{"id":"1a6cf9cc-4231-476e-ae42-6d5c89a7f16c","arxiv_id":"2411.11147","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"The random-field Ising model's SUSY/DR fixed point annihilates with an unstable fixed point at d_DR ≈ 5.11 ± 0.09, so no SUSY/DR critical point survives below this dimension in the nonperturbative FRG.","lead":"This paper uses a functional renormalization group calculation to argue that the supersymmetry and dimensional-reduction fixed point of the random-field Ising model disappears at about 5.11 spatial dimensions, rather than merely becoming unstable as earlier perturbative work suggested. The result matters because it explains why the standard epsilon expansion misses this mechanism, and it gives simulations a concrete target for where SUSY/DR remnants should vanish.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The disappearance claim rests on the generic-coalescence assumption; a nongeneric cancellation would leave the SUSY/DR fixed point present but unstable below d_DR.","rationale":"The reader's weakest_assumption identifies precisely the generic-coalescence condition as the load-bearing assumption, and my reading agrees. The paper is otherwise internally consistent: the FRG truncations preserve SUSY Ward identities, the four approximation levels bracket d_DR around 5.11, and the perturbative 2-loop Lambda_2 is reproduced near d=6. The RFO(N>2)M detour provides a controlled setting where the annihilation mechanism is verified exactly at one loop and at two loops (Appendix B), which supports the plausibility of the analogous mechanism in the RFIM. However, the extrapolation to the RFIM is not exact: the RFIM fixed-point equations are solved only within truncated function space, and the proof that the annihilation is generic rather than accidental relies on nonzero coefficients that are not explicitly evaluated. The paper's own appendices admit the heuristic character of the argument. The requested test—an independent direct search for the annihilation dimension in DE4, plus evaluation of the coefficient product—would settle whether the central claim is a robust property of the FRG equations or an artifact of the truncation and the chosen d_DR definition. Since this is the same concern the reader raised, the verdict remains CONDITIONAL: believable within the approximation scheme, but requiring independent confirmation or released exact coefficient data before full acceptance.","tokens_in":46119,"tokens_out":3109,"duration_ms":33889,"concrete_test":"Using the DE4 flow equations (38-43), compute the coefficient A_*(phi) and the analogous linear/cubic coefficients at d_DR = 5.0678 and check that the discriminant-like product (e.g., A_*(phi) times the derivative of the fixed-point equation with respect to delta_*2) is nonzero over the field range. Then solve the full fixed-point equations for the six functions delta_*2, x_a;*2, x_e;*2, x_f;*2, s_*2, s_*3 directly, without invoking the linearized eigenvalue, and locate the dimension d_annihil at which real solutions cease to exist. If d_annihil differs from the zero of Lambda_2 by more than the numerical precision (currently claimed to be 5 digits), the coincidence claim fails. The same test can be run at DE2, and for the RFO(N)M the analogous coefficient product should be checked analytically at two loops as a controlled benchmark.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central nonperturbative claim is that when the eigenvalue Lambda_2 associated with Feldman's operator F4 vanishes at d_DR about 5.1, the SUSY/DR fixed point disappears by coalescing with an unstable partner, rather than merely becoming unstable as KRT argue. The evidence for this is the observed square-root collapse of the stable and unstable eigenvalues in LPA', LPA'', DE2, and DE4 (Fig. 6). The mechanism, however, is established only through the generic-coalescence argument spelled out in Appendices B, E, and G: one assumes that the coefficients multiplying the nonlinear terms in the fixed-point equation for delta_*2 (e.g., A_*(phi) in Eq. (24), or R_NDR(X*0)Q''_NDR(X*0) in Appendix B) are nonzero and have the required signs. If these coefficients vanish accidentally or by a hidden symmetry, the two fixed-point branches cross instead of annihilating, and the SUSY/DR fixed point would exist below d_DR as an unstable fixed point—exactly the KRT scenario the paper rejects. For the RFO(N>2)M the authors verify the generic condition explicitly at one and two loops, but for the RFIM no such verification is provided; the conclusion relies on the numerical solutions of the truncated equations near d_DR. The authors themselves label the Appendix E/G arguments 'not rigorous' and 'indicative'. Because d_DR is defined as the zero of Lambda_2, the apparent coincidence of this zero with the disappearance is partly built into the analysis; an independent determination of the annihilation dimension is needed to confirm that the coincidence is not a truncation artifact.