χ-CCM / aiccm2026dev-b: an independent finite-torus derivation

Status and scope

χ-CCM[1] is the prose name for the aiccm2026dev-b development line: the finite-translation-group character approach to the variational finite-BvK-torus CCM. It is an experimental, independently derived restricted and unrestricted Hartree–Fock and Kohn–Sham method with finite-torus MP2 and coupled-cluster extensions. Its sibling is Γ-CCM, selected as aiccm2026dev-a, which uses the union-and-weight/Wigner–Seitz integral-weighting construction. χ-CCM coexists with the earlier CCM implementation and is selected with

D89 makes the B identity explicit in every finite-torus convention record: ccm_approach="chi-ccm", ccm_construction="finite-translation-group-character", and evaluation_representation="gamma-centred-character-mesh". The first two identify the approach and mathematical construction; the third identifies how that χ-defined Hamiltonian is evaluated. Its real-Gamma form is an exact finite-Fourier representation control but is not assigned a Γ-CCM identity. The fields are immutable and are carried through B diagnostics/results, fleet finite_torus_convention, QVF vendor metadata, and .system run metadata. QVF wrapper schema version 2 retains the legacy version 1 representation string for compatibility; evaluation_representation is normative. Pre-D89 records are not numerically retracted solely because these fields are absent, but they cannot support a Γ/χ construction-comparison claim.

result = vibeqc.run_periodic_job(
    system,
    basis,
    method="RHF",
    jk_method="aiccm2026dev-b",
    aiccm_lattice_extension=(4, 4, 4),
    aiccm_backend="four_center",  # or "ri" / "rijcosx"
)

The implementation does not use vibeqc.periodic.ccm and does not copy the historical program. The earlier implementation, the historical program, and published CCM formulae are comparison targets. The present definition is the finite Born–von Karman (BvK) torus Hamiltonian restricted to translation-invariant determinants. It is evaluated in the finite translation group’s reciprocal irreducible representations. This is an exact change of basis, not an appeal to reciprocal-space convergence.

The implemented phase is deliberately narrow:

• RHF, RKS, UHF, and UKS (including hybrid functionals), integer occupations, neutral primitive cell;

• finite-group and Wigner–Seitz constructions in 1D, 2D, and 3D, while all production SCF backends currently fail closed below 3D until the shared wire/slab Coulomb convention described below is derived;

• direct corrected-gauge four-centre, pair-resolved RI, and RIJCOSX electron-repulsion backends;

• the existing periodic integral, GDF, Ewald, DIIS, basis, and lattice infrastructure;

• 3D canonical RHF RI-MP2, unrestricted RI-MP2, full-domain UCCSD(T), and restricted/unrestricted local-PNO routes on the exact real torus;

• gauge-invariant SCF one-particle properties; no charged-cell background, total analytic gradients, Berry-phase polarization, or relaxed correlated density matrices yet. Independently checkable 3D Ewald nuclear-repulsion, fixed-density Ewald electron-nuclear, fixed-density kinetic, fixed energy-weighted overlap, and restricted/unrestricted active BvK seam derivative components are exposed only as derivation aids.

The principal source studied was Peintinger and Bredow, J. Comput. Chem. 35, 839 (2014), DOI 10.1002/jcc.23550, together with Chapter 8 of Peintinger’s thesis and the earlier CCM papers cited there. The KS-ADFT CCM of Janetzko, Köster, and Salahub, J. Chem. Phys. 128, 024102 (2008), DOI 10.1063/1.2817582, was audited separately because it introduces two- and three-center multiplicative weights in deMon2k. Those sources motivate the questions; they do not define the implementation below.

1. Finite cyclic translation group

Let the primitive direct lattice be

[ A=(\mathbf a_1,\mathbf a_2,\mathbf a_3), \qquad \mathbf R_{\mathbf n}=A\mathbf n, ]

with only the first (d) directions active. The mathematical matrix (A) has lattice vectors as columns, exactly as PeriodicSystem.lattice does. Thus Cartesian translations are (\mathbf R_{\mathbf n}=A\mathbf n), and the BvK supercell matrix is

[ A_N=A\operatorname{diag}(N_1,N_2,N_3). ]

The finite-torus descriptor records this executable contract as lattice_vector_convention="columns", and successful fleet payloads record primitive_lattice_bohr. The B fleet audit binds the exact (A\operatorname{diag}(N)) matrix to those fields; the Γ/χ comparator defines no comparison when the binding is absent or inconsistent.

D88 supersedes the false row-lattice premises in D17 and D81. The article theory already uses the column convention correctly; the repair aligns the code and its fingerprints with that derivation and changes neither the Coulomb kernel nor the signed-Madelung convention. On the skew audit

[ A=\begin{pmatrix} 7&0.4&0.2\ 0.3&8&0.5\ 0.1&0.6&9 \end{pmatrix}, \qquad N=(2,3,1), ]

the corrected positive code convention is (\xi_M=0.138352993811598). The pre-D88 fitted helper returned (0.143621616230995), and the pre-D88 four-center probe returned (0.138913224426180). The fitted error corresponds to 5.268622 mHa/cell of overbinding for a two-electron RHF seam. Because the Madelung, probe-charge, and fleet helpers are shared, all affected pre-D88 Γ-CCM and χ-CCM records require rerun under the fingerprint rule; no missing convention field is inferred. A symmetric lattice that commutes with diag(mesh) is numerically outside this defect class, but its old record is not newly attested. The shared builders for graphene, mgo-slab, ice-ih, co2-dryice, and sio2-quartz require fresh Γ and χ fingerprints. Ordinary pre-D88 periodic GDF exact-exchange and BIPOLE J/K records for affected lattice/mesh combinations also require audit and rerun because they traverse the shared helpers. A concrete ordinary-GDF control confirms the impact: the c-diamond/sto-3g (2,1,1) fixed point moves from the pre-D88 post-D78 pin -74.6405167828 Ha to -74.4119904741 Ha. The old and corrected positive Madelung constants are 0.5078067935392725 and 0.46971907524744966; multiplying their difference by six occupied closed-shell bands predicts the measured +0.228526309751 Ha shift. D89 records that Γ-CCM and χ-CCM are distinct approaches compared at a declared common exchange-q=0 convention; their names do not select different Coulomb kernels.

For positive integers (N_i), with (N_i=1) on inactive directions, define

[ \mathcal T_N = \mathbb Z_{N_1}\times\mathbb Z_{N_2} \times\mathbb Z_{N_3}, \qquad N_c=N_1N_2N_3. ]

The BvK identifications are

[ \mathbf R_{\mathbf n+N_i\mathbf e_i}\equiv\mathbf R_{\mathbf n}. ]

This quotient is the cyclic cluster. It is not a finite open cluster and it has no surface. Its superlattice vectors are (N_i\mathbf a_i). AO labels are ((\mu,\mathbf n)), with (\mu) in the primitive cell and (\mathbf n\in\mathcal T_N).

The primary user parameter is the real-space tuple ((N_1,N_2,N_3)), called the lattice extension. Alternatively, a shell radius (s_i) requests the odd extension

[ N_i=2s_i+1, ]

whose representatives run from (-s_i) through (+s_i). The Wigner–Seitz cell of the superlattice has a half-extent (N_i/2) in primitive lattice coordinates. This is analogous to k sampling because the reciprocal net below is its exact character group. It is not an additional real-space truncation: the selected Coulomb kernel remains BvK-periodic, and its Ewald or fitted long-range terms are not discarded outside a per-atom sphere.

The characters of this finite Abelian group are

[ \chi_{\mathbf m}(\mathbf n) =\exp\left(2\pi i\sum_i \frac{m_i n_i}{N_i}\right), \qquad m_i=0,\ldots,N_i-1, ]

corresponding to the Γ-centred mesh

[ \mathbf k_{\mathbf m}=\sum_i \frac{m_i}{N_i}\mathbf b_i \pmod{\mathcal L^*}. ]

An even-mesh half shift is a different boundary condition and is therefore not permitted. Symmetry reduction is also not used: all (N_c) characters belong to the exact transform.

For any block-circulant AO matrix,

[ X_{\mu\mathbf0,\nu\mathbf R}=X_{\mu\nu}(\mathbf R), ]

the finite Fourier pair is

[ X_{\mu\nu}(\mathbf k) =\sum_{\mathbf R\in\mathcal T_N} e^{+i\mathbf k\cdot\mathbf R}X_{\mu\nu}(\mathbf R), ]

[ X_{\mu\nu}(\mathbf R) =\frac1{N_c}\sum_{\mathbf k} e^{-i\mathbf k\cdot\mathbf R}X_{\mu\nu}(\mathbf k). ]

Consequently, a Gamma-point calculation on the (N_c)-cell BvK supercell, constrained to commute with primitive translations, and a primitive-cell calculation on the above full mesh are unitarily identical. This equality holds for every finite (N); it is not merely a large-cluster limit.

2. Wigner–Seitz representatives and boundary weights

For centres (A) and (B) with intra-cell displacement (\boldsymbol\delta_{AB}=\boldsymbol\tau_B-\boldsymbol\tau_A), an element ([\mathbf n]\in\mathcal T_N) has infinitely many representatives

[ \boldsymbol\delta_{AB}+A\left(\mathbf n+ \operatorname{diag}(N_1,N_2,N_3)\mathbf z\right), \qquad \mathbf z\in\mathbb Z^d. ]

Let (M_{AB}([\mathbf n])) be the set that minimises this Cartesian distance from centre (A). If (m_{AB,[\mathbf n]}=|M_{AB}([\mathbf n])|), the weight of each tied representative is

[ w_{AB}(\mathbf r\mid[\mathbf n])=\frac1{m_{AB,[\mathbf n]}},\qquad \sum_{\mathbf r\in M_{AB}([\mathbf n])} w_{AB}(\mathbf r\mid[\mathbf n])=1. ]

For coincident basis offsets in a one-dimensional four-cell cluster, residue 2 has representatives (+2) and (-2), each with weight (1/2). In skew cells there may be higher-multiplicity edge or vertex ties. The code solves the closest-vector problem and stops only when a smallest-singular-value bound proves that no unexamined image can tie or improve the minimum. The implementation accepts the explicit centre-pair offset, so a non-Bravais basis is not reduced to the coincident-centre special case.

Where the weights enter

The weights select a representative of a translation equivalence class. They do not multiply an AO merely because its atom lies on a geometric boundary. For a periodised one-electron kernel,

[ h^N_{\mu\nu}([\mathbf R]) =\sum_{\mathbf r\in M([\mathbf R])} w(\mathbf r\mid[\mathbf R])h^N_{\mu\nu}(\mathbf r). ]

All terms on the right are identical under a superlattice translation, so the weights form a partition of unity. Similarly, after fixing the first AO at the origin, a periodised four-centre integral is

[ (\mu\mathbf0,\nu[\mathbf R]\mid \lambda[\mathbf S],\sigma[\mathbf T])N =\sum{\mathbf r,\mathbf s,\mathbf t} w_{\mathbf R}(\mathbf r)w_{\mathbf S}(\mathbf s)w_{\mathbf T}(\mathbf t) (\mu\mathbf0,\nu\mathbf r\mid \lambda\mathbf s,\sigma\mathbf t)_N. ]

Again, this is an average over equivalent representatives of the same periodised integral. It therefore leaves the value unchanged. Applying the same class rule to every symmetry-related term preserves

[ (12\mid34)=(21\mid34)=(12\mid43)=(34\mid12)^*. ]

These identities are necessary for the Coulomb and exchange matrices to be functional derivatives of one scalar RHF energy.

This differs fundamentally from multiplying an ordinary free-space ERI by products of pair-dependent boundary factors. Pair-product factors generally change under ((12\mid34)\leftrightarrow(34\mid12)) and therefore do not, without an additional derivation or symmetrisation, define the usual RHF energy functional.

