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 pdftitle={The Ab Initio Cyclic Cluster Model Reconsidered},
 pdfauthor={vibe-qc method development},
 pdfkeywords={cyclic cluster model, periodic Hartree--Fock, Kohn--Sham DFT,
 Born--von Karman boundary conditions, RIJCOSX}
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\title{The Ab Initio Cyclic Cluster Model Reconsidered:\\
an Independent Finite-Torus Formulation, a Diagnosis of Historical
Boundary Weights, and Comparison with Reciprocal-Space Periodic LCAO}
\author{vibe-qc method-development manuscript}
\date{21 June 2026}

\begin{document}
\maketitle

\begin{abstract}
The cyclic cluster model replaces an infinite crystal by a finite set of
primitive cells with Born--von Karman identifications.  Its attraction is
that a periodic problem can be expressed with molecular-style real-space
objects.  Its danger is that a finite, converged self-consistent-field
calculation can conceal a Hamiltonian that is not the derivative of a scalar
energy.  We reconstruct the 2014 ab initio cyclic cluster model (AICCM),
including its two-, three-, and four-centre boundary factors, and compare it
with an independently derived implementation, \texttt{aiccm2026dev-b}.  The new
definition is a neutral finite electrostatic torus, restricted to
translation-invariant restricted or unrestricted HF/KS projectors.
The complete unreduced Gamma-centred reciprocal mesh is the exact character
representation of this finite group; it is not an additional approximation.
Wigner--Seitz fractions select tied representatives of a translation class
and sum to unity.  They are not multiplied into non-periodised four-centre
integrals.  This distinction preserves electron-repulsion-integral
permutation symmetry and makes the Fock matrix the functional derivative of
one energy.

The historical four-centre factor
$x_{lm}(x_{lq}+x_{mq})x_{qr}/2$ is not invariant under bra--ket interchange
unless an extra equality, absent in a general non-Bravais environment, holds.
Consequently the published weighted tensor does not by itself establish the
usual Hartree--Fock variational functional.  This is a mathematical
diagnosis, not a claim that every historical number is poor.  Indeed, old and
new results can be close after cancellation.  Measured H$_4$/STO-3G values
from the implementation artifacts approach one another to
\SI{3.301}{\kilo\joule\per\mole} at a four-cell mesh.  A compact two-cell
H$_2$/STO-3G control gives direct four-centre, RI, and RIJCOSX energies within
\SI{0.115}{\kilo\joule\per\mole} for both RHF and PBE0.  Conversely, the
vacuum-padded H$_4$ two-cell direct-versus-RI difference is
\SI{58.296}{\kilo\joule\per\mole}; it remains an open truncation and embedding
question.  A separate 3D RI-MP2 control agrees with out-of-process PySCF
KRHF/KMP2 below one nanohartree per cell.  An exact inverse transform of the
same fitted finite Hamiltonian supports real-torus local-PNO MP2 and
CCSD(T); their no-truncation limits agree with canonical finite-torus
oracles through the nonzero triples term.  An unrestricted Li/STO-3G control
closes both spin-projector invariants and the UMP2/UCCSD(T) PNO exact limit.
These internal coupled-cluster
oracles are not mislabeled as infinite-crystal benchmarks. Post-SCF Wannier
and IAO gauges, PAO/PNO construction, translation-pair orbits, and the exact
cyclic-cluster-compatible space-group subgroup are derived. Pair and integral
symmetry reduction remains disabled until amplitude and Fock parity is proved.
The related 2008 deMon2k KS-ADFT CCM is also audited.  Its two-centre
reference argument and weighted RI adjoint relation are not proved in general,
but its published three-centre definition explicitly averages both AO-centre
orderings; it is therefore not assigned the 2014 four-centre symmetry defect
without further evidence.
\end{abstract}

\noindent\textbf{Keywords:} cyclic cluster model; periodic Hartree--Fock;
Kohn--Sham density-functional theory; Born--von Karman boundary conditions;
Gaussian LCAO; Wigner--Seitz cell; Ewald summation; density fitting; RIJCOSX;
local correlation; coupled cluster

\section{Introduction}

An electronic-structure model of a perfect crystal has to reconcile three
facts: nuclei and electrons interact through the long-ranged Coulomb law,
one-electron states obey translation symmetry, and practical calculations
use finite basis sets and finite reciprocal meshes.  Periodic Gaussian LCAO
programs such as CRYSTAL solve this problem by forming Bloch sums, evaluating
matrix blocks at reciprocal-space points, and treating electrostatics with a
balanced Ewald and multipolar construction
\cite{pisani1988,saunders1992,dovesi2014}.  The cyclic cluster model (CCM)
takes a different route.  It closes a finite cluster by Born--von Karman
identifications and seeks to retain molecular-style real-space algebra
\cite{bredow2000,bredow2001,janetzko2002,janetzko2008}.

The ab initio Hartree--Fock form published in 2014 introduced explicit
Wigner--Seitz boundary factors for integrals involving as many as four
centres \cite{peintinger2013thesis,peintinger2014}.  That construction gave
plausible convergence and, for the reported hydrogen chain, agreement with a
periodic Hartree--Fock sequence.  Such agreement is evidence, but not a proof
that the weighted Fock is the derivative of one energy.  The implementation
work reported here was therefore conducted as an independent derivation,
with the historical code and the separate in-repository CCM implementation
used only as comparison targets.

The gap closed here is conceptual.  We state the finite Hamiltonian, its
Coulomb gauge, and the admissible variational space before choosing an
algorithm.  We then ask whether each proposed boundary operation respects the
symmetries required of that Hamiltonian.  The result is the separately
selectable \texttt{aiccm2026dev-b} method.  The old implementation remains
unchanged so that disagreements remain observable rather than being merged
away.

All dimensional equations below are in SI units.  Numerical electronic
structure programs conventionally store atomic-unit values; wherever a raw
Hartree value is retained for traceability, its SI molar conversion uses
$1\,\Ha=\SI{2625.4996394799}{\kilo\joule\per\mole}$ and
$a_0=\SI{5.29177210903e-11}{\metre}$.

\section{Reference framework: periodic LCAO in reciprocal space}
\label{sec:reference}

\subsection{Hamiltonian and translation symmetry}

For clamped nuclei at positions $\bm R_A$ with charges $Z_Ae$, the
non-relativistic electronic Hamiltonian is
\begin{align}
 \hat H &=-\sum_{i=1}^{N_e}\frac{\hbar^2}{2m_e}\nabla_i^2
 -\sum_{iA}\frac{Z_Ae^2}{4\pi\epsilon_0|\bm r_i-\bm R_A|}
 +\sum_{i<j}\frac{e^2}{4\pi\epsilon_0|\bm r_i-\bm r_j|}
 +E_{NN},                                                     \label{eq:ham}\\
 E_{NN}&=\frac12\sum_{A\ne B}\frac{Z_AZ_Be^2}
 {4\pi\epsilon_0|\bm R_A-\bm R_B|}.                         \label{eq:enn}
\end{align}
The sums in a crystal include lattice images and are not separately
absolutely convergent.  Translation by
$\R_{\bm n}=\sum_i n_i\bm a_i$ commutes with $\hat H$, so an eigenfunction
may be chosen to satisfy Bloch's theorem,
\begin{equation}
 \psi_{p\kvec}(\bm r+\R_{\bm n})
 =e^{i\kvec\cdot\R_{\bm n}}\psi_{p\kvec}(\bm r).             \label{eq:bloch}
\end{equation}

Let $\chi_\mu(\bm r-\R_{\bm n})$ be a normalised Gaussian AO, with SI
dimension $\si{\metre^{-3/2}}$.  On a finite set of $N_c$ Born--von Karman
cells its Bloch sum is
\begin{equation}
 \phi_{\mu\kvec}(\bm r)=\frac1{\sqrt{N_c}}
 \sum_{\bm n}e^{i\kvec\cdot\R_{\bm n}}
 \chi_\mu(\bm r-\R_{\bm n}).                                \label{eq:blochsum}
\end{equation}
The crystalline orbital is
\begin{equation}
 \psi_{p\kvec}(\bm r)=\sum_\mu C_{\mu p}(\kvec)
 \phi_{\mu\kvec}(\bm r).                                   \label{eq:co}
\end{equation}

\subsection{Density and Brillouin-zone quadrature}

For a closed-shell insulator, the spin-summed one-particle density is
\begin{align}
 \rho(\bm r)&=\frac{2}{\Omega_{\mathrm{BZ}}}
 \sum_{p\in\mathrm{occ}}\int_{\mathrm{BZ}}
 |\psi_{p\kvec}(\bm r)|^2\,\dd^3k,                           \label{eq:rho_bz}\\
 D_{\mu\nu}(\R)&=\frac{2}{\Omega_{\mathrm{BZ}}}
 \sum_{p\in\mathrm{occ}}\int_{\mathrm{BZ}}
 e^{-i\kvec\cdot\R}C_{\mu p}(\kvec)C^*_{\nu p}(\kvec)
 \,\dd^3k.                                                   \label{eq:dreal_bz}
\end{align}
A uniform mesh replaces the integral by
\begin{equation}
 \frac1{\Omega_{\mathrm{BZ}}}\int_{\mathrm{BZ}}f(\kvec)\dd^3k
 \simeq\sum_{\kvec}w_{\kvec}f(\kvec),\qquad
 \sum_{\kvec}w_{\kvec}=1,                                  \label{eq:kquad}
\end{equation}
with Monkhorst--Pack meshes being the standard special-point construction
\cite{monkhorst1976}.  In the finite torus derived later, an unreduced
Gamma-centred mesh is not a quadrature approximation: it is the complete set
of group characters.

