MCBP: Multi-Domain Technical Constructs

• MCBP is an overloaded acronym used in atomic physics, geometric machine learning, graphical-model inference, and LLM accelerator design, with each domain employing distinct computational workflows.
• In atomic physics, the multi-configuration Breit–Pauli method models dielectronic recombination through configuration-interaction and perturbation theory to achieve plasma rate accuracies.
• In machine learning and inference, MCBP encompasses mean curvature boundary detection, MCMC assisted by belief propagation, and memory-compute co-design for LLM acceleration with optimized hardware.

MCBP is an overloaded acronym that denotes several unrelated technical constructs in current research literature. In the sources considered here, it refers to the multi-configuration Breit–Pauli formalism in atomic collision theory, Mean Curvature Boundary Points in geometric machine learning, MCMC assisted by Belief Propagation in graphical-model inference, and a Memory-Compute co-design exploiting Bit-slice sparsity and Repetitiveness for large-language-model inference acceleration. The shared acronym does not imply conceptual continuity across these domains; each usage has its own mathematical objects, computational workflow, and validation regime (Novotný et al., 2012, Levada, 5 May 2026, Ahn et al., 2016, Wang et al., 12 Sep 2025).

1. Disambiguation and research domains

In the cited literature, MCBP appears in four principal senses.

Expansion	Domain	Representative source
Multi-configuration Breit–Pauli	Atomic physics; dielectronic recombination	(Novotný et al., 2012)
Mean Curvature Boundary Points	Geometric machine learning; unsupervised learning	(Levada, 5 May 2026)
MCMC assisted by Belief Propagation	Probabilistic inference in graphical models	(Ahn et al., 2016)
Memory-Compute co-design exploiting Bit-slice sparsity and Repetitiveness	LLM accelerator architecture	(Wang et al., 12 Sep 2025)

The ambiguity is especially relevant because two of these senses are currently active in high-dimensional data analysis: Mean Curvature Boundary Points as an unsupervised-learning method, and a later acceleration paper that explicitly targets the curvature computation used by that method (Levada, 5 May 2026, Levada, 4 Jun 2026). By contrast, the atomic-physics sense predates them and is embedded in the AUTOSTRUCTURE/IPIRDW tradition for recombination-rate calculations (Nikolić et al., 2010, Kaur et al., 2018).

2. MCBP as multi-configuration Breit–Pauli theory

In atomic physics, MCBP is an atomic-structure and collision methodology in which the non-relativistic Hamiltonian is supplemented by Breit–Pauli operators, bound and autoionizing levels are represented by configuration-interaction expansions, and scattering and radiative transitions are computed in lowest-order perturbation theory (Kaur et al., 2018). In intermediate coupling, the Hamiltonian is written as

$$H_{\rm BP}=H_{\rm NR}+H_{\rm MV}+H_{\rm D}+H_{\rm SO}+H_{\rm SS}+H_{\rm SOO},$$

with mass-velocity, Darwin, spin–orbit, spin–spin, and spin–other-orbit contributions explicitly included (Nikolić et al., 2010). The corresponding ionic states are expanded as

$\Psi_i=\sum_k c_k^{(i)}\Phi_k,$

and the coefficients are obtained by diagonalizing the Breit–Pauli Hamiltonian in a CSF basis (Nikolić et al., 2010, Kaur et al., 2018).

The method is used extensively for dielectronic recombination (DR). In the independent-processes, isolated-resonance, distorted-wave approximation, the DR cross section is represented as a sum over isolated resonances whose widths are determined by autoionization and radiative rates. For argon-like ions, the partial DR cross section is written in Lorentzian form and Maxwellian averaging gives

$$\alpha^{\rm DR}(T)\approx \Bigl(\tfrac{8}{\pi m_e}\Bigr)^{1/2}(k_BT)^{-3/2} \int_0^\infty E\,\sigma^{\rm DR}(E)e^{-E/k_BT}\,dE,$$

with resonance sums carried explicitly to $n=1000$ and $\ell=10$, followed by a hydrogenic top-up (Nikolić et al., 2010). For the silicon isoelectronic sequence, the partial DR rate from initial $i$ to final $f$ is expressed as

$$\alpha_{if}(T)=\Bigl(\frac{4\pi a_0^2 I_H}{k_BT}\Bigr)^{3/2} \sum_d \frac{\omega_d}{2\omega_i}e^{-E_c/k_BT} \frac{A^a_{d\to i}A^r_{d\to f}} {\sum_h A^r_{d\to h}+\sum_m A^a_{d\to m}},$$

with both $\Delta n_c=0$ and $\Delta n_c=1$ core excitations included (Kaur et al., 2018).