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper revisits the Parisi-Sourlas supersymmetry (SUSY) and dimensional-reduction (DR) properties of random-field models, focusing on the random-field Ising model (RFIM) and the random-field O(N) model (RFO(N)M). The central claim is that, within the authors' nonperturbative functional renormalization group (FRG) framework, the SUSY/DR fixed point of the RFIM does not merely become unstable below some dimension but disappears entirely by coalescing with an unstable SUSY/DR fixed point at d_DR ≈ 5.11 ± 0.09. This disappearance is associated with the vanishing of the eigenvalue Λ2 of Feldman's operator F4. The paper also analyzes the analogous mechanism in the RFO(N>2)M near d=4, where the SUSY/DR fixed point disappears at N=18 at one-loop order. The authors compare their results with the perturbative ϵ=6−d calculations of Kaviraj, Rychkov, and Trevisani (KRT), argue that the perturbative expansion cannot capture the annihilation mechanism, and provide an error bar for d_DR from four truncation levels of their FRG scheme (LPA', LPA\", DE2, and DE4).","tokens_in":46400,"tokens_out":3519,"duration_ms":36191,"significance":"If the central claim is correct, the paper resolves a long-standing question about the mechanism of SUSY/DR breakdown in random-field systems: below d_DR there is no SUSY/DR fixed point at all, so no fine-tuning of the disorder distribution can restore dimensional reduction. The paper provides a concrete, falsifiable prediction (d_DR ≈ 5.11 ± 0.09) and supports it by comparing several successive truncation levels and by matching the exact two-loop perturbative results near d=6 to within 20–30% at the DE4 level. The RFO(N)M analysis is particularly valuable because there the coalescence mechanism can be checked explicitly at one- and two-loop order. However, the generality of the mechanism for the RFIM rests on a generic-coalescence assumption that the authors themselves label as 'not rigorous' and 'indicative' in Appendices E and G, and no explicit verification of the required non-vanishing coefficients is provided for the RFIM. This limits the certainty of the central disappearance claim, although the numerical evidence from multiple truncations is substantial.","major_comments":[{"comment":"The claim that the SUSY/DR fixed point disappears for d < d_DR depends on the generic-coalescence assumption that certain coefficient functions do not vanish accidentally or by symmetry. In Appendix E, the argument assumes A*(φ) ≠ 0 and in Appendix G the absence of 'accidental (or symmetry-induced) cancellation' is assumed. The authors explicitly state in Appendix E that the argument 'is not rigorous but is indicative of what the generic behavior should be,' and Appendix G similarly relies on generic expectations. For the RFO(N)M, Appendix B verifies R_NDR(X*0)Q''_NDR(X*0) ≠ 0 at one- and two-loop order, but no analogous explicit check is performed for the RFIM. If a nongeneric cancellation occurred, the SUSY/DR fixed point would survive below d_DR as an unstable fixed point, which is precisely the scenario of KRT that the paper rejects. This is a load-bearing point that needs to be strengthened, for example by numerically verifying that the relevant products of coefficients are nonzero with controlled error bars in the DE2 and DE4 truncations near d_DR, or by providing an independent argument ruling out such cancellations.","section":"Sec. IV.C and Appendices E, G"},{"comment":"The critical dimension d_DR is determined as the location where the eigenvalue Λ2(d) vanishes, and the disappearance of the fixed point is then identified with this same value. This makes the 'coincidence' of the vanishing of Λ2 and the annihilation of the fixed point partly built into the procedure. An independent determination would be preferable: for instance, one could solve the fixed-point equations for δ*,2(φ) (and the additional functions at DE4) directly and demonstrate that no real solution exists below some dimension, or measure the divergence of the Larkin time as d → d_DR⁻. The authors mention this alternative in Sec. IV.C but say it is 'less accurate' than the eigenvalue