{"id":"8f87dc60-a13e-4cb7-b5e3-cab03362228a","arxiv_id":"2608.02640","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A new Fortran package computes SU(3)-basis Talmi-Moshinsky brackets using isofactors from the external Su3cgvcs library, matching an independent reference to machine precision through E=12.","lead":"This paper releases a Fortran 2008 package that computes harmonic-oscillator transformation brackets, a standard ingredient in nuclear shell-model calculations, using SU(3) symmetry and an external Clebsch-Gordan library. It validates every bracket against an independent reference code to machine precision, so codes that already use the SU(3) library can adopt it.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (5)'s phase fix is asserted, not derived, and is validated only up to E=12 on mid-L blocks; orthonormality is phase-blind, so the abstract's unqualified machine-precision claim is unsupported for larger E / high-n states.","rationale":"The reader's weakest_assumption is the same as the one I would single out: the universal validity of the phase relation in Eq. (5) is not established beyond the tested quantum numbers. I agree this is the load-bearing point, because orthonormality is explicitly phase-insensitive and therefore cannot substitute for an absolute test. My stress-test pass sharpens the concern in one respect: the absolute comparison in Table 2 is not only limited to E<=12 but also to the mid-L blocks L=E/2, so states with maximal radial quantum numbers are never checked even at E<=12. I also note that the paper's explanation for stopping at E≈12 is somewhat inconsistent with its own estimate that the reference binomial tables are exact up to n≈57; this makes the unvalidated range more, not less, concerning. However, none of this contradicts the evidence within the tested range, and the package may well be correct. The reader already issued CONDITIONAL, and my analysis supports that rather than moving to REJECT or ACCEPT. Hence the verdict should remain unchanged. I mark agreement as partial because I add the mid-L/radial-quantum scope limitation and query the stated precision-floor justification, while endorsing the core phase-convention worry.","tokens_in":7071,"tokens_out":7302,"duration_ms":96219,"concrete_test":"Compute the full E=14 and E=16 blocks for every L with Eq. (4), both with and without the phase of Eq. (5), and compare every bracket against the Buck--Merchant closed form evaluated in 100-digit MPFR arithmetic (or against the Kamuntavičius HOTB code run in extended precision). Specifically include the L=0 block, where n1+n2 is maximal. If the with-phase values match the high-precision reference exactly and the no-phase values differ exactly by (-1)^{n1+n2} for every entry, Eq. (5) survives; if any entry deviates, the proposed phase map is incomplete and needs an additional state-dependent factor.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires Eq. (5) to be an exact universal map between the Su3cgvcs VCS convention and the HO isofactor convention. This is the only nontrivial bridge from a general CG library to physically correct brackets, and it is asserted rather than derived from the phase conventions of Refs. [2,3]. Orthonormality, reported in Sec. 6, is invariant under per-state sign flips: a sign attached to row i of C multiplies both H and H^T in H H^T, so the phase-blind 'PASS' cannot detect a wrong phase. The absolute element-by-element test that could detect a phase error, Table 2, covers only E=2,...,12 and only the mid-L blocks L=E/2. States with the largest radial quantum numbers—e.g., low-l states at small L with n_i = (e_i-l_i)/2 as large as E/2—are therefore never compared with an independent reference. The stated reason for stopping at E≈12 is also not clean: the reference's raw binomial tables remain exact up to n≈57, so the 10^-11 deviation seen already at E=12 is not obviously attributable to the reference's precision floor. If the true VCS/HO phase contains an additional factor that is trivial on the tested mid-L/low-n window, the package would pass every reported test while returning sign-wrong brackets outside that window. Thus the universal validity of Eq. (5) is the load-bearing unverified premise behind the headline 'bracket for bracket to machine precision' claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript presents SU3HOB-cgvcs, a Fortran 2008 package for computing general Talmi–Moshinsky harmonic-oscillator transformation brackets in an SU(3)-coupled