AUTOSTRUCTURE is the principal implementation platform in the cited work. In the Fe XII study, all structure, autoionization, and radiative data generation was done with AUTOSTRUCTURE, using scaled Thomas–Fermi–Dirac–Amaldi potentials, CI+BP Hamiltonians with several thousand CSFs, and continuum functions in analytic Coulomb form (Novotný et al., 2012). The same paper compared merged-beams recombination rate coefficients measured at the TSR heavy-ion storage ring with MCBP theory and found significant differences in resonance energies and strengths at the MBRRC level, including $\Psi_i=\sum_k c_k^{(i)}\Phi_k,$0 ranging from $\Psi_i=\sum_k c_k^{(i)}\Phi_k,$1 to $\Psi_i=\sum_k c_k^{(i)}\Phi_k,$2 across different energy intervals (Novotný et al., 2012). Yet the Maxwellian-averaged plasma rate coefficient was much more robust: the MCBP PRRC agreed with experiment to within $\Psi_i=\sum_k c_k^{(i)}\Phi_k,$3 at photoionized-plasma temperatures and within the $\Psi_i=\sum_k c_k^{(i)}\Phi_k,$4 experimental uncertainty at collisionally ionized-plasma temperatures (Novotný et al., 2012).

A recurring conclusion in the atomic-physics literature is that accurate threshold positions and CI completeness are decisive at low temperature. For argon-like ions, the lowest $\Psi_i=\sum_k c_k^{(i)}\Phi_k,$5 thresholds control the strong low-$\Psi_i=\sum_k c_k^{(i)}\Phi_k,$6 DR enhancement, and the reported MCBP rates at $\Psi_i=\sum_k c_k^{(i)}\Phi_k,$7 K are larger by factors of $\Psi_i=\sum_k c_k^{(i)}\Phi_k,$8–$\Psi_i=\sum_k c_k^{(i)}\Phi_k,$9 than widely used empirical formulas (Nikolić et al., 2010). For silicon-like ions, older recommended fits miss low-$$\alpha^{\rm DR}(T)\approx \Bigl(\tfrac{8}{\pi m_e}\Bigr)^{1/2}(k_BT)^{-3/2} \int_0^\infty E\,\sigma^{\rm DR}(E)e^{-E/k_BT}\,dE,$$0 fine-structure DR, whereas MCBP with full fine-structure mixing and $$\alpha^{\rm DR}(T)\approx \Bigl(\tfrac{8}{\pi m_e}\Bigr)^{1/2}(k_BT)^{-3/2} \int_0^\infty E\,\sigma^{\rm DR}(E)e^{-E/k_BT}\,dE,$$1 coverage yields data intended for generalized collisional-radiative modelling frameworks and OPEN-ADAS archiving (Kaur et al., 2018).

3. MCBP as Mean Curvature Boundary Points

In geometric machine learning, MCBP denotes Mean Curvature Boundary Points, a boundary-detection framework for unsupervised learning that uses local mean curvature as a descriptor of boundary structure (Levada, 5 May 2026). The method starts from the differential-geometric observation that the shape operator

$$\alpha^{\rm DR}(T)\approx \Bigl(\tfrac{8}{\pi m_e}\Bigr)^{1/2}(k_BT)^{-3/2} \int_0^\infty E\,\sigma^{\rm DR}(E)e^{-E/k_BT}\,dE,$$2

encodes second-order bending, and that the mean curvature can be written as

$$\alpha^{\rm DR}(T)\approx \Bigl(\tfrac{8}{\pi m_e}\Bigr)^{1/2}(k_BT)^{-3/2} \int_0^\infty E\,\sigma^{\rm DR}(E)e^{-E/k_BT}\,dE,$$3

