{"id":"4f7d8c37-328d-4be6-9a15-b1856cef6e92","arxiv_id":"2607.14151","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Candia-v2 implements (approximate) N3LO x-space DGLAP evolution — exact four-loop non-singlet splitting functions, envelope-approximated singlet inputs, three-loop heavy-quark matching — as a public C++ package with GMVFN subtraction-PDF tools.","lead":"Candia-v2 is an open-source C++/Fortran package that evolves the proton's quark and gluon densities under the DGLAP equations to the fourth order in QCD, directly in the momentum-fraction (x) variable. It matters for precision LHC, HL-LHC, and EIC phenomenology — and its credibility hinges on validation against existing benchmarks, which the paper does not show.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central N3LO-accuracy claim rests on an unvalidated numerical core: all convolutions use a fixed, hand-chosen Gauss-Legendre grid (Sec. 5.4.2) with no grid-refinement or external benchmark check shown, so Table 2 could conceal numerically induced errors.","rationale":"The reader's conditional verdict is well aligned with my reading. The strongest claim — that Candia-v2 correctly performs x-space (a)N3LO DGLAP evolution — requires that the numerical convolution machinery faithfully represents the continuum convolutions. That condition is least secure precisely because the paper gives no convergence study and no external comparison. The grid defaults (Sec. 5.4.2) are presented as static choices, and the only 'tests' are explicitly non-standard benchmarking scripts (Sec. 5.2). Prior bugs (Sec. 5.5) further undermine the assumption that implementation details are correct until demonstrated otherwise. I find no internal inconsistency in the recursion relations themselves: the algorithm is a known, published approach (Ref. [6]) and the paper transparently credits the exact NS results and approximate singlet inputs. The overstatement of 'N3LO' in the abstract versus 'approximate N3LO' in the conclusions is a labeling issue, not the central technical risk. The concrete test I propose would resolve the numerical-convergence concern by direct grid refinement and benchmark comparison; if the results are stable, the central claim would be significantly strengthened. Because the evidence is absent rather than contradictory, the appropriate verdict remains CONDITIONAL, which matches the reader's original assessment — hence UNCHANGED.","tokens_in":26689,"tokens_out":3005,"duration_ms":32001,"concrete_test":"Run the Les Houches benchmark evolution of Table 2 with the default grid (100/50/25 nodes, 50 Gauss points, 4 interpolation points) and with a refined grid (e.g., 200/100/50 nodes, 100 and 200 Gauss points, 8 interpolation points), keeping all physics inputs fixed. Compare the evolved PDFs (xuv, xg, xc+, xb+) at x = 1e-5, 1e-3, 0.1, 0.5, 0.9 at Q=100 GeV for each perturbative order. Also compare the same quantities at LO/NLO/NNLO against the published Les Houches benchmark values (Ref. [101]) and at aN3LO against the evolution comparison of Ref. [75]. If any value shifts by more than 1% under grid refinement (or 0.1% for x≥0.1), or disagrees with the benchmarks beyond the quoted precision, the numerical core is not converged and the N3LO-accuracy claim needs qualification.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that Candia-v2 solves the DGLAP equations at (a)N3LO accuracy in x-space. Every convolution is evaluated by Gauss-Legendre quadrature on a fixed three-segment grid with defaults of 100/50/25 nodes, 50 Gauss points, and 4 interpolation points (Sec. 5.4.2). The integrands include plus-distributions and delta-functions, and the small-x region is particularly sensitive to discretization. The paper shows no grid-refinement, quadrature-order, or iteration-convergence study (Sec. 3, Figs. 1-3), and the repository's tests are explicitly 'not standard tests' but benchmarking/plot files (Sec. 5.2). The disclosed v1 bugs in the matching procedure and scale-ratio handling (Sec. 5.5) show that implementation details are error-prone. If the discretized recursion relations do not reproduce the continuum DGLAP solution to well below the N3LO perturbative corrections, then Table 2 and Figs. 1-3 are not reliable N3LO results even if the analytic algorithm is correct. This is load-bearing because the paper offers no evidence that the numerical error is smaller than the claimed N3LO accuracy.