criterion; however, given that the central claim is about existence of the fixed point, an independent existence check, even if less precise, would provide decisive evidence that the disappearance is not an artifact of the chosen diagnostic.","section":"Sec. V.C and Eq. (44)"},{"comment":"The convergence evidence is presented as 'apparent rapid convergence,' but the numerical sequence d_DR = 5.2005 (LPA'), 5.0180 (LPA\"), 5.1307 (DE2), 5.0678 (DE4) oscillates rather than monotonically converging, and the two-loop coefficients at DE4 differ from the exact values by about 20% for Λ2 and Λ3 and about 30% for Λ3/2. The quoted error bar of ±0.09 in Eq. (44) is based on the spread of these four values, but the regulator dependence is stated to be within this range rather than being explicitly demonstrated. Since the central quantitative prediction is d_DR ≈ 5.11 ± 0.09, the paper should either provide a more transparent error estimate that separates truncation and regulator uncertainties, or explicitly state that the error bar is a heuristic estimate rather than a rigorous bound.","section":"Sec. V.C, Table I"}],"minor_comments":[{"comment":"The discussion of the discrepancy in canonical dimensions of the F_{2p} operators for p ≥ 3 is clear, but the sentence 'which differs by an additive term −2(p−2) from the result of [1]' would benefit from an explicit statement that for p=2 both formalisms agree, since that agreement is crucial for the main conclusions.","section":"Sec. IV.A"},{"comment":"The dashed curve is described as a '2-loop calculation in d=6−ϵ, together with a plausible extrapolation,' but the extrapolation procedure is not specified. Please state how the extrapolated curve is obtained, since the comparison with the nonperturbative results depends on this choice.","section":"Fig. 1 caption"},{"comment":"There is a typo in the phrase 'the explanantion of the SUSY/DR breakdown' in the first sentence of Sec. V.A; it should read 'explanation.'","section":"Sec. V.A"},{"comment":"The text states that a Mathematica notebook 'can be made available' for the DE4 flow equations. Since the equations are too long to display, the paper should either include the notebook as supplementary material or provide a stable repository link, so that the numerical results can be reproduced independently.","section":"Appendix F"}],"recommendation":"major_revision","confidential_remarks":"The paper is substantially based on the authors' own prior FRG formalism, and the self-citation density is high. However, the independent two-loop perturbative comparison and the explicit RFO(N)M checks provide genuine anchors. The main reservation for acceptance is the gap between the numerical evidence and the rigorous status of the coalescence mechanism for the RFIM; if the authors can provide an independent existence check or an explicit non-vanishing-coefficient verification, the paper would be significantly stronger. The comparison with KRT is fair overall, though the tone is at times assertive about conclusions that are explicitly labeled non-rigorous in the appendices."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The real news here is the DE4 computation and the error bar: d_DR ≈ 5.11 ± 0.09 assembled from four truncation levels, plus the clean demonstration in the RFO(N>2)M that the SUSY/DR fixed point disappears exactly when the Λ2 eigenvalue vanishes. The paper also makes a solid structural point: a perturbative expansion in ϵ = 6−d or 1/N cannot see a fixed-point annihilation caused by nonlinearity, so KRT's 'unstable but still present' scenario is not directly testable in that framework.\n\nWhat I credit: near d=6 the DE4 eigenvalues match the exact 2-loop results to roughly 20–30%, which gives an external anchor the earlier FRG papers lacked. The four-approximation spread is a reasonable way to estimate error, and the RFO(N)M case is nearly airtight because the authors explicitly verify the generic-coalescence condition at 1- and 2-loop order.\n\nWhere it gets softer: for the RFIM itself, they do not verify that the products like R_NDR(X*0)Q''_NDR(X*0) are nonzero in the same explicit way; they rely on numerical fixed-point solutions plus Appendices E and G, which they themselves describe as 'not rigorous' and 'indicative.' If a nongeneric cancellation occurs, the SUSY/DR fixed point would survive below d_DR as an unstable fixed point—exactly the KRT scenario the paper rejects. That concern is real, and the stress-test note lands. Also, because d_DR is defined as the zero of Λ2, the coincidence of that zero with the annihilation is partly built into the analysis; an independent determination of the annihilation dimension would strengthen the claim. No code is shipped and the full DE4 beta functions are omitted (a notebook is promised), which makes independent replication harder than it needs to be.