basis with an arbitrary mass-ratio parameter d. The method follows the authors' earlier SU(3) reformulation, reducing the bracket to a single sum over Wigner d-functions weighted by products of SU(3)⊃SO(3) isofactors. The new contribution is the interface to the external Su3cgvcs Clebsch–Gordan library, which requires a state-dependent sign fix (Eq. 5) and an analytic treatment of the (0,0) scalar oscillator factor. The package is validated by orthonormality tests and by element-by-element comparison with the independent HOTB code of Kamuntavičius et al. for selected (E,L) blocks up to E=12. The central claim is that the package reproduces the reference brackets to machine precision (~1e-13) and provides a template for building SU(3)-scheme HOB codes on top of a general-purpose SU(3) CG library.","tokens_in":7501,"tokens_out":9882,"duration_ms":112507,"significance":"If the central claim is fully supported, this is a useful contribution to the computational nuclear-structure toolbox. The 'prepare once, evaluate many' factorization is attractive: isofactor towers are d-independent and built once per (E,L) block, while subsequent evaluations only require small Wigner d-functions and matrix products. The use of an external, separately published CG library rather than a new bespoke coefficient generator is a commendable design choice, and the comparison against an independent coordinate-space code is the right kind of validation. The paper also supplies explicit multiplicity checks, a public repository, and a build-and-check workflow, which enhance reproducibility. However, the validity of the phase relation in Eq. (5) is load-bearing for the universal 'bracket-for-bracket' claim, and the reported validation leaves gaps that need to be closed before the claim is fully credible.","major_comments":[{"comment":"The phase relation (−1)^{n1+n2} is asserted, not derived or traced to the explicit phase conventions of Refs. [2] and [3]. This is the only nontrivial bridge between the external VCS-based library and the harmonic-oscillator isofactor convention, and it is exactly the kind of step that can fail in a state-dependent way. As the manuscript itself notes, orthonormality is invariant under per-state sign flips, so the orthonormality tests cannot detect an error in Eq. (5). The absolute element-by-element test in Table 2 covers only L=E/2 and even E up to 12; low-L blocks and odd-E blocks are not reported. The universal statement 'makes Eq. (1) reproduce the physical Talmi–Moshinsky bracket exactly' is therefore stronger than the evidence. I recommend either deriving Eq. (5) from the phase conventions of the two libraries or extending the absolute comparison to all L values (and, where feasibl","section":"Sec. 3.2, Eq. (5)"},{"comment":"Table 2 shows max|SU3HOB − HOTB| = 1.4×10^{-11} at E=12, while the abstract and summary claim agreement to ~10^{-13}. This discrepancy is attributed to 'the double-precision floor of the reference's raw binomial tables,' but the stated reason is not convincing: the text says those tables are exact only while C(n,m)<2^53 (n≲57), whereas E=12 involves n values far below that threshold. If the reference is still reliable at E=12, then 1.4×10^{-11} is a genuine disagreement with the machine-precision claim; if it is not reliable, the authors need to provide a concrete error analysis (e.g., cancellation in the HOTB summation) rather than the current binomial-overflow argument. This point directly affects the central numerical claim and should be resolved.","section":"Sec. 6, Table 2"},{"comment":"The introduction states that the computed brackets agree with the reference 'for every (E,L) block and mass ratio d,' but the quantitative validation presented in Table 2 is limited to the representative mid-L blocks L=E/2 for E=2,4,...,12, with d=1 tabulated and d=1/2,2 only described as differing in the last digit. No table or supplementary output is provided for all L, all odd-E blocks, or the full d range. The 'PASS' statement in the Makefile check may cover more, but the manuscript should show or cite the complete validation data. At minimum, the claims in the abstract and introduction should match the range actually demonstrated, or the full sweep should be reported.","section":"Sec. 1 and Sec. 6"}],"minor_comments":[{"comment":"Typos: 'These brackets arte constructed' should be 'are constructed'; the phrase 'In this library they are computed there' is redundant.","section":"Abstract"},{"comment":"'ab initiono-core' should be 'ab