3. Finite-torus Coulomb Hamiltonian

The Coulomb kernel is periodised on the BvK supercell. In three dimensions it is the zero-average solution

[ -\nabla^2G_N(\mathbf r) =4\pi\left(\sum_{\mathbf L_N}\delta(\mathbf r-\mathbf L_N) -\frac1{\Omega_N}\right), ]

or, equivalently,

[ G_N(\mathbf r)=\frac{4\pi}{\Omega_N} \sum_{\mathbf G_N\ne\mathbf0} \frac{e^{i\mathbf G_N\cdot\mathbf r}}{|\mathbf G_N|^2}. ]

The (\mathbf G=0) coefficient is fixed to zero. The constant background in the Poisson equation makes each separated charge component finite; for a neutral electron-plus-nucleus cell the arbitrary constant cancels in the total energy. The current implementation rejects charged primitive cells rather than silently choosing a jellium convention.

The electron–nucleus, nucleus–nucleus, and Hartree terms must use this same gauge. The implementation obtains all three from vibe-qc’s common periodic Ewald/GDF path. Exchange uses the same periodised interaction. Its finite mesh (q=0) singular channel is treated with the BvK-supercell Madelung (exxdiv="ewald") correction already implemented in the periodic GDF backend. Using the primitive-cell Madelung constant here would define a different finite Hamiltonian and is incorrect.

Every executable χ-CCM result now records this finite-N convention explicitly: coulomb_kernel="3d-periodic-g0", exchange_q0="bvk-ewald", exchange_q0_applicability, the boundary model, the full character mesh, and the BvK supercell used for the Madelung constant. Applicability is active only when the actual operator has a nonzero full-range exact-exchange arm. It is therefore inactive for pure functionals and for HSE06, whose exact exchange is the finite short-range erfc kernel and has no singular q=0 seam. A fleet record is reportable only when its live applicability matches the independent route contract: RHF and post-HF routes and PBE0 are active, while PBE is inactive. Both audit_b.py and compare_b.py enforce that match; serializing the field without validating it is not sufficient evidence. The separate B-owned AICCM2026DevBExactExchangeAssembly records the route-resolved c_full, c_sr, and physical omega_screen_bohr_inv obtained from the shared live periodic exchange resolver, together with the fixed v1 schema, resolver, screened applicability, and screened assembly provenance. It belongs to route-resolved operator provenance rather than to the finite-torus family convention: exchange_q0_applicability is active exactly when c_full is nonzero, while an HSE-type screened arm has c_full=0, c_sr>0, positive omega_screen_bohr_inv, screened applicability active, and the implemented short-range-direct assembly. The active screened label is admitted only after matching branch-level BIPOLE evidence; it is not an independent numerical-support attestation or matrix oracle. A matched descriptor is necessary for any future Γ-CCM/χ-CCM comparison, but it is not sufficient to establish equality of the two constructions. The current reportable approach map is empty. A strict-zero-mode full-range exchange reference is a different finite-N Hamiltonian with the same thermodynamic target, not a hidden gauge choice. The familiar rank-1 Madelung expression is only the leading molecular-limit piece of the exchange seam. At finite solid-like cells the remaining non-rank-1 contribution is ordinary finite-size physics of the same (v_E) kernel; it is not an RI error and not a gauge freedom that can be discarded independently.

The 3D direct four-center realization also requires a coherent real-space cell set for neutral charge. When the Ewald J split is active, the nuclear real-space cutoff may not extend beyond the electronic J/K cutoff: otherwise nuclear point-charge images are included without their compensating electronic charge. The current default therefore resolves from 15 bohr electronic and 25 bohr nuclear to 15/15 bohr. Fleet records serialize these post-driver values as direct_lattice_cutoffs; the field is null for RI and RIJCOSX because those numbers are not their two-electron direct-lattice cutoffs. Four-center records made before the shared clamp in 6fed8620 are revision-bound and require a rerun before quantitative use.

The finite-group construction itself applies unchanged in one, two, or three periodic dimensions. In 3D the four-center route uses the neutral Ewald J split and the Ewald exact-exchange correction above. In 1D and 2D the available direct BIPOLE fallback is a direct-truncated active-lattice kernel, not the neutral wire/slab finite-torus Green function. aiccm2026dev-b therefore blocks the lower-dimensional four-center route before SCF. The shared lower-dimensional neutral-RI/GDF mesh is also not a Coulomb kernel: pinning every transverse reciprocal component to zero removes transverse Coulomb structure and gives a sheet-like term with vacuum-padding artifacts. aiccm2026dev-b therefore blocks every lower-dimensional SCF backend before SCF. Archived 1D/2D RI and RIJCOSX numbers are failure evidence, not 3D-periodic-in-vacuum model results that may be reported as absolute energies. A common 1D wire or 2D slab Green function must be established before any backend can be re-enabled and backend differences can be called fitting errors.

Low-dimensional shared-kernel stub

The fail-closed production path is intentional. The next lower-dimensional kernel module should be shared by four-center, RI, RIJCOSX, HF, KS, and post-HF routes, and should provide:

• a 2D slab Green function with isolated transverse boundary;

• a 1D wire Green function with isolated transverse boundary;

• the matching self-potential and exchange q=0 seam derived from the same kernel;

• tests proving that this mixed-boundary kernel, not the 3D exchange_q0="bvk-ewald" convention, is the finite-N Hamiltonian being correlated.

Until those items exist, isolated wire/slab absolute-energy claims remain out of scope.

4. RHF energy, Fock matrix, and double counting

Use a spin-summed density (D(\mathbf k)). With uniform weight (w_{\mathbf k}=1/N_c), the electron count per primitive cell is

[ N_e=\sum_{\mathbf k}w_{\mathbf k} \operatorname{Tr}[D(\mathbf k)S(\mathbf k)]. ]

The closed-shell projector condition is

[ D(\mathbf k)S(\mathbf k)D(\mathbf k)=2D(\mathbf k). ]

Define the periodised Coulomb and exchange contractions in the conventional way,

[ J_{12}[D]=\sum_{34}D_{34}(12\mid34)N, \qquad K{12}[D]=\sum_{34}D_{34}(13\mid24)_N. ]

Translation conservation makes these block diagonal in (\mathbf k), with the usual momentum-transfer sums implicit in the GDF contraction. The energy per primitive cell is

[ \mathcal E_N[D]=E_{NN}^N +\sum_{\mathbf k}w_{\mathbf k} \operatorname{Tr}[D(\mathbf k)h(\mathbf k)] +\frac12\sum_{\mathbf k}w_{\mathbf k} \operatorname{Tr}\left[D(\mathbf k) \left(J(\mathbf k)-\frac12K(\mathbf k)\right)\right]. ]

Its derivative is

[ F(\mathbf k)=h(\mathbf k)+J(\mathbf k)-\frac12K(\mathbf k). ]

The factors (1/2) and (1/2) are respectively the pair-counting factor and closed-shell exchange factor. No separate “central cell” energy is added, and no Wigner–Seitz factor is applied a second time. Each finite-group translation class occurs exactly once; boundary representative weights sum to one. This is the double-counting argument.

For RKS, let (E_{\mathrm{xc}}[\rho_D]) be the libxc functional and (\alpha) the functional’s exact-exchange fraction. The finite-torus energy is

[ \mathcal E_N^{\mathrm{RKS}}[D]=E_{NN}^N +\operatorname{Tr}_w[Dh] +\frac12\operatorname{Tr}_w[DJ] -\frac{\alpha}{4}\operatorname{Tr}w[DK] +E{\mathrm{xc}}[\rho_D], ]

where (\operatorname{Tr}_w) includes the uniform finite-character weights. Its derivative is

[ F^{\mathrm{RKS}}=h+J-\frac{\alpha}{2}K+V_{\mathrm{xc}}. ]

Thus pure semilocal DFT has (\alpha=0), whereas hybrids use the same periodised exchange tensor and finite-size convention as RHF.

5. Variational statement

For a fixed real-space extension, basis, periodised Coulomb gauge, and nuclear geometry, aiccm2026dev-b minimises the appropriate SCF functional over

[ \mathcal V_N={D:\ D^\dagger=D,\ DSD=2D,
\operatorname{Tr}(DS)=N_cN_e,\ [D,T_{\mathbf R}]=0}. ]

Equivalently, it minimises independently represented occupied subspaces at all finite-group (\mathbf k), subject to the common electron count. The generalised Roothaan equations are

[ F(\mathbf k)C(\mathbf k) =S(\mathbf k)C(\mathbf k)\varepsilon(\mathbf k). ]

DIIS and damping may alter the path to a stationary point but not the functional. A converged solution is stationary in (\mathcal V_N); as with ordinary RHF or approximate KS-DFT, SCF alone does not prove it is the global minimum. A fully unconstrained Gamma-supercell calculation has a larger determinant space and may break primitive translation symmetry. Such a broken-symmetry result is not required to equal the primitive-cell mesh result.

SCF path controls are numerical provenance rather than terms in the finite Hamiltonian. The B fleet therefore keeps the requested Fock-mixing fraction in scf_options.fock_mixing, while AICCM2026DevBDiagnostics.fock_mixing and convergence_diagnostics.fock_mixing record the executed effective weight of the previous Fock matrix after backend defaults are resolved. In the four-center KS backend, a request of 0.0 can execute as 0.30 when DIIS is disabled. D86 changes only the accuracy of that provenance record; it does not change the SCF algorithm, energy, or finite Hamiltonian. Pre-D86 DIIS-off four-center KS records are not convergence-fingerprint-complete.

D87 resolves the direct-call input before backend dispatch. A non-None fock_mixing= keyword overrides options.fock_mixing; otherwise the options field supplies the request. The resolver validates [0, 1) and does not rewrite the caller’s mixing field. Every four_center route and multi-cell fitted RHF/RKS can execute nonzero mixing. Three-dimensional Gamma-only RI RHF accepts resolved zero but fails closed on nonzero rather than switching to the legacy molecular-limit GDF operator; the other one-cell fitted restrictions remain unchanged. UHF and UKS can execute nonzero mixing on four_center, including both phases of a spin schedule, but fitted RI/RIJCOSX UHF/UKS fail closed because their multi-k open-shell GDF loop has no previous-Fock update. An explicit keyword zero therefore overrides a nonzero options value; on a DIIS-off four-center KS route, that requested zero remains subject to the separate automatic 0.30 execution rule. On supported execution routes this can change the SCF trajectory or stationary basin, but not the finite Hamiltonian or backend mixing formula. The Gamma explicit-zero correction deliberately restores the declared operator instead of preserving the old route-selection bug. Successful pre-D87 direct calls with differing keyword/options values, plus fitted RI/RIJCOSX UHF/UKS calls with a nonzero options-only request, have incomplete request fingerprints. Pre-D87 Gamma-only RI RHF calls where either input source was nonzero, including an explicit-zero keyword over nonzero options, followed the legacy operator and are not χ-CCM-B results. The closed-shell fleet is options-only and its records are unchanged. D72 and the analytic total-gradient fail-close remain in force. The shared D72 dimension guard precedes every backend-specific Gamma and mixing guard; this closes the former 1D one-cell RI-RHF escape, whose absolute-energy values are invalid and unreportable.

D97 applies the same execution discipline to unrestricted level shifting. For a restricted density (D=2P), the virtual-space shift has the form (F_b=F+bS-(b/2)SDS). For a unit-occupation unrestricted spin projector, the corresponding form is (F^\sigma_b=F^\sigma+bS-bSP^\sigma S). The shared four-center UHF/UKS loop currently applies the restricted /2 coefficient to each spin density, while the fitted multi-k UHF/UKS loop does not execute the request. The B entry points therefore reject a nonzero options.level_shift or any nonzero options.level_shift_schedule entry for every unrestricted backend before dispatch. run_periodic_job(..., level_shift=...) propagates its resolved value to the B options object before dispatch, rather than dropping a public request. Empty and all-zero schedules remain valid. The common D72 guard still runs first, so this convergence-control gate cannot expose a lower-dimensional energy. Restricted RHF/RKS driver semantics are unchanged, and the forwarding repair makes their supported public requests execute as declared.