\n\nNone of this is fatal. The paper is honest about its assumptions, the approximation scheme is systematic, and the external anchor near d=6 is genuine evidence. The central claim is believable within the FRG scheme but not yet proven. A serious referee should engage with it, and the authors should be pushed to release the code and to test the generic-coalescence condition more directly. If that check survives, this will be the reference for the d_DR ≈ 5.1 claim.\n\nRecommendation: send to peer review.","headline":"A credible, well-scoped FRG case that the RFIM SUSY/DR fixed point annihilates at d≈5.11±0.09, but the RFIM annihilation mechanism still depends on a generic-coalescence assumption that the authors honestly label non-rigorous.","tokens_in":46980,"tokens_out":1822,"would_cite":true,"duration_ms":18978,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["82B27","82B28","82B44"],"pacs":["11.10.Hi","75.40.Cx"],"model":"deepseek-v4-flash","headline":"The paper argues that the SUSY/DR controlling fixed point of the random-field Ising model disappears at $d_{\\mathrm{DR}}\\approx 5.11\\pm 0.09$ by annihilating with an unstable twin fixed point, and that the random-field $O(N>2)$ model…","keywords":["random-field Ising model","Parisi-Sourlas supersymmetry","dimensional reduction","functional renormalization group","Feldman operators","random-field O(N) model","fixed-point annihilation","cuspy fixed point"],"falsifier":"Evaluate the product $R_{N_{\\mathrm{DR}}}(X_{\\ast 0})\\,Q''_{N_{\\mathrm{DR}}}(X_{\\ast 0})$ at the candidate $N_{\\mathrm{DR}}$ (or its DE4 analogue at the candidate $d_{\\mathrm{DR}}$): if it vanishes, the fixed points cross rather than annihilate and the SUSY/DR fixed point could survive below the threshold. A direct numerical search for a cuspless fixed-point solution $\\delta_{\\ast,2}(\\varphi)$ in $d=5$ below $d_{\\mathrm{DR}}$, with the full functional dependence of the cumulants retained, would also decide the issue: finding one would falsify the claim, while in $d=4$ observing true asymptotic SUSY/DR scaling in a simulation with fine-tuned disorder would contradict the no-fixed-point conclusion.","tokens_in":45870,"feed_emoji":"🧲","tokens_out":11482,"duration_ms":131029,"temperature":0.7,"pith_summary":"This paper aims to establish that the breakdown of Parisi-Sourlas supersymmetry (SUSY) and dimensional reduction (DR) in random-field models is a fixed-point annihilation rather than a mere instability. The nonperturbative functional renormalization group (FRG) described here predicts that the SUSY/DR fixed point of the random-field Ising model disappears at $d_{\\mathrm{DR}}\\approx 5.11\\pm 0.09$, at the exact dimension where the eigenvalue $\\Lambda_2$ associated with Feldman's operator $F_4$ hits zero; below that dimension only a ``cuspy'' fixed point with broken SUSY and DR exists. The same mechanism is exhibited analytically for the random-field $O(N>2)$ model near $d=4$, where the SUSY/DR fixed point vanishes at $N=18$ at one-loop order. The annihilation follows from the nonlinearity of the FRG fixed-point equation for the second cumulant, which is why the perturbative $\\epsilon=6-d$ and $1/N$ expansions cannot see it. If the prediction is correct, simulations in $d=4$ and $d=5$ should show at most transient SUSY/DR-like remnants, never an asymptotic SUSY/DR critical point.","feed_headline":"Random-field Ising SUSY fixed point vanishes at d≈5.11","feed_subtitle":"Functional RG: the cuspless fixed point annihilates with an unstable twin; epsilon expansion cannot see it.","key_machinery":"The load-bearing object is Feldman's operator $F_4$, the $p=2$ term in the expansion of the second cumulant in powers of $(\\varphi_a-\\varphi_b)^2$, whose 1-PI counterpart is the function $\\delta_2(\\varphi)$. Its fixed-point equation is nonlinear in $\\delta_2(\\varphi)$ itself---$A_*(\\varphi)\\,\\delta_2(\\varphi)^2 + L_*(\\varphi,\\partial_\\varphi,\\partial_\\varphi^2)\\,\\delta_2(\\varphi)+B_*(\\varphi)=0$ at DE2, with up to cubic nonlinearities at DE4---while all higher-order $F_{2p}$ equations are linear. The eigenvalue $\\Lambda_2$ is obtained by linearizing this equation around the fixed point, and the vanishing of $\\Lambda_2$ coincides with the merging of two solutions and the disappearance of the