Literature-to-code assessment, October 2026¶

The baseline assessment below preceded the complex Abelian increment. The architecture and status remain the authority for production scope. No external QC implementation or library data is bundled. The operator-contract and operator-consumer designs are now integrated. Their original workstream versions and the MANTA engine/scientific designs were initially inspected read-only; the interface mapping in the shared symmetry design follows those contracts without duplicating implementations. The coordinated MANTA P0 libint reference provider is now integrated for the subsequent quartet consumer.

Literature inspected	Mathematical addition	Existing foundation / immediate consumer	Missing admission and evidence
Melega et al., 2026, published article, Sections 2.1, 2.3-2.6, 3 and 4.3; geometry supplement	CD/RI derivative reformulation, Lambda and orbital Z-vector response, symmetry-blocked density intermediates	Shared retained-auxiliary audits and correlation tensors can supply representations	No shared implementation of the relaxed CD-CCSD gradient; differentiated metric, fitting and support terms and displaced-geometry validation remain method-owned
Hellmann and Gauss, 2026, published article, Sections 2.3-2.7 and 3	Resolve `E x E = A1 + A2 + E`; convert reduced/full four-index tensors for CC contractions, permutations and spin adaptation	Compact noncommuting actions and retained-space group audits; future CC residual consumer	Demonstrated closed-shell C3v CCSD pilot, not a general non-Abelian implementation; partner gauges, multiplicity spaces and recoupling are absent
Kitsaras and Stopkowicz, journal DOI, inspected arXiv:2602.12457v1 preprint, Sections II A-C and III	Complex-conjugate irreps in double-coset integrals, creation/annihilation selection rules and block contractions	Complex AO/Bloch actions exist; molecular finite-ERI J/K is an immediate execution consumer	Explicit conjugate/product labels, tensor variance and source-qualified sparse execution are missing; antiunitary operations need separate admission
Goodlett et al., MolSym 2024, accepted manuscript; representation and RHF supplements	Complete matrix projectors and consistently oriented partners, including repeated irreps; nonredundant RHF blocks	Existing character tables, molecular rotations and metric/subspace audits	Character projection alone does not resolve partners. RHF degeneracy factors assume an invariant state; neither geometry repair nor a character table qualifies arbitrary-density Fock production
Johansson and Veryazov, libmsym 2017, pp. 5-7	Trace projection plus subgroup subduction and partner alignment for real spherical harmonics	Geometry and pure-shell representation construction	This is a representation reference, not a certificate of finite-source symmetry or an integral/contraction engine; its realification is insufficient for complex Abelian tensor labels

The selected increment is now implemented as symmetry_abelian.py: an opt-in, bounded complex-Abelian ERI block consumer for J/K. It inspects a complete supplied finite tensor before omitting forbidden blocks, retains its numerical identity and omitted norm, and accepts arbitrary complex trial densities without state projection. Actual counts and measurements appear in the status. Integral evaluation is unchanged, and no production driver is enabled.

The next gap was integral work: the block consumer computes all ERIs before reducing contractions. symmetry_quartets.py now adds real molecular shell representatives and full angular stabilizer checks over the MANTA P0 tile provider. It audits the complete finite source once, then requests only representatives on repeated arbitrary-density J/K builds. This implements the petite-list execution lesson of Dupuis and King without assuming a symmetric state. Full C3v AO block actions are supported; irreducible-product resolution, CC recoupling, complex integral sources and producer-side screening proofs remain missing. Measurements separate integral calls, unchanged contraction work, transport, setup and memory.

The subsequent opt-in native executor compiles that four-leg transport and J/K digestion without changing its mathematical or source-admission scope. Dupuis and King’s nonunitary AO ordering is retained explicitly; real-ERI permutations act on indices, while complex trial densities keep their conjugate/index-variance convention. It is an execution improvement, not new admission of complex integral kernels, antiunitary symmetry or CC reduced products. Native/Python comparisons and the retained audit cost are recorded in the status page.