initio no-core'.","section":"Sec. 1"},{"comment":"The notation n_i = (e_i − l_i)/2 implicitly assumes e_i and l_i have the same parity; this is standard for HO states but should be stated explicitly for readers outside the field.","section":"Sec. 3.2, Eq. (5)"},{"comment":"The column layout of Table 2 is hard to read because the entries are packed together; adding explicit column separators or aligning the numbers would improve clarity. Also, the table caption should state the ranges of e1,e2 and l1,l2 included in 'every bracket' for each block.","section":"Sec. 6, Table 2"},{"comment":"The phrase 'not guarded against it' is informal; suggest a more precise statement, e.g., 'the routine does not handle the degenerate scalar-factor case.'","section":"Sec. 3.3"}],"recommendation":"major_revision","confidential_remarks":"The paper is appropriate for a computational physics software journal, and the design of building an SU(3)-scheme HOB code on an external CG library is genuinely useful. The main risk is the unverified universality of the phase fix, combined with validation evidence that is narrower than the headline claim. If the authors can provide a derivation of Eq. (5) or a full validation over all L and a defensible error analysis for the E=12 deviation, I would support publication. I do not see a fatal flaw in the method itself, but the claims need to be brought into line with the evidence."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nI read the SU3HOB-cgvcs paper. It is a straightforward, useful software contribution. The new pieces are the explicit phase relation between Su3cgvcs's VCS convention and the HO isofactors (Eq. 5), the analytic treatment of the (0,0) scalar factor, and the package itself. The validation against the independent HOTB code is the right thing to do: element-by-element agreement, not just orthonormality. Within the tested range, the agreement is convincing, with deviations around 10^-13 to 10^-11.\n\nThe soft spots are real but minor. The abstract's unqualified 'bracket for bracket to machine precision' overstates what is shown. Table 2 covers only E=2..12 and only mid-L blocks L=E/2. Orthonormality, tested for all blocks, is phase-blind, so it cannot catch a wrong sign in Eq. (5). The largest radial-quantum states, at low L and small e_i, are never compared with the reference. So the universal validity of the phase fix rests on a narrower test than the abstract suggests. Also, the explanation for stopping at E=12—the reference's binomial tables—does not hold up on inspection: for E=12 the binomial arguments are small, so the 1e-11 deviation at E=12 is not obviously a precision floor. The authors should either explain that deviation or extend the test with another reference.\n\nNone of this makes me doubt the package. If Eq. (5) were wrong, the tested elements would show sign errors of order one, not 10^-11 noise. The authors clearly know what they are doing. The missing repository link (the URL stops at 'Augustinaz') is a trivial fix.\n\nWho is this for? Anyone computing general Talmi-Moshinsky brackets in the SU(3) basis, especially no-core shell-model codes that already link Su3cgvcs. I'd send it to a referee. My recommendation: accept after minor revisions, with the authors qualifying the machine-precision claim in the abstract, fixing the repository link, and either deriving Eq. (5) or validating it on low-L blocks. The package itself is a solid contribution.","headline":"Useful SU(3)-scheme HOB package with a new phase bridge to Su3cgvcs; the abstract's unqualified machine-precision claim outruns the actual validation window.","tokens_in":7964,"tokens_out":4491,"would_cite":true,"duration_ms":49204,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["22E70","81R05"],"pacs":["21.60.Cs","21.60.Fw"],"model":"deepseek-v4-flash","headline":"A Fortran 2008 package computes Talmi–Moshinsky harmonic-oscillator brackets in an SU(3) basis, taking the needed isofactors from a general-purpose SU(3) Clebsch–Gordan library and bridging conventions with a single state-dependent sign; th","keywords":["Talmi–Moshinsky brackets","harmonic-oscillator transformation brackets","SU(3) symmetry","Clebsch–Gordan coefficients","vector-coherent-state method","no-core shell model","Fortran 2008 package","isofactors"],"falsifier":"Take a block beyond the validated range, e.g. E=16, L=8 with d=2, and compare every bracket from this package against an arbitrary-precision direct evaluation of the Talmi–Moshinsky integral (or an independent