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript presents Candia-v2, an open-source C++/Fortran reimplementation of the x-space Candia algorithm for DGLAP evolution. It claims to generalize the previous version to (approximate) N3LO accuracy by implementing four-loop non-singlet splitting functions, state-of-the-art approximations for singlet splitting functions, three-loop operator-matrix-element matching via libome, and optional LHAPDF interfaces. The algorithm is based on logarithmic expansions and recursion relations for coefficients in x-space. Results are shown for the Les Houches toy model evolved from 2 GeV^2 to 10^4 GeV^2 in the VFNS, including scale dependence and heavy-quark subtraction/residual PDFs. The paper also provides a detailed program description, API overview, and a list of bug fixes from Candia-v1.","tokens_in":26876,"tokens_out":12297,"duration_ms":126250,"significance":"If the numerical implementation is correct, Candia-v2 would be a useful independent x-space evolution tool at approximate N3LO, with practical features such as subtraction PDFs, LHAPDF I/O, parallelization, and modular design. The paper explicitly uses exact four-loop non-singlet splitting functions, state-of-the-art singlet approximations, and libome for three-loop OMEs, and it openly discusses known v1 bugs. These are strengths that make the package potentially valuable to the community. The main risk is the absence of any quantitative validation of the numerical core, which underlies all reported N3LO results.","major_comments":[{"comment":"The recursion relations in Eq. (24) contain the undefined symbol gamma in the denominators of Z21, Z22, Z31, Z32, and Z43. Since these relations are the core of the N3LO implementation, the algorithm is not self-contained as written. Please define gamma, presumably gamma = sqrt(4cbar - bbar^2) as suggested by Eq. (21), or explicitly refer to the corresponding definition in Ref. [6].","section":"Eq. (24)"},{"comment":"No external validation is provided for the central N3LO-accuracy claim. The results are compared only across LO/NLO/NNLO/N3LO within the same code. The paper cites the aN3LO benchmarking paper Ref. [75] but does not compare against it or any other evolution code (e.g., PEGASUS, HOPPET, APFEL, EKO). The repository's tests are explicitly 'not standard tests' (Sec. 5.2), and the disclosed v1 bugs (Sec. 5.5) show that implementation errors are a real risk. A direct numerical comparison with at least one independent evolution code is essential.","section":"Sec. 3, Table 2"},{"comment":"The convolution quadrature and the iteration/truncation of the logarithmic series are not validated. All convolutions use a fixed three-segment grid with 100/50/25 points, 50 Gauss-Legendre nodes, and 4 interpolation points (Sec. 5.4.2). The example uses iterations=10, trunc_idx=10, with no convergence study in these parameters. Since the integrands involve plus-distributions and delta-functions, and the small-x region is highly sensitive, the numerical error could be comparable to the N3LO corrections. Grid-refinement and iteration-convergence tests are needed to show the numerical error is below the target accuracy.","section":"Sec. 5.4.2, Sec. 3"},{"comment":"The implementation of heavy-quark threshold matching inside the solver is not described. The AlphaS class handles n_f changes and libome provides OMEs, but the paper does not explain how the OMEs are applied at each threshold within the recursion framework. Given that v1 had a bug in the matching procedure (Sec. 5.5), the N3LO charm and bottom results in Fig. 2 cannot be assessed. A concise description of the matching step and a threshold-crossing check (e.g., PDF continuity) would address this.","section":"Sec. 5.4.3, Sec. 5.5"}],"minor_comments":[{"comment":"The text 's=n+m+ell+k' uses an undefined k (and ell is not used in the sums). This appears to be a typographical error.","section":"Eq. (22)"},{"comment":"The notation R(2), R(1), R(0) should be R2, R1, R0 for consistency with Eq. (25).","section":"Eq. (29)"},{"comment":"The citation 'GMVFN schemes currently in use [104? ? ?–113]' contains placeholder question marks and is broken.","section":"Sec. 4"},{"comment":"The abstract states 'N3LO accuracy in QCD', while the conclusions and Sec. 3 correctly use 'approximate N3LO' (aN3LO). Please make the wording consistent, since the singlet splitting functions are approximations.","section":"Abstract / Sec. 6"},{"comment":"The variables order, iterations, and trunc_idx are declared as const double but represent integer counts; using an unsigned integer type would be clearer.","section":"Listing 1"},{"comment":"The description of the 'tests' as 'not standard tests' is confusing; consider renaming them to 'benchmark and plot-generation programs'.","section":"Sec. 5.2"}],"recommendation":"major_revision","confidential_remarks":"The manuscript describes a potentially useful tool, but the missing validation and the undefined gamma in Eq. (24) are blocking. I would encourage the editor to require an independent benchmark comparison and a convergence study before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a public, modernized C++ reincarnation of the Candia x-space DGLAP solver, with exact four-loop non-singlet splitting functions, the current approximate singlet inputs, libome-based three-loop matching, LHAPDF I/O, and a new subtraction/residual PDF module. The genuinely new content is the code itself and its N3LO output; the recursion relations were published by the same authors in Ref. [6], and the paper says so. The software is real, public, GPLv3, builds through CMake, and the bug-fix list from v1 is unusually candid.