The density-batch increment reuses each admitted petite-list tile and angular rotation across independent spin or response densities. This follows the linearity of the existing contraction, not a new state selection rule. Arbitrary transition densities retain their conjugation/index variance; source work is shared while contractions and output storage still grow with the number of channels. The immediate consumer boundary is spin J/K and density-response application. Producer-qualified screening, reduced CC products and method-specific derivative response remain missing.

Selective batch targets now omit algebraically unneeded J or K contractions and their output panels. This applies the existing density/index convention without introducing an irrep selection rule: an unrestricted caller can request one total-density J and two spin-resolved K targets. Source admission, non-Abelian full-AO transport and complex trial densities are unchanged. Producer screening, complex integral kernels and reduced CC tensor products still require the separate qualifications described below.

The next execution increment specializes selected native contractions by resolving member permutations and matrix addresses before scalar work. No new mathematical symmetry is claimed. Dupuis-King AO transport, the complete P0 source audit and arbitrary complex density conventions are retained. The current consumer is the same opt-in quartet batch API; the method-owned evaluated Fock-state adapter remains a separate draft. Same-source timing, counts and memory are measured independently of the unchanged admission cost. Periodic finite domains, source-qualified screening, complex integral kernels, partner/multiplicity resolution and derivative acceptance remain missing production capabilities.

Source qualification and screening increment¶

Mathematical capability	Already implemented	Immediate consumer	Still missing
Fixed omission of complete shell orbits, with an aggregate finite-tensor norm budget	Full ordered P0 source audit, inverse AO transport, representative/stabilizer schedules and arbitrary-density J/K	Stored-plan `jk_batch`, including selected total-J/spin-K channels	Scalable producer bounds, streaming certificate storage, method-owned dispatch and acceptance on other sources
Distinguish actual mask identity from screening-policy identity and density-specific error bounds	Immutable native producer/session and complete finite-source audit	MANTA receipts and the separate Fock evaluated-state source record	Agreed general producer schema for kernels, finite supports, auxiliary cuts and BUPO trees

Maurer et al. (2012), Section II, Eq. (1), gives a far-field multipole bound with extent and separation conditions. Its QQR Eq. (2) substitutes Schwarz factors and is explicitly an estimate, not a rigorous bound. Section II also explains why a small individual-integral cutoff alone is not a useful total-energy error bound. Thompson and Ochsenfeld (2017), Eqs. (6)-(8), combines the ordinary Schwarz product and two distance-sensitive products using their minimum. The required positive kernel/inner-product conditions matter. Eqs. (9)-(12) instead define tighter nonrigorous CSA estimates. Eqs. (18)-(19) multiply integral estimates by density entries and use separate J/K thresholds; an arbitrary-density equivariance certificate cannot silently replace that rule by a fixed-state bound.

This increment implements none of those kernel estimates. MANTA owns them. It uses measured full-orbit norms from the already mandatory finite-source audit and Cauchy-Schwarz contraction bounds. A fixed whole-orbit mask is compatible with Dupuis-King transport and Davidson/Pitzer orbit reduction without requiring an invariant electronic state. All actual members enter the norm: normalized Cartesian actions are metric-unitary, so multiplying one representative’s Euclidean norm by the square root of its multiplicity is not generally valid. Numerical norm and contraction roundoff remain outside the reported exact-arithmetic bound. The measured results and explicit unsupported domains are recorded in the status.

Neither this bound nor a tighter integral estimate supplies the Melega response/gradient terms, Hellmann-Gauss reduced CC products, complex-kernel conjugation rules, periodic finite-domain proof or BUPO tree covariance. Those earlier representation and source qualifications remain unchanged.

Producer-envelope increment¶

The next coherent increment consumes MANTA’s existing full-space product mass/potential envelopes. The new mathematics at the symmetry boundary is |(ab|cd)| <= min(Q_AB V_CD, Q_CD V_AB) followed by a Frobenius bound over all AO entries and every ordered member of an orbit. This is a direct positive-Coulomb inequality, not the distance-dependent QQR/CSA estimates above. Symmetry implements no new envelope kernel or BUPO hierarchy.