For a point cloud $$\alpha^{\rm DR}(T)\approx \Bigl(\tfrac{8}{\pi m_e}\Bigr)^{1/2}(k_BT)^{-3/2} \int_0^\infty E\,\sigma^{\rm DR}(E)e^{-E/k_BT}\,dE,$$4, explicit manifold parametrization is avoided by estimating a local patch $$\alpha^{\rm DR}(T)\approx \Bigl(\tfrac{8}{\pi m_e}\Bigr)^{1/2}(k_BT)^{-3/2} \int_0^\infty E\,\sigma^{\rm DR}(E)e^{-E/k_BT}\,dE,$$5 from $$\alpha^{\rm DR}(T)\approx \Bigl(\tfrac{8}{\pi m_e}\Bigr)^{1/2}(k_BT)^{-3/2} \int_0^\infty E\,\sigma^{\rm DR}(E)e^{-E/k_BT}\,dE,$$6-nearest neighbors, computing the empirical covariance

$$\alpha^{\rm DR}(T)\approx \Bigl(\tfrac{8}{\pi m_e}\Bigr)^{1/2}(k_BT)^{-3/2} \int_0^\infty E\,\sigma^{\rm DR}(E)e^{-E/k_BT}\,dE,$$7

approximating the first fundamental form by $$\alpha^{\rm DR}(T)\approx \Bigl(\tfrac{8}{\pi m_e}\Bigr)^{1/2}(k_BT)^{-3/2} \int_0^\infty E\,\sigma^{\rm DR}(E)e^{-E/k_BT}\,dE,$$8, and forming a discrete proxy $$\alpha^{\rm DR}(T)\approx \Bigl(\tfrac{8}{\pi m_e}\Bigr)^{1/2}(k_BT)^{-3/2} \int_0^\infty E\,\sigma^{\rm DR}(E)e^{-E/k_BT}\,dE,$$9 for the second fundamental form from a local quadratic fit (Levada, 5 May 2026).

The resulting shape-operator approximation is

$n=1000$0

and the raw mean-curvature score is

$n=1000$1

These scores are normalized to

$n=1000$2

after which boundary points are extracted through percentile thresholding: $n=1000$3 The authors describe this as an adaptive percentile-based thresholding scheme enabling multiscale boundary extraction without ad hoc density parameters (Levada, 5 May 2026).

The geometric interpretation is broader than classical contour detection. High-curvature regions are said to correspond to transitions between clusters, geometric irregularities, and low-density interfaces; the same curvature field therefore yields a unified interpretation of boundary points, outliers, and transition points (Levada, 5 May 2026). The method also induces a curvature-driven decomposition into a smooth set

$n=1000$4

and a boundary set

$n=1000$5

with the smooth subset functioning as a non-linear geometric filter for downstream clustering (Levada, 5 May 2026).

The reported complexity is dominated by ambient dimension. Building the $n=1000$6-nearest-neighbor graph costs $n=1000$7, while per-point covariance estimation, eigendecomposition, and local quadratic fitting lead to a total complexity of $n=1000$8; a prior PCA to $n=1000$9 reduces this to $\ell=10$0 (Levada, 5 May 2026). Empirically, synthetic experiments on Gaussian blob, two-blobs, anisotropic ellipses, and two-moons datasets showed curvature peaks at outer contours and inter-cluster interfaces, and experiments on 25 real datasets reported average clustering improvements after filtering out the top 25% curvature points: Silhouette Coefficient increased by $\ell=10$1, Calinski–Harabasz by $\ell=10$2, and Davies–Bouldin decreased by $\ell=10$3 (Levada, 5 May 2026).

4. Scalable curvature computation for MCBP

A later paper addresses the principal computational bottleneck of the Mean Curvature Boundary Points pipeline: local mean-curvature computation in high ambient dimension (Levada, 4 Jun 2026). The original estimator constructs, for each neighborhood, a covariance matrix

$\ell=10$4

computes its eigendecomposition $\ell=10$5, and then forms a feature matrix $\ell=10$6 with $\ell=10$7 columns from squared and cross products of eigenvectors (Levada, 4 Jun 2026). The discrete shape-operator estimator is

$\ell=10$8

and the mean-curvature estimate is

$\ell=10$9

Forming $i$0 costs $i$1 per point, which the paper identifies as the dominant obstacle to using MCBP on data with more than a few dozen features (Levada, 4 Jun 2026).

The first acceleration is an exact algebraic identity: $i$2 where $i$3 is the columnwise element-wise square of $i$4. This removes explicit construction of $i$5 from the trace computation and yields

$i$6

with $i$7 (Levada, 4 Jun 2026). After eigendecomposition, the trace computation drops to $i$8.

The second acceleration exploits the fact that $i$9 has rank at most $f$0. Replacing full eigendecomposition with a truncated SVD of the centered $f$1 data matrix reduces the cost to $f$2, and an analytical approximation for the null-space contribution is derived from the expected outer product of null-space eigenvectors under the Haar measure (Levada, 4 Jun 2026). The resulting total complexity is

$f$3

For fixed small $f$4, the dominant term is $f$5 (Levada, 4 Jun 2026).