Unrestricted variational space

Let (P^\alpha) and (P^\beta) be spin densities with unit occupation and let (P=P^\alpha+P^\beta). The finite-torus UHF functional per primitive cell is

[ \mathcal E_N^{\mathrm{UHF}}[P^\alpha,P^\beta] =E_{NN}^N+\sum_{\sigma\in{\alpha,\beta}} \operatorname{Tr}_w[P^\sigma h] +\frac12\operatorname{Tr}w[PJ[P]] -\frac12\sum\sigma\operatorname{Tr}_w[P^\sigma K[P^\sigma]]. ]

Its independent functional derivatives are

[ F^\sigma=h+J[P]-K[P^\sigma]. ]

The minimized set is

[ \mathcal V_N^{\mathrm U}={(P^\alpha,P^\beta): (P^\sigma)^\dagger=P^\sigma,\quad P^\sigma S P^\sigma=P^\sigma,\quad [P^\sigma,T_{\mathbf R}]=0,\quad \operatorname{Tr}(P^\sigma S)=N_c n_\sigma}. ]

Here (n_\alpha=(N_e+2S)/2) and (n_\beta=(N_e-2S)/2) are integers fixed by the primitive-cell multiplicity. This is a genuine unrestricted variation: alpha and beta orbitals are not constrained to share a spatial subspace. UHF stationarity does not prove spin purity. The implementation therefore reports both (\langle S^2\rangle) and the ideal (S(S+1)).

For UKS with a hybrid fraction (a_x), replace the UHF exchange energy by

[ -\frac{a_x}{2}\sum_\sigma \operatorname{Tr}w[P^\sigma K[P^\sigma]] +E{\mathrm{xc}}[\rho_\alpha,\rho_\beta], ]

which gives

[ F^\sigma_{\mathrm{UKS}}=h+J[P]-a_xK[P^\sigma] +V_{\mathrm{xc}}^\sigma. ]

No new Wigner–Seitz weight appears in these equations. Both spin projectors use the same declared finite Hamiltonian, including exchange_q0="bvk-ewald" when exact exchange is present, and the same translation-class partition.

6. Algorithm

1. Validate neutral, integer-occupation, zero-temperature RHF/RKS/UHF/UKS input in three periodic dimensions. For 1D/2D inputs, stop before SCF until one shared neutral wire/slab Coulomb kernel exists for every backend.

2. Form (\mathcal T_N), its exact Wigner–Seitz representative partition, and the unreduced Γ-centred character mesh.

3. Build lattice-summed overlap and one-electron matrices with the common periodic gauge.

4. Build the periodised J and K matrices with one of the three backends defined below. Apply the BvK-supercell exchange-divergence correction only in the 3D Ewald gauge.

5. Solve the per-k generalised eigenproblems using the existing canonical linear-dependence handling and SCF accelerator.

6. Return energy per primitive cell and measure (\max_k\lVert D_kS_kD_k-2D_k\rVert_F), electron-count error, Wigner–Seitz partition error, and the imaginary residue of the inverse Bloch density.

The inverse Bloch transform stored in the implementation is also the bridge for future explicitly real-space post-HF work.

Integral backends

All backends act on the same finite-character density at the declared coulomb_kernel="3d-periodic-g0" and exchange_q0="bvk-ewald" convention. Backend differences are then representations of the electron-repulsion tensor, not silent Hamiltonian changes.

1. four_center uses the periodic BIPOLE engine in 3D: direct four-centre short-range J/K plus the reciprocal long-range Hartree term and BvK exchange correction. The 1D/2D four-centre route fails closed because the current direct-truncated fallback is not the neutral wire/slab finite-torus Green function and over-binds chain benchmarks by a Madelung-scale shift. D105 explicitly disables BIPOLE’s quartet far-field prototype on every χ dispatch. The shared drivers now fail closed on an explicit request because the prototype does not preserve the exact three-translation Fock domain. D93 attests the direct M5 image domain, not far-field activation, order, penetration dispatch, or approximation error.

2. ri uses pair-resolved RSGDF or MDF. With auxiliary functions (P,Q) and Coulomb metric (V_{PQ}=(P|Q)),

   [ (\mu\nu|\lambda\sigma){\mathrm{RI}} =\sum{PQ}(\mu\nu|P)(V^{-1})_{PQ}(Q|\lambda\sigma). ]

   The fit is applied consistently to J and K.

3. rijcosx uses the same pair-resolved RI J and evaluates exact exchange by periodic chain-of-spheres quadrature. It is therefore a controlled three-centre/seminumerical approximation, not a different cyclic Hamiltonian.

Fitted nuclear assembly and inherited support guards

D99 pins the shared fitted-route behavior at the χ-CCM-B selector. Since ddbe859d1, every supported 3D fitted Gamma-only path and every multi-k restricted or open-shell route evaluates nucleus–nucleus repulsion with ewald_nuclear_repulsion rather than nuclear_repulsion_per_cell. This pins routing only. D102 records the historical defect: the shared helper first preselected translations by an unshifted norm and could omit a shifted pair inside the requested cutoff. The four_center route used the same affected helper. Same-helper equality therefore proved assembly, not convergence, and every successful or non-converged χ-B absolute-energy row from that interval was quarantined. Here Gamma-only identifies a one-character evaluation path inside χ-CCM; it does not assign Γ-CCM construction identity.

That historical quarantine is permanently bound to exact vibeqc.aiccm2026dev-b.ewald-shifted-pair-support/v1 evidence. Version 1 requires the affected implementation label, qualification="not-qualified", and repair_commit=null; it has no qualified state and is never upgraded.

D104 binds the repaired shared helper independently. Commit e578b86c00268a172b651cae29c7845836e31e17 enumerates pairs from the centered source-observer displacement with exact interplanar bounds. Before entering SCF, the producer verifies repair ancestry and runs an 18-bohr MgO point-charge canary that shifts the oxygen basis charge by 37 a1. The stale core differs by about 15.08 Ha; the repaired core gives zero difference against a 1e-10 Ha tolerance. Failure stops before any absolute energy is assembled. The cached canary is then bound after calculation to finalized D77 identity in exact vibeqc.aiccm2026dev-b.ewald-shifted-pair-support/v2 evidence: implementation label, repair SHA, full producer commit, core build ID, probe-attestation ID, verified ancestry, and the exact canary payload. The audit and comparator cross-bind those identities to provenance and the nested attestation. A fresh v2 row may clear this shared-Ewald component only; D93, D98, and D103 remain independent.

Since 6ce142339, the shared GDF layer rejects compact dense-core MDF systems in the MgO/STO-3G class, and that failure propagates through the B selector. The separately validated vacuum-padded MDF envelope is unchanged, but D93 continues to mark every fitted B row not-qualified. Commit ad96a2f63 also gives the fitted B routes a bounded shared-q auxiliary Fourier-transform cache. That inheritance is bit-identical and changes memory/performance only. Every fitted B RHF/RKS/UHF/UKS dispatch additionally pins ibz_native=False and compute_gradient=False. Future generic IBZ or GDF-gradient defaults therefore cannot replace the complete unreduced character mesh or open a total-gradient route. None of these shared changes widens B: 1D/2D SCF and all analytic total-gradient routes still fail closed, and D83, D88, and D98 are unchanged. The run_ccm_rhf_gdf campaign producer is a neutral fitted-torus Bloch/GDF representation control, not Γ-CCM production, and separately recomputes its executed nuclear scalar through the same shared Ewald helper. It emits D104 v2 for that helper but always remains quantitatively not-qualified because it is a Bloch/GDF representation control with unqualified fitted support. It is not the mapped real-Gamma control. The mapped A-owned aiccm-hf-direct producer currently emits no D104 v2 evidence, so the optional control remains unavailable. The dedicated D101 run_case_cmp_b.py producer is instead the high-level χ-CCM RI route-control diagnostic assigned to this line. Neither producer realizes the union-and-weight/Wigner–Seitz Γ-CCM construction, so their labels and common lower-level ancestry cannot establish a Γ-CCM/χ-CCM approach result.

For global hybrids such as PBE0, all three backends use the same full-range exchange convention described above. The 3D four-center RKS/UKS route also supports HSE06 as (0.25K_{\operatorname{erfc}}(\omega=0.11)), with no full-range exchange seam. RI and RIJCOSX currently fail closed for HSE06 and all other range-separated functionals because the χ fitted-backend contract has no B-owned screened-COSX validation or provenance. The shared generic COSX backend can build HSE exchange, but this does not widen the χ construction by inheritance. This backend split prevents the former silent HSE06 to PBE0 substitution; it is not a Γ-CCM/χ-CCM construction or approach distinction. The four-center HSE algebra now uses the shared M5 padded screened-exchange traversal. D98 records this as short-range-direct only when the K-erfc branch emits matching execution evidence; it never infers the label from the functional name. HSE remains outside the B fleet and still needs route-specific quantitative validation before its absolute energies are reportable.

All three backends are currently 3D-only. Lower-dimensional inputs fail before the backend build; RI and RIJCOSX do not retain a 3D-periodic-in-vacuum exception.

The q-only compcell fit is rejected: on tight cells it does not represent one consistent four-centre finite-torus tensor and is known to admit non-physical converged fixed points. The present native bridges also reject one-cell RI/RIJCOSX RKS and one-cell RIJCOSX RHF rather than silently substituting a different Gamma-only algorithm. In 3D, one-cell RI RHF is supported only at resolved fock_mixing=0: a nonzero request would make the shared dispatcher substitute the legacy molecular-limit GDF exchange operator for the declared pair-resolved exxdiv="ewald" route, so χ-CCM-B fails closed.

7. Canonical finite-torus RI-MP2

MP2 is built on the converged χ-CCM RI-RHF determinant, not on the legacy Gamma-supercell GDF driver. That distinction is numerical as well as formal: on the two-cell 8 x 12 x 12 bohr H2 control, the legacy Gamma-supercell HF reference differs from χ-CCM RI-HF by 0.0046408868 Ha per cell. Reusing it would change the zeroth-order Hamiltonian. The difference is not a gauge-invariant post-HF offset. The exchange q=0 seam shifts the HF orbital energies, so the MP2 denominator [ \Delta_{ij}^{ab}=\varepsilon_i+\varepsilon_j-\varepsilon_a-\varepsilon_b ] inherits the declared convention. External KMP2 or CCSD(T) parity is meaningful only when the reference and the correlated calculation both use exchange_q0="bvk-ewald" or both use some explicitly labelled alternative.

Let (i,j) denote occupied bands, (a,b) virtual bands, and let (N_k=N_c). The pair-resolved canonical auxiliary factor is

[ \widetilde L^{\mathbf k_i\mathbf k_a}{P\mu\nu} =\left[U{\mathbf q}\lambda_{\mathbf q,+}^{-1/2} U_{\mathbf q}^{\dagger}T_{\mathbf q}\right]_{P\mu\nu}, \qquad \mathbf q=\mathbf k_a-\mathbf k_i. ]

Here only metric eigenvalues above the declared threshold enter. Keeping the factor in the original auxiliary-AO basis is important: the compact factor (\lambda^{-1/2}U^{\dagger}T) is invariant in an SCF contraction with its own adjoint, but independently diagonalised (\mathbf q) and (-\mathbf q) blocks have arbitrary eigenvector phases. Those phases need not cancel in an MP2 contraction between two blocks. The canonical matrix function above is unique on the retained subspace and obeys the required conjugation relation.