The existing native envelopes, immutable molecular source, Davidson-style pair-stabilizer schedule and Dupuis-King AO transport supply the foundations. The current consumer is compact screened molecular J/K and its unchanged native RHF/UHF FockBuild bridge. Whole-orbit omission now precedes integral reads; retained members and stabilizers still require the numerical audit. Bounds do not certify equivariance, and this partial plan cannot reconstruct an unmasked source. Measured acceptance demonstrates actual avoided calls on separated fragments and the lack of pruning on NH3.

Missing production capabilities remain tighter/hierarchical producer bounds, producer-issued equivariance and finite-support qualification, a cost-aware public dispatch policy, complex integral-source admission and the non-Abelian partner/product/multiplicity and derivative machinery identified above. The new molecular Q/V receipt does not qualify a periodic source, auxiliary rank cut, local projector, gradient or stress. Roundoff remains separate from the analytic integral bound.

Equations and qualifications that affect the implementation¶

Kitsaras and Stopkowicz’s preprint Eq. (2) conjugates the character in the projector; Eqs. (8)-(10) retain small angular rotation matrices inside the double-coset sums. Replacing those matrices by signs is restricted to the real Abelian case discussed there. Eqs. (16)-(21) distinguish creation from annihilation characters, reference-state from excitation symmetry, and transition from expectation densities. Eq. (25)’s roughly quadratic contraction saving assumes two symmetry-selected tensors with balanced irrep populations. Arbitrary-density J/K has only one selected operand, so that estimate does not apply. Small blocks can be slower, as their timing discussion and our measurements both show. Their implementation treats finite-field complex Abelian groups of order at most eight; importing the mathematical selection rule does not implement London orbitals or magnetic groups here.

Hellmann and Gauss’s Eq. (4) resolves the C3v E x E product. Eqs. (5)-(8) give normalized partner combinations, including the antisymmetric A2 and two E partners. Eqs. (9)-(12) show why reduction followed by naive index permutation is wrong: a resort can require omitted partner elements. Sections 2.6-2.7 also require full/reduced conversion for spin adaptation and different degeneracy factors in lower-order contractions. Their closed-shell CCSD pilot uses Cs integrals, aligns E partner phases and partly hard-codes C3v. The NH3/PH3 operation-count tables are not general wall-time guarantees. The paper explicitly excludes interpreting realified complex C3 irreps as this same construction. Generalization requires the representation and recoupling contract in the design.

Melega et al.’s Eq. (6) differentiates the RI-like representation using two three-center derivatives and a negative metric-derivative term. Eqs. (9)-(14) introduce the stationary CC Lagrangian and Lambda equations; Eq. (29) supplies orbital Z-vector response. Eqs. (40)-(43) include relaxed density, overlap (Pulay) and fitted two-electron terms. Eqs. (44)-(49) contract lower-rank intermediates with derivative integrals, avoiding a stored full AO two-body density. Their two-step CD chooses pivots before constructing full vectors; this is not the eigenthreshold-based SR/LR GDF producer. Symmetry-adapted displacements and integral derivatives are additional representations. The method is closed-shell CD-CCSD, not an open-shell, triples or stress implementation. A source-aware gradient must retain rank/pivot policy and all response terms even when the energy contracts correctly.

The Melega geometry supplement was inspected as data: initial/final coronene ZMATs (C24H12), initial/final hexabenzocoronene ZMATs (C42H18), and the 24-atom planar azobenzene XYZ. The ZMAT dummy center, variable references and starred optimization variables matter. For example, coronene’s R1 changes from 1.4 to approximately 1.41310473 in the supplied files. These are optimization inputs/results, not a displacement-resolved analytic-gradient oracle. Future reproduction must preserve declared units, basis, frozen-core and CD thresholds from the paper; it must compare displaced energies and gradients, rather than just compare the final coordinates. No supplement geometry or third-party program is bundled or executed by this increment.

MolSym’s main Eq. (3) uses matrix elements of each irrep to produce similarly oriented partners across repeated copies. The representation supplement lists complex C3 rows separately, which provides a useful independent check of conjugate labels. The RHF supplement Eqs. (3)-(4) weight nonredundant one-electron blocks by irrep dimension and use a per-spin density convention (F = H + 2J - K). Its Table S1 measures Fock builds averaged over 50 iterations, not complete job time. Its smallest NH3 example already shows that full symmetry need not beat C1. Our J/K adapter leaves density normalization to the caller and never uses degeneracy weights to reconstruct a broken-symmetry density. libmsym’s pp. 5-6 explain why a trace projector loses partner information and use subgroup subduction to recover it. Both sources motivate explicit partner gauges, rather than more character tables.