\n\nWhat the paper does well: it gives a concrete path to (a)N3LO evolution in x-space with modern infrastructure, documents the numerical grid and quadrature setup honestly, and ships example programs. The GMVFN subtraction/residual module is a useful and distinct addition. For practitioners who need an x-space evolution engine they can inspect and modify, this is a serious candidate.\n\nThe soft spots are real but fixable. There is no external validation anywhere: no comparison against the Les Houches benchmark numbers from Ref. [101], no cross-check with EKO, Pegasus, or the aN3LO benchmarking in Ref. [75], and no grid-refinement or iteration-convergence study. The repository tests are explicitly 'not standard tests' but benchmarking/plot files. That matters because the convolutions are the numerical core and the integrands are singular; a fixed 100/50/25 Gauss-Legendre grid with 50 points per segment can be knocked off silently. The disclosed v1 bugs in the matching procedure and the mu_R^2/mu_F^2 handling show that implementation details have bitten before. I also want a pinned commit hash so the claims refer to a frozen artifact.\n\nTwo smaller points. The abstract's unqualified 'N3LO accuracy' overstates the singlet sector, which uses envelope approximations; the text discloses this in Secs. 1 and 3, so the abstract should be brought into line. And in Table 2, xc+ at x=0.7 flips sign between mu_r^2=0.5 mu_f^2 and the other two scales; that could just be scale variation near a zero, but it would be nice to see it explained.\n\nNet: the work is coherent, the code is public, and the omissions are the kind that a referee report can fix. Send it to review, and require a benchmark table plus a convergence study before acceptance. I wouldn't cite the numbers until that's done.","headline":"Worth refereeing as a software release, but the N3LO numbers in Table 2 are not backed by any external check.","tokens_in":27533,"tokens_out":3044,"would_cite":false,"duration_ms":31084,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["12.38.-t","12.38.Bx"],"model":"deepseek-v4-flash","headline":"Candia-v2 solves the DGLAP evolution equations at N3LO accuracy in QCD, directly in x-space.","keywords":["DGLAP evolution","parton distribution functions","N3LO QCD","x-space","logarithmic expansions","recursion relations","heavy-quark matching","subtraction PDFs"],"falsifier":"Run Candia-v2 on the Les Houches toy-model input at mu0^2 = 2 GeV^2 with the same aN3LO splitting functions and operator matrix elements, and compare its evolved xg and xuv at mu = 100 GeV, plus the c(-) and b(-) asymmetries, against an independent evolution code using a different discretization, such as Mellin-space with contour inversion. If the difference at the tabulated x points is much larger than the claimed tolerance, or if doubling the Gauss nodes or grid segments materially changes the output, the numerical claim fails even though the analytic recursion may be correct.","tokens_in":26419,"feed_emoji":"🧮","tokens_out":5670,"duration_ms":56527,"temperature":0.7,"pith_summary":"Candia-v2 claims to solve the DGLAP integro-differential equations that govern the scale evolution of proton parton densities directly in Bjorken-x space, at approximate N3LO accuracy in QCD. The method writes the solution as an exponential-type logarithmic power series and determines the unknown coefficient functions through recursion relations, avoiding Mellin-space inversion. The release adds exact four-loop non-singlet splitting functions, state-of-the-art approximations for the four-loop singlet sector, and three-loop operator matrix elements for heavy-quark matching. If the implementation is correct, the package is a production-grade evolution tool for high-precision collider phenomenology and supplies heavy-quark subtraction and residual PDFs for general-mass variable-flavor-number schemes.","feed_headline":"Candia-v2 pushes parton evolution to N3LO in x-space","feed_subtitle":"Four-loop QCD splitting functions plus recursion relations solve PDF scale evolution directly in x-space.","key_machinery":"The central object is the logarithmic-expansion ansatz with the L_i logs: one log of the alpha_s ratio, one log of a quadratic in alpha_s, one log of a linear root term, and an arctangent coming from the roots of the four-loop beta-function. The coefficient functions a_i(x) are built from convolutions and fixed by the four recursion relations, which generate each higher-order coefficient from lower-index convolutions