith_summary":"This paper studies a two-parameter family of classical fracton Hamiltonians, $H_{\\alpha,\\beta}=\\sum_{i<j}|p_i-p_j|^\\alpha |x_i-x_j|^\\beta$, whose phase-space flow is Hamiltonian yet whose reduced dynamics can form attractors. Separating the overall scale from the shape coordinates yields autonomous shape equations whose radial fixed points are exactly the central configurations of a Riesz potential with exponent $q=\\alpha+\\beta$, and at such points the system expands homothetically as $R(t)\\propto |t|^{\\alpha/(\\alpha-\\beta)}$. The paper singles out $(\\alpha,\\beta)=(-2,1)$: its fixed-point equation is the equal-mass Newtonian central-configuration equation, its expansion is Einstein-de Sitter ($R\\propto t^{2/3}$), its large-$N$ profile is a homogeneous solid ball, and its homothetic trajectories carry a zero-energy Newtonian gravitational dual. The reason to care is that the paper argues these cosmological features — flatness, homogeneity, central-configuration shape, and an arrow of time — are attractor properties reached from random initial data, rather than fine-tuned initial conditions, within the basins studied analytically and numerically.","feed_headline":"One model makes Einstein-de Sitter expansion the late-time attractor","feed_subtitle":"In one dipole-conserving Hamiltonian, a homogeneous expanding ball and an arrow of time emerge from random data.","key_machinery":"The load-bearing object is the dynamical-similarity reduction: coordinates are rewritten as an overall scale $R=\\sqrt{\\sum_i|x_i-X_C|^2}$ and centered, shape-normalized variables $s_i=(x_i-X_C)/R$, $u_i=R^{\\beta/\\alpha}(p_i-P_C)$, with rescaled time $d\\tau=dt/R^{(\\alpha-\\beta)/\\alpha}$. The resulting shape dynamics is autonomous and is not symplectic in the reduced variables, which is how attractors can coexist with phase-space volume preservation (Liouville's theorem) for the full Hamiltonian flow. On the radial ansatz $u_i=\\gamma s_i$ the fixed-point equations collapse to $H_i^q=S_q s_i$ with $H_i^q=\\sum_{j\\ne i}(s_i-s_j)/|s_i-s_j|^{2-q}$, the central-configuration equation of the Riesz potential $\\Xi_q=\\sum_{i<j}|s_i-s_j|^q$, where $q=\\alpha+\\beta$. Stability is carried by the damped-oscillator equation $\\ddot v+3S_q\\dot v-2C_qQ_q v=0$, in which $Q_q$ is the constrained Hessian of $\\Xi_q$ and $C_q=L-2P+S_q I$ is the auxiliary form whose positivity is the paper's main technical hypothesis. The large-$N$ distributions come from the equilibrium measures of the corresponding power-law interaction: weighted balls for $-3<q<1$, a homogeneous ball at $q=-1$, and spherical shells for $1\\le q<2$ and $2<q<4$.","core_discovery":"The paper establishes that radial fixed points of the reduced shape dynamics are central configurations of the Riesz shape potential, and on the Einstein-de Sitter branch $\\alpha=-2\\beta$ the expanding fixed points are linearly asymptotically stable under a positivity condition on the auxiliary quadratic form $C_q$, proven for the equilateral three-body configuration and checked numerically for larger $N$. For the distinguished model $(\\alpha,\\beta)=(-2,1)$ this means: the fixed-point equation coincides with the equal-mass Newtonian central-configuration equation; the regular large-$N$ distribution is a homogeneous expanding ball; the scale obeys $R(t)\\propto |t|^{2/3}$; and every homothetic fixed-point trajectory admits an exact zero-energy Newtonian gravitational dual whose coupling is generated by the fracton data. The paper further claims that random initial conditions at moderate $N$ are attracted to this state, so the Einstein-de Sitter structure is selected without tuning. At large $N$ the attractors instead form bound clusters of nearly fixed physical size whose centers obey an unequal-mass Newtonian central-configuration equation (Conjecture 2), preserving large-scale homogeneity. Trajectories generically show a bidirectional arrow of time from a Janus point (a moment of minimal scale), with the Boltzmann entropy growing logarithmically in the scale as $S_B\\sim k_B(9N-20)\\log R/2$.","pith_inferences":["The authors' own stability criterion suggests a finite-$N$ numerical programme the paper only begins: computing the lowest eigenvalue of $C_q$ on the physical tangent space for regular minima of $\\Xi_{-1}$ across a sequence of $N$ would either upgrade Conjecture 1 to a proven fact or produce an explicit counterexample.","Conjecture 2 carries a quantitative prediction the paper does not test: the intra-cluster to inter-cluster length-scale ratio should decay with a definite law, and measuring its approach to zero would show whether the unequal-mass Newtonian dual is reached asymptotically or only approximated over accessible times.","The matching condition $q=\\alpha+\\beta$ suggests a broader prescription: any conservative many-body system with a mechanical-similarity symmetry whose reduced shape flow admits central-configuration fixed points may generate such cosmological attractors; the distinguished model is the case where the reduced fixed-point problem coincides with the desired dual theory.","A natural next step the paper does not take is relativistic: whether an analogous attracting dynamics can live inside a causal spacetime setting, rather than on the fixed Euclidean background used here, separates the toy analogue from a mechanism relevant to gravitational physics."],"forward_implications":["Within the studied basins, the Einstein-de Sitter state is an attractor of a microscopic conservative system: generic sampled initial data flow to a homothetic $t^{2/3}$ expansion with no imposed central-configuration, flatness, or homogeneity conditions.","The zero-energy Newtonian dual means effective flatness is automatic along fixed-point trajectories: the critical Newtonian energy and the dual gravitational constant are outputs of the fracton solution, not inputs.","The large-$N$ profile is model-selective: the distinguished model forms an expanding homogeneous ball of radius $\\sim t^{2/3}N^{-1/2}$, whereas models with $q=1$, such as $(\\alpha,\\beta)=(2,-1)$, collapse onto an expanding spherical shell.","Under Conjecture 2, large-$N$ attractors support bound clusters of nearly fixed physical size whose centers separate as $t^{2/3}$ and satisfy an unequal-mass Newtonian central-configuration equation, so small-scale structure coexists with large-scale homogeneity.","Time-reversal invariance notwithstanding, generic trajectories exhibit a bidirectional arrow of time: scale expansion and shape complexity grow away from a Janus point, while the coarse-grained Boltzmann entropy grows as $k_B(9N-20)\\log(R/R_*)/2$ in each direction."],"supporting_citations":[{"why":"Supplies the dynamical-similarity or contact-reduction framework that produces the non-symplectic shape flow in which attractors can exist.","marker":"[4]"},{"why":"Establishes that classical fractonic Hamiltonians show configuration-space attractors and Machian behaviour, the phenomenon the paper makes analytically tractable.","marker":"[6]"},{"why":"Introduces classical nonrelativistic fracton dynamics, the Hamiltonian family from which the scale-invariant models here are drawn.","marker":"[7]"},{"why":"Provides the arrow-of-time and ergodicity-breaking results for classical fractons, and the momentum-complexity observable used in the entropy computation.","marker":"[8]"},{"why":"Supplies the central-configuration theory, including the equal-mass Newtonian central-configuration equation and its link to large-N homogeneity, that the fixed-point analysis builds on.","marker":"[10]"},{"why":"Classifies global extrema of attractive-repulsive power-law interaction energies, the source of the large-N ball and shell profiles in Theorem 1.","marker":"[13]"},{"why":"Provides the discrete-to-continuum particle-approximation convergence that carries the continuum equilibrium profiles back to finite-N fixed-point distributions.","marker":"[14]"},{"why":"Supplies the Janus-point construction of a bidirectional gravitational arrow of time that Section 5.3 adapts to the fracton trajectories.","marker":"[18]"},{"why":"Defines the discrete Newtonian cosmology benchmark (homothetic solutions from central configurations, the Friedmann equation) whose fine-tuning conditions the paper claims to replace with attractor properties.","marker":"[19]"},{"why":"Adds the Newtonian perturbation analysis and the low-entropy past-hypothesis discussion that define the arrow-of-time condition the Janus-point dynamics avoids imposing.","marker":"[20]"}],"fun_headline_variants":["Einstein-de Sitter expansion emerges as attractor in fracton model","Fracton model produces cosmological expansion without fine-tuning","One Hamiltonian gives Einstein-de Sitter growth from random data","Dipole-conserving model converges to homogeneous expanding ball"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise, not yet proven in general, is that the auxiliary quadratic form $C_q$ is strictly positive on the physical perturbation space of every regular extremum central configuration: the authors prove this only for the equilateral $N=3$ configuration and check it numerically for selected $N=75$ configurations, leaving the general case as Conjecture 1, and the large-$N$ clustered picture separately depends on the unproven scale-separation conjecture (Conjecture 2); if either premise fails, the corresponding no-fine-tuning conclusion collapses.","fun_headline_variants_meta":{"raw":{"variants":["Einstein-de Sitter expansion emerges as attractor in fracton model","Fracton model produces cosmological expansion without fine-tuning","One Hamiltonian gives Einstein-de Sitter growth from random data","Dipole-conserving model converges to homogeneous expanding ball"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000734,"raw_usage":{"total_tokens":3389,"prompt_tokens":1162,"completion_tokens":2227,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":778,"completion_tokens_details":{"reasoning_tokens":2158}},"tokens_in":778,"tokens_out":2227,"duration_ms":17733,"temperature":1.0,"reasoning_tokens":2158,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T00:25:12.948044+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the smallest eigenvalue of the auxiliary form $C_q$ restricted to the physical tangent space (after quotienting translations, rotations, and the $\\gamma$-family zero mode) for regular extrema of the shape potential $\\Xi_{-1}$ at, say, $N=100,200,500$, using the paper's own finite-difference projection procedure (its Appendix E). A negative eigenvalue for any regular minimum would contradict Conjecture 1 and remove the linear stability on which the no-fine-tuning claim rests; a positive gap persisting as $N$ grows would support it. Independently, in large-$N$ runs ($N=1000$ to $15000$), measure the ratio of intra-cluster to inter-cluster length scales: Conjecture 2 predicts this ratio tends to zero with the cluster centers satisfying the unequal-mass central-configuration equation, so a trajectory in which clusters fail to shrink relative to their separations, or in which the renormalized center diagnostics do not vanish, would falsify the clustered-attractor picture.","supporting_citations":[{"cited_title":"Scaling Symmetries, Contact Reduction and Poincar\\'e's dream","cited_arxiv_id":"2206.09911","evidence_quote":"Supplies the dynamical-similarity or contact-reduction framework that produces the non-symplectic shape flow in which attractors can exist."},{"cited_title":"Prakash, Y","cited_arxiv_id":null,"evidence_quote":"Establishes that classical fractonic Hamiltonians show configuration-space attractors and Machian behaviour, the phenomenon the paper makes analytically tractable."},{"cited_title":"Prakash, A","cited_arxiv_id":null,"evidence_quote":"Introduces classical nonrelativistic fracton dynamics, the Hamiltonian family from which the scale-invariant models here are drawn."},{"cited_title":"Classical Fractons: Local chaos, global broken ergodicity and an arrow of time","cited_arxiv_id":"2501.12445","evidence_quote":"Provides the arrow-of-time and ergodicity-breaking results for classical fractons, and the momentum-complexity observable used in the entropy computation."},{"cited_title":"Central Configurations in Three Dimensions","cited_arxiv_id":"hep-th/0201101","evidence_quote":"Supplies the central-configuration theory, including the equal-mass Newtonian central-configuration equation and its link to large-N homogeneity, that the fixed-point analysis builds on."},{"cited_title":"Frank and R.W","cited_arxiv_id":null,"evidence_quote":"Classifies global extrema of attractive-repulsive power-law interaction energies, the source of the large-N ball and shell profiles in Theorem 1."},{"cited_title":"Carazzato, A","cited_arxiv_id":null,"evidence_quote":"Provides the discrete-to-continuum particle-approximation convergence that carries the continuum equilibrium profiles back to finite-N fixed-point distributions."},{"cited_title":"Discrete Newtonian Cosmology","cited_arxiv_id":"1308.1852","evidence_quote":"Defines the discrete Newtonian cosmology benchmark (homothetic solutions from central configurations, the Friedmann equation) whose fine-tuning conditions the paper claims to replace with attractor properties."},{"cited_title":"Discrete Newtonian Cosmology: Perturbations","cited_arxiv_id":"1409.0395","evidence_quote":"Adds the Newtonian perturbation analysis and the low-entropy past-hypothesis discussion that define the arrow-of-time condition the Janus-point dynamics avoids imposing."}],"review_version":1}