The empirical consequences are substantial. On 40 OpenML datasets with $f$6 from 4 to 279, the combined exact/fast implementation achieved median Spearman $f$7, Chatterjee $f$8 after normalization, and median normalized MAE below $f$9. Median wall-clock time was $$\alpha_{if}(T)=\Bigl(\frac{4\pi a_0^2 I_H}{k_BT}\Bigr)^{3/2} \sum_d \frac{\omega_d}{2\omega_i}e^{-E_c/k_BT} \frac{A^a_{d\to i}A^r_{d\to f}} {\sum_h A^r_{d\to h}+\sum_m A^a_{d\to m}},$$0 s versus $$\alpha_{if}(T)=\Bigl(\frac{4\pi a_0^2 I_H}{k_BT}\Bigr)^{3/2} \sum_d \frac{\omega_d}{2\omega_i}e^{-E_c/k_BT} \frac{A^a_{d\to i}A^r_{d\to f}} {\sum_h A^r_{d\to h}+\sum_m A^a_{d\to m}},$$1 s for the original on low dimensions, while for $$\alpha_{if}(T)=\Bigl(\frac{4\pi a_0^2 I_H}{k_BT}\Bigr)^{3/2} \sum_d \frac{\omega_d}{2\omega_i}e^{-E_c/k_BT} \frac{A^a_{d\to i}A^r_{d\to f}} {\sum_h A^r_{d\to h}+\sum_m A^a_{d\to m}},$$2–$$\alpha_{if}(T)=\Bigl(\frac{4\pi a_0^2 I_H}{k_BT}\Bigr)^{3/2} \sum_d \frac{\omega_d}{2\omega_i}e^{-E_c/k_BT} \frac{A^a_{d\to i}A^r_{d\to f}} {\sum_h A^r_{d\to h}+\sum_m A^a_{d\to m}},$$3 the reported speedups were $$\alpha_{if}(T)=\Bigl(\frac{4\pi a_0^2 I_H}{k_BT}\Bigr)^{3/2} \sum_d \frac{\omega_d}{2\omega_i}e^{-E_c/k_BT} \frac{A^a_{d\to i}A^r_{d\to f}} {\sum_h A^r_{d\to h}+\sum_m A^a_{d\to m}},$$4–$$\alpha_{if}(T)=\Bigl(\frac{4\pi a_0^2 I_H}{k_BT}\Bigr)^{3/2} \sum_d \frac{\omega_d}{2\omega_i}e^{-E_c/k_BT} \frac{A^a_{d\to i}A^r_{d\to f}} {\sum_h A^r_{d\to h}+\sum_m A^a_{d\to m}},$$5; specific examples included USPS ($$\alpha_{if}(T)=\Bigl(\frac{4\pi a_0^2 I_H}{k_BT}\Bigr)^{3/2} \sum_d \frac{\omega_d}{2\omega_i}e^{-E_c/k_BT} \frac{A^a_{d\to i}A^r_{d\to f}} {\sum_h A^r_{d\to h}+\sum_m A^a_{d\to m}},$$6) from $$\alpha_{if}(T)=\Bigl(\frac{4\pi a_0^2 I_H}{k_BT}\Bigr)^{3/2} \sum_d \frac{\omega_d}{2\omega_i}e^{-E_c/k_BT} \frac{A^a_{d\to i}A^r_{d\to f}} {\sum_h A^r_{d\to h}+\sum_m A^a_{d\to m}},$$7 s to $$\alpha_{if}(T)=\Bigl(\frac{4\pi a_0^2 I_H}{k_BT}\Bigr)^{3/2} \sum_d \frac{\omega_d}{2\omega_i}e^{-E_c/k_BT} \frac{A^a_{d\to i}A^r_{d\to f}} {\sum_h A^r_{d\to h}+\sum_m A^a_{d\to m}},$$8 s and Arrhythmia ($$\alpha_{if}(T)=\Bigl(\frac{4\pi a_0^2 I_H}{k_BT}\Bigr)^{3/2} \sum_d \frac{\omega_d}{2\omega_i}e^{-E_c/k_BT} \frac{A^a_{d\to i}A^r_{d\to f}} {\sum_h A^r_{d\to h}+\sum_m A^a_{d\to m}},$$9) from $\Delta n_c=0$0 s to $\Delta n_c=0$1 s (Levada, 4 Jun 2026). Exact mode is algebraically identical, with errors of order $\Delta n_c=0$2, whereas the fast mode introduces $\Delta n_c=0$3 bias that is described as negligible for $\Delta n_c=0$4 and rank-order preserving (Levada, 4 Jun 2026).