\subsection{Periodic Fock and Kohn--Sham equations}

The reciprocal-space Roothaan equation is
\begin{equation}
 \bm F(\kvec)\bm C(\kvec)=\bm S(\kvec)\bm C(\kvec)
 \bm\varepsilon(\kvec),                                      \label{eq:roothaan}
\end{equation}
where
\begin{align}
 F_{\mu\nu}^{\mathrm{RHF}}(\kvec)
 &=h_{\mu\nu}(\kvec)+J_{\mu\nu}(\kvec)
 -\frac12K_{\mu\nu}(\kvec),                                \label{eq:fock_rhf}\\
 F_{\mu\nu}^{\mathrm{RKS}}(\kvec)
 &=h_{\mu\nu}(\kvec)+J_{\mu\nu}(\kvec)
 +V_{\mu\nu}^{xc}(\kvec)-\frac{\alpha}{2}K_{\mu\nu}(\kvec).
                                                                    \label{eq:fock_rks}
\end{align}
Here $\alpha$ is the exact-exchange fraction.  In SI units
\begin{align}
 (12|34)&=\iint \chi_1^*(\bm r)\chi_2(\bm r)
 \frac{e^2}{4\pi\epsilon_0|\bm r-\bm r'|}
 \chi_3^*(\bm r')\chi_4(\bm r')\dd^3r\dd^3r',              \label{eq:eri_si}\\
 J_{12}[D]&=\sum_{34}D_{34}(12|34),\qquad
 K_{12}[D]=\sum_{34}D_{34}(13|24).                           \label{eq:jk}
\end{align}
The Kohn--Sham potential is the functional derivative
$V^{xc}(\bm r)=\delta E_{xc}[\rho]/\delta\rho(\bm r)$
\cite{hohenberg1964,kohn1965}.  Dispersion and geometrical counterpoise
corrections, such as D3 and gCP \cite{grimme2010,kruse2012}, are external
energy models and were not enabled in the benchmarks below.

\subsection{Infinite Coulomb series, Ewald gauge, and penetration}

Introduce the identity
\begin{equation}
 \frac1r=\frac{\operatorname{erfc}(\eta r)}r
 +\frac{\operatorname{erf}(\eta r)}r.                        \label{eq:ewald_split}
\end{equation}
The first term decays in real space; the second has Fourier coefficient
$4\pi\exp[-G^2/(4\eta^2)]/G^2$.  For charges $q_A$ in a cell of volume
$\Omega$, the Ewald energy is
\begin{align}
 E_{\mathrm E}={}&\frac{1}{8\pi\epsilon_0}
 \sum_{AB\bm n}'\frac{q_Aq_B\operatorname{erfc}
 (\eta|\bm r_{AB}+\R_{\bm n}|)}{|\bm r_{AB}+\R_{\bm n}|}
 \nonumber\\
 &+\frac{1}{2\epsilon_0\Omega}\sum_{\G\ne0}
 \frac{e^{-G^2/(4\eta^2)}}{G^2}
 \left|\sum_Aq_Ae^{i\G\cdot\bm r_A}\right|^2
 -\frac{\eta}{4\pi^{3/2}\epsilon_0}\sum_Aq_A^2
 -\frac{Q^2}{8\epsilon_0\Omega\eta^2},                      \label{eq:ewald}
\end{align}
where $Q=\sum_Aq_A$ and the prime omits self interaction.  Equation
\eqref{eq:ewald} is independent of $\eta$ when both sums converge
\cite{ewald1921}.  The last term is the uniform-background contribution.
For a neutral total cell, $Q=0$.  Electron--electron,
electron--nuclear, and nuclear--nuclear pieces must use the same convention;
otherwise finite constants fail to cancel.

CRYSTAL accelerates Gaussian electrostatics by separating local charge
penetration from the far multipolar field \cite{saunders1992}.  If two charge
distributions overlap, point multipoles do not reproduce their Coulomb
integral and the penetration contribution must remain explicit.  At long
range, charge, dipole, quadrupole, and higher moments provide the rapidly
convergent complement.  The $\G=0$ coefficient fixes the arbitrary constant
of the periodic potential.  Exact exchange has a related finite-mesh
$q=0$ channel; a BvK-supercell Madelung term must enter the exchange matrix,
not merely the printed total energy.

\section{The cyclic cluster as a finite translation group}
\label{sec:concept}

Choose positive integers $N_i$ and identify
$\bm n\sim\bm n+N_i\bm e_i$.  The finite translation group is
\begin{equation}
 \Tgroup=\mathbb Z_{N_1}\times\mathbb Z_{N_2}\times\mathbb Z_{N_3},
 \qquad N_c=N_1N_2N_3.                                      \label{eq:tgroup}
\end{equation}
It is a ring in one active dimension, a torus in two, and a three-torus in
three.  It has no physical surface.  Its characters are
\begin{equation}
 \chi_{\bm m}(\bm n)=\exp\left(2\pi i\sum_j\frac{m_jn_j}{N_j}\right),
 \qquad m_j=0,\ldots,N_j-1.                                 \label{eq:characters}
\end{equation}
The primary numerical parameter is therefore the real-space lattice extension
$(N_1,N_2,N_3)$.  A radius-style input $s_i$ is defined without ambiguity by
$N_i=2s_i+1$, so that representatives range from $-s_i$ to $+s_i$.  The
Wigner--Seitz cell of the cyclic superlattice has half-extent $N_i/2$ in
primitive-vector coordinates.  The reciprocal net derived below is the exact
character group of this real-space quotient; it is not a second, independently
chosen convergence parameter.  Nor is the extension a hard Coulomb cutoff:
the chosen BvK-periodic Green function retains its long-range Ewald or fitted
part.
Character orthogonality follows from the finite geometric series:
\begin{align}
 \frac1{N_c}\sum_{\bm m}\chi_{\bm m}(\bm n)
 \chi^*_{\bm m}(\bm n')
 &=\prod_j\left[\frac1{N_j}\sum_{m_j=0}^{N_j-1}
 e^{2\pi im_j(n_j-n_j')/N_j}\right] \nonumber\\
 &=\prod_j\delta_{n_jn_j'}=\delta_{\bm n\bm n'}.             \label{eq:orthogonality}
\end{align}
Thus every block-circulant AO matrix has the exact transform
\begin{align}
 X_{\mu\nu}(\kvec_{\bm m})&=\sum_{\R\in\Tgroup}
 e^{i\kvec_{\bm m}\cdot\R}X_{\mu\nu}(\R),                  \label{eq:forward}\\
 X_{\mu\nu}(\R)&=\frac1{N_c}\sum_{\bm m}
 e^{-i\kvec_{\bm m}\cdot\R}X_{\mu\nu}(\kvec_{\bm m}),    \label{eq:inverse}
\end{align}
where $\kvec_{\bm m}=\sum_j(m_j/N_j)\bm b_j$.  Substituting
\eqref{eq:forward} and using Eq.~\eqref{eq:orthogonality} proves
Eq.~\eqref{eq:inverse}.  A Gamma-point calculation on the BvK supercell,
constrained to commute with primitive translations, therefore decomposes
exactly into the full unreduced Gamma-centred mesh.  A half-shifted even mesh
imposes a different boundary condition.

\subsection{Wigner--Seitz representatives}

For centres $A$ and $B$ with intra-cell offset
$\bm\delta_{AB}=\bm\tau_B-\bm\tau_A$, a residue $[\bm n]\in\Tgroup$ has
representatives
\begin{equation}
 \bm d_{AB}(\bm n,\bm z)=\bm\delta_{AB}
 +\bm A\left(\bm n+\operatorname{diag}(N_1,N_2,N_3)\bm z\right),
 \qquad\bm z\in\mathbb Z^3.                                 \label{eq:representatives}
\end{equation}
Let $M_{AB}([\bm n])$ be the set minimising $|\bm d_{AB}|$.  If its
multiplicity is $m_{AB,[\bm n]}$, define
\begin{equation}
 w_{AB}(\bm d|[\bm n])=\frac1{m_{AB,[\bm n]}},\qquad
 \sum_{\bm d\in M_{AB}([\bm n])}w_{AB}(\bm d|[\bm n])=1.    \label{eq:partition}
\end{equation}
This is the exact role of the Wigner--Seitz boundary weight in the new
formulation: it partitions tied drawings of one quotient-group element.  For
an even one-dimensional cluster, the half-cell residues $+N/2$ and $-N/2$
each carry weight $1/2$.  A closest-vector search with a smallest-singular-
value stopping bound handles skew lattices and non-Bravais offsets.
Here $\bm A$ contains lattice vectors as columns.  The repository lattice
object stores them as rows, so the implemented closest-vector generator is
its transpose.  This convention is tested on a skew two-dimensional cell;
orthogonal cells alone cannot expose a transposition error.

\section{Historical AICCM formulation}
\label{sec:old}

This section reconstructs Eqs.~(4)--(25) of the 2014 paper rather than
retroactively rewriting them in the new notation \cite{peintinger2014}.  Let
$x_{lm'}=1/n_{m'}$ be the inverse number of occurrences of the translated
centre $m'$ on the Wigner--Seitz boundary.  The cyclic overlap and kinetic
blocks are
\begin{align}
 S^{\old}_{lm}&=\sum_{m'}x_{lm'}\langle l|m'\rangle,           \label{eq:old_s}\\
 T^{\old}_{lm}&=\sum_{m'}x_{lm'}\left\langle l\left|
 -\frac{\hbar^2}{2m_e}\nabla^2\right|m'\right\rangle.       \label{eq:old_t}
\end{align}
For a nuclear centre $C$, the proposed three-centre factor is
\begin{equation}
 x_{lmC}=x_{lm}\frac{x_{lC}+x_{mC}}2,                        \label{eq:old_3c}
\end{equation}
and the electron--nuclear contribution is assembled as
\begin{equation}
 V^{\old}=-\sum_{lm'C}P_{lm'}x_{lm'C}
 \left\langle lm'\left|\frac{Z_Ce^2}{4\pi\epsilon_0r_C}
 \right.\right\rangle.                                      \label{eq:old_vne}
\end{equation}

For a Coulomb ERI $(lm|qr)$, the historical four-centre factor is
\begin{equation}
 x_{lmqr}=x_{lm}\frac{x_{lq}+x_{mq}}2x_{qr}.                 \label{eq:old_4c}
\end{equation}
The analogous factors are inserted into separate Coulomb and exchange
contractions,
\begin{align}
 J^{\old}_{lm}&=\sum_{qr}P_{qr}x_{lmqr}(lm|qr),               \label{eq:old_j}\\
 K^{\old}_{lm}&=\sum_{qr}P_{qr}x_{lrqm}(lr|qm),               \label{eq:old_k}\\
 F^{\old}&=h^{\old}+\sum_n\left(2J_n^{\old}-K_n^{\old}\right),
                                                                    \label{eq:old_f}
\end{align}
where the final expression uses the paper's one-spin density convention.
The interaction domain is enlarged through unions of centre-specific
Wigner--Seitz cells, reaching translated copies beyond the original cluster.
Long-range Coulomb contributions may then be added with an Ewald field
\cite{janetzko2002,peintinger2014}.  The construction is operationally clear;
what is not supplied is a proof that the resulting weighted four-index tensor
has all symmetries of Eq.~\eqref{eq:eri_si} and hence defines one scalar RHF
energy.