Transform one occupied and one virtual index,

[ L^{\mathbf k_i\mathbf k_a}{Pia} =\sum{\mu\nu}C_{\mu i}^{*}(\mathbf k_i) \widetilde L^{\mathbf k_i\mathbf k_a}{P\mu\nu} C{\nu a}(\mathbf k_a). ]

For every triple ((\mathbf k_i,\mathbf k_a,\mathbf k_j)), the fourth character is fixed exactly by

[ \mathbf k_b=\mathbf k_i-\mathbf k_a+\mathbf k_j+\mathbf G, ]

where (mathbf G) returns the point to the finite reciprocal group. The RI two-electron block in the normalization used here is

[ V_{ij}^{ab}(\mathbf k_i,\mathbf k_j,\mathbf k_a) =\frac1{N_k}\sum_P L^{\mathbf k_i\mathbf k_a}{Pia} L^{\mathbf k_j\mathbf k_b}{Pjb}. ]

With

[ \Delta_{ij}^{ab}=\varepsilon_{i\mathbf k_i} +\varepsilon_{j\mathbf k_j}-\varepsilon_{a\mathbf k_a} -\varepsilon_{b\mathbf k_b}, ]

the closed-shell spin components per primitive cell are

[ E_{\mathrm{OS}}^{(2)}=\frac1{N_k} \sum_{\mathbf k_i\mathbf k_j\mathbf k_a}\sum_{ijab} \frac{|V_{ij}^{ab}(\mathbf k_a)|^2}{\Delta_{ij}^{ab}}, ]

[ E_{\mathrm{SS}}^{(2)}=\frac1{N_k} \sum_{\mathbf k_i\mathbf k_j\mathbf k_a}\sum_{ijab} \frac{|V_{ij}^{ab}(\mathbf k_a)|^2 -V_{ij}^{ab}(\mathbf k_a)^{*}V_{ij}^{ba}(\mathbf k_b)} {\Delta_{ij}^{ab}}, ]

and (E_{\mathrm{MP2}}^{(2)}=E_{\mathrm{OS}}^{(2)}+ E_{\mathrm{SS}}^{(2)}). MP2 is not variational: it is the second-order Rayleigh–Schrodinger correction for the finite RI Hamiltonian around the stationary RHF determinant. The finite-group Fourier transform nevertheless makes it exactly equivalent to canonical RI-MP2 in the corresponding BvK supercell when the same Coulomb kernel, exchange q=0 convention, and fit are used.

The implemented 3D H2/STO-3G two-point mesh agrees with an out-of-process PySCF KRHF/KMP2 calculation to (5.9\times10^{-10}) Ha in HF and (2.4\times10^{-10}) Ha in correlation energy per cell. The machine-readable record is docs/manuscripts/aiccm_comparison/data/h2_mp2_2026-06-21.json. MP2 currently fails closed in 1D and 2D until O1 supplies a common lower-dimensional Coulomb gauge.

Unrestricted MP2

For an unrestricted reference, occupied and virtual labels carry an explicit spin. With antisymmetrized same-spin integrals and ordinary opposite-spin integrals, the second-order correction is

[ E_{\mathrm{UMP2}}^{(2)}=E_{\alpha\alpha}^{(2)} +E_{\beta\beta}^{(2)}+E_{\alpha\beta}^{(2)}, ]

[ E_{\sigma\sigma}^{(2)}= \frac14\sum_{ij\in\sigma}\sum_{ab\in\sigma} \frac{|\langle i_\sigma j_\sigma\Vert a_\sigma b_\sigma\rangle|^2} {\varepsilon_i^\sigma+\varepsilon_j^\sigma -\varepsilon_a^\sigma-\varepsilon_b^\sigma}, ]

[ E_{\alpha\beta}^{(2)}= \sum_{i a\in\alpha}\sum_{j b\in\beta} \frac{|(i_\alpha a_\alpha|j_\beta b_\beta)|^2} {\varepsilon_i^\alpha+\varepsilon_j^\beta -\varepsilon_a^\alpha-\varepsilon_b^\beta}. ]

Every sum also contains the finite characters, with the same momentum conservation and one final division by (N_c) as the restricted expression. The implementation evaluates this equation after the exact inverse transform of the χ-CCM RI Hamiltonian. Zero PNO threshold and canonical occupieds give the full UMP2 result. Independent unitary localization of the alpha and beta occupied projectors leaves both spin densities invariant; the coupled local MP2 equations are then required because each localized occupied Fock block is not diagonal.

8. Real-torus local correlation and coupled cluster

Exact inverse transform of the fitted Hamiltonian

Let the normalized finite-group Bloch AO be

[ |\mu\mathbf{k}\rangle =N_c^{-1/2}\sum_{\mathbf R\in\mathcal T_N} e^{+i\mathbf k\cdot\mathbf R}|\mu\mathbf R\rangle . ]

For any translation-invariant one-particle matrix, the full finite-torus AO matrix is therefore

[ X_{\mu\mathbf R,\nu\mathbf S} =\frac{1}{N_c}\sum_{\mathbf k} e^{+i\mathbf k\cdot\mathbf R}X_{\mu\nu}(\mathbf k) e^{-i\mathbf k\cdot\mathbf S}. ]

This formula is used for both overlap and Fock. It is a unitary change of representation, not a new Gamma-point SCF.

The RI factors need one more character. Let (\widetilde L^{\mathbf k_i\mathbf k_a}_{P,\mu\nu}) be the canonical auxiliary-AO factor of Section 7 and (\mathbf q=\mathbf k_a-\mathbf k_i). Define

[ B_{P\mathbf T,\mu\mathbf R,\nu\mathbf S} =\frac{1}{N_c^2}\sum_{\mathbf k_i\mathbf k_a} e^{+i\mathbf k_i\cdot\mathbf R} e^{-i\mathbf k_a\cdot\mathbf S} e^{+i\mathbf q\cdot\mathbf T} \widetilde L^{\mathbf k_i\mathbf k_a}_{P,\mu\nu}. ]

The corresponding real-torus MO coefficient is

[ C_{\mu\mathbf R,p\mathbf k} =N_c^{-1/2}e^{+i\mathbf k\cdot\mathbf R}C_{\mu p}(\mathbf k). ]

Contracting the previous two equations gives

[ B_{P\mathbf T,i\mathbf k_i,a\mathbf k_a} =N_c^{-1}e^{+i\mathbf q\cdot\mathbf T} L^{\mathbf k_i\mathbf k_a}_{Pia}. ]

Thus a product of factors with transfers (\mathbf q) and (-\mathbf q) satisfies

[ \sum_{P\mathbf T}B_{P\mathbf T,ia}B_{P\mathbf T,jb} =\frac{1}{N_c}\sum_P L_{Pia}L_{Pjb}, ]

while any nonconserving transfer vanishes by character orthogonality. This is exactly the normalization in the momentum-space MP2 integral. The real and character representations therefore describe the same χ-defined fitted finite Hamiltonian at the recorded Coulomb and exchange q=0 convention. This internal Fourier equivalence does not assert equality with the union-and-weight Γ-CCM approach.

Local MP2 and CCSD(T)

The present local implementation deliberately builds the whole finite torus. It includes every occupied pair and every ordered occupied triple required by the spin-integrated equations, obtains a total correlation energy (E_{\mathrm{corr},N}), and reports

[ E_{\mathrm{corr}}^{\mathrm{cell}} =\frac{E_{\mathrm{corr},N}}{N_c}. ]

There is no additional boundary or pair multiplicity. A future translation- reduced implementation may retain one orbit under simultaneous translation of all occupied indices, but then its multiplicity must follow the orbit-stabilizer theorem. Selecting visually unique pairs and multiplying by an occurrence factor would recreate the historical ambiguity.

Occupied orbitals are localized with the Pipek–Mezey objective built from Mulliken populations. This is invariant to the choice of position origin and does not use the ill-defined ordinary position operator on a torus. Molecular Foster–Boys localization is rejected. The occupied rotation is unitary, so with complete pair domains, no PNO truncation, no pair screening, and complete occupied coupling, local MP2 and local CCSD reproduce their canonical finite-torus limits. With complete triple virtual spaces the same statement holds for the perturbative triples correction.

The current defaults allow PNO truncation but keep all occupied pairs and all occupied couplings. Distance-based pair screening and local auxiliary fitting are disabled because their existing molecular implementations use Euclidean distances across the cell boundary. They require minimum-image Wannier and auxiliary domains before they can be enabled without breaking translation symmetry.

The local-correlation result records the same finite-torus convention as its HF reference. This is required because the semicanonical occupied and virtual energy blocks used in MP2, CCSD, triples, and future DLPNO denominators inherit the exchange q=0 seam from that reference.

The gauge-invariant SCF property object, the finite-torus band-structure wrapper, and the finite-torus Mayer bond-order analysis also record that convention. These analyses are derived from the same one-particle Fock, density, and overlap matrices; the descriptor is provenance for the finite Hamiltonian, not an additional factor in the population formula or band interpolation.

Neither MP2 nor CCSD(T) is variational. CCSD is stationary only through its left-right coupled-cluster Lagrangian, and the perturbative triples energy has no upper-bound property. PNO/domain truncations add controlled numerical approximations, not a new Hamiltonian or a variational subspace theorem.

On the H2/STO-3G two-cell control, no-truncation real-torus local MP2 equals the character-space result within (7\times10^{-18}) hartree per cell. On the LiH/STO-3G two-cell control, no-truncation local and canonical finite-torus DF-CCSD correlation energies differ by (6.9\times10^{-12}) hartree total; their nonzero triples corrections differ by (1.4\times10^{-12}) hartree. These are finite-Hamiltonian exact-limit tests, not external infinite-crystal CCSD(T) benchmarks.

For UCCSD, use spin orbitals and

[ |\Psi_{\mathrm{CCSD}}\rangle=e^{T_1+T_2}|\Phi_0\rangle, \qquad \langle\Phi_\mu|e^{-T}He^T|\Phi_0\rangle=0 ]

for every single and double excitation (\mu). The energy is

[ E_{\mathrm{CCSD}}=\langle\Phi_0|e^{-T}He^T|\Phi_0\rangle. ]

The unrestricted PNO pilot expands each pair amplitude into the full finite-torus spin-orbital virtual space, evaluates the complete projected residual, and projects back. With zero PNO threshold this projection is a unitary coordinate change and recovers full-domain UCCSD. Perturbative triples are evaluated from the converged amplitudes. This establishes the equations and exact PNO limit, but the current implementation remains an explicitly cost-capped O(N^6) oracle: it does not yet skip translation-equivalent pairs or triples and therefore does not yet establish reduced scaling.

Post-SCF occupied localization

Let the complete finite character group contain (N_c) points. For each isolated occupied band, the canonical back transform is

[ |w_{n\mathbf R}^{0}\rangle =\frac{1}{\sqrt{N_c}}\sum_{\mathbf k} e^{-i\mathbf k\cdot\mathbf R}|\psi_{n\mathbf k}\rangle . ]

All (N_c n_{\mathrm{occ}}) functions are rotated together, (C^{\mathrm{loc}}=C^{0}U), with (U^\dagger U=I). Consequently

[ C^{\mathrm{loc}}C^{\mathrm{loc}\dagger} =C^0UU^\dagger C^{0\dagger}=C^0C^{0\dagger}. ]

This proves invariance of the finite-torus density and every RHF or RKS energy term. Localizing independent k blocks would not produce the required real-space family.