Connections to the established literature¶

Davidson 1975, Eqs. (11)-(14): the double-coset degeneracy is |H intersect g K g^-1|, generally dependent on the representative. The Abelian simplification |H intersect K| must not migrate into a non-Abelian planner. Existing Orbit stabilizers are reused by the new capped full shell-orbit scheduler. A scalable pair-stabilizer iterator now avoids retaining that full ordered inventory. It enumerates H_a orbits in G/H_b, resolves exchange using inverse transporters as required after Eq. (17), and keeps all angular matrices. This index-space implementation does not implement the paper’s complete symmetry-orbital integral formulas. Complete qualification visits every ordered quartet. The new Q/V route avoids reads only for bounded omitted orbits; hierarchical producer admission remains absent.
Pitzer 1973, symmetry-adapted charge distributions and coupling-coefficient orthogonality: resolving pair products is different from assigning one label to a non-Abelian pair. The paper explicitly notes the need for extra notation in nonsimply reducible groups. Modern multiplicity labels make that requirement concrete.
Dupuis and King 1977, Eqs. (8) and (11): density and operator transformations have opposite ordering in a nonunitary AO representation. Petite-list shell orbits can reduce integral work without first constructing SALCs. Paper II, 1978, around Eqs. (33)-(34), states the symmetric-density assumption, separately for both UHF spins. Derivative-center orbits are not scalar energy multiplicities.
Stanton et al. 1991, Eqs. (14)-(15): direct-product pair dimensions determine actual contraction costs. The real D2h multiplication simplification is replaced by explicit dual characters here. Future CC residual integration must retain its different pairings, spin adaptation and occupied/virtual masks.
Dovesi 1986, Eqs. (30)-(36): orbital rotations act together with distinct integer shifts of each center and of the finite summation set. Finite Coulomb/exchange truncation is part of the operator, not an implementation detail to average away.
Zicovich-Wilson and Dovesi, 1998 I, Eqs. (31)-(33) and concluding discussion, requires aligned partners, k-dependent actions and projective little-group representations at some nonsymmorphic boundary points. 1998 II applies SACO blocking to the generalized SCF eigenproblem. An ordinary molecular Abelian character table is insufficient at those k points.
Casassa et al. 2006, Eq. (3): localized Wannier subsets mix through matrices as well as translations. PAO/PNO/TNO projectors must transform with those matrices; a pair permutation is only a special case.
Rusakov et al. 2013, Eqs. (5)-(7) and Algorithm 1: reanchor every transformed pair using the first center’s full image shift. Translational density invariance is an explicit premise; a universal reduced list of right-hand pairs is generally unavailable.
Orlando et al. 2014, Eqs. (1)-(5): combine sparse direct-space reconstruction, Fourier transformation and SACO transformation instead of materializing intermediate full matrices. The useful lesson is the execution schedule and screening interaction, not simply the theoretical number of symmetry orbits.

Validation boundary¶

The new tests evaluate native full molecular ERIs, J/K and trial-density RHF energy functionals, native converged RHF Fock comparisons and density-energy directional finite differences, including complex nonsymmetric/non-Hermitian densities, nonorthogonal Cartesian AOs, dense gauges within repeated irreps, separate scalar conjugation and displaced-source rejection. Their complex C3 action is tested on C3v ammonia, without claiming full non-Abelian reduction. Existing shared/GDF/periodic tests supply independent coverage of retained auxiliary factors, fractional translations, Gamma and multi-k transport, odd/even stabilizers and incompatible finite domains. These existing tests do not turn the new molecular block consumer into a periodic tensor engine.

No CC residual, analytic gradient or stress code changes in this increment. Consequently no numerical acceptance for those quantities is claimed. Full reduced-versus-unreduced CC residuals, relaxed energies and displaced finite-difference derivative tests remain required before enabling those consumers. No tolerance or legacy fallback has been changed.