fixed point below $d_{\\mathrm{DR}}$. In the RFO($N>2$)M the same mechanism is carried by the polynomial $Q_{N,j}(R''(1))$ whose degree grows with loop order and whose derivative at the fixed point is $\\Lambda_2(N)$. The functional, nonperturbative character of the flow is what allows the fixed point itself to be followed; the perturbative expansion around $d=6$ treats only the marginal coupling constant and cannot see the nonlinear coalescence.","core_discovery":"The central claim is that a SUSY/DR fixed point exists only above a critical dimension (or above a critical $N$ for the $O(N)$ model) and ceases to exist below it. In the FRG, the second cumulant of the renormalized random field is expanded around equal field arguments; the coefficient of the $(\\varphi_1-\\varphi_2)^2$ term, $\\delta_2(\\varphi)$, satisfies a fixed-point equation that is nonlinear in $\\delta_2$ itself, with the nonlinearity growing from quadratic at DE2 to cubic at DE4. The stability eigenvalue $\\Lambda_2$ is the derivative of that nonlinear function at the fixed point. Generically, when the eigenvalue vanishes the stable and unstable fixed-point branches merge with a square-root behavior and no real solution remains below $d_{\\mathrm{DR}}$, so the full fixed point---including all cumulants at nonequal field arguments---disappears. The cuspy fixed point that replaces it has a nonanalytic dependence of the second cumulant on the field difference, and this cusp is the signature of broken SUSY and DR. For the RFO($N>2$)M in $d=4+\\epsilon$ the same logic is carried by a polynomial equation for $R''(1)$, and the SUSY/DR fixed point disappears at $N=18$ at one-loop order.","pith_inferences":["I infer that the annihilation mechanism implies an order-of-limits subtlety: extrapolating the 2-loop $\\epsilon=6-d$ result gives $\\Lambda_2=0$ near $d\\approx 4.6$, but the nonperturbative zero sits at $d\\approx 5.11$ because the eigenvalue has a square-root branch point.","A testable extension suggested by the same logic is to study long-range RFIM variants in which $d_{\\mathrm{DR}}$ depends on the interaction range and could be shifted into dimensions accessible to simulation.","The authors leave open whether another unstable cuspy fixed point exists below $d_{\\mathrm{DR}}$; finding it would complete the fixed-point diagram and sharpen the prediction for corrections to scaling.","I infer that the discrepancy with KRT about canonical dimensions of $F_6$ and higher operators affects subleading corrections but not the main conclusion, since all derivations agree on $F_4$."],"forward_implications":["If the claim is correct, no fine-tuning of the bare random-field distribution can produce a SUSY/DR critical point in $d=4$, because the fixed point does not exist there.","In $d=5$, only slightly above $d_{\\mathrm{DR}}\\approx 5.11$, simulations should see SUSY/DR-like behavior over finite sizes, but the asymptotic critical regime must be governed by the cuspy, SUSY/DR-broken fixed point.","The avalanche eigenvalue $\\Lambda_{3/2}$ is still irrelevant at $d_{\\mathrm{DR}}$, so the disappearance is not caused by avalanches; avalanches become the controlling mechanism only below $d_{\\mathrm{DR}}$.","In the RFO($N>2$)M near $d=4$, the SUSY/DR fixed point ceases to exist for $N<18$ at one loop, so the cuspy fixed point is the only critical fixed point in that regime.","Near $d=6$ the nonperturbative FRG reproduces the 2-loop perturbative eigenvalues, so the two calculations agree where perturbation theory is controlled; they differ by the square-root annihilation below $d_{\\mathrm{DR}}$. "],"supporting_citations":[{"why":"KRT perturbative RG study that identifies F_4 and F_6 as dangerous SUSY-destabilizing operators and proposes that the SUSY fixed point becomes unstable; serves as the main alternative scenario the paper rejects.","marker":"[1]"},{"why":"Rychkov's lecture course presenting the unstable-but-present interpretation and questions about the nonperturbative FRG that this paper addresses.","marker":"[2]"},{"why":"Feldman's construction of the F_{2p} operators and the two-loop anomalous dimensions; supplies the eigenvalue formula for F_4 used throughout.","marker":"[50]"},{"why":"Derivation of nonperturbative FRG flow equations for the 1-PI cumulants with SUSY Ward identities and the zero-temperature fixed point; the central formalism.","marker":"[19]"},{"why":"Earlier nonperturbative FRG work establishing the cuspless SUSY/DR fixed point and its Ward identities; basis for the present fixed-point analysis.","marker":"[26]"},{"why":"Previous