\section{The deMon2k KS-ADFT CCM}
\label{sec:demon}

Janetzko, K\"oster, and Salahub developed a related CCM for the auxiliary-density
Kohn--Sham method in deMon2k \cite{janetzko2008}.  It must be audited separately
from the 2014 four-centre construction because its highest stored tensors are
three-centre quantities.  If $M,N,C$ carry AO functions $\mu,\nu$ and a
nuclear or auxiliary centre, respectively, their two-centre occurrence factor
and kinetic energy are
\begin{align}
 \omega_{MN}&=n_{MN}^{-1},                                      \label{eq:demon_w2}\\
 T^{\mathrm{CCM}}&=\sum_{\mu\nu}P_{\mu\nu}^{\mathrm{CCM}}
 \omega_{MN}\sum_{\nu'}T_{\mu\nu'}.                            \label{eq:demon_t}
\end{align}
The assertion that either AO centre may be the reference requires
\begin{equation}
 \omega_{MN}\sum_{\nu'} I_{\mu\nu'}
 =\omega_{NM}\sum_{\mu'} I_{\nu\mu'},                          \label{eq:demon_herm}
\end{equation}
which is stronger than the free-space identity
$I_{\mu\nu}=I_{\nu\mu}$.  Different centre-specific WS enumerations do not
make Eq.~\eqref{eq:demon_herm} automatic.

For a three-centre integral, however, the paper explicitly averages both
nested AO-centre orderings.  In compact notation its Eq.~(20) is
\begin{equation}
 I_{\mu\nu C}^{\mathrm{CCM}}=\frac{1}{n_{MNC}}
 \left[\sum_{M'\in W_C}\sum_{N'\in W_{M'}}I_{\mu'\nu'C}
 +\sum_{N'\in W_C}\sum_{M'\in W_{N'}}I_{\mu'\nu'C}\right],      \label{eq:demon_3c}
\end{equation}
with
\begin{equation}
 n_{MNC}=n_{MN}(n_{MC}+n_{NC}),\qquad
 \omega_{MNC}=n_{MNC}^{-1}.                                   \label{eq:demon_w3}
\end{equation}
Equation~\eqref{eq:demon_3c} is invariant under simultaneous interchange
$M,\mu\leftrightarrow N,\nu$ when both enumerations are complete.  Therefore
the simple bra--ket counterexample to Eq.~\eqref{eq:old_4c} cannot honestly be
copied to this three-centre tensor: the 2008 construction already contains a
specific symmetrisation.

Writing its weighted three-centre map as $B^w$ and weighted auxiliary Coulomb
metric as $V^w$, the fitted Coulomb part has the generic stationary form
\begin{equation}
 E_J^{\mathrm{ADFT}}[P,\bm x]
 =\bm P^{\mathsf T}B^w\bm x-\frac12\bm x^{\mathsf T}V^w\bm x,
 \qquad V^w\bm x=(B^w)^{\mathsf T}\bm P.                       \label{eq:demon_fit}
\end{equation}
This is variational only if $B^w$ and $(B^w)^{\mathsf T}$ are adjoints under
one finite-cluster inner product and $V^w$ is symmetric in that same metric.
The published energy and KS derivative show the intended stationary fitting,
but no quotient-group proof of these adjoint identities is given for arbitrary
non-Bravais or skew clusters.

The exchange--correlation quadrature introduces a separate approximation.  A
reference-cluster grid is used, translated auxiliary functions reconstruct the
density, and the auxiliary lattice sum is truncated after one neighbouring
shell.  The authors explicitly state that their finite CCM is not fully
equivalent to the periodic calculation because interactions outside the WS
regions are omitted.  Thus this deMon2k formulation can converge with cluster
size, but it is not the exact finite-$N$ BvK/character identity proved in
Sec.~\ref{sec:new}.  The multiplicative weights deserve symmetry tests; the
present audit does not label Eq.~\eqref{eq:demon_3c} broken where its two-order
average already enforces AO-pair symmetry.

\section{Independent \texttt{aiccm2026dev-b} formulation}
\label{sec:new}

\subsection{Finite Hamiltonian and electrostatic gauge}

The new method first defines the BvK-supercell Green function as the
zero-average solution
\begin{align}
 -\epsilon_0\nabla^2G_{\bm N}(\bm r)
 &=\sum_{\bm L_{\bm N}}\delta(\bm r-\bm L_{\bm N})
 -\frac1{\Omega_{\bm N}},                                   \label{eq:poisson_torus}\\
 G_{\bm N}(\bm r)&=\frac1{\epsilon_0\Omega_{\bm N}}
 \sum_{\G_{\bm N}\ne0}\frac{e^{i\G_{\bm N}\cdot\bm r}}
 {G_{\bm N}^2}.                                              \label{eq:green_torus}
\end{align}
The $\G=0$ coefficient is zero.  The same gauge is used for
electron--electron, electron--nuclear, and nuclear--nuclear interactions.
The primitive cell is required to be neutral.  Charged cells are rejected
rather than assigned an implicit jellium convention.

Equations~\eqref{eq:poisson_torus}--\eqref{eq:green_torus} state the current
3D gauge.  The finite translation-group algebra is dimension independent, and
the implementation also executes 1D and 2D cells.  In those dimensions the
four-centre route uses a symmetric direct-truncated active-lattice kernel,
whereas RI uses its dimension-aware fitted metric; no 3D spheropole or Ewald
exchange-divergence term is applied.  Each selected kernel defines a scalar
functional of the form below, but the lower-dimensional four-centre and RI
absolute energies are not claimed to represent the same Green function.

The periodised two-electron integral is
\begin{equation}
 (12|34)_{\bm N}=\iint\chi_1^*(\bm r)\chi_2(\bm r)
 e^2G_{\bm N}(\bm r-\bm r')
 \chi_3^*(\bm r')\chi_4(\bm r')\dd^3r\dd^3r'.               \label{eq:periodic_eri}
\end{equation}
It obeys
\begin{equation}
 (12|34)_{\bm N}=(21|34)^*_{\bm N}=(12|43)^*_{\bm N}
 =(34|12)_{\bm N}.                                          \label{eq:eri_symmetry}
\end{equation}
If tied Wigner--Seitz representatives are written explicitly, their factors
from Eq.~\eqref{eq:partition} average translationally equivalent values of
Eq.~\eqref{eq:periodic_eri}; their product sums to one and does not change the
integral.  No second, centre-dependent multiplier is applied.

\subsection{RHF functional and variational statement}

With spin-summed $D(\kvec)$ and uniform $w_{\kvec}=1/N_c$,
\begin{align}
 N_e&=\sum_{\kvec}w_{\kvec}\Tr[D(\kvec)S(\kvec)],            \label{eq:ne}\\
 D(\kvec)S(\kvec)D(\kvec)&=2D(\kvec).                       \label{eq:idempotency}
\end{align}
The energy per primitive cell is
\begin{align}
 \mathcal E_{\bm N}^{\mathrm{RHF}}[D]
 ={}&E_{NN}^{\bm N}+\sum_{\kvec}w_{\kvec}\Tr[D(\kvec)h(\kvec)]
 \nonumber\\
 &+\frac12\sum_{\kvec}w_{\kvec}\Tr\left[D(\kvec)
 \left(J(\kvec)-\frac12K(\kvec)\right)\right].             \label{eq:new_energy}
\end{align}
Varying $D\mapsto D+\delta D$ and using the bra--ket symmetry
Eq.~\eqref{eq:eri_symmetry} gives
\begin{align}
 \delta E_J&=\frac12\Tr[\delta D J]+
 \frac12\Tr[D J[\delta D]]=\Tr[\delta D J],                 \label{eq:var_j}\\
 \delta E_K&=-\frac14\Tr[\delta D K]-
 \frac14\Tr[D K[\delta D]]=-\frac12\Tr[\delta D K].        \label{eq:var_k}
\end{align}
Therefore
\begin{equation}
 \frac{\delta\mathcal E_{\bm N}^{\mathrm{RHF}}}{\delta D(\kvec)}
 =w_{\kvec}\left[h(\kvec)+J(\kvec)-\frac12K(\kvec)\right].
                                                                    \label{eq:functional_derivative}
\end{equation}
The admissible set is
\begin{equation}
 \mathcal V_{\bm N}=\{D:D^\dagger=D,\ DSD=2D,\
 \Tr(DS)=N_cN_e,\ [D,T_{\R}]=0\}.                            \label{eq:var_space}
\end{equation}
The method minimises Eq.~\eqref{eq:new_energy} over
$\mathcal V_{\bm N}$.  A converged SCF is a stationary point in this set;
ordinary RHF caveats about local rather than global minima remain.

\subsection{Closed-shell Kohn--Sham extension}

For RKS the finite-torus functional is
\begin{align}
 \mathcal E_{\bm N}^{\mathrm{RKS}}[D]
 ={}&E_{NN}^{\bm N}+\Tr_w[Dh]+\frac12\Tr_w[DJ]
 -\frac{\alpha}{4}\Tr_w[DK]+E_{xc}[\rho_D],                 \label{eq:rks_energy}\\
 \rho_D(\bm r)&=\sum_{\kvec}w_{\kvec}
 \sum_{\mu\nu}D_{\mu\nu}(\kvec)
 \phi^*_{\mu\kvec}(\bm r)\phi_{\nu\kvec}(\bm r),        \label{eq:rks_density}
\end{align}
where $\Tr_w$ denotes the weighted k sum.  Differentiation gives
Eq.~\eqref{eq:fock_rks}.  The implementation uses libxc through the existing
periodic quadrature.  Closed-shell LDA, GGA, meta-GGA, and hybrid machinery is
available, but the present numerical cross-backend result uses PBE0 only.
Finite temperature, charged cells, gradients, and a matched common 1D/2D
Coulomb kernel across all backends are not claimed.