high-precision HOB code). If the restored-phase brackets do not agree to the package's own precision, the claimed universal sign bridge—and with it the general claim of machine-precision agreement—is falsified. A second check: compare the external library's isofactors for a high (λ,μ) such as (16,0) with those from an independently derived SU(3) CG algorithm.","tokens_in":7001,"feed_emoji":"⚛️","tokens_out":8613,"duration_ms":84193,"temperature":0.7,"pith_summary":"Talmi–Moshinsky harmonic-oscillator brackets re-express a product of two oscillator states in one coordinate frame as a sum over products in a rotated frame; they are the workhorse of translationally invariant few-body and no-core shell-model calculations. This paper presents a Fortran 2008 package that computes these brackets in an SU(3)-coupled basis by the method of Ref. [1], with the required SU(3)⊃SO(3) isofactors taken from an existing general-purpose SU(3) Clebsch–Gordan library rather than built from scratch. Bridging the two requires exactly two fixes: a state-dependent sign (−1)^(n1+n2) that converts the library's coefficient convention to the harmonic-oscillator convention, and an analytic treatment of the trivial (0,0) oscillator factor. After those fixes the package reproduces the established general Talmi–Moshinsky bracket code element by element to machine precision (~10^-13) throughout the tested range E≤10, and at ~10^-11 at E=12. The result is a concrete demonstration that the SU(3)-scheme HOB can be assembled on top of a separately developed SU(3) coupling library, with the d-independent isofactor towers built once and reused for every mass ratio d.","feed_headline":"Oscillator brackets match the standard code to 10^-13","feed_subtitle":"A single state-dependent sign adapts an external SU(3) library to compute Talmi–Moshinsky brackets for any mass ratio.","key_machinery":"The central object is the SU(3)-basis expression of Eq. (1): the bracket is a single sum over J of small Wigner d-functions of the reflection-containing Talmi–Moshinsky matrix, weighted by products of SU(3)⊃SO(3) isofactors of U(6)⊃U(3)×U(2). The load-bearing bridge is Eq. (5), a state-dependent sign (−1)^(n1+n2) that maps the external library's vector-coherent-state convention coefficients to the harmonic-oscillator isofactor convention, plus an analytic override for the (0,0) representation. Because the isofactor towers in C are d-independent, they are calculated once per (E,L) and reused for all mass ratios d—that factorization is what makes the \"prepare once, evaluate many\" design work.","core_discovery":"The paper's central claim is that the general Talmi–Moshinsky bracket factorizes as H(d)=C D(d) C^T per (E,L) block: a d-independent matrix C of SU(3)⊃SO(3) isofactors, and a d-dependent diagonal-in-J matrix D of small Wigner d-functions of the reflection-containing Talmi–Moshinsky transformation. The isofactors can be read directly from a general-purpose SU(3) Clebsch–Gordan library, provided the user reinserts one sign per state, (−1)^(n1+n2), and supplies the scalar (0,0) factor analytically. With these conventions restored, the computed brackets agree with the coordinate-space Talmi–Moshinsky code of Ref. [3] to machine precision. The paper further notes that the assembled bracket is inv","pith_inferences":["The same phase bridge likely applies to other SU(3)⊃SO(3) recoupling coefficients drawn from a vector-coherent-state library, so quantities such as SU(3) Racah factors may need only the same sign fix; this is testable but not claimed in the paper.","Since the phase depends only on radial parity, the method may tolerate convention changes inside the external library that do not alter that parity; conversely, if a future library version flips the VCS phase convention, the bridge would silently break—a version-pinning concern.","The restriction to E≤12 is a validation-range limit, not a method limit; extending the comparison with arbitrary-precision reference brackets for E>12 would directly test the universality of the sign mapping.","The factorization H=CDC^T suggests a route to generalized brackets where each Jacobi coordinate has its own mass ratio (multiple d parameters), since C is unchanged and only D would gain additional structure."],"forward_implications":["For any (E,L) block, all brackets for all mass ratios d are obtained cheaply after one isofactor preparation; this makes wide scans over d practical in few-body calculations.","The multiplicity-basis invariance means a code