D100 evaluates the corresponding numerical audits without forming either full AO projector. For (P_i=C_iC_i^\dagger),

[ |P_1-P_0|_F^2 =|C_1^\dagger C_1|_F^2+|C_0^\dagger C_0|_F^2 -2|C_1^\dagger C_0|_F^2, ]

and the one-particle check uses

[ \operatorname{Tr}[(P_1-P_0)F] =\operatorname{Tr}(C_1^\dagger F C_1) -\operatorname{Tr}(C_0^\dagger F C_0). ]

For a cyclic AO-row permutation (T), translation covariance is checked from

[ |TP_1T^\dagger-P_1|_F^2 =2|C_1^\dagger C_1|_F^2 -2|(TC_1)^\dagger C_1|_F^2. ]

The native kernel keeps only coefficient and occupied-space work arrays, reducing diagnostic scratch from (O(n_{\mathrm{AO}}^2)) to (O(n_{\mathrm{AO}}n_{\mathrm{occ}}+n_{\mathrm{occ}}^2)). The earlier dense projector equations remain a private parity oracle. When the Gram-norm subtraction is close enough to exact cancellation to lose relative precision, the native path streams the AO-pair projector difference and accumulates its norm without storing the matrix. This changes only how the three scalar diagnostics are evaluated; the localization rotation and every downstream PNO or correlated quantity are unchanged.

Exact Marzari–Vanderbilt localization requires cross-k matrix elements of the exponential position operator. Those integrals are not exposed by the current native API. The implemented Wannier route jointly diagonalizes Lowdin-projected AO-centre operators (\cos(2\pi r_\alpha/L_\alpha)) and (\sin(2\pi r_\alpha/L_\alpha)). If

[ z_{n\alpha}=\langle w_n|e^{2\pi i r_\alpha/L_\alpha}|w_n\rangle, \qquad \Omega_{n\alpha} =-\left(\frac{L_\alpha}{2\pi}\right)^2\log|z_{n\alpha}|^2, ]

the latter is the reported circular spread. It has the correct torus topology but is labelled a projected circular AO-centre approximation. Exact continuum spreads and Wannier90 parity remain open.

The IAO alternative constructs polarized intrinsic atomic orbitals from a minimal reference and maximizes squared atomic populations. Its present cross overlap is an explicit-supercell integral rather than a periodized torus integral. A four-cell H2 run gives translation covariance residuals of (1.17\times10^{-13}) for the Wannier path and (7.08\times10^{-6}) for IAO. Density invariance remains at machine precision. IAO translation maps are therefore diagnostic and are not used to skip correlated pairs.

Wraparound is flagged when the antipodal-shell probability exceeds 0.05 or when (\sqrt{\Omega_n}) exceeds one quarter of the shortest cyclic-cluster vector. A nonpositive indirect gap fails closed. Metallic-band disentanglement is not implemented.

PAOs, PNOs, and pair orbits

For an (S)-orthonormal occupied coefficient matrix (C_o), the coefficient projector out of the occupied space is

[ Q=I-C_oC_o^\dagger S,\qquad C_o^\dagger S Q=0. ]

If (E_D) selects AOs in a pair domain, let (\widetilde V_D=QE_D) and diagonalize its Gram matrix:

[ G_D=\widetilde V_D^\dagger S\widetilde V_D=XgX^\dagger, \qquad V_D=\widetilde V_DX_+g_+^{-1/2}. ]

Then (V_D^\dagger S V_D=I) and (C_o^\dagger S V_D=0). This removes both occupied leakage and redundant PAOs.

For pair amplitude matrix (T_{ij}), the model pair density is

[ D_{ij}=\frac{T_{ij}T_{ij}^\dagger+T_{ij}^\dagger T_{ij}} {1+\delta_{ij}}. ]

It is Hermitian positive semidefinite because (x^\dagger D_{ij}x=(|T_{ij}^\dagger x|^2+ |T_{ij}x|^2)/(1+\delta_{ij})\ge0). Its eigenvectors are PNOs. A zero threshold retains the complete PAO span, and positive thresholds give nested ranks. This proves rank nesting, not monotonic MP2 energy error: MP2 is nonvariational and convergence of the energy must be measured.

Translations act simultaneously on both indices of an unordered occupied pair. For measured localization permutations (p_g),

[ \mathcal O(i,j)={\operatorname{sort}(p_g(i),p_g(j)):g\in G_N}. ]

The per-cell coefficient is (|\mathcal O|/N_c), including smaller orbits with nontrivial stabilizers such as half-cell pairs on an even ring. The code verifies a disjoint partition of all unordered pairs, but it still evaluates all pairs. The orbit reduction factor is not a timing claim.

At complete PAO domains and zero PNO thresholds, the B MP2 wrapper also evaluates a rotation-invariant full-space DF-MP2 contraction. The raw iterative local-pair energy and the correction are both reported. Before the real local solver, a full-rank real time-reversal gauge is constructed and its occupied projector is checked against the complex localized projector. This avoids silently discarding imaginary coefficients. No correction is applied to truncated calculations.

Finite-cluster space-group symmetry

Write primitive lattice vectors as columns of (A), and let (N=\operatorname{diag}(N_1,N_2,N_3)). A crystallographic operation (g={W|\mathbf w}) preserves the finite torus exactly when

[ N^{-1}WN\in\mathbb Z^{3\times3}. ]

For primitive atom (a), retain both the atom permutation and integer image shift,

[ W\mathbf f_a+\mathbf w=\mathbf f_{p_g(a)}+\mathbf q_{g,a}, ]

so ((a,\mathbf r)) maps to

[ \left(p_g(a),W\mathbf r+\mathbf q_{g,a}\pmod{\mathbf N}\right). ]

The image shift is essential for screw and glide operations. Reciprocal characters transform as (\mathbf k’=W^{-T}\mathbf k), and a k-orbit has weight (|\mathcal O_k|/N_c). spglib supplies the crystallographic group; the B module builds its exact cluster-compatible subgroup.

Symmetry is diagnostic only. At Gamma, AO matrices can be checked with the Reynolds projector

[ \mathcal P_G[M]=\frac{1}{|G_N|}\sum_g U_g M U_g^T, ]

whose idempotency follows from group closure. At general k, nonsymmorphic sewing phases depend on each backend’s Bloch convention. Therefore symmetry_mode="integrals" fails closed until representative shell-pair and quartet scattering passes full-build Fock and energy parity. Primitive shell pairs and intrinsically eightfold-unique shell quartets are already partitioned into exact point-group orbits and reported as reduction counts; no libint call is skipped from those counts yet.

9. Limits and convergence

Exact finite-size identity

For the same declared finite-torus Hamiltonian, in particular the same exchange_q0="bvk-ewald" finite-size convention, and the same translation-invariant variational space,

[ E_N^{\Gamma\text{-supercell}}/N_c =E_N^{\Gamma\text{-centred mesh}} ]

up to numerical error. This is the strongest internal reference and should hold at every mesh size.

Infinite-crystal limit

As all active (N_i\to\infty), the character sum approaches the Brillouin zone integral and the BvK Green’s function approaches the chosen infinite periodic Coulomb convention. Insulators with localised density matrices are expected to converge rapidly in short-range pieces; exact exchange and the Coulomb finite-size terms can converge algebraically. The sequence need not be monotone because each (N) defines a different finite Hamiltonian.

Molecular limit

For a one-cell cluster with increasing vacuum, image interactions and the BvK Madelung correction vanish, and the result approaches molecular RHF or RKS in the same basis. Basis linear dependence is handled per k by canonical orthogonalisation; dropping different ranks at different k is diagnosed and can spoil smooth convergence.

BSSE is not repaired by the cyclic construction. Counterpoise or basis extrapolation is a separate modelling choice.

10. Assumptions and explicit divergences from historical CCM

Audit of the 2008 deMon2k KS-ADFT CCM

Janetzko, Köster, and Salahub define the two-center occurrence weight

[ \omega_{MN}=\frac{1}{n_{MN}},\qquad T^{\mathrm{CCM}}=\sum_{\mu\nu}P^{\mathrm{CCM}}{\mu\nu} \omega{MN}\sum_{\nu’}T_{\mu\nu’}. ]

For a three-center quantity they do not merely multiply one selected integral. Their Eq. (20) averages both nested enumerations (M’\to N’) and (N’\to M’), divided by

[ n_{MNC}=n_{MN}(n_{MC}+n_{NC}),\qquad \omega_{MNC}=n_{MNC}^{-1}. ]

That explicit two-order average matters. It means the simple (M\leftrightarrow N) objection to the 2014 four-center factor cannot be transferred unchanged to the deMon2k three-center tensor. If both enumerations are complete, Eq. (20) is symmetric in the two AO centers by construction.

The two-center formula still requires the stronger Hermiticity identity

[ \omega_{MN}\sum_{\nu’}I_{\mu\nu’} =\omega_{NM}\sum_{\mu’}I_{\nu\mu’}. ]

The free-space identity (I_{\mu\nu}=I_{\nu\mu}), cited in the paper, does not by itself prove this equality for two different center-specific Wigner– Seitz enumerations. Likewise, variational auxiliary fitting requires the weighted three-center map and weighted auxiliary Coulomb metric to be adjoints under one finite inner product. The paper supplies a fitted-energy expression and its derivative, so the intended scalar functional is clearer than in the 2014 four-center derivation, but it does not prove this quotient-space adjoint condition for arbitrary non-Bravais or skew clusters.

There is also a separate finite-size issue. The paper explicitly states that its CCM and the corresponding periodic calculation are not fully equivalent because interactions outside the Wigner–Seitz regions are omitted. Its auxiliary-density lattice sum is truncated after one neighboring shell, and the XC quadrature is evaluated on a reference-cluster grid with translated auxiliary functions. Those can converge with cluster size, but they are not an exact finite-(N) BvK identity. Thus the deMon2k multiplicative weights remain a symmetry and variational audit target; the present derivation does not label them broken where Eq. (20) already enforces the relevant AO-pair symmetry.

1. Finite torus, not a weighted open cluster. The method is defined by a BvK Hamiltonian first. Wigner–Seitz geometry is a representation of its translation classes.

2. Boundary weights are a partition of unity. They average only tied representatives of the same periodised object. They are not attached to atoms, shells, or arbitrary AO pairs.

3. No historical product-weighted ERI. The published pair-product four-centre factors are not invariant under all ERI permutations in their displayed form. Without a separate scalar functional derivation, using them directly would make the variational and double-counting status unclear. aiccm2026dev-b therefore does not use them.

4. No Fock symmetrisation as a repair. Hermiticity follows from the Hamiltonian. A non-Hermitian raw Fock is treated as a bug.

5. Long range is part of the Hamiltonian. It is not an after-the-fact Madelung energy added to a short-range SCF.

6. Charged cells fail closed. No implicit uniform-background energy is reported.

7. Exchange is not independently truncated. It uses the finite-torus periodised kernel and BvK-size divergence correction.

8. The reciprocal implementation is exact, not a reference shortcut. It is the diagonal representation of the finite cyclic translation group.

9. Source agreement is secondary. Historical numerical agreement does not override idempotency, permutation symmetry, variational consistency, gauge consistency, or finite-supercell/mesh equivalence.

11. Validation ladder

Fixed-density kinetic derivative component

For a caller-supplied real-torus density whose cell list is fixed by declared lattice_options, the kinetic contribution is

[ E_T[D;R] = \sum_g \operatorname{Tr}[D(g) T(g;R)]. ]

compute_aiccm2026dev_b_fixed_density_kinetic_energy(...) evaluates this scalar, while compute_aiccm2026dev_b_fixed_density_kinetic_gradient(...) evaluates (\partial E_T/\partial R_A) from analytic AO-centre derivatives at fixed (D(g)). The derivative is independently checked against central differences of the matching scalar after rebuilding the displaced basis. Both paths require exactly the same lattice-cell indices and Cartesian translations, AO dimension, and block shapes.

There is no SCF convenience wrapper for this component. The current SCF result does not attest the resolved one-electron lattice cutoff, so reconstructing an operator from a guessed or default cutoff could silently compare a different cell support. The stationary energy-weighted density construction and Pulay/adjoint assembly, BvK exchange-q=0 seam or screened-exchange derivative, RI/RIJCOSX three-centre and metric response, DFT exchange-correlation quadrature/grid derivatives where applicable, and post-HF response terms remain open. The public total-gradient route therefore still fails closed. Γ-CCM and χ-CCM are distinct approaches compared at a declared common exchange-q=0 convention; their names do not select different Coulomb kernels.

Fixed energy-weighted overlap derivative component

For a caller-supplied real-torus energy-weighted density (W(g)), define the fixed overlap-constraint scalar

[ \mathcal L_S[W;R] =-\sum_{g\mu\nu}W_{\mu\nu}(g)S_{\mu\nu}(g;R) =-\sum_g\operatorname{Tr}[W(g)^{\mathsf T}S(g;R)]. ]

The transpose in the final expression is important: individual real-space residue blocks need not be symmetric even when the complete finite-character object obeys Hermiticity and time reversal. At fixed numerical (W(g)) and fixed lattice,

[ \left.\frac{\partial\mathcal L_S}{\partial R_{A\alpha}}\right|{W,\mathrm{lattice}} =-\sum{g\mu\nu}W_{\mu\nu}(g) \frac{\partial S_{\mu\nu}(g;R)}{\partial R_{A\alpha}}. ]

compute_aiccm2026dev_b_fixed_energy_weighted_overlap_lagrangian(...) evaluates the scalar, and compute_aiccm2026dev_b_fixed_energy_weighted_overlap_gradient(...) evaluates the analytic AO-centre derivative. A 3D odd-character conjugate pair produces real but individually nonsymmetric (W(g)) blocks, pinning the Frobenius orientation and sign against central differences of the public scalar. Both helpers require identical ordered cell indices, Cartesian translations, AO dimension, and block shapes.

This scalar is a Lagrangian constraint companion, not a separately additive SCF energy. The helpers do not construct the stationary χ-CCM energy-weighted density from orbitals, occupations, spin channels, backend, or exchange convention. They also do not include the separate explicit derivative of the BvK exchange seam or screened-exchange kernel. Consequently, they prove only the overlap skeleton; B-owned adjoint construction and variational assembly remain open, and the total gradient still fails closed. Γ-CCM and χ-CCM are distinct approaches compared at a declared common exchange-q=0 convention; their names do not select different Coulomb kernels.

Stationary energy-weighted-density binding contract

D85 specifies the object that must eventually connect the fixed-W D80 skeleton to a stationary SCF Lagrangian. For character (q) and spin (\sigma), rebuild the final physical variational Fock (F^\sigma_{\mathrm{var}}(q)=\partial E/\partial D^\sigma(q)) without DIIS mixing, damping, level shift, clipping, or a stale pre-final operator. With an orthonormal occupied block (C_o^{\sigma\dagger}S C_o^\sigma=I), define

[ \Lambda^\sigma=C_o^{\sigma\dagger}F^\sigma_{\mathrm{var}}C_o^\sigma, \qquad W(q)=\begin{cases} 2C_o\Lambda C_o^\dagger,&\text{RHF/RKS},\ \sum_\sigma C_o^\sigma\Lambda^\sigma C_o^{\sigma\dagger}, &\text{UHF/UKS}. \end{cases} ]

This definition is invariant under occupied rotations and phases. The first contract version accepts only the existing zero-temperature integer occupations. Unrestricted spin blocks remain separately attested; only their sum enters the spin-independent overlap term. The contract inverse-transforms the verified time-reversal-consistent character blocks as (W(g)=\sum_q w_q e^{-2\pi i q\cdot g}W(q)); it must not repair a failed sewing relation by taking an unexplained real part.

The immutable binding includes the atom order and effective charges/ECPs, basis shell and AO order, finite-torus descriptor, retained AO projectors at every character, canonical digests of (S,F_{\mathrm{var}},D,C_o), occupations and (W), the resolved one-electron cell support, and the route-specific two-electron and XC-grid support. A rebuilt-Fock stationarity and directional variational-closure check is required for each backend. RIJCOSX fails closed unless its approximate exchange energy and Fock satisfy that closure or a derived adjoint supplies the missing response.

Under D95, the result-owned exact-exchange assembly records its full-range and screened coefficients, physical screening parameter, and consistency with q=0 applicability. Every active BvK seam and screened-exchange term enters the stationary (F_{\mathrm{var}}), hence (W). Their explicit fixed-density derivatives remain separate: the BvK seam derivative is present only when its applicability is active, while the finite HSE erfc exchange derivative uses the same screening parameter and numerical domain as the SCF. Neither is hidden in or counted twice with the overlap derivative. This first contract covers atomic displacements at fixed primitive lattice; stress remains open.

The seam record stores an effective operator coefficient (\eta) rather than relying on a signed-Madelung naming convention. The raw full-range exchange operator and its restricted variational-Fock and energy contributions are

[ \Delta K_{\mathrm{raw}}^R=\xi_{\mathrm{BvK}}SDS, \qquad \Delta F_{\mathrm{seam}}^R=-\frac{\eta}{2}SDS, \qquad E_{\mathrm{seam}}^R=-\frac{\eta}{4} \operatorname{Tr}[(DS)^2], ]

and for unrestricted spin densities,

[ \Delta K_{\sigma,\mathrm{raw}}=\xi_{\mathrm{BvK}}SD_\sigma S, \qquad \Delta F_{\sigma,\mathrm{seam}}=-\eta SD_\sigma S, \qquad E_{\mathrm{seam}}^U=-\frac{\eta}{2}\sum_\sigma \operatorname{Tr}[(D_\sigma S)^2]. ]

Here (\eta=\xi_{\mathrm{BvK}}) for RHF/UHF, (\eta=c_{\mathrm{full}}\xi_{\mathrm{BvK}}) for a global hybrid, and (\eta=0) when applicability is inactive. HSE06 instead records c_full=0, c_sr=0.25, and omega_screen_bohr_inv=0.11. Its fixed-density exchange functional is (E_{x,\mathrm{sr}}^R=-c_{\mathrm{sr}}\operatorname{Tr}[D K_{\mathrm{erfc}}[D]]/4), or the corresponding unrestricted spin sum with coefficient (-c_{\mathrm{sr}}/2). No q=0 seam is attached to that finite screened kernel. Force derivation must differentiate these exact recorded functionals using the same numerical support as the SCF.

In these equations (\xi_{\mathrm{BvK}}>0) is the code’s probe-charge Madelung coefficient. A theory notation that instead calls the signed self-potential (\xi_N<0) maps to this operator coefficient as (\eta=-\xi_N) for RHF/UHF and (\eta=-c_{\mathrm{full}}\xi_N) for a global hybrid. Mixing those sign conventions without this map reverses both the seam scalar and derivative.

Each approach must map its own stationary state into the binding contract and derive the fixed-density seam or screened terms against that contract. Within χ-CCM, the character and real-Gamma forms meet by the exact finite Fourier transform. Cross-approach equality is a separate result to establish with operator-specific and finite-difference gates under the common exchange-q=0 convention.

Route-resolved exact-exchange assembly provenance

D95 introduces the immutable B-owned AICCM2026DevBExactExchangeAssembly with fields c_full, c_sr, omega_screen_bohr_inv, and fixed resolver provenance resolver="vibeqc.periodic_screened_exchange.resolve_periodic_exchange". D98 extends that historical unversioned shape to the first formal nested schema, schema="vibeqc.aiccm2026dev-b.exact-exchange-assembly/v1", with screened_exchange_applicability and screened_exchange_assembly. The object is attached as result.aiccm2026dev_b.exact_exchange_assembly; the result also exposes result.exact_exchange_assembly as the same object. Fleet records serialize the exact seven-field v1 object under top-level exact_exchange_assembly, alongside rather than inside finite_torus_convention. The separation is normative: the latter declares the exchange-q=0 family, whereas the former records which full-range and finite screened exact-exchange arms the selected route resolved and how an active screened arm was assembled.

The values come from that shared live resolver, not from a second B functional-name table. The supported contracts are:

Route	c_full	c_sr	omega_screen_bohr_inv	screened applicability	screened assembly
RHF/UHF	1	0	0	inactive	not-applicable
pure DFT	0	0	0	inactive	not-applicable
global hybrid	nonzero resolved fraction	0	0	inactive	not-applicable
HSE type	0	positive resolved fraction	positive physical value	active	short-range-direct

omega_screen_bohr_inv is in inverse bohr and is zero when the screened arm is absent. Finite values, fixed resolver provenance, and the route-specific zero/nonzero pattern are part of the descriptor contract. D83 remains the independent convention-use flag and must agree exactly: exchange_q0_applicability="active" if and only if c_full is nonzero. A screened-only HSE-type record is therefore inactive at the full-range seam even though c_sr, its separate screened applicability, and its physical omega are active. The schema vocabulary distinguishes short-range-direct from full-range-minus-long-range, which are different finite-N operators with the same thermodynamic target. Current χ-CCM-B implements only short-range-direct.

For an active screened arm, the four-center RKS/UKS Fock branch emits vibeqc.pbc-bipole.screened-exchange-execution/v1 only after the direct K-erfc matrix has been folded, symmetrized where needed, and energy contracted. The B wrapper requires the emitted coefficient and omega to equal the live resolver. The shared corrected-gauge route may still record exchange_ewald_split=True and exchange_exxdiv="ewald" when c_full=0; those selector labels do not mean a full-range K correction or seam contributed. D83 therefore remains coefficient-derived rather than inferred from raw route flags. Missing, stray, or contradictory evidence fails before diagnostics attach. This is branch-level action evidence, not a matrix digest, independent numerical oracle, or domain qualification. D93 support qualification remains separate. D92 and D94 continue to require an explicit positive eta because their caller-supplied densities, BvK self-potential, basis, and overlap support are not SCF-bound. D98 changes no Hamiltonian and adds no screened-exchange derivative or total-gradient route. It also proves no cross-approach identity: Γ-CCM remains the union-and-weight/Wigner–Seitz integral-weighting construction, χ-CCM remains the finite-translation-group character construction, and a common declared exchange-q=0 convention does not identify them.

Fleet acceptance of exact-exchange assembly provenance

D96 made the D95 object normative for stored quantitative-status fleet records. D98 supersedes its payload shape with exact v1 keys. The producer still obtains coefficients from the live periodic exchange resolver. Independently, b_routes.py declares the exact expected (c_full,c_sr,omega_screen_bohr_inv) tuple and screened pair for every registered route and requires the fixed schema and resolver identifiers. The current route oracle is

Fleet route family	c_full	c_sr	omega_screen_bohr_inv	screened pair
RHF and post-HF on the RHF reference	1	0	0	inactive / not-applicable
RKS/PBE	0	0	0	inactive / not-applicable
RKS/PBE0	0.25	0	0	inactive / not-applicable

The stored object must have exactly the seven D98 v1 keys. Coefficients must be finite JSON numbers rather than booleans. The route declaration must agree with D83 for c_full, while screened applicability must be active exactly when c_sr is nonzero and the assembly label must match the independent route oracle. All current fleet routes are screened-inactive; HSE remains outside the fleet. Legacy unversioned four-field records, extra fields, an active screened label on a current route, and full-range-minus-long-range on a current route fail closed rather than being backfilled. The audit exposes a specific failure reason; the comparator refuses the row before emitting an energy or representation-control table. This requirement applies to ok and not_converged records. unsupported and error records remain failure evidence and do not need to claim a successful assembly.

The fleet declaration is an independent metadata oracle. For current four-center HSE results outside that fleet, the B wrapper additionally consumes the branch-emitted execution value described above. Neither layer proves the numerical matrix independently. D93 numerical-support qualification, D85 state binding, screened-exchange derivatives, HSE quantitative validation, and the total-gradient fail-close remain separate.