with the splitting-function kernels. Numerically, convolutions are evaluated by Gauss-Legendre quadrature on a three-segment x-grid, with singular plus-distribution and delta-function pieces factored out and precomputed per active flavor number.","core_discovery":"The paper argues that the DGLAP solution can be built in x-space from the ansatz f(x, mu2) = product over i of [sum_n a_i(x)^n/n! L_i^n] convoluted with the initial PDF, with the logarithmic functions L_i depending only on alpha_s, the starting coupling, and the roots of the four-loop beta-function, and with the coefficient functions fixed by recursion relations. Candia-v2 implements this at N3LO: exact four-loop splitting functions in the non-singlet sector, the currently best available approximations in the singlet sector, and three-loop operator matrix elements for heavy-quark threshold matching. The paper presents evolved gluon and valence densities, heavy-flavor asymmetries, and subtrac","pith_inferences":["Editorial: the paper does not report a grid-refinement or quadrature-order convergence study, so a user should verify that the default 100/50/25 grid and 50 Gauss points are sufficient for the x region of interest before trusting the tabulated digits.","Editorial: because the repository's 'tests' are described as benchmarking and plotting files rather than standard tests, an independent comparison of Candia-v2 with a differently discretized evolution code on identical splitting-function inputs is the direct way to check the N3LO numbers.","Editorial: the same logarithmic-expansion machinery should carry over to polarized parton densities and to QED corrections, since the recursion relations depend only on the beta-function and splitting-function inputs.","Editorial: the exact-ansatz recursion cost scales exponentially with perturbative order, so the truncated ansatz is the practical route at higher orders; the code already exposes both, suggesting a path toward N4LO when the required kernels exist."],"forward_implications":["Evolved densities at approximate N3LO with all currently known perturbative ingredients become available in a single open-source package, suitable for LHC and HL-LHC scale predictions.","Heavy-flavor asymmetries c(-) and b(-), which first appear at NNLO, now include N3LO evolution and matching corrections, and Fig. 2 shows their size.","Subtraction and residual PDFs for charm and bottom can be generated on demand, giving reusable, process-independent pieces for general-mass variable-flavor-number schemes.","When exact four-loop singlet splitting functions become available, the recursion structure allows them to be inserted without restructuring the algorithm.","LHAPDF input and output let the code be embedded in larger fitting and prediction environments as an x-space alternative to Mellin-space evolution."],"fun_headline_variants":["Candia-v2 reaches N3LO in x-space DGLAP","Open-source Candia-v2: N3LO DGLAP in x-space","Candia-v2: four-loop splitting functions for DGLAP","Candia-v2: efficient N3LO parton evolution in x-space","N3LO DGLAP evolution via Candia-v2 in x-space"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is that discretizing the convolutions by Gauss-Legendre quadrature on a hand-chosen three-segment grid reproduces the continuum DGLAP solution to the implicit precision, despite there being no grid-refinement or convergence study in the paper and the repository's 'tests' being benchmarking files rather than standard tests.","fun_headline_variants_meta":{"raw":{"variants":["Candia-v2 reaches N3LO in x-space DGLAP","Open-source Candia-v2: N3LO DGLAP in x-space","Candia-v2: four-loop splitting functions for DGLAP","Candia-v2: efficient N3LO parton evolution in x-space","N3LO DGLAP evolution via Candia-v2 in x-space"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000839,"raw_usage":{"total_tokens":3442,"prompt_tokens":641,"completion_tokens":2801,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":385,"completion_tokens_details":{"reasoning_tokens":2699}},"tokens_in":385,"tokens_out":2801,"duration_ms":17905,"temperature":1.0,"reasoning_tokens":2699,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T06:15:12.442095+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run Candia-v2 on the Les Houches toy-model input at mu0^2 = 2 GeV^2 with the same aN3LO splitting functions and operator matrix elements, and compare its evolved xg and xuv at mu = 100 GeV, plus the c(-) and b(-) asymmetries, against an independent evolution code using a different discretization, such as Mellin-space with contour inversion. If the difference at the tabulated x points is much larger than the claimed tolerance, or if doubling the Gauss nodes or grid segments materially changes the output, the numerical claim fails even though the analytic recursion may be correct.","supporting_citations":[],"review_version":1}