study identifying d_DR, the eigenvalues Lambda_2 and Lambda_3/2, and the coalescence/cusp scenario for both the RFO(N)M and the RFIM; provides the starting point.","marker":"[27]"},{"why":"DE2-level nonperturbative FRG computation of d_DR and corrections to scaling in d=5; used alongside the new DE4 calculation for the error bar.","marker":"[10]"},{"why":"Two-loop perturbative FRG in 6-epsilon computing the avalanche eigenvalue Lambda_3/2; shows the cuspy perturbation is irrelevant when Lambda_2 vanishes.","marker":"[39]"},{"why":"Two-loop perturbative FRG results for the RFO(N)M in d=4+epsilon, giving N_DR=18-(49/5)epsilon; supports the detour analysis.","marker":"[46]"}],"fun_headline_variants":["SUSY fixed point vanishes at d≈5.11 in RFIM","Random-field models: SUSY breaks below critical dimension","FRG shows why epsilon expansion misses SUSY breakdown","Cusp fixed point wins: no SUSY below d≈5.11","Nonlinear FRG: SUSY fixed point dies at d≈5.11"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the coefficients controlling the fixed-point equation for $\\delta_2(\\varphi)$ (or for $R''(1)$ in the $O(N)$ model) do not vanish accidentally or by symmetry at the critical point, so the two fixed-point branches genuinely merge and disappear; if such a nongeneric cancellation occurred, the SUSY/DR fixed point could survive below $d_{\\mathrm{DR}}$ as an unstable fixed point, which is the scenario the paper rejects. The rapid apparent convergence of the truncated FRG hierarchy is a separate additional assumption.","fun_headline_variants_meta":{"raw":{"variants":["SUSY fixed point vanishes at d≈5.11 in RFIM","Random-field models: SUSY breaks below critical dimension","FRG shows why epsilon expansion misses SUSY breakdown","Cusp fixed point wins: no SUSY below d≈5.11","Nonlinear FRG: SUSY fixed point dies at d≈5.11"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000291,"raw_usage":{"total_tokens":1772,"prompt_tokens":1086,"completion_tokens":686,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":702,"completion_tokens_details":{"reasoning_tokens":594}},"tokens_in":702,"tokens_out":686,"duration_ms":7105,"temperature":1.0,"reasoning_tokens":594,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T18:51:50.753859+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the product $R_{N_{\\mathrm{DR}}}(X_{\\ast 0})\\,Q''_{N_{\\mathrm{DR}}}(X_{\\ast 0})$ at the candidate $N_{\\mathrm{DR}}$ (or its DE4 analogue at the candidate $d_{\\mathrm{DR}}$): if it vanishes, the fixed points cross rather than annihilate and the SUSY/DR fixed point could survive below the threshold. A direct numerical search for a cuspless fixed-point solution $\\delta_{\\ast,2}(\\varphi)$ in $d=5$ below $d_{\\mathrm{DR}}$, with the full functional dependence of the cumulants retained, would also decide the issue: finding one would falsify the claim, while in $d=4$ observing true asymptotic SUSY/DR scaling in a simulation with fine-tuned disorder would contradict the no-fixed-point conclusion.","supporting_citations":[{"cited_title":"Balog and G","cited_arxiv_id":null,"evidence_quote":"Feldman's construction of the F_{2p} operators and the two-loop anomalous dimensions; supplies the eigenvalue formula for F_4 used throughout."},{"cited_title":"Tissier and G","cited_arxiv_id":null,"evidence_quote":"Derivation of nonperturbative FRG flow equations for the 1-PI cumulants with SUSY Ward identities and the zero-temperature fixed point; the central formalism."},{"cited_title":"Tissier and G","cited_arxiv_id":null,"evidence_quote":"Earlier nonperturbative FRG work establishing the cuspless SUSY/DR fixed point and its Ward identities; basis for the present fixed-point analysis."},{"cited_title":"Tissier and G","cited_arxiv_id":null,"evidence_quote":"Previous study identifying d_DR, the eigenvalues Lambda_2 and Lambda_3/2, and the coalescence/cusp scenario for both the RFO(N)M and the RFIM; provides the starting point."},{"cited_title":"Angelini, C","cited_arxiv_id":null,"evidence_quote":"DE2-level nonperturbative FRG computation of d_DR and corrections to scaling in d=5; used alongside the new DE4 calculation for the error bar."},{"cited_title":"Le Doussal and K","cited_arxiv_id":null,"evidence_quote":"Two-loop perturbative FRG in 6-epsilon computing the avalanche eigenvalue Lambda_3/2; shows the cuspy perturbation is irrelevant when Lambda_2 vanishes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Two-loop perturbative FRG results for the RFO(N)M in d=4+epsilon, giving N_DR=18-(49/5)epsilon; supports the detour analysis."}],"review_version":1}