5. MCBP as MCMC assisted by Belief Propagation

In graphical-model inference, MCBP stands for MCMC assisted by Belief Propagation (Ahn et al., 2016). The framework addresses the contrast between BP, which is typically fast but approximate on loopy graphs, and MCMC, which is asymptotically exact but may mix exponentially slowly. The formal starting point is loop calculus for a pairwise binary Markov random field

$\Delta n_c=0$5

together with the identity

$\Delta n_c=0$6

where $\Delta n_c=0$7 is the set of generalized loops and $\Delta n_c=0$8 is the Bethe approximation produced by BP (Ahn et al., 2016).

A central truncation keeps only 2-regular loops,

$\Delta n_c=0$9

leading to the 2-loop series

$\Delta n_c=1$0

For planar pairwise binary graphical models this truncated sum is computable in polynomial time by Pfaffian methods; the paper’s contribution is to approximate it in general graphs by MCMC (Ahn et al., 2016). The proposed sampler uses the worm algorithm on an enlarged state space containing both 2-regular loops and subgraphs with exactly two odd-degree vertices, plus a rejection scheme that turns endpoint states into samples from the target distribution over 2-regular loops (Ahn et al., 2016).

For the full loop series, the paper introduces a rejection-free chain based on a cycle basis and a fixed path set. Any generalized loop is decomposed as an XOR over elements of $\Delta n_c=1$1, and the Markov chain proposes $\Delta n_c=1$2 for a uniformly selected basis element $\Delta n_c=1$3, accepting with probability $\Delta n_c=1$4 whenever $\Delta n_c=1$5 remains a generalized loop (Ahn et al., 2016). Both truncated and full-series estimators are embedded in a simulated-annealing schedule $\Delta n_c=1$6 to estimate partition-function ratios stage by stage.

The theoretical guarantees focus on polynomial-time approximation under explicit assumptions. The worm chain is shown to mix in time

$\Delta n_c=1$7

and the paper states that relative-error estimation of the truncated or full loop series is polynomial in $\Delta n_c=1$8, $\Delta n_c=1$9, $\Psi_i=\sum_k c_k^{(i)}\Phi_k,$00, and $\Psi_i=\sum_k c_k^{(i)}\Phi_k,$01, provided $\Psi_i=\sum_k c_k^{(i)}\Phi_k,$02 and the sign imbalance remain at least inverse-polynomial (Ahn et al., 2016). Empirically, on $\Psi_i=\sum_k c_k^{(i)}\Phi_k,$03 grid Ising models with and without random external field, and on the hard-core model, the MCBP variants reduced the $\Psi_i=\sum_k c_k^{(i)}\Phi_k,$04 error in $\Psi_i=\sum_k c_k^{(i)}\Phi_k,$05 relative to BP and standard Gibbs-MCMC, often by an order of magnitude, while incurring comparable chain-run cost (Ahn et al., 2016).

6. MCBP as a memory-compute efficient LLM accelerator

In computer architecture, MCBP denotes a Memory-Compute co-design exploiting Bit-slice sparsity and Repetitiveness for decoder-only LLM inference (Wang et al., 12 Sep 2025). The architecture is organized around three bottlenecks—GEMM computation, weight loading, and KV-cache loading—and seeks to optimize all three simultaneously by operating at the bit-slice level rather than the value level (Wang et al., 12 Sep 2025). Quantized 8-bit weights are stored in off-chip HBM in a bit-slice–first layout, decoded into on-chip SRAM, processed by BRCR units for GEMM reduction, and combined with BGPP units that progressively prune KV accesses during attention (Wang et al., 12 Sep 2025).

The first mechanism, BS-Repetitiveness-Enabled Computation Reduction (BRCR), exploits repetition patterns among columns of an $\Psi_i=\sum_k c_k^{(i)}\Phi_k,$06 bit-slice block. A naive bit-plane GEMV requires

$\Psi_i=\sum_k c_k^{(i)}\Phi_k,$07

where $\Psi_i=\sum_k c_k^{(i)}\Phi_k,$08 is average bit-sparsity. BRCR rewrites a group multiply as

$\Psi_i=\sum_k c_k^{(i)}\Phi_k,$09

so that the addition count per bit-plane becomes

$\Psi_i=\sum_k c_k^{(i)}\Phi_k,$10

and over $\Psi_i=\sum_k c_k^{(i)}\Phi_k,$11 bit-planes

$\Psi_i=\sum_k c_k^{(i)}\Phi_k,$12

(Wang et al., 12 Sep 2025).