\subsection{Unrestricted HF and spin-polarized KS}

Let $P^\alpha$ and $P^\beta$ have unit spin occupation and
$P=P^\alpha+P^\beta$.  Direct counting of electron pairs gives
\begin{align}
\mathcal E_{\bm N}^{\rm UHF}
={}&E_{NN}^{\bm N}+\sum_\sigma\Tr_w[P^\sigma h]
+\frac12\Tr_w[PJ[P]]
-\frac12\sum_\sigma\Tr_w[P^\sigma K[P^\sigma]],             \label{eq:uhf_energy}\\
F^\sigma&=h+J[P]-K[P^\sigma].                               \label{eq:uhf_fock}
\end{align}
The unrestricted variational set is
\begin{align}
\mathcal V_{\bm N}^{\rm U}=\{(P^\alpha,P^\beta):\;&
(P^\sigma)^\dagger=P^\sigma,\quad
P^\sigma SP^\sigma=P^\sigma,\quad [P^\sigma,T_{\R}]=0,
\nonumber\\
&\Tr(P^\sigma S)=N_cn_\sigma\}.                             \label{eq:uhf_space}
\end{align}
Here $n_\alpha=(N_e+2S)/2$ and $n_\beta=(N_e-2S)/2$.  Thus UHF
minimizes Eq.~\eqref{eq:uhf_energy} over two independent translation-invariant
spin projectors.  Spin restriction is not reintroduced by the cyclic
boundary.  The stationary determinant need not be an eigenfunction of
$\hat S^2$, so the implementation reports the measured and ideal values.

For UKS with exact-exchange fraction $a_x$,
\begin{align}
\mathcal E_{\bm N}^{\rm UKS}
={}&E_{NN}^{\bm N}+\sum_\sigma\Tr_w[P^\sigma h]
+\frac12\Tr_w[PJ[P]]
-\frac{a_x}{2}\sum_\sigma\Tr_w[P^\sigma K[P^\sigma]]
+E_{xc}[\rho_\alpha,\rho_\beta],                            \label{eq:uks_energy}\\
F^\sigma_{\rm UKS}&=h+J[P]-a_xK[P^\sigma]+V^\sigma_{xc}.
                                                                    \label{eq:uks_fock}
\end{align}
No additional occurrence weight follows from spin.  Both projectors use the
same translation quotient, Green function, and Wigner--Seitz representative
partition.

\subsection{Three integral backends}

All backends target Eqs.~\eqref{eq:new_energy} and
\eqref{eq:rks_energy}:
\begin{enumerate}
 \item \texttt{four\_center}: direct short-range four-centre J/K, analytic
 long-range Ewald J, reciprocal long-range exchange, and the BvK-supercell
 exchange correction in the corrected CRYSTAL gauge;
 \item \texttt{ri}: pair-resolved periodic density fitting.  With auxiliary
 functions $P,Q$ and Coulomb metric $V_{PQ}=(P|Q)$,
 \begin{align}
  (\mu\nu|\lambda\sigma)&\simeq
  \sum_{PQ}(\mu\nu|P)(V^{-1})_{PQ}(Q|\lambda\sigma),          \label{eq:ri}\\
  L^P_{\mu\nu}&=\sum_Q(\mu\nu|Q)(V^{-1/2})_{QP},             \label{eq:lthree}\\
  (\mu\nu|\lambda\sigma)&\simeq\sum_PL^P_{\mu\nu}L^P_{\lambda\sigma};
                                                                    \label{eq:ri_factored}
 \end{align}
 momentum-transfer-resolved three-centre tensors are retained
 \cite{sun2017};
 \item \texttt{rijcosx}: the same RI Coulomb build with chain-of-spheres
 quadrature for exact exchange \cite{neese2009}.  It is therefore not a pure
 three-centre approximation to K; ``RIJ+COSX'' is the accurate description.
\end{enumerate}
The q-only compensated-cell fit is rejected because it does not define one
pair-resolved four-index tensor on tight cells.  One-cell RKS RI and one-cell
RIJCOSX are also rejected because the current native paths would silently
fall back to a differently labelled Gamma algorithm.

\subsection{Canonical finite-group RI-MP2}

The first post-HF extension is canonical RI-MP2
\cite{moller1934,feyereisen1993}.  It uses the converged B-stream RI-RHF
orbitals, not the existing legacy Gamma-supercell helper.  On the compact
two-cell H$_2$ control those two HF references differ by
$0.0046408868\,\Ha$ per cell, so substituting the latter would change the
zeroth-order Hamiltonian.

Let $i,j$ be occupied bands, $a,b$ virtual bands, and $N_k=N_c$.  For
$\bm q=\kvec_a-\kvec_i$, diagonalise the auxiliary metric
$M_{\bm q}=U_{\bm q}\lambda_{\bm q}U_{\bm q}^{\dagger}$ and define the
canonical retained-subspace factor
\begin{equation}
 \widetilde L^{\kvec_i\kvec_a}_{P\mu\nu}=
 \left[U_{\bm q}\lambda_{\bm q,+}^{-1/2}
 U_{\bm q}^{\dagger}T_{\bm q}\right]_{P\mu\nu}.             \label{eq:canonical_lpq}
\end{equation}
Unlike a compact eigenmode factor, Eq.~\eqref{eq:canonical_lpq} is expressed
in the original auxiliary-AO basis.  Arbitrary eigenvector phases and
degenerate-subspace rotations chosen independently at $\bm q$ and $-\bm q$
therefore cannot contaminate a cross-block contraction.

The occupied--virtual transform is
\begin{equation}
 L^{\kvec_i\kvec_a}_{Pia}=\sum_{\mu\nu}
 C_{\mu i}^{*}(\kvec_i)\widetilde L^{\kvec_i\kvec_a}_{P\mu\nu}
 C_{\nu a}(\kvec_a).                                       \label{eq:lov}
\end{equation}
For each $(\kvec_i,\kvec_a,\kvec_j)$ the finite group fixes
\begin{equation}
 \kvec_b=\kvec_i-\kvec_a+\kvec_j+\G,                       \label{eq:kconserv}
\end{equation}
and the fitted integral block is
\begin{equation}
 V_{ij}^{ab}(\kvec_i,\kvec_j,\kvec_a)=\frac1{N_k}\sum_P
 L^{\kvec_i\kvec_a}_{Pia}L^{\kvec_j\kvec_b}_{Pjb}.        \label{eq:mp2_v}
\end{equation}
With $\Delta_{ij}^{ab}=\varepsilon_{i\kvec_i}+\varepsilon_{j\kvec_j}
-\varepsilon_{a\kvec_a}-\varepsilon_{b\kvec_b}$, the spin components per
primitive cell are
\begin{align}
 E_{\rm OS}^{(2)}&=\frac1{N_k}\sum_{\kvec_i\kvec_j\kvec_a}\sum_{ijab}
 \frac{|V_{ij}^{ab}(\kvec_a)|^2}{\Delta_{ij}^{ab}},        \label{eq:mp2_os}\\
 E_{\rm SS}^{(2)}&=\frac1{N_k}\sum_{\kvec_i\kvec_j\kvec_a}\sum_{ijab}
 \frac{|V_{ij}^{ab}(\kvec_a)|^2-
 V_{ij}^{ab}(\kvec_a)^*V_{ij}^{ba}(\kvec_b)}
 {\Delta_{ij}^{ab}}.                                       \label{eq:mp2_ss}
\end{align}
Thus $E_{\rm MP2}^{(2)}=E_{\rm OS}^{(2)}+E_{\rm SS}^{(2)}$.  MP2 is not
variational; it is Rayleigh--Schr\"odinger perturbation theory through second
order about the stationary finite RI-RHF determinant.  Equations
\eqref{eq:kconserv}--\eqref{eq:mp2_ss} are exactly the character
representation of canonical RI-MP2 on the same BvK torus.  This section is a
new result of the B implementation, not a reconstruction of historical AICCM.

\subsection{Unrestricted MP2}

For UMP2 the spatial occupied and virtual spaces differ by spin.  Separating
same- and opposite-spin contractions gives
\begin{align}
E_{\rm UMP2}^{(2)}
&=E_{\alpha\alpha}^{(2)}+E_{\beta\beta}^{(2)}
+E_{\alpha\beta}^{(2)},                                    \label{eq:ump2_sum}\\
E_{\sigma\sigma}^{(2)}
&=\frac14\sum_{ijab\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},              \label{eq:ump2_ss}\\
E_{\alpha\beta}^{(2)}
&=\sum_{\substack{ia\in\alpha\\jb\in\beta}}
\frac{|(i_\alpha a_\alpha|j_\beta b_\beta)|^2}
{\varepsilon_i^\alpha+\varepsilon_j^\beta
-\varepsilon_a^\alpha-\varepsilon_b^\beta}.               \label{eq:ump2_os}
\end{align}
Every orbital sum in Eqs.~\eqref{eq:ump2_ss} and \eqref{eq:ump2_os} includes
finite characters constrained by Eq.~\eqref{eq:kconserv}, and the total is
normalized once per primitive cell.  The implementation obtains the same
contractions from the exact real-torus factors below.  Zero PNO threshold and
canonical occupieds define the full UMP2 limit.

\subsection{Exact real-torus transform and local correlation}

Local correlation requires a real finite-torus Hamiltonian, not the legacy
Gamma-supercell approximation.  Define normalized finite-group AOs and their
inverse transform by
\begin{align}
 |\mu\kvec\rangle
 &=N_c^{-1/2}\sum_{\R\in\Tgroup}e^{+i\kvec\cdot\R}|\mu\R\rangle,
                                                               \label{eq:bloch_normalized}\\
 X_{\mu\R,\nu\bm S}
 &=\frac1{N_c}\sum_{\kvec}e^{+i\kvec\cdot\R}
 X_{\mu\nu}(\kvec)e^{-i\kvec\cdot\bm S}.                    \label{eq:real_matrix}
\end{align}
Equation~\eqref{eq:real_matrix} is applied to overlap and Fock.  For the
canonical pair factor of Eq.~\eqref{eq:canonical_lpq}, with
$\bm q=\kvec_a-\kvec_i$, define
\begin{equation}
 B_{P\bm T,\mu\R,\nu\bm S}
 =\frac1{N_c^2}\sum_{\kvec_i\kvec_a}
 e^{+i\kvec_i\cdot\R}e^{-i\kvec_a\cdot\bm S}
 e^{+i\bm q\cdot\bm T}
 \widetilde L^{\kvec_i\kvec_a}_{P\mu\nu}.                  \label{eq:real_cderi}
\end{equation}
The real-torus MO coefficient is
$C_{\mu\R,p\kvec}=N_c^{-1/2}e^{+i\kvec\cdot\R}C_{\mu p}(\kvec)$.
Direct contraction then yields
\begin{align}
 B_{P\bm T,i\kvec_i,a\kvec_a}
 &=N_c^{-1}e^{+i\bm q\cdot\bm T}
 L^{\kvec_i\kvec_a}_{Pia},                                  \label{eq:real_lov}\\
 \sum_{P\bm T}B_{P\bm T,ia}B_{P\bm T,jb}
 &=\frac1{N_c}\sum_P L_{Pia}L_{Pjb},                         \label{eq:real_eri_identity}
\end{align}
when the two transfers sum to a reciprocal vector; otherwise character
orthogonality makes the result zero.  Equations
\eqref{eq:real_cderi}--\eqref{eq:real_eri_identity} prove that the real and
character algorithms use one fitted finite Hamiltonian.