Fixed-density restricted active BvK seam derivative component

D92 implements the restricted full-range seam part of the preceding contract as a fixed-input component audit. On the complete character net, with the spin-summed restricted density, primitive lattice, and positive finite operator coefficient (\eta) held fixed,

[ E_{\mathrm{seam}}^R[D;R] =-\frac{\eta}{4}\sum_q w_q \operatorname{Tr}[D(q)S(q;R)D(q)S(q;R)], ]

and therefore

[ \left.\frac{\partial E_{\mathrm{seam}}^R}{\partial R_{A\alpha}} \right|{D,\eta,\mathrm{lattice}} =-\frac{\eta}{2}\sum_q w_q \operatorname{Tr}\left[D(q)S(q)D(q) \frac{\partial S(q)}{\partial R{A\alpha}}\right]. ]

The implementation forms (M(q)=\eta D(q)S(q)D(q)/2), inverse-transforms it with the exact finite-character convention, rejects a non-real residue, and evaluates (-\sum_g M_{\mu\nu}(g),\partial S_{\mu\nu}(g)) with the native overlap AO-centre derivative. The factor (1/2) in (M), rather than the energy’s (1/4), is required because the scalar contains two overlap factors. An odd three-character, skew-cell restricted projector makes the real-space blocks individually nonsymmetric and pins the transform orientation, sign, factor of two, linearity in (\eta), and translational sum. Central differences of the public scalar agree with the analytic component below (10^{-7}) Ha/bohr.

Both public helpers require a 3D bvk-ewald descriptor whose exchange_q0_applicability is active, a restricted RHF/RKS record, the complete unreduced uniform unshifted character net, and finite Hermitian, time-reversal-consistent density blocks. operator_coefficient_eta is a required positive input. D95 records the route-resolved full-range fraction, but does not bind the caller’s density, the BvK self-potential, basis, or overlap support into the numeric (\eta). The helper therefore does not infer eta from the descriptor or a method label. Inactive pure and HSE records refuse the seam component rather than returning a zero that could be mistaken for the screened-exchange derivative. Negative full-range fractions are outside this first validated component contract.

The result supplies only the declared χ convention and executed character net. It does not attest that the caller’s density, (\eta), basis, or lattice_options reproduce the final SCF realization. There is consequently no SCF convenience wrapper and no stationary-force claim. The separate unrestricted fixed-density seam pair is described by D94 below. HSE screened-exchange derivatives, D85 state and support binding, RI/RIJCOSX response, and the total χ gradient remain fail-closed.

Fixed-density unrestricted active BvK seam derivative component

D94 extends the same fixed-input audit to the active unrestricted full-range seam without forming a spin-summed exchange density. For independently supplied alpha and beta character densities,

[ E_{\mathrm{seam}}^U[D_\alpha,D_\beta;R] =-\frac{\eta}{2}\sum_q w_q\sum_{s\in{\alpha,\beta}} \operatorname{Tr}[D_s(q)S(q;R)D_s(q)S(q;R)], ]

so at fixed spin densities, eta, and primitive lattice,

[ \left.\frac{\partial E_{\mathrm{seam}}^U}{\partial R_{A\alpha}} \right|{D\alpha,D_\beta,\eta,\mathrm{lattice}} =-\eta\sum_q w_q\sum_{s\in{\alpha,\beta}} \operatorname{Tr}\left[D_s(q)S(q)D_s(q) \frac{\partial S(q)}{\partial R_{A\alpha}}\right]. ]

The corresponding per-spin Fock shift and overlap-response weight are

[ \Delta F_s(q)=-\eta S(q)D_s(q)S(q),\qquad M(q)=\eta\left[D_\alpha(q)S(q)D_\alpha(q) +D_\beta(q)S(q)D_\beta(q)\right]. ]

The implementation inverse-transforms (M(q)) onto the exact overlap support and uses the native overlap AO-centre derivative. Alpha and beta are contracted separately: replacing them by their sum would introduce unphysical cross-spin exchange. The closed-shell substitution (D_\alpha=D_\beta=D/2) gives (M=\eta DSD/2) and reduces both the scalar and derivative exactly to D92. An odd three-character skew-cell broken-spin fixture uses distinct alpha and beta projectors to compare against a separate-spin oracle and to prove that a total-density contraction gives a different, incorrect answer. The same gate checks linearity in eta, the translational sum, and the analytic derivative against central differences below (10^{-7}) Ha/bohr; an independent closed-shell fixture pins the exact D92 reduction.

Both unrestricted helpers require a 3D bvk-ewald descriptor with active exchange_q0_applicability, method="UHF" or a nonempty method="UKS/<functional>", and the complete unreduced uniform unshifted character net. Alpha and beta blocks are validated independently for count, shape, finiteness, Hermiticity, and time reversal; a zero beta density is a valid saturated-spin input. operator_coefficient_eta remains a required positive finite input. Active applicability declares that the route resolves a nonzero full-range exact-exchange arm, and D95 records that arm’s fraction, but neither binds the caller’s numeric eta or its remaining state and support inputs. Restricted records fail the method guard. Production pure-functional and HSE records carry inactive applicability and fail the applicability guard; the helper does not recompute applicability from a functional label.

As in D92, this is a fixed-input component audit. The result does not bind the supplied spin densities, eta, basis identity, or resolved overlap support to the SCF state. There is no SCF wrapper, D85 stationary-state binding, screened-exchange derivative, density or orbital response, RI/RIJCOSX three-center or metric response, or total-gradient claim.

Gradient-audit finite-torus binding

Every component helper now checks that its result descriptor and active system identify the same finite torus. For the present direct-product, Γ-centred construction,

[ N_{\mathrm{result}}=N_{\mathrm{character}} =N_{\mathrm{BvK}}, \qquad A_{\mathrm{BvK}}=A_{\mathrm{system}}\operatorname{diag}(N), ]

where PeriodicSystem.lattice stores primitive vectors as columns. The recorded BvK lattice must match within (10^{-12}) bohr. A skew, unequal-mesh test distinguishes this column scaling from the incorrect row-scaled expression (\operatorname{diag}(N)A_{\mathrm{system}}). Diagnostics and top-level convention descriptors must agree, and the boundary model must be explicitly 3d-periodic.

SCF-density folding additionally validates the complete unreduced, unshifted Γ-centred character set. Fractional labels are reduced modulo the integer mesh, so equivalent representatives such as (-1/3) and (2/3) are accepted, but duplicate, incomplete, shifted, or nonuniformly weighted nets fail before the inverse transform.

This is a provenance guard, not another force term. It binds the torus size and lattice but does not attest the atoms, basis identity, resolved one-electron cutoff, or the source of caller-supplied (D(g)) or (W(g)). Atomic displacements at fixed lattice remain valid for finite-difference component checks; stresses and cell derivatives are outside this contract. The equality above would also need revision for a future twisted boundary or non-diagonal supercell construction.

Two-electron numerical-support qualification

The declared 3D Hamiltonian does not by itself prove that a finite numerical realization has converged all integration domains. The combined MgO audits identified two independent support problems. Direct BIPOLE J/K needs an internal ket-image traversal padded by the smeared AO-pair kernel range even when the physical density and nuclear cutoffs are coherent. Fitted RI and RIJCOSX avoid that traversal, but default RSGDF KE=200 has a distinct dense-core high-G limitation. The split audit at 51e3b250 establishes the direct issue; the dense-tail audit at 2355f52e, pinned by tests/test_rsgdf_dense_mesh_tail.py, establishes the fitted issue.

The shared BIPOLE workstream owns the direct fix. M4a defined an explicit absolute-radius oracle. M4b supplied the separation-aware QQR bound and pair-resolved compositions, and M5 made sr_image_precision=1e-6 the default for every erfc short-range build. The driver now resolves an absolute radius larger than the physical electronic cutoff and stores it as sr_image_extent_bohr. Pure restricted semilocal RKS has used the same padded SR+LR default since the 2026-07-18 exact-FT retirement; the finite-KE analytic-FT route remains only an explicit oracle. The shared HSE traversal and unrestricted spin recursion are padded as well. HSE is not a fleet route and still requires its own quantitative validation. Commit 2fd23eff further keeps the physical Bloch density unmasked when pair-resolved SYM3b is active, applying the pair mask only inside the private direct-SR tensor build. Symmetry-enabled four-center records made before that repair are revision-bound; default symmetry-off radial runs are outside that defect. The quartet far-field prototype is separately outside this contract. D105 pins use_multipole_far_field=False at all four χ wrappers, and the shared drivers reject explicit True before setup. Records made while χ inherited the unrecorded shared default are revision-bound. Any future route needs a three-translation domain identity plus executed-path telemetry and matched-support quantitative validation before it can extend D93.

D93 implements the reportability transport as vibeqc.aiccm2026dev-b.two-electron-support/v1. A current fleet four-center RHF, PBE, or PBE0 row is qualified only when its record contains the preserved low-level pbc-bipole backend, exact executed M5 sr_image_precision=1e-6 policy, electronic and nuclear cutoffs, and a finite resolved absolute radius strictly larger than the electronic cutoff. The nested cutoffs must match the independent direct_lattice_cutoffs serialization. This consumes the shared BIPOLE result and does not copy its kernel into the χ line. RI, RIJCOSX, MDF, and post-HF rows serialize their active base/runtime settings but remain not-qualified: no accepted high-G RSGDF tail, COSX exchange-domain, MDF, or correlation-factor support contract exists. Missing, malformed, internally contradictory, route-inconsistent, or unqualified quantitative rows fail independently in audit_b.py and compare_b.py; unsupported and error rows remain visible failure evidence. Existing fitted SCF convergence and route-plumbing records remain useful but are not external absolute-energy validation. These are numerical-support conditions inside the declared convention, not new gauges, Coulomb kernels, or Γ/χ construction physics.

D103 adds an independent direct-fold contract, vibeqc.aiccm2026dev-b.overlap-fold-support/v1. Every corrected-gauge BIPOLE result preserves the executed maximum overlap-fold drift over the character mesh. A drift above 1e-2 fails before the first Fock build; 1e-4 < drift <= 1e-2 retains the shared note-only execution path but is not-qualified; and drift <= 1e-4 meets the quantitative fold target. A direct fleet payload fixes diagnostic="max-abs-element-overlap-bloch-fold-drift/v1" and reference_cutoff_factor=1.5, then binds the drift to the executed electronic cutoff, character mesh, two fixed thresholds, and its four_center route. Its cutoff must equal the independent direct_lattice_cutoffs value and its mesh must equal the row mesh. RI, RIJCOSX, MDF, and post-HF payloads use the same exact schema as not-applicable, with null drift, cutoff, reference factor, and mesh. Missing, malformed, nonfinite, negative, cutoff-inconsistent, mesh-inconsistent, or moderate direct evidence fails in both audit_b.py and compare_b.py.

D93 and D103 are independent necessary support gates. Neither establishes the other. D104 permits a fresh exact-v2 record to clear only the old shared-Ewald hold; every v1 record remains permanently quarantined. These are numerical-support conditions within the χ-CCM construction at the declared exchange-q=0 convention. They neither identify Γ-CCM with χ-CCM nor imply different Coulomb kernels.

Pinned nonquantitative χ route-control producer

D101 implements the χ-owned route in aiccm-2026/COMPARISON_1D2D3D_2026-07-15.md through the public high-level entry point. run_case_cmp_b.py calls only run_periodic_job(..., method="RHF", functional=None, jk_method="aiccm2026dev-b", aiccm_backend="ri"). The chain and monolayer labels are embedded objects in vacuum, but their PeriodicSystem instances remain deliberately declared dim=3, like the crystal. The producer binds the exact campaign lattices, atoms, meshes, STO-3G basis, resolved auxiliary basis, RSGDF KE cutoff, HCORE guess, zero smearing, zero static and dynamic damping, zero level shift, DIIS settings, convergence tolerances, and requested and executed zero Fock mixing. Explicit inputs, source-resolved defaults, and executed evidence are serialized separately so a default cannot masquerade as an executed pin. The result’s GDF method and cutoff fields are explicitly labelled as B-selector-resolved settings forwarded to the fitted driver, not backend-owned execution telemetry or D93 qualification. Likewise, declared_model="3d-periodic-in-vacuum" describes a vacuum-padded geometry, while boundary_model="3d-periodic" names the fully 3D periodic Green function. The pair is intentional and does not select an isolated wire/slab kernel or bypass D72/O1.

Acceptance requires contract checks of the complete character residue product, reciprocal Cartesian consistency, the D83 exchange-q=0 applicability, the D88 column-vector BvK lattice and positive code Madelung coefficient, the route-resolved exact-exchange assembly, the shared Ewald nuclear scalar, and the total/electronic/nuclear energy decomposition. That last energy check is boolean assembly evidence only. Because D93 leaves RI support not-qualified, neither total nor per-atom energy is serialized in the D101 campaign JSON; ordinary runner and SCF-log energy output remains a nonreportable diagnostic artifact. The Madelung and nuclear values are separately recomputed through the same shared helpers used by SCF. They provide dispatch and assembly evidence, not an independent kernel or Ewald implementation.

The nuclear comparison is intentionally same-helper only. D102 records the historical shifted-pair omission, and D104 records its pair-complete repair and same-process canary. The D101 equality still establishes route and assembly consistency rather than an independent Ewald implementation. D101 remains usable as a nonquantitative diagnostic because it emits no energy and D93 still leaves the fitted route unqualified.

The 3D crystal must complete first. Each embedded-object run consumes the exact contract-valid nonquantitative 3D JSON by file SHA256 and records its vq job, Slurm job, and producer-attestation identifiers. All three runs must share the stable D77 source/version/core/attestor/library/payload tuple; the Slurm node hostname and full per-job attestation digest may differ. The producer rejects bundle provenance, checks exact current origin/main plus the required repair ancestors, and compares the tracked producer, probe, test-set, route-registry, and launcher bytes before and after SCF. The normative D91 bundle contract exists, but its transport/producer/auditor implementation remains prospective. The current Marvin managed runtime is bundle-shaped, but its version-and-short-SHA artifact path and sibling checksum are replaceable and its launcher does not bind a literal digest. It is not an accepted immutable D91 artifact. A routine bundle refresh cannot satisfy D101 even at the right SHA. An accepted run needs either a separate D77-clean Git checkout or a future implemented D91 contract with no-clobber publication, graph-proven ancestry, and digest-bound launch. Successful JSON always combines status="ok", evidence_role="route-plumbing-diagnostic", quantitative_status="not-qualified", and comparison_status="no-gamma-chi-construction-comparison-defined". audit_cmp_b.py revalidates the exact schema without printing an energy. A 1d or 2d audit requires the exact fetched 3D JSON so its file SHA and parent identifiers can be recomputed rather than trusted from the child record. Here status="ok" means only that execution and the internal contract audit succeeded. The asserted vq_target_host="marvin" and numeric Slurm job id record operator intent and scheduler context; vq currently exports no field that cryptographically proves the target alias.

D101 deliberately emits neither the historical v2 comparison_input nor a comparison_contract. It defines no paired representation result or Γ/χ approach pairing and does not assert identical discretized Hamiltonians merely because convention labels match. It changes no construction, Coulomb convention, backend support, or lower-dimensional and total-gradient fail-close.

The committed tests cover the selector, exact mesh, Wigner–Seitz ties, partition unity, inverse Fourier transform, metric idempotency, fail-closed domain, RHF/RKS runner dispatch, native 3D SCF runs through all three integral backends, explicit 1D/2D rejection across every backend, and χ-CCM native post-HF setup kernels. The native OpenMP kernels are parity-tested against the earlier NumPy equations for the one-body inverse transform, pair-resolved RI-factor inverse transform, and three-index AO-to-MO contractions. The benchmark harness adds:

1. the archived pre-guard 1D alternating hydrogen-chain ladder as failure and historical comparison evidence, not as reportable absolute energies;

2. direct comparison with both the historical union12 and separately developed symmetric aiccm2026dev-a weights in the in-repo CCM on identical geometry;

3. exact comparison with the native Γ-centred k-mesh representation;

4. energy, convergence, idempotency, electron count, wall time, and implementation-to-implementation differences;

5. later 3D LiH, MgO, NaCl, Al2O3, and diamond ladders only after each basis, gauge, internal traversal, and reciprocal tail is pinned to an out-of-process reference input.

The reportable Γ/χ approach map is empty. The A RI and RIJCOSX harness routes call the same multi-k GDF SCF drivers as B, so agreement between those routes tests their shared implementation and cannot isolate an approach change. B rhf-ri versus A aiccm-hf-direct is a neutral-torus representation control, not evidence for the union-and-weight Γ-CCM approach. The direct solver assembles and minimizes a real Γ-supercell problem, but its folded cderi and one-electron matrices intentionally reuse the common per-q RSGDF fits and lattice-sum primitives. The control therefore checks Γ-supercell versus character/Bloch assembly, not an independent RI implementation or a Γ-CCM versus χ-CCM approach delta. It is still not reportable because comparator contract aiccm2026-gamma-chi/v2 is only an incomplete validation scaffold and must not be emitted unchanged. A superseding contract needs the operator and retained-space evidence below.

Contract v2 requires the canonical comparison_input to be embedded beside its exact comparison_contract.input_sha256 hash, validates its required physical/numerical fields, and reconciles them with the result and contract. It distinguishes the AO linear-dependence threshold, currently 1e-7, from the auxiliary-metric linear-dependence threshold, currently 1e-9. It fingerprints and reconciles requested and reported SCF smearing and accepts only the zero-temperature setting (smearing_temperature=0.0 Ha). Contract v2 does not bind Fock mixing, and D86 does not change or upgrade that incomplete comparator contract. A superseding contract must bind requested and executed accelerator controls before using them as comparison evidence. Contract v2 also requires matched clean source, native-core, host, package-version, and successful composite producer-process attestation. The cached aiccm-host-probe/v2 preflight remains the trusted dispatch gate, but it is not accepted as result evidence by itself. Each producer binds that preflight to its own current identity and a canonical manifest of its copied producer, probe, test-set, launcher, and stream-specific route files. In the producer process it records linked native-library versions and always runs a cheap shifted-mesh native-versus-Python AO-pair Fourier-transform canary. If the native-core path SHA256 changed after preflight, it reruns the full v2 host checks in process, including API shape, reciprocal cutoff, and LiH direct-versus-GDF checks; the cheap canary alone is not sufficient for a changed core.

Immediately before result serialization the producer re-reads its clean source, host, package, native-core path SHA256, native-library versions, and payload and requires them to be unchanged. The SHA256 describes bytes currently at the imported module’s filesystem path, not bytes already mapped into the process. The same-process numerical checks therefore attest loaded behavior, but do not claim a cryptographic identity for the mapped image. The comparator independently validates the composite and nested canonical digests, exact schemas, check versions, and tolerances. It compares A and B using their current producer identities; different valid preflight histories and the intentionally different stream payload manifests need not match. A sidecar added during later curation cannot establish any of these facts retrospectively, and old rows are not upgraded by the new schema.

D91 defines the requirements for one possible future alternative on an immutable-bundle host where a clean Git checkout is structurally unavailable. A bundle-attested producer would have to bind the checksum-verified artifact to a full source SHA proven resolvable on origin/main, record package, native-core, and native-library identities, run the full versioned numerical checks and loaded-core canary in process, bind its copied producer payload, and recheck every identity layer before serialization. The row would record attestation="bundle"; a source reconstructed as a clean checkout records attestation="git-checkout". The current B producer, curator, and comparator accept only the D77 clean-checkout schema, so bundle rows remain fail-closed until an authoritative immutable-artifact manifest is created, its full source SHA is proven resolvable on origin/main, and the second schema and its tests land. D95 exchange-assembly provenance supplies none of those identity proofs. Bundle attestation cannot repair old rows, replace the numerical-input or two-electron-support contracts, or turn a finite-torus route control into a Γ-CCM/χ-CCM approach comparison.

The factor audit also exposed a finite-cutoff reciprocal-support seam. An ordered character pair can report a raw transfer q + G while the neutral-torus real-Gamma fold uses the equivalent finite-group transfer q. The ket Bloch cell phase is unchanged, but shifting one fixed finite |G| ball by those two raw labels selected different boundary crescents. On the compact H2 (3,1,1) control, the relative fitted-Gram residual was 9.73e-8 at 40 Ha and 2.46e-11 at 200 Ha. A half-open centered representative fixes that odd-mesh case but breaks time reversal at an even-mesh Nyquist transfer; keeping both Nyquist signs restores time reversal but not reciprocal-label invariance. The shared Bloch RSGDF builder now enumerates the physical shifted sphere 0 < |G+q| <= sqrt(2 E_cut) directly. That support is both modulo-G invariant and inversion covariant. On skew H2 (2,1,1) and (3,1,1) controls, the canonical χ-CCM factor inverse transform now matches the neutral-torus folded fitted-Gram control at the 1e-14 numerical floor. Separate gates pin reciprocal relabelling of the compact fitted Gram and canonical factor, plus canonical-frame Nyquist time reversal. This is a numerical representation fix inside the same finite-torus Hamiltonian and Coulomb convention. The q=0 support and accumulation order are byte-preserved. Finite nonzero-q RSGDF support changed, so pre-D78 exact-GDF-K RHF/UHF and hybrid RI, neutral-torus fold, and RI post-HF/full-pair records are revision-bound until rerun. Semilocal RI and RIJCOSX SCF build only q=0 RSGDF J factors and are numerically unaffected by this support change.

Numerical-input emission remains disabled pending the remaining contract corrections. The legacy real-Gamma control builder blocks use the canonical auxiliary-AO embedding, but the final control cderi is an unfolded-k Fourier fold followed by a real q/-q stack and a numerical-null-row prune. B SCF instead self-contracts one compact metric-eigenmode factor per ordered character pair. Raw factors need not match. Acceptance must name the common Bloch pair-density phase, record the route-specific factor pipelines, and establish q-resolved agreement of the retained auxiliary projectors together with the gauge-invariant fitted Gram operators. Matching metric ranks, thresholds, or projectors alone is insufficient.

A also applies an absolute Γ-supercell overlap guard while B performs normalized per-character canonical orthogonalization. Equal threshold numbers do not prove equal retained spaces. For the current control route, acceptance requires full/no-drop at every B character together with the finite-Fourier overlap residual, so every character retains all primitive-cell AO directions. If a future comparison permits projected spaces, it must compare Fourier-related retained-space projectors rather than rank counts alone. No Γ/χ approach delta can be emitted until these conditions are specified and tested and a route actually realizing the union-and-weight Γ-CCM construction is paired with χ-CCM. Equality between the distinct Γ-CCM and χ-CCM approaches for a specified operator and route remains evidence to establish under the declared common exchange-q=0 convention.

External CRYSTAL or PySCF comparisons remain out of process and are weak signals unless their cell, Coulomb, exchange-divergence, basis, and k-mesh conventions are matched explicitly.

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Γ-CCM and χ-CCM are distinct CCM approaches. Γ-CCM uses the union-and-weight/Wigner–Seitz integral-weighting construction; χ-CCM uses the finite-translation-group character construction. They are compared at a declared common exchange-q=0 convention, currently exchange_q0="bvk-ewald" for the production χ-CCM path. Their distinction is not a choice of Coulomb kernels, and equality for a specified operator/route is evidence to establish, not a naming premise.