using any SU(3) coupling library, regardless of its convention for degenerate α-states, can produce the same physical HOBs without re-orthogonalizing.","The validated agreement for E≤10 (and near-machine-precision at E=12) supports use of SU(3)-scheme HOBs as a drop-in replacement for coordinate-space Talmi–Moshinsky brackets in existing shell-model workflows.","The package provides a call-through template for embedding SU(3)-scheme HOBs in codes that already link the external SU(3) coupling library."],"supporting_citations":[{"why":"Supplies the SU(3)-basis HOB method this package implements, including the single-sum formula of Eq. (1).","marker":"[1]"},{"why":"The external SU(3)⊃SO(3) Clebsch–Gordan library whose VCS-convention isofactors are read via readfact, readtab, cgu3hw, cgu3o3.","marker":"[2]"},{"why":"The general Talmi–Moshinsky bracket code and closed-form expression used as the independent, coordinate-space validation reference.","marker":"[3]"},{"why":"Introduces the harmonic-oscillator transformation brackets whose definition underlies the whole calculation.","marker":"[4]"},{"why":"Provides the equivalent simple expression for the general oscillator bracket, used as a second coordinate-space reference.","marker":"[5]"}],"fun_headline_variants":["SU(3) brackets from external library with one sign fix","Talmi-Moshinsky brackets via SU(3) library, exact to 1e-13","One sign reinserted: SU(3) library yields HOB to machine precision","Brackets factorize: d-independent SU(3) isofactors times d-dependent matrix","Reuse SU(3) CG library for HOB, fix sign per state"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The bridge relies on the claim that one state-dependent sign, (−1)^(n1+n2), completely converts the external library's coefficient convention to the harmonic-oscillator convention for every (E,L) block; this is verified only up to E=12 because the independent reference code loses numerical reliability above that.","fun_headline_variants_meta":{"raw":{"variants":["SU(3) brackets from external library with one sign fix","Talmi-Moshinsky brackets via SU(3) library, exact to 1e-13","One sign reinserted: SU(3) library yields HOB to machine precision","Brackets factorize: d-independent SU(3) isofactors times d-dependent matrix","Reuse SU(3) CG library for HOB, fix sign per state"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000924,"raw_usage":{"total_tokens":3883,"prompt_tokens":916,"completion_tokens":2967,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":660,"completion_tokens_details":{"reasoning_tokens":2857}},"tokens_in":660,"tokens_out":2967,"duration_ms":20317,"temperature":1.0,"reasoning_tokens":2857,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T04:19:57.501201+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a block beyond the validated range, e.g. E=16, L=8 with d=2, and compare every bracket from this package against an arbitrary-precision direct evaluation of the Talmi–Moshinsky integral (or an independent high-precision HOB code). If the restored-phase brackets do not agree to the package's own precision, the claimed universal sign bridge—and with it the general claim of machine-precision agreement—is falsified. A second check: compare the external library's isofactors for a high (λ,μ) such as (16,0) with those from an independently derived SU(3) CG algorithm.","supporting_citations":[{"cited_title":"Calculation of harmonic oscillator brackets in SU(3) basis , journal =","cited_arxiv_id":null,"evidence_quote":"Supplies the SU(3)-basis HOB method this package implements, including the single-sum formula of Eq. (1)."},{"cited_title":"Computer Physics Communications , volume =","cited_arxiv_id":null,"evidence_quote":"The external SU(3)⊃SO(3) Clebsch–Gordan library whose VCS-convention isofactors are read via readfact, readtab, cgu3hw, cgu3o3."},{"cited_title":", title =","cited_arxiv_id":null,"evidence_quote":"The general Talmi–Moshinsky bracket code and closed-form expression used as the independent, coordinate-space validation reference."},{"cited_title":"The general harmonic-oscillator brackets: compact expression, symmetries, sums and","cited_arxiv_id":null,"evidence_quote":"Introduces the harmonic-oscillator transformation brackets whose definition underlies the whole calculation."},{"cited_title":"and Merchant, A","cited_arxiv_id":null,"evidence_quote":"Provides the equivalent simple expression for the general oscillator bracket, used as a second coordinate-space reference."}],"review_version":1}