The current algorithm includes all occupied pairs and all ordered occupied
triples on the finite torus, computes $E_{\mathrm{corr},N}$, and reports
\begin{equation}
 E_{\mathrm{corr}}^{\mathrm{cell}}=E_{\mathrm{corr},N}/N_c.  \label{eq:local_per_cell}
\end{equation}
There is no occurrence weight.  Translation reduction may later retain one
orbit under simultaneous translation of every occupied index, but its
multiplicity must follow the orbit--stabilizer theorem.

Pair domains and PNOs follow local MP2 and DLPNO constructions
\cite{pinski2015,riplinger2013,nejad2025}; CCSD and perturbative triples use
the standard connected equations and local triple spaces
\cite{purvis1982,raghavachari1989,riplingertriples2013,guo2018}.
At zero domain and PNO thresholds, with all pairs and complete occupied
coupling, the canonical occupied route reproduces finite-torus MP2, CCSD,
and exact (T). Truncation removes that equality. MP2 and CCSD(T) are
nonvariational; CCSD is stationary only through its left--right Lagrangian.

For the unrestricted route, the spin-orbital ansatz and projected equations
are
\begin{align}
|\Psi_{\rm UCCSD}\rangle&=e^{T_1+T_2}|\Phi_{\rm UHF}\rangle,
                                                               \label{eq:ucc_ansatz}\\
\langle\Phi_\mu|e^{-T}He^T|\Phi_{\rm UHF}\rangle&=0,
\qquad \mu\in\{\text{singles,doubles}\},                     \label{eq:ucc_residual}\\
E_{\rm UCCSD}&=\langle\Phi_{\rm UHF}|e^{-T}He^T
|\Phi_{\rm UHF}\rangle .                                    \label{eq:ucc_energy}
\end{align}
The current unrestricted PNO engine expands each pair amplitude into the full
finite-torus spin-orbital virtual space, evaluates
Eq.~\eqref{eq:ucc_residual}, and projects back.  At zero PNO threshold this
projection is a unitary coordinate change and recovers full-domain UCCSD.
The perturbative triples correction is then evaluated from the converged
amplitudes.  This is an explicitly cost-capped $O(N^6)$ correctness oracle.
It does not yet exploit translation-equivalent pair representatives, so no
reduced-scaling claim follows from the DLPNO label.

\subsubsection{Wannier and IAO occupied gauges}

For an isolated occupied manifold, the discrete back transform is
\begin{equation}
 |w^0_{n\R}\rangle=\frac{1}{\sqrt{N_c}}\sum_{\kvec}
 e^{-i\kvec\cdot\R}|\psi_{n\kvec}\rangle .                \label{eq:wannier_back}
\end{equation}
One unitary rotation is applied to the complete $N_c n_{\rm occ}$ space,
$C^{\rm loc}=C^0U$. Hence
\begin{equation}
 C^{\rm loc}C^{{\rm loc}\dagger}=C^0UU^\dagger C^{0\dagger}
 =C^0C^{0\dagger}.                                        \label{eq:loc_projector}
\end{equation}
Equation~\eqref{eq:loc_projector}, not constancy of a plausible scalar
energy, is the invariance proof for RHF and RKS.  For UHF/UKS the alpha and
beta projectors are localized independently, and Eq.~\eqref{eq:loc_projector}
must hold in each spin space.

Exact Marzari--Vanderbilt localization uses cross-k exponential-position
matrix elements \cite{marzari1997}. The current integral API lacks them.
The B implementation instead jointly diagonalizes Lowdin-projected circular
AO-centre operators. With $L_\alpha$ in metres,
\begin{align}
 z_{n\alpha}&=\langle w_n|e^{2\pi i r_\alpha/L_\alpha}|w_n\rangle,
                                                               \label{eq:circular_z}\\
 \Omega_{n\alpha}&=-\left(\frac{L_\alpha}{2\pi}\right)^2
 \ln|z_{n\alpha}|^2.                                      \label{eq:circular_spread}
\end{align}
Thus $\Omega$ has SI unit m$^2$. It is a projected circular AO-centre
approximation, not an exact continuum MLWF spread. The IAO alternative uses
a minimal atomic reference and squared intrinsic populations
\cite{knizia2013}. Its present target--reference overlap is evaluated in an
explicit supercell and is not periodized across the torus boundary.

The localized orbitals are mapped under each primitive translation by their
finite-metric overlaps. That mapping is not assumed from column order. A
wrap warning is emitted when antipodal probability exceeds 0.05 or the
localization length exceeds one quarter of the shortest cluster vector.
Metallic and entangled manifolds fail closed because disentanglement is not
implemented.

\subsubsection{Projected atomic and pair natural orbitals}

For $C_o^\dagger S C_o=I$, the coefficient projector and raw domain PAOs are
\begin{align}
 Q&=I-C_oC_o^\dagger S,                                    \label{eq:pao_projector}\\
 \widetilde V_D&=QE_D,                                     \label{eq:pao_raw}
\end{align}
where $E_D$ selects domain AOs. If
$\widetilde V_D^\dagger S\widetilde V_D=XgX^\dagger$, then
\begin{equation}
 V_D=\widetilde V_DX_+g_+^{-1/2},\quad
 V_D^\dagger S V_D=I,\quad C_o^\dagger S V_D=0.           \label{eq:pao_orthogonal}
\end{equation}
Canonical orthogonalization therefore removes PAO redundancy without
reintroducing occupied components.

For pair amplitudes $T_{ij}$, define
\begin{equation}
 D_{ij}=\frac{T_{ij}T_{ij}^\dagger+T_{ij}^\dagger T_{ij}}
 {1+\delta_{ij}}.                                          \label{eq:pno_density}
\end{equation}
For every vector $x$, $x^\dagger D_{ij}x$ is a sum of squared norms, so
$D_{ij}$ is positive semidefinite. Its eigenvectors are PNOs. Tightening an
occupation threshold produces nested ranks and the zero threshold retains
the complete PAO span. It does not prove monotonic energy convergence because
MP2 is nonvariational.

Translations act simultaneously on both members of an unordered occupied
pair. For measured permutations $p_g$,
\begin{equation}
 {\cal O}(i,j)=\{\operatorname{sort}[p_g(i),p_g(j)]:g\in G_N\},\qquad
 w_{ij}^{\rm cell}=|{\cal O}(i,j)|/N_c.                    \label{eq:pair_orbit}
\end{equation}
This orbit--stabilizer weight correctly handles fixed half-cell pairs on an
even cyclic ring. The implementation enumerates and checks the partition but
does not yet skip nonrepresentative amplitudes.

The current route disables Euclidean pair-distance screening, finite
occupied-coupling radii, and local auxiliary fitting. Their molecular
domains are not invariant across a torus boundary. At the complete-domain,
zero-PNO-threshold MP2 limit, a separate rotation-invariant full-space DF
contraction audits the raw local-pair energy. Both the raw energy and the
complete-space correction are reported. A real time-reversal gauge is first
constructed and its occupied projector is checked, because the inherited
local solver uses real equations. Truncated calculations receive no such
correction.

\subsubsection{Space-group action on the finite torus}

Let $N=\operatorname{diag}(N_1,N_2,N_3)$ and
$g=\{W|\bm w\}$. The operation descends to the cyclic quotient if and only if
\begin{equation}
 N^{-1}WN\in\mathbb Z^{3\times3}.                           \label{eq:cluster_compat}
\end{equation}
For atom $a$,
\begin{equation}
 W\bm f_a+\bm w=\bm f_{p_g(a)}+\bm q_{g,a},                \label{eq:atom_image}
\end{equation}
and cell $\bm r$ maps to
$W\bm r+\bm q_{g,a}\pmod{\bm N}$. The integer image shift must be retained
for screw and glide operations. Reciprocal characters transform as
$\kvec'=W^{-T}\kvec$ and orbit weights are $|{\cal O}_{\kvec}|/N_c$.

spglib supplies the crystallographic operations. The current B route builds
their exact cluster-compatible subgroup and reports irreducible k orbits,
plus primitive shell-pair and eightfold shell-quartet orbit partitions, but
it remains diagnostic. General-k AO sewing phases depend on the Bloch
convention used by each backend. Representative shell-pair and quartet
scattering is therefore disabled until symmetry-on/off Fock and energy parity
is established.

\section{Term-by-term comparison}

\begin{table}[htbp]
\centering
\footnotesize
\caption{Historical AICCM and \texttt{aiccm2026dev-b}.}
\label{tab:comparison}
\begin{tabular}{p{0.18\linewidth}p{0.36\linewidth}p{0.36\linewidth}}
\toprule
Term & Historical AICCM & \texttt{aiccm2026dev-b}\\
\midrule
Finite object & Cluster plus centre-specific Wigner--Seitz interaction
regions & Quotient group $\Tgroup$ and one explicitly periodised Hamiltonian\\
One electron & Two-centre occurrence factors; three-centre averaged factors
for nuclear attraction & Block-circulant periodised $h$ in one Ewald gauge;
WS ties only choose equivalent representatives\\
Coulomb & $x_{lm}(x_{lq}+x_{mq})x_{qr}/2$ on free-space ERIs & Symmetric
periodic ERI tensor or a controlled RI factorisation of that tensor\\
Exchange & Separate centre-dependent product inserted into exchange
contraction & Same symmetric tensor, permuted indices; corrected BvK
long-range exchange\\
$\G=0$ & WSSC neutrality; Ewald may be added, but a single functional gauge
is not derived in the 2014 equations & Zero-average torus Green function;
neutral cells only; common electron/nuclear gauge\\
Energy & Molecular-looking weighted J/K expression & Explicit scalar
functional, Eqs.~\eqref{eq:new_energy} and \eqref{eq:rks_energy}\\
Variational status & Not established for the displayed four-centre weights &
Stationary over Eq.~\eqref{eq:var_space}; global-minimum stability remains open\\
Algorithms & Historical direct four-centre implementation & Direct corrected
four-centre, pair-resolved RI, or RIJCOSX; same cyclic mesh\\
\bottomrule
\end{tabular}
\end{table}

The other active development line now also exposes
\texttt{method="aiccm2026dev-a"}, which replaces the historical bridge by an
explicitly symmetric four-centre bridge while retaining the weighted-cluster
construction. It is neither silently substituted for the historical column
nor merged into \texttt{aiccm2026dev-b}. The benchmark archive reports all
three. Their equality for the present 1D chain is not evidence of equality
for general 3D boundary multiplicities.

\section{What is mathematically unresolved in the old weighting}
\label{sec:diagnosis}

\subsection{Explicit loss of bra--ket symmetry}

Apply bra--ket interchange to Eq.~\eqref{eq:old_4c}.  The original factor is
\begin{equation}
 W_{lm,qr}=x_{lm}x_{qr}\frac{x_{lq}+x_{mq}}2.                 \label{eq:w_original}
\end{equation}
After $(lm|qr)=(qr|lm)$ it becomes
\begin{equation}
 W_{qr,lm}=x_{qr}x_{lm}\frac{x_{ql}+x_{rl}}2
 =x_{lm}x_{qr}\frac{x_{lq}+x_{rl}}2.                        \label{eq:w_swapped}
\end{equation}
Equality requires
\begin{equation}
 x_{mq}=x_{rl}.                                               \label{eq:extra_condition}
\end{equation}
Equation~\eqref{eq:extra_condition} is not a geometric identity.  It can hold
in special symmetric configurations or when all factors are unity, which
explains why small high-symmetry examples may look sound.  It is not true for
a general four-centre non-Bravais configuration.  Thus the effective tensor
$\widetilde{(lm|qr)}=W_{lm,qr}(lm|qr)$ need not satisfy
Eq.~\eqref{eq:eri_symmetry}.

\subsection{Why this breaks the usual Fock derivative}

Suppose a nominal Coulomb energy is
\begin{equation}
 \widetilde E_J[D]=\frac12\sum_{lmqr}D_{lm}D_{qr}
 W_{lm,qr}(lm|qr).                                           \label{eq:weighted_energy}
\end{equation}
Differentiation gives
\begin{equation}
 \frac{\partial\widetilde E_J}{\partial D_{ab}}=
 \frac12\sum_{qr}D_{qr}\left[W_{ab,qr}(ab|qr)
 +W_{qr,ab}(qr|ab)\right].                                  \label{eq:sym_derivative}
\end{equation}
The derivative contains the \emph{symmetrised} weight.  The directly assembled
old Coulomb matrix, Eq.~\eqref{eq:old_j}, contains only the first term.  Unless
Eq.~\eqref{eq:extra_condition} holds, the old Fock is not the derivative of
Eq.~\eqref{eq:weighted_energy}.  The same issue propagates to exchange.  One
could define a new method by explicitly symmetrising all weighted tensors,
but that would be a third formulation, not the equations published in 2014.

\subsection{Double counting, central cells, and long range}

The old factors are motivated by avoiding repeated boundary atoms.  That is
correct for counting geometric representatives.  The unproved step is
turning a representative count into a centre-pair multiplier on a
non-periodised four-centre kernel.  In the quotient-group construction each
translation class appears once, and tied representative weights sum to one;
there is no additional ``central cell'' energy.  Applying another factor
changes the Hamiltonian rather than merely preventing double counting.

The 2014 article states that long-range Coulomb terms may be included by
Ewald summation.  It does not derive one electron--electron,
electron--nuclear, nuclear--nuclear, and exchange gauge whose derivative is
the displayed Fock.  This is classified here as an ambiguity, not proof that
every historical Ewald implementation is wrong.  The implementation finding
is narrower: a scalar correction added after SCF is insufficient whenever
its density derivative is nonzero.

\subsection{Why convergence can conceal the problem}

DIIS solves a fixed-point equation; it does not test whether that equation is
Euler--Lagrange for a scalar energy.  A nonsymmetric effective tensor can
still generate a Hermitian-looking Fock after partial cancellation, and its
SCF can converge smoothly.  Boundary factors differ from unity only on a
small subset of terms, while exchange is short ranged in insulators.  Errors
can therefore decrease with cluster size and produce credible totals.  These
facts explain ``sort of works'' without either dismissing the historical
calculations or accepting them as a variational proof.

\section{Relation to full reciprocal-space periodic boundary conditions}

Equations~\eqref{eq:orthogonality}--\eqref{eq:inverse} prove the finite-size
identity
\begin{equation}
 \frac{E_{\bm N}^{\Gamma\text{-supercell}}}{N_c}
 =E_{\bm N}^{\text{full Gamma-centred character mesh}}        \label{eq:finite_identity}
\end{equation}
provided both sides use the same finite-torus Green function, basis, and
translation-invariant variational space.  This does not prove equality to an
arbitrary CRYSTAL calculation: CRYSTAL normally represents the infinite
periodic Hamiltonian and approximates the Brillouin-zone integral, whereas
Eq.~\eqref{eq:finite_identity} defines a finite BvK Hamiltonian.  The
algorithms meet in the limit
\begin{equation}
 \lim_{N_1,N_2,N_3\to\infty}E_{\bm N}
 =\frac1{\Omega_{\mathrm{BZ}}}\int_{\mathrm{BZ}}E(\kvec)\dd^3k,
                                                                    \label{eq:infinite_limit}
\end{equation}
when the same Coulomb convention is approached and the sequence of SCF
solutions remains in the same physical basin.

Finite-size errors have several sources: discretisation of occupied
subspaces, algebraically convergent exact exchange, finite electrostatic
self-image terms, real-space truncation in direct builds, auxiliary-basis and
plane-wave cutoffs in RI, and numerical XC grids in KS.  Insulating density
matrices decay exponentially under standard gap and analyticity conditions,
but the Coulomb kernel does not.  Monotone convergence is therefore not
guaranteed.  Basis linear dependence is handled per character by canonical
orthogonalisation; changing retained rank across the mesh can create
non-smooth sequences.  BSSE is a basis issue and is not cured by cyclic
closure; pob basis sets \cite{peintinger2013pob} and explicit counterpoise
strategies remain separate choices.

\section{Numerical results}
\label{sec:numerics}

\subsection{Provenance and limits of the evidence}

Every number in Tables~\ref{tab:h4} and \ref{tab:backend} comes from the
implementation artifacts dated 20--21 June 2026. The H$_4$ machine-readable
archive is distributed beside this source as
\nolinkurl{data/h4_ri_2026-06-21.json}. The old column is the
separate in-repository CCM implementation with its historical
``\texttt{union12}'' weight. The same harness also runs that implementation's
new symmetric ``\texttt{aiccm2026dev-a}'' weight; the archived JSON retains
the pre-rename field name \texttt{peer\_aiccmdev}.  The peer is kept distinct and
is identical to the old column to the displayed digits for this 1D sequence.
The ``character mesh'' column is
a direct call to the same full Gamma-centred reciprocal representation as the
new RI route and is expected to be identical by construction; it is an
internal identity test, not an independent program.  No new CRYSTAL or PySCF
job was run.  The 2014 article reports, for the same nominal alternating
hydrogen-chain/STO-3G model, a dense CRYSTAL value of
$-0.542875\,\Ha$ per atom at SHRINK 16 and tight integral tolerances
\cite{peintinger2014}.  That published value is contextual because executable
input and all modern backend cutoffs were not matched in the present run.

\begin{table}[htbp]
\centering
\small
\caption{Alternating H$_4$/STO-3G chain, \SI{4.0e-10}{\metre} repeat and
\SI{2.1167e-9}{\metre} transverse cell (40 bohr).  Energies are per primitive
cell.  SI values are primary; Hartree values in parentheses preserve the raw
artifact.  The new route used pair-resolved RI.}
\label{tab:h4}
\begin{tabular}{rrrrr}
\toprule
$N_1$ & \shortstack{New /\\\si{\kilo\joule\per\mole}} &
\shortstack{Character mesh /\\\si{\kilo\joule\per\mole}} &
\shortstack{Old /\\\si{\kilo\joule\per\mole}} &
\shortstack{New-old /\\\si{\kilo\joule\per\mole}}\\
\midrule
1 & -5443.652 (-2.0733777009) & -5443.652 & -5675.271 (-2.1615967328) & +231.619\\
2 & -5683.693 (-2.1648043358) & -5683.693 & -5676.013 (-2.1618792916) & -7.680\\
4 & -5702.483 (-2.1719609271) & -5702.483 & -5699.181 (-2.1707034746) & -3.301\\
\bottomrule
\end{tabular}
\end{table}

\begin{figure}[htbp]
\centering
\begin{tikzpicture}
\begin{axis}[
 width=0.83\linewidth,height=0.48\linewidth,
 xlabel={cyclic length $N_1$},ylabel={energy / \si{\kilo\joule\per\mole}},
 xtick={1,2,4},grid=both,legend pos=south east]
 \addplot+[mark=o,thick] coordinates {(1,-5443.652406) (2,-5683.693003) (4,-5702.482631)};
 \addlegendentry{\texttt{aiccm2026dev-b} RI}
 \addplot+[mark=square,thick] coordinates {(1,-5675.271443) (2,-5676.013301) (4,-5699.181190)};
 \addlegendentry{historical CCM (union12)}
\end{axis}
\end{tikzpicture}
\caption{Measured H$_4$ sequence.  The first point shows why a plausible
single cluster is not a validation criterion.  Lines guide the eye and do not
assert a convergence law.}
\label{fig:h4}
\end{figure}

The new wall times for $N_1=1,2,4$ were 22.58, 20.61, and
\SI{97.23}{\second}; the four-cell increase reflects all-k-pair exchange
fitting. The corresponding historical/symmetric-peer times were
0.03/0.02, 0.01/0.01, and \SI{0.07}{\second}/\SI{0.06}{\second}.
These single-process wall times are implementation-cost observations, not
hardware-normalised scaling exponents.

\subsection{Direct, RI, and RIJCOSX control}

\begin{table}[htbp]
\centering
\small
\caption{H$_2$/STO-3G in an $8\times12\times12$ bohr
($\SI{4.233e-10}{\metre}\times\SI{6.350e-10}{\metre}\times
\SI{6.350e-10}{\metre}$) cell at a $(2,1,1)$ cyclic mesh.  All calculations
converged and their densities satisfied the nonorthogonal idempotency test.}
\label{tab:backend}
\begin{tabular}{lrrr}
\toprule
Method & Four-centre & RI & RIJCOSX\\
 & \multicolumn{3}{c}{energy / \si{\kilo\joule\per\mole} (raw $\Ha$)}\\
\midrule
RHF & -2944.106 (-1.1213506456) & -2944.162 (-1.1213722182) & -2944.220 (-1.1213943367)\\
PBE0 & -3033.557 (-1.1554208287) & -3033.657 (-1.1554589526) & -3033.672 (-1.1554644822)\\
\bottomrule
\end{tabular}
\end{table}

\begin{table}[htbp]
\centering
\small
\caption{Executable lower-dimensional H$_2$/STO-3G controls at a $(2,1,1)$
cyclic mesh. Energies are per primitive cell in \si{\kilo\joule\per\mole}.
All densities had idempotency residual below $3\times10^{-15}$.  Four-centre
and RI use different current long-range gauges, so their gap is not a fitting
error.}
\label{tab:lowerdim}
\begin{tabular}{llrrr}
\toprule
Dimension & Method & Four-centre & RI & RIJCOSX\\
\midrule
1D & RHF & -4028.015 & -5998.179 & -5998.237\\
1D & PBE & -4120.268 & -6085.952 & -6085.952\\
2D & RHF & -9059.675 & -13694.260 & -13694.319\\
2D & PBE & -8499.154 & -13777.487 & -13777.487\\
\bottomrule
\end{tabular}
\end{table}

The RHF and PBE0 spreads are each approximately
\SI{0.115}{\kilo\joule\per\mole}.  This shows that all requested paths run and
agree on a compact control.  It is not a global accuracy bound.  On the
two-cell H$_4$/40-bohr model, four-centre RHF gave
\SI{-5741.989}{\kilo\joule\per\mole} ($-2.1870080428\,\Ha$), whereas RI gave
\SI{-5683.693}{\kilo\joule\per\mole} ($-2.1648043358\,\Ha$): a
\SI{-58.296}{\kilo\joule\per\mole} difference.  Its origin -- direct
truncation, vacuum embedding, or another backend convention -- is unresolved.
No conclusion about the dense limit is drawn from it.

Table~\ref{tab:lowerdim} establishes that every requested 1D/2D RHF and KS
route executes and converges.  It simultaneously exposes the next theory
gate: the four-centre/RI lower-dimensional offsets are hundreds of
\si{\kilo\joule\per\mole}.  They are expected from the unmatched direct-
truncated and fitted long-range kernels and forbid an accuracy claim.

The committed fleet layer now supplies identical A/B geometries for 18
systems and explicit B inputs for RHF, PBE, and PBE0 with four-centre, RI, and
RIJCOSX backends.  It additionally supplies 3D RI-MP2, local-PNO MP2, CCSD,
and CCSD(T) inputs.  The compact coverage profile contains 31 runnable jobs
distributed over 1D H-chain, 2D graphene, and 3D LiH controls.  At the time of
writing, the requested LiH, MgO, NaCl, and diamond fleet results and matched
CRYSTAL inputs have not been executed.  They remain open acceptance work, not
hidden supplementary results.  Likewise, no timing scaling law can be fitted
from three non-monotone wall times and one compact backend control.

\subsection{Canonical RI-MP2 reciprocal-space parity}

The first B-stream post-HF benchmark is H$_2$/STO-3G in a
$20\times20\times6$ bohr cell at a $(1,1,2)$ character mesh.  The external
reference is PySCF 2.13.1 KRHF with density fitting followed by KMP2, executed
out of process.  Table~\ref{tab:mp2} reports energy per primitive cell.

\begin{table}[htbp]
\centering
\small
\caption{Canonical 3D RI-MP2 parity in \si{\kilo\joule\per\mole} per cell.
The machine-readable Hartree values and provenance are archived in
\texttt{data/h2\_mp2\_2026-06-21.json}.}
\label{tab:mp2}
\begin{tabular}{lrrr}
\toprule
Contribution & \texttt{aiccm2026dev-b} & external KRHF/KMP2 & B $-$ reference\\
\midrule
HF & -2935.926214487 & -2935.926216046 & $+0.000001558$\\
RI-MP2 correlation & -34.191094559 & -34.191093946 & $-0.000000614$\\
Total & -2970.117309047 & -2970.117309991 & $+0.000000945$\\
\bottomrule
\end{tabular}
\end{table}

The maximum imaginary residue of the accumulated MP2 energy terms is
$8.7\times10^{-19}\,\Ha$, and the finite-group momentum residual is zero to
the reported precision.  The agreement below one nanohartree per cell tests
the normalization in Eqs.~\eqref{eq:mp2_v}--\eqref{eq:mp2_ss}, not merely
SCF convergence.  By contrast, the older Gamma-supercell HF helper differs
from B RI-HF by \SI{12.185}{\kilo\joule\per\mole} on the compact two-cell
control and is therefore rejected as a B post-HF reference.  One- and
two-dimensional MP2 remain disabled until the lower-dimensional Coulomb
gauge is common to the reference and correlation tensors.

\subsection{Real-torus local-correlation exact limits}

The first inverse-transform control reuses the H$_2$/STO-3G system of
Table~\ref{tab:mp2}.  With full PAO domains, no PNO truncation, no pair
screening, and canonical occupied orbitals, real-torus local MP2 gives
$-0.013022700154029399\,\Ha$ per cell.  The character-space result is
$-0.013022700154029405\,\Ha$ per cell, a difference of
$6.9\times10^{-18}\,\Ha$.

A nonzero-triples control uses LiH/STO-3G in a
$20\times20\times7$ bohr cell at the $(1,1,2)$ mesh.  A separately
contracted canonical DF-CCSD and exact (T) calculation uses the real factors
from Eq.~\eqref{eq:real_cderi}; the no-truncation local solver uses the same
factors.  Table~\ref{tab:localcc} reports energies per primitive cell.

\begin{table}[htbp]
\centering
\small
\caption{LiH finite-torus exact-limit test in \si{\kilo\joule\per\mole} per
cell.  The Hartree values and transform residuals are archived in
\texttt{data/lih\_dlpno\_ccsdt\_2026-06-21.json}.}
\label{tab:localcc}
\begin{tabular}{lrrr}
\toprule
Contribution & canonical finite torus & no-truncation local & local $-$ canonical\\
\midrule
DF-CCSD correlation & -39.707443419799 & -39.707443410743 & $+0.000000009056$\\
(T) & -0.122614924121 & -0.122614922321 & $+0.000000001800$\\
\bottomrule
\end{tabular}
\end{table}

The total-torus Hartree differences are $6.9\times10^{-12}$ for CCSD and
$1.4\times10^{-12}$ for (T).  The maximum imaginary, AO-factor symmetry, and
one-particle-matrix residuals of the inverse transform are respectively
$2.2\times10^{-8}$, $4.3\times10^{-8}$, and $5.6\times10^{-9}$.  This proves
the implemented no-truncation limit for the finite fitted Hamiltonian.  It is
not an external periodic CCSD(T) benchmark and does not validate a PNO
threshold for an extended solid.

The same H$_2$ control was repeated after the B-specific Wannier rotation.
The measured translation covariance residual is $3.3\times10^{-16}$, the
full PAO rank is two, and translations reduce three unordered occupied pairs
to two diagnostic orbits. The raw local correlation energy is
$-34.191094559387\,\si{\kilo\joule\per\mole}$ per cell. The independent
full-space audit is $-34.191094559459\,\si{\kilo\joule\per\mole}$ per cell,
a correction of $-7.17\times10^{-11}\,\si{\kilo\joule\per\mole}$ per cell.
The underlying Hartree values are archived in
\texttt{data/h2\_wannier\_pno\_2026-06-21.json}. The orbit count is not a
speedup result because all three pairs were still evaluated.

\subsection{Unrestricted finite-torus control}

A neutral Li doublet with STO-3G in a cubic 15-bohr cell and lattice
extension $(1,1,1)$ provides the first unrestricted transform test.
Table~\ref{tab:unrestricted} records the calculation run on 2026-06-22.
It is intentionally a one-cell algebraic control, not a converged Li crystal.

\begin{table}[htbp]
\centering
\small
\caption{Unrestricted B-stream control in
\si{\kilo\joule\per\mole} per primitive cell.}
\label{tab:unrestricted}
\begin{tabular}{lr}
\toprule
Contribution & Value\\
\midrule
RI-UHF & -19236.251051970\\
UMP2 correlation & -0.638678183\\
UCCSD correlation & -0.676877715\\
(T) & $-5.5\times10^{-31}$\\
UCCSD(T) total & -19236.927929685\\
\bottomrule
\end{tabular}
\end{table}

The two spin projectors have electron counts 2 and 1, idempotency residual
$5.1\times10^{-16}$, and $\langle S^2\rangle=0.7500000000000001$.
The inverse RI factor has maximum imaginary and permutation residuals
$2.31\times10^{-16}$ and $7.11\times10^{-15}$.  Independent alpha/beta
Wannier rotation changes the density by $4.3\times10^{-34}$; at zero PNO
threshold its UCCSD(T) total agrees with the canonical occupied gauge within
$5\times10^{-14}\,\Ha$.  These internal identities validate the
unrestricted finite-Hamiltonian transform and PNO exact limit.  No external
periodic UMP2 or UCCSD(T) comparison is yet available.

\section{Discussion}

The finite-group viewpoint makes the useful part of the CCM precise.  A
cyclic cluster is not a molecular fragment with patched edge atoms; it is a
finite periodic quotient.  Its Gamma-supercell and complete character-mesh
representations are unitarily equivalent.  The real-space view is attractive
for local correlation, defects that fit naturally in a large cell, sparse
density matrices, and reuse of molecular integral technology.  Reciprocal
space is attractive for primitive-cell efficiency, symmetry reduction,
metals, and systematic Brillouin-zone convergence.  Neither representation
is intrinsically more exact.

The old AICCM's main weakness is not that it always returns a large numerical
error.  It is that its displayed four-centre weighting does not establish the
tensor symmetry needed for a conventional energy derivative.  Special
symmetry, short-ranged exchange, and decreasing boundary fraction explain
why the method can nevertheless converge toward periodic values.  The new
method replaces that empirical hope with a defined Hamiltonian and explicit
invariants: weight partition, electron count, idempotency, and inverse-Bloch
residual.

The new implementation is still experimental.  Four-centre versus RI
agreement is system dependent in the current measurements.  The direct route
uses the corrected CRYSTAL-gauge Fock infrastructure, not the separate old
AICCM core.  RI and RIJCOSX introduce auxiliary-basis, Fourier cutoff, and
quadrature errors.  RKS adds XC-grid error.  The deMon2k audit finds a real
unproved two-centre Hermiticity condition and unproved RI adjoint consistency,
but also finds that Eq.~\eqref{eq:demon_3c} explicitly symmetrises the two AO
orderings.  Calling that tensor broken solely because it has a multiplicative
normalisation would overstate the evidence.  The present work therefore proves
the algebraic finite-torus construction and demonstrates executable RHF/PBE0
paths; it does not certify production accuracy for ionic solids.

Canonical 3D RI-MP2 is now formulated with momentum-conserving finite-group
amplitudes and validated against reciprocal-space KMP2.  The exact inverse
transform now supports local-PNO MP2, CCSD, and CCSD(T), and the complete-
domain limits agree with canonical finite-torus oracles.  The implementation
avoids an unproved ``unique pair'' weight by summing the full finite torus and
dividing once by $N_c$.  Its present local route is not yet linear scaling:
minimum-image pair screening, finite occupied-coupling domains, local fitting,
and translation-orbit reduction remain open. The B implementation now
enumerates translation-pair orbits and space-group-compatible k orbits, but
does not turn a formal count into a timing claim. UHF and UKS now use two
independently idempotent spin projectors and no new boundary rule; unrestricted
MP2 and the full-domain/PNO UCCSD(T) oracle use the same transformed
Hamiltonian.  Orbital stability, ECP charge
bookkeeping, a common analytical
1D/2D Coulomb kernel, gradients, and matched external benchmark decks remain
open.

\section{Conclusion}

The sound foundation of a cyclic cluster is the finite Born--von Karman
translation group and its periodised Coulomb Hamiltonian.  Wigner--Seitz
fractions are a partition of equivalent representatives, not arbitrary
weights on free-space ERIs.  With that distinction, the Gamma-supercell and
full Gamma-centred character mesh are exactly equivalent, restricted and
unrestricted HF/KS have explicit scalar functionals, and direct, RI, and
RIJCOSX algorithms can be
compared without changing the method.

The historical 2014 four-centre factor fails a general bra--ket symmetry test
and therefore does not, as printed, prove the usual variational Fock.  This
explains how convergence and believable energies could coexist with a subtle
formal defect.  The numerical evidence is encouraging but incomplete:
compact H$_2$ backend agreement is at the sub-
\SI{0.12}{\kilo\joule\per\mole} level, old/new H$_4$ agreement improves with
cluster size, and a separate H$_4$ direct/RI discrepancy remains open.  A
matched 3D RI-MP2 control agrees with out-of-process KRHF/KMP2 below one
nanohartree per cell.  No-truncation real-torus local MP2 agrees with that
canonical finite Hamiltonian to $7\times10^{-18}\,\Ha$ per cell, and the LiH
CCSD/(T) exact-limit discrepancies are below $7\times10^{-12}\,\Ha$ total.
Dense SCF references, an external periodic CC benchmark, and the four
requested 3D solid ladders remain scientific gates.

The new post-SCF localization preserves the occupied projector by one global
finite-torus unitary rotation. PAO orthogonality, positive-semidefinite PNO
densities, translation-pair orbit partitions, and nonsymmorphic atom/cell
maps have explicit invariant tests. Exact continuum MV spreads, periodized
IAO cross overlaps, representative-only correlation amplitudes, and general-k
space-group integral scattering remain open and fail closed where exposed.

The 2008 deMon2k KS-ADFT formulation is subtler: its two-centre reference
argument and fitted-metric adjoint relation remain unproved, while its
three-centre definition explicitly averages both AO-centre orderings and is
not condemned by the four-centre counterexample.

\appendix
\section{Density idempotency in a nonorthogonal basis}

Let occupied coefficients satisfy $C_o^\dagger SC_o=I$ and define the
spin-summed density $D=2C_oC_o^\dagger$.  Then
\begin{align}
 DSD&=4C_oC_o^\dagger SC_oC_o^\dagger
 =4C_oIC_o^\dagger=2D.                                      \label{eq:idempotency_proof}
\end{align}
The implementation reports the maximum Frobenius residual
$\max_{\kvec}\|D(\kvec)S(\kvec)D(\kvec)-2D(\kvec)\|_F$.

\section{No-double-counting proof for quotient classes}

Let $f([\bm n])$ be any superlattice-periodic one- or two-electron matrix
element.  Every $\bm d\in M([\bm n])$ represents the same quotient element,
so $f(\bm d)=f([\bm n])$.  Equation~\eqref{eq:partition} gives
\begin{equation}
 \sum_{\bm d\in M([\bm n])}w(\bm d|[\bm n])f(\bm d)
 =f([\bm n])\sum_{\bm d}w(\bm d|[\bm n])=f([\bm n]).         \label{eq:no_double_count}
\end{equation}
For a four-centre term, the product of three independent representative
partitions also sums to one.  The central class $[\bm0]$ occurs once.  Thus no
separate central-cell correction is required.

\section{Notation and SI units}

\begin{longtable}{p{0.17\linewidth}p{0.55\linewidth}p{0.18\linewidth}}
\caption{Notation.}\label{tab:notation}\\
\toprule
Symbol & Meaning & SI unit\\
\midrule
\endfirsthead
\toprule
Symbol & Meaning & SI unit\\
\midrule
\endhead
$\bm a_i,\R,\bm r$ & Direct lattice vectors, translations, positions & \si{\metre}\\
$\bm b_i,\kvec,\G$ & Reciprocal vectors and wave vectors & \si{\per\metre}\\
$N_i,N_c$ & Cyclic dimensions and cell count & dimensionless\\
$\Omega,\Omega_{\mathrm{BZ}}$ & Direct and reciprocal cell volumes & \si{\metre^3}, \si{\per\metre^3}\\
$e,m_e,\hbar,\epsilon_0$ & Elementary charge, electron mass, reduced Planck constant, vacuum permittivity & \si{\coulomb}, \si{\kilogram}, \si{\joule\second}, \si{\farad\per\metre}\\
$\chi_\mu,\phi_{\mu\kvec}$ & Local Gaussian AO and Bloch sum & \si{\metre^{-3/2}}\\
$S,h,F,J,K$ & Overlap, core, Fock, Coulomb, exchange matrices & $S$: 1; others: \si{\joule}\\
$D,P$ & Spin-summed or historical one-spin density matrix & dimensionless\\
$\rho$ & Electron number density & \si{\per\metre^3}\\
$G_{\bm N}$ & Zero-average periodic electrostatic Green function & \si{\per\farad}\\
$E,\mathcal E$ & Energy or energy functional per primitive cell & \si{\joule}; tables: \si{\kilo\joule\per\mole}\\
$w_{\kvec},w(\bm d|[\bm n])$ & k weight and representative weight & dimensionless\\
$x_{ab},x_{abcd}$ & Historical occurrence-derived boundary factors & dimensionless\\
$\eta$ & Ewald splitting parameter & \si{\per\metre}\\
$\alpha$ & Exact-exchange fraction & dimensionless\\
$\widetilde L,B$ & Character-space and real-torus RI factors & $\si{\joule^{1/2}}$ in SI-normalized orbitals\\
$w_{n\R}^0,C^{\rm loc},U$ & Canonical Wannier function, localized coefficients, occupied-space unitary & \si{\metre^{-3/2}}, \si{\metre^{-3/2}}, dimensionless\\
$L_\alpha,z_{n\alpha},\Omega_{n\alpha}$ & Cyclic length, circular position moment, spread & \si{\metre}, dimensionless, \si{\metre^2}\\
$Q,E_D,V_D,G_D$ & Virtual projector, domain selector, orthonormal PAOs, PAO Gram matrix & dimensionless\\
$T_{ij},D_{ij}$ & Pair amplitudes and pair density & dimensionless\\
$g=\{W|\bm w\},p_g,\bm q_{g,a}$ & Space-group operation, atom permutation, integer image shift & dimensionless\\
$N$ & Diagonal cyclic-shape matrix & dimensionless\\
$\mathcal O(i,j),\mathcal O_{\kvec}$ & Occupied-pair and reciprocal-character orbits & dimensionless sets\\
$E_{\mathrm{corr},N}$ & Correlation energy of the complete finite torus & \si{\joule}\\
\bottomrule
\end{longtable}

\section{Assumptions and open validation items}

\begin{enumerate}
 \item Nuclei are clamped and non-relativistic; the all-electron benchmarks
 use STO-3G.
 \item The primitive cell is neutral and at zero electronic temperature.
 Restricted controls are closed shell; unrestricted controls have an
 integer multiplicity.
 \item The 3D route uses the neutral Ewald gauge.  The 1D/2D four-centre route
 uses a direct-truncated active-lattice kernel and RI uses a fitted dimensional
 metric; backend absolute energies are not compared until those gauges match.
 \item The variational space preserves primitive translation symmetry.
 Broken-symmetry supercell determinants lie outside it.
 \item No orbital-Hessian stability test was run.
 \item The H$_4$ direct-versus-RI discrepancy is unresolved.
 \item Matched external CRYSTAL solid-state runs and LiH, MgO, NaCl, and
 diamond mesh ladders are missing and must not be inferred from the internal
 identity tests.  The sole new external reference is the H$_2$ KRHF/KMP2
 control.
 \item SCF gradients, lower-dimensional post-HF, representative-only
 local correlation, general-k symmetry integral reduction, and external
 periodic CCSD(T) remain future work. Canonical RI-MP2 and full-torus
 restricted/unrestricted MP2/CCSD(T) PNO routes are implemented in 3D.
 The unrestricted CC engine is an $O(N^6)$ exact-limit oracle, not yet a
 reduced-scaling production solver.
\end{enumerate}

\bibliographystyle{unsrtnat}
\bibliography{aiccm_comparison}

\end{document}