The second mechanism, BS-Sparsity-Enabled Two-State Coding (BSTC), compresses each bit-slice group columnwise into either a zero-block or a non-zero block with a one-bit state tag and, for nonzero entries, the raw $\Psi_i=\sum_k c_k^{(i)}\Phi_k,$13 bits (Wang et al., 12 Sep 2025). The paper reports that on LLaMA-7B, high-order bit-slices with $\Psi_i=\sum_k c_k^{(i)}\Phi_k,$14 exhibit $\Psi_i=\sum_k c_k^{(i)}\Phi_k,$15, yielding compression ratios of approximately $\Psi_i=\sum_k c_k^{(i)}\Phi_k,$16–$\Psi_i=\sum_k c_k^{(i)}\Phi_k,$17, and that overall weight traffic is reduced by $\Psi_i=\sum_k c_k^{(i)}\Phi_k,$18 on average in decoding (Wang et al., 12 Sep 2025). The third mechanism, Bit-Grained Progressive Prediction (BGPP), performs progressive top-$\Psi_i=\sum_k c_k^{(i)}\Phi_k,$19 attention filtering from MSB to LSB using thresholds

$\Psi_i=\sum_k c_k^{(i)}\Phi_k,$20

with empirical radius $\Psi_i=\sum_k c_k^{(i)}\Phi_k,$21. Candidates whose partial score plus the maximum remaining-bit contribution cannot exceed the threshold are pruned early; the reported effect is up to $\Psi_i=\sum_k c_k^{(i)}\Phi_k,$22 reduction in KV-cache accesses and $\Psi_i=\sum_k c_k^{(i)}\Phi_k,$23 reduction in attention computation with negligible quality loss below $\Psi_i=\sum_k c_k^{(i)}\Phi_k,$24 for $\Psi_i=\sum_k c_k^{(i)}\Phi_k,$25 around $\Psi_i=\sum_k c_k^{(i)}\Phi_k,$26–$\Psi_i=\sum_k c_k^{(i)}\Phi_k,$27 (Wang et al., 12 Sep 2025).

The accelerator instantiates these mechanisms with HBM2 channels, dedicated SRAMs, PE clusters for BRCR, lightweight BSTC codecs, and BGPP units running asynchronously with the main pipeline (Wang et al., 12 Sep 2025). On 26 benchmarks, the standard design achieved $\Psi_i=\sum_k c_k^{(i)}\Phi_k,$28 speedup over Nvidia A100 INT8 TensorRT and $\Psi_i=\sum_k c_k^{(i)}\Phi_k,$29 GOPS/W, while the aggressive design achieved $\Psi_i=\sum_k c_k^{(i)}\Phi_k,$30 speedup and $\Psi_i=\sum_k c_k^{(i)}\Phi_k,$31 GOPS/W; the abstract additionally reports $\Psi_i=\sum_k c_k^{(i)}\Phi_k,$32 higher energy efficiency than the A100 and energy savings of $\Psi_i=\sum_k c_k^{(i)}\Phi_k,$33, $\Psi_i=\sum_k c_k^{(i)}\Phi_k,$34, and $\Psi_i=\sum_k c_k^{(i)}\Phi_k,$35 relative to SpAtten, FACT, and SOFA, respectively (Wang et al., 12 Sep 2025). Area and power figures at 28 nm and 1 GHz are given as $\Psi_i=\sum_k c_k^{(i)}\Phi_k,$36 and $\Psi_i=\sum_k c_k^{(i)}\Phi_k,$37 W, excluding the note that the HBM interface accounts for about $\Psi_i=\sum_k c_k^{(i)}\Phi_k,$38 of power (Wang et al., 12 Sep 2025). The limitations explicitly identified are dependence of BGPP on $\Psi_i=\sum_k c_k^{(i)}\Phi_k,$39, the fixed group size $\Psi_i=\sum_k c_k^{(i)}\Phi_k,$40, BSTC’s one-bit overhead for each non-zero block, and the need to extend the co-design to mixed precision, INT4/FP8, and broader system-level co-optimization (Wang et al., 12 Sep 2025).