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arxiv: 2606.18802 · v2 · pith:2EZVPUIDnew · submitted 2026-06-17 · ✦ hep-th · gr-qc· hep-lat· math-ph· math.MP

Mutation and crossover of simplicial complexes

Boyu Li , Kohta Hatakeyama , Matsuo Sato , Yuji Sugimoto , Gota Tanaka This is my paper

Pith reviewed 2026-06-26 20:09 UTC · model grok-4.3

classification ✦ hep-th gr-qchep-latmath-phmath.MP
keywords simplicial complexespseudomanifoldscolored graphsmutation and crossovergenetic algorithmsmatrix representationsbubble graphstopological diversity
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The pith

Mutation and crossover operations defined on colored graphs extend to simplicial complexes of pseudomanifolds through matrix representations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that colored graphs correspond bijectively to simplicial complexes of pseudomanifolds, with bubble graphs matching subsimplices. Matrix representations are introduced for both, enabling the transfer of mutation and crossover from graphs to these complexes and matrices. A genetic algorithm is then applied to generate matrices that produce pseudomanifolds with varied topologies. Additional procedures decompose the matrices, reconstruct the complexes, and calculate geometric features such as simplex volumes. This framework turns evolutionary operations into a tool for exploring pseudomanifold structures systematically.

Core claim

Through the established correspondence among simplicial complexes, colored graphs, and simplicial-complex matrices, mutation and crossover operations defined on colored graphs extend to simplicial complexes and simplicial-complex matrices, and a genetic algorithm performing these operations on the matrices produces pseudomanifolds exhibiting diverse topologies.

What carries the argument

The bijective correspondence between colored graphs (with bubble graphs) and simplicial complexes of pseudomanifolds (with subsimplices), together with their simplicial-complex matrix representations, which carries the definition of mutation and crossover.

If this is right

  • The genetic algorithm systematically generates pseudomanifolds with a wider range of topologies than manual construction.
  • Decomposition of generated matrices into simplex matrices allows reconstruction of the full simplicial complex.
  • Geometric quantities including volume, circumcenter, and dual-simplex volume become computable directly from the matrix entries.
  • Subsimplex matrices derived from bubble graphs support local modifications while preserving the overall correspondence.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same matrix operations could be applied to search for pseudomanifolds that minimize or maximize specific geometric invariants.
  • If the generated topologies include known manifolds, the method supplies an automated check on whether evolutionary search recovers standard examples.
  • Extending the matrices to include additional labels might allow incorporation of matter fields or other structures common in quantum-gravity models.

Load-bearing premise

The bijective correspondence between colored graphs and simplicial complexes of pseudomanifolds remains valid after mutation and crossover, so that the resulting matrices always encode legitimate pseudomanifolds.

What would settle it

An explicit matrix obtained after one mutation step whose reconstructed graph fails to be a valid colored graph or whose associated complex fails to be a pseudomanifold.

Figures

Figures reproduced from arXiv: 2606.18802 by Boyu Li, Gota Tanaka, Kohta Hatakeyama, Matsuo Sato, Yuji Sugimoto.

Figure 1
Figure 1. Figure 1: Schematic depiction of positive (black) and negative (white) vertices in a ( [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: illustrates, for the case D = 3, a (D + 1)−colored graph G together with one of its 3-bubbles, B ˆ1 vv¯ . G = v v¯ , 0 1 2 3 B ˆ1 vv¯ = v v¯ 0 2 3 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: A 4-colored graph G consