{"id":"f387cbc9-a591-4563-9c75-0e0ddb502427","arxiv_id":"2506.04114","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":2.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A review talk arguing that hot QCD hosts an 'IR phase' in which Anderson-like localization of Dirac modes structures the thermal transition.","lead":"This conference talk summarizes evidence for a newly proposed phase of hot nuclear matter, called the infrared phase, where quark modes become trapped by disorder much like electrons in dirty materials. It argues that this phase, sitting just above the usual quark-gluon plasma crossover, could change how we think about the transition from hadrons to plasma.","discovery_kind":"review","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The power-law exponent p in Eq. (1) is inferred from cumulative spectral data over a finite window; the λ→0 limit and the phase classification rest on an extrapolation that has not been demonstrated to be free of finite-volume or discretization contamination.","rationale":"The reader's weakest_assumption is precisely the power-law behavior surviving to λ→0 in the continuum limit. My concern matches and sharpens it: the phase label depends on the sign of p, and p = -1+δ is very close to the B/IR boundary p=0 only in the sense that p<0 is required; actually p=-1 is far from p=0, but the claimed δ is 'very small', so the distinction between p=-1+δ and p=-1 is not what decides IR vs B — any p<0 gives IR. However, the more serious issue is whether the apparent p<0 is an artifact: finite volume can deplete or enhance the lowest eigenvalues and change the apparent power law. The text's own Fig. 7 shows the power-law region extending over three orders of magnitude in T/λ, but the smallest λ values are near the mode-counting floor; without a volume test, the extrapolation is not secure. The continuum extrapolation in Fig. 6 checks the peak strength, not the exponent. Thus the strongest claim (new IR phase with T_IR between 200 and 230 MeV) is plausible but not yet established by this talk's evidence. Verdict CONDITIONAL is appropriate because the underlying published papers may contain the missing volume tests, but this standalone text does not demonstrate them.","tokens_in":9814,"tokens_out":1651,"duration_ms":15470,"concrete_test":"From existing ensembles, compute the direct spectral density ρ(λ) (not cumulative) at fixed lattice spacing and fixed T/T_IR for at least three volumes (e.g., 32^3×8, 48^3×8, 64^3×8 at a≈0.085 fm). Fit log ρ vs log λ over a fixed decade in λ; if the fitted p changes by more than ~0.1 as the volume doubles, the λ→0 power law is not yet established and the classification is conditional. Additionally, repeat the fit with the lowest one or two λ bins removed to test window stability.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central classification (Eq. 1) and the claimed IR phase hinge on the near-singular power law ρ(λ) ∝ λ^p with p = -1 + δ, δ ≥ 0, as λ → 0. The evidence in Figs. 5–7 is cumulative density σ(λ,T) over a finite interval, shown for L = 2.0 fm (pure glue) and L = 3.4–5.0 fm (real-world). Neither the text nor the figures establish (a) that the inferred power law is stable as the volume increases at fixed lattice spacing, (b) that the exponent δ is independent of the fitting window as the lower-λ cutoff is lowered, or (c) that discretization errors are controlled in the deep-IR region where the mode density is largest. The continuum extrapolation in Fig. 6 addresses only the strength of the IR peak (ρ at fixed small D), not the exponent p that defines the phase. Since p = -1 is a critical borderline (p < 0 classifies IR phase), a tiny shift in δ from finite-volume effects could move the classification from IR to B. This is the load-bearing assumption exactly matching the reader's 'weakest_assumption'.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript, a proceedings contribution from QCHSC24, argues that thermal QCD possesses a previously unrecognized 'IR phase' between the hadronic/crossover regime and the weakly coupled QGP. The phase is defined by the infrared behavior of the Dirac spectral density, ρ(λ) ∝ λ^p with p < 0 (near p = −1), and is characterized by proliferation of deep-IR modes, IR-bulk separation, IR scale invariance, non-analyticity, and infinite glue screening lengths. The paper combines numerical lattice evidence from pure-glue and Nf = 2+1 QCD with an Anderson-localization picture in which two mobility edges, λ_A > 0 and λ_IR = 0, delimit the phase; it further appeals to an effective-dimension formalism to argue that near-zero modes have IR dimension d_IR = 2, leading to non-analytic behavior at T_IR. The paper explicitly states that the transition may be a true phase transition and gives the real-world estimate 200 MeV < T_IR < 230 MeV.","tokens_in":10016,"tokens_out":10953,"duration_ms":97513,"significance":"If established, the IR phase and the metal-to-critical scenario would substantially revise the standard picture of the QCD phase diagram: they predict a distinct thermal regime above the chiral crossover, an Anderson-like mobility edge at zero eigenvalue, a specific near-singular spectral shape ρ(λ) ∝ λ^{-1+δ}, and a non-trivial IR dimension d_IR = 2 for the deep-IR Dirac modes. The paper deserves credit for stating these claims in falsifiable form—Eq. (1) is a crisp classification, Eq. (2) gives a concrete temperature window, and Fig. 6 provides a continuum extrapolation of the IR peak strength—and for framing the phase diagram in a way that invites quantitative checks. However, the significance of the contribution as a standalone paper is limited by the fact that most of the load-bearing evidence is inherited from previous publications and from an unpublished reference [14]; the present manuscript does not itself contain the analysis needed to establish the power-law exponent.","major_comments":[{"comment":"The phase classification in Eq. (1) hinges on the infrared exponent p, but the manuscript does not demonstrate that the near-pure power law ρ(λ) ∝ λ^p with p = −1 + δ is stable under the uncontrolled systematic variations. Figure 7 shows cumulative spectral densities σ(λ,T) over a finite window without error bars or fit ranges: for pure glue the window covers about three orders of magnitude in T/λ, and for real-world QCD the window is shorter; neither volume dependence at fixed lattice spacing nor the dependence of the fitted δ on the lower-λ cutoff is shown. The continuum extrapolation in Fig. 6 addresses only the integrated strength of the IR peak at fixed small D, not the exponent p that defines the phase. Because p = −1 is the boundary between phases B and IR, a small finite-volume or discretization shift in δ could move the classification; this is the load-bearing assumption behind the existence of the IR phase and needs to be confronted directly.","section":"§2.1, Eq. (1), Figs. 5–7"},{"comment":"The pure-glue evidence for the IR phase at large volumes, shown in Fig. 5 (left), is taken from an unpublished work [14], and the continuum scaling shown in Fig. 6 (left) reproduces only the abundance at D = 4 MeV rather than the full spectral shape. As a result, the central spectral-density claim cannot be independently checked from the published record. The manuscript should either include the relevant numerical results (fitted p, δ, and their systematic errors) or make the unpublished data available and explain why the conclusions do not rest on that reference alone.","section":"§2.1, Ref. [14], Figs. 5–6"},{"comment":"There is a circularity in the scale-invariance argument. In §2.1 the near-pure power-law spectral density is presented as evidence that the IR glue is scale invariant ('in the spirit of the inverse scattering problem'), while in §3.3 the metal-to-critical scenario explains that same power law as a consequence of the mobility edge λ_IR = 0. As written, the power law is both evidence for and consequence of the same phenomenon, so the explanatory claim is not falsifiable by the data shown. A discriminating test is needed, for example a prediction for the exponent δ or for a two-point correlation function that differs between the 'scale-invariant glue' and 'Anderson criticality at λ_IR' interpretations.","section":"§3.3, §2.1"},{"comment":"The statement that the change at T_IR 'may be a true phase transition' is not supported by the evidence presented. The real-world estimate in Eq. (2) is based on finite lattices (L = 3.4–5.0 fm, a = 0.099–0.123 fm), and the manuscript shows no finite-size scaling of T_IR, no extrapolation of the transition sharpness to the thermodynamic limit, and no estimate of the systematic uncertainty in Eq. (2). If the phase-transition interpretation is to be retained, it needs at least a scaling analysis of the IR-regime onset with volume; otherwise the claim should be labeled as a conjecture distinct from the verified existence of a new regime.","section":"§2.2, Eq. (2), Fig. 4"}],"minor_comments":[{"comment":"In §2.1 the text compares T = 0.98 T_IR with T = 1.12 T_IR, while the top panels of Fig. 7 are labeled T = 0.98 T_c and T = 1.12 T_c; since the text also states that T_IR coincides with the Polyakov-line T_c, the notation should be made consistent to avoid confusion.","section":"§2.1, Fig. 7"},{"comment":"Because the horizontal axis is log10(T/λ), the statement that 'approaching deeper IR means moving to the right' is correct but may confuse readers who expect λ to decrease from left to right; consider annotating the axis with λ as well.","section":"Fig. 7"},{"comment":"Fig. 5 and Fig. 7 do not show error bars; for a quantity as central as the IR spectral density, at least representative error bars are necessary to judge the significance of the power-law behavior.","section":"Figs. 5, 7"},{"comment":"The 'unique notion' of effective dimension from Refs. [26,27] is asserted without stating the axioms or the uniqueness theorem; a short summary or an explicit statement of the relevant result would help readers assess the d_IR = 2 claim.","section":"§3.4, Eq. (4)"},{"comment":"The axis label 'Nf' is missing its subscript and the 16.5 asymptotic-freedom boundary is not marked on the axis; please improve the figure labeling.","section":"Fig. 3 (right panel)"},{"comment":"The paper repeatedly cites its own prior publications (Refs. [1]–[4], [15], [16]) for central results; while this is natural in a proceedings, the reader would benefit from explicit statements of which numerical results are new in the present work.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"This is a proceedings/talk manuscript, and the reviewing standard may differ from that for an archival research article. The central evidence for the IR phase is concentrated in the author's own earlier publications and in one unpublished reference [14]; if the journal expects a self-contained contribution, the evidentiary gap is substantial. I would suggest asking the authors to make the unpublished data available or to add a supplementary analysis of the power-law exponent. There is also a notable circularity in the scale-invariance argument that should be addressed in revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe takeaway: this is a conference proceedings paper, not a new research result. It restates Horvath's IR-phase scenario for QCD and the Anderson metal-to-critical picture, drawing almost entirely on the author's own prior work. If you haven't followed that series, it's a readable summary of the claims; if you have, there is nothing new here.\n\nWhat it does well: the logic of the phase classification in Eq. (1) is presented transparently, and the paper is honest about what is established by lattice simulation versus what is conjectural. The connection between Anderson-like mobility edges and IR-bulk separation is explained without hand-waving, and the author explicitly notes that full depletion of the spectrum between IR and bulk has not been verified. It also gives a fair nod to the competing metal-to-insulator scenario.\n\nThe soft spots are real but mostly inherited from the original papers. The central classification hinges on the exponent p in rho(lambda) ~ lambda^p being slightly negative in the deep IR. The evidence in Fig. 7 is cumulative spectral density over a finite window, and the paper does not show that the fitted p is stable as the volume grows or as the lower fitting cutoff drops. The continuum extrapolation in Fig. 6 addresses the strength of the IR peak, not the exponent p itself. Since p = -1 is the critical borderline separating the IR and B phases, a small finite-volume or discretization shift could flip the classification. The stress-test note lands squarely here. Also, one key reference ([14]) is listed as unpublished, which hampers independent checking. There is a mild circularity in Sec. 3.3: the near-power-law rho(lambda) is offered as evidence for IR scale invariance, and then scale invariance is invoked to explain the power law. The author doesn't hide this, but it is a conceptual weakness rather than a fatal one.\n\nThat said, the underlying scenario is plausible, and the original papers survived peer review in good journals. This proceedings version is an honest overview that flags its own limitations. It would serve someone entering the area, but it should be read alongside the primary sources. If submitted as a research article, I would want a referee to demand a careful analysis of the exponent extrapolation before accepting the phase classification. As a proceedings contribution, it is serviceable, but it does not constitute new evidence.\n\nRecommendation: worth engaging with if you work on QCD phase structure or Anderson localization in gauge theories. I'd let it through a proceedings-level review, but I would not treat it as a primary reference.","headline":"A clear proceedings summary of the IR-phase scenario, but it adds no new evidence and the phase classification rests on a power-law extrapolation that the paper never directly validates.","tokens_in":10595,"tokens_out":3487,"would_cite":false,"duration_ms":34292,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper argues that thermal QCD, just above the chiral crossover, enters a distinct infrared phase whose deep Dirac modes follow a near-critical power law set by Anderson-like mobility edges.","keywords":["IR phase","thermal QCD","Anderson localization","Dirac spectral density","mobility edge","metal-to-critical transition","lattice QCD","IR dimension"],"falsifier":"Compute the continuum-extrapolated deep-infrared spectral density for $N_f = 2+1$ QCD at physical quark masses at several temperatures between 200 and 230 MeV on lattices with $L \\ge 5\\ \\mathrm{fm}$ and multiple lattice spacings, and fit the exponent $p$ from $\\rho(\\lambda)$ for $\\lambda/T$ near zero; if $p$ tends to 0 or positive as the volume grows, or if the change across $T_{\\mathrm{IR}}$ does not sharpen with volume, the proposed IR phase is a lattice artifact rather than a true transition.","tokens_in":9512,"feed_emoji":"🔥","tokens_out":13311,"duration_ms":107875,"temperature":0.7,"pith_summary":"The paper argues that thermal QCD does not simply shed its infrared content upon heating; at a temperature $T_{\\mathrm{IR}}$ with $200\\ \\mathrm{MeV} < T_{\\mathrm{IR}} < 230\\ \\mathrm{MeV}$, just above the chiral crossover near $T_A \\approx 155\\ \\mathrm{MeV}$, the theory enters a distinct IR phase. In that phase the Dirac spectral density grows toward zero eigenvalue as $\\rho(\\lambda) \\sim \\lambda^p$ with $p$ close to $-1$, meaning deep-infrared modes proliferate instead of being depleted. The phase is defined by four features: separation of the infrared component from the bulk, scale-invariant glue in that component, non-analytic spectral behavior that may make the transition a true phase transition, and infinite glue screening lengths. The author ties these features to Anderson localization through two mobility edges, $\\lambda_A > 0$ and $\\lambda_{\\mathrm{IR}} = 0$, making the transition a metal-to-critical transition rather than a metal-to-insulator one. If the picture holds, hot QCD's long-range physics is controlled by critical near-zero Dirac modes, with possible connections to the near-perfect fluid behavior seen in heavy-ion experiments.","feed_headline":"A new thermal phase of QCD may appear between 200 and 230 MeV","feed_subtitle":"Deep infrared quark modes proliferate and turn critical, tying the quark-gluon plasma to Anderson localization.","key_machinery":"The central object is the Dirac spectral density $\\rho(\\lambda)$, the average number of Dirac eigenmodes per unit four-volume and unit spectral interval, whose deep-infrared behavior supplies the phase classification. The argument is carried by Anderson-like mobility edges: $\\lambda_A > 0$, already found in hot QCD, and $\\lambda_{\\mathrm{IR}} = 0$, proposed as a new critical point. These edges divide the spectrum into localized and critical regions, produce the non-analyticities that enforce IR-bulk separation, and shield the IR component from renormalization-group running. The newly introduced IR dimension, an effective spatial dimension obtained by counting how a mode's effective measure responds to an increasing infrared cutoff, provides the dimensional signal: discontinuities at the mobility edges and $d_{\\mathrm{IR}} = 2$ for near-zero critical modes.","core_discovery":"The central claim is that thermal SU(3) gauge theories with fundamental quarks are classified by the infrared exponent $p$ in $\\rho(\\lambda) \\propto \\lambda^p$: $p = 0$ is the IR-broken hadronic phase, $p < 0$ is the IR-symmetric phase (the IR phase), and $p > 0$ is the IR-trivial ultraviolet phase. For real-world $N_f = 2+1$ QCD at physical quark masses, the theory enters the IR phase at $T_{\\mathrm{IR}}$ satisfying $200\\ \\mathrm{MeV} < T_{\\mathrm{IR}} < 230\\ \\mathrm{MeV}$. The IR phase is characterized by proliferation of deep-IR Dirac modes, IR-bulk separation, scale-invariant IR glue, non-analyticity in spectral quantities, and infinite glue screening lengths. The metal-to-critical scenario adds a second Anderson-like mobility edge at $\\lambda_{\\mathrm{IR}} = 0$ alongside the known $\\lambda_A > 0$, and the effective IR dimension of the lowest near-zero modes is $d_{\\mathrm{IR}} = 2$ while exact zero modes have $d_{\\mathrm{IR}} = 3$.","pith_inferences":["If $\\lambda_{\\mathrm{IR}} = 0$ is a true mobility edge, then the IR phase transition is a realization of a zero-energy critical point in a chiral four-dimensional gauge theory; the same spectral diagnostic could be applied to other strongly coupled gauge theories to look for analogous phases.","The conjectured phase boundary in the mass-temperature plane predicts that $T_{\\mathrm{IR}}$ decreases as light quark masses are lowered toward the chiral limit; this could be tested with $N_f = 2$ or $N_f = 2+1$ simulations at pion masses below the physical one.","The difference between exact zero modes ($d_{\\mathrm{IR}} = 3$) and near-zero critical modes ($d_{\\mathrm{IR}} = 2$) suggests two distinct universality classes coexist at $\\lambda = 0$; a multifractal analysis of eigenmode intensities would test whether the topological zero-mode sector and the critical near-zero sector are truly independent.","If critical near-zero modes enhance long-range correlations, transport-like observables such as quark-number susceptibility or dilepton rates could show non-monotonic temperature dependence near 200-230 MeV; heavy-ion data at those temperatures might be reexamined for such a signal."],"forward_implications":["The chiral crossover at about 155 MeV is not the only thermal structure: QCD gains a second transition, possibly a true phase transition, at $T_{\\mathrm{IR}}$ between 200 and 230 MeV.","In the IR phase, the deep infrared sector behaves as an autonomous component with scale-invariant glue and infinite screening lengths, so low-energy observables receive long-range contributions from near-zero Dirac modes.","The mobility edges $\\lambda_A > 0$ and $\\lambda_{\\mathrm{IR}} = 0$ make the quark-gluon plasma a metal-to-critical system: modes between the edges are localized, while modes exactly at $\\lambda_{\\mathrm{IR}}$ are critical with $d_{\\mathrm{IR}} = 2$.","The classification extends across the space of SU(3) theories with fundamental quarks: increasing temperature, increasing flavor number, or decreasing quark masses all drive the sequence B $\\to$ IR $\\to$ UV, linking the thermal IR phase to the conformal window.","Exact zero modes have $d_{\\mathrm{IR}} = 3$ while the lowest near-zero modes have $d_{\\mathrm{IR}} = 2$, so the $d_{\\mathrm{IR}} = 2$ sector dominates the deep-IR action density and its temperature dependence."],"supporting_citations":[{"why":"Defines the IR phase, the power-law classification by p, and the transition temperatures.","marker":"[1]"},{"why":"Provides the first evidence of anomalous deep-IR Dirac mode accumulation and shows it survives the continuum limit in pure-glue QCD.","marker":"[2]"},{"why":"Establishes the unusual features of IR-phase low-energy modes, including IR dimension d_IR = 2 for near-zero modes.","marker":"[3]"},{"why":"Formulates the metal-to-critical scenario and the phase diagram with mobility edges lambda_A and lambda_IR.","marker":"[4]"},{"why":"Supplies the Anderson localization phenomenon and the notion of mobility edge on which the QCD analogy rests.","marker":"[5]"},{"why":"Identifies the Anderson-like mobility edge lambda_A > 0 in the quark-gluon plasma.","marker":"[9]"},{"why":"Provides real-world N_f = 2+1 QCD evidence for IR-bulk separation and d_IR = 2 near-zero modes at 234 MeV.","marker":"[15]"},{"why":"Confirms continuum scaling of the deep-IR mode abundance at T = 230 MeV for real-world QCD, supporting the T_IR bound.","marker":"[16]"},{"why":"Constructs effective counting measures on which the IR dimension is based.","marker":"[26]"},{"why":"Defines the IR dimension and shows it is the unique effective dimension probing increasing infrared cutoffs.","marker":"[27]"}],"fun_headline_variants":["QCD's new IR phase emerges between 200 and 230 MeV","Anderson localization shapes a novel QCD thermal phase","IR phase of QCD: critical modes link to Anderson localization","QCD gets an infrared phase, driven by Anderson-like localization"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument rests on the assumption that the near-pure power law $\\rho(\\lambda) \\sim \\lambda^p$ with $p = -1 + \\delta$, seen on finite lattices and extrapolated to the continuum in Fig. 6, is the true infinite-volume continuum behavior all the way down to $\\lambda = 0$; if finite-volume or discretization effects contaminate that extrapolation, the phase classification in Eq. (1) and the inferred mobility edges collapse.","fun_headline_variants_meta":{"raw":{"variants":["QCD's new IR phase emerges between 200 and 230 MeV","Anderson localization shapes a novel QCD thermal phase","IR phase of QCD: critical modes link to Anderson localization","QCD gets an infrared phase, driven by Anderson-like localization"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000104,"raw_usage":{"total_tokens":976,"prompt_tokens":834,"completion_tokens":142,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":450,"completion_tokens_details":{"reasoning_tokens":81}},"tokens_in":450,"tokens_out":142,"duration_ms":2558,"temperature":1.0,"reasoning_tokens":81,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T10:47:28.446597+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the continuum-extrapolated deep-infrared spectral density for $N_f = 2+1$ QCD at physical quark masses at several temperatures between 200 and 230 MeV on lattices with $L \\ge 5\\ \\mathrm{fm}$ and multiple lattice spacings, and fit the exponent $p$ from $\\rho(\\lambda)$ for $\\lambda/T$ near zero; if $p$ tends to 0 or positive as the volume grows, or if the change across $T_{\\mathrm{IR}}$ does not sharpen with volume, the proposed IR phase is a lattice artifact rather than a true transition.","supporting_citations":[{"cited_title":"Possible New Phase of Thermal QCD.Phys","cited_arxiv_id":null,"evidence_quote":"Defines the IR phase, the power-law classification by p, and the transition temperatures."},{"cited_title":"Phases of SU(3) Gauge Theories with Fundamental Quarks via Dirac Spectral Density.Phys","cited_arxiv_id":null,"evidence_quote":"Provides the first evidence of anomalous deep-IR Dirac mode accumulation and shows it survives the continuum limit in pure-glue QCD."},{"cited_title":"Unusual Features of QCD Low-Energy Modes in the Infrared Phase.Phys","cited_arxiv_id":null,"evidence_quote":"Establishes the unusual features of IR-phase low-energy modes, including IR dimension d_IR = 2 for near-zero modes."},{"cited_title":"Anderson metal-to-critical transition in QCD.Phys","cited_arxiv_id":null,"evidence_quote":"Formulates the metal-to-critical scenario and the phase diagram with mobility edges lambda_A and lambda_IR."},{"cited_title":"Kovacs and Ferenc Pittler","cited_arxiv_id":null,"evidence_quote":"Identifies the Anderson-like mobility edge lambda_A > 0 in the quark-gluon plasma."},{"cited_title":"Separation of Infrared and Bulk in Thermal QCD.JHEP, 2024(12):101, 2024","cited_arxiv_id":null,"evidence_quote":"Provides real-world N_f = 2+1 QCD evidence for IR-bulk separation and d_IR = 2 near-zero modes at 234 MeV."},{"cited_title":"Dirac spectral density in Nf=2+1 QCD at T=230 MeV.Phys","cited_arxiv_id":null,"evidence_quote":"Confirms continuum scaling of the deep-IR mode abundance at T = 230 MeV for real-world QCD, supporting the T_IR bound."},{"cited_title":"Effective Number Theory: Counting the Identities of a Quantum State.Entropy, 22:1273, 2020","cited_arxiv_id":null,"evidence_quote":"Constructs effective counting measures on which the IR dimension is based."},{"cited_title":"Counting-Based Effective Dimension and Discrete Regularizations.Entropy, 25(3):482, 2023","cited_arxiv_id":null,"evidence_quote":"Defines the IR dimension and shows it is the unique effective dimension probing increasing infrared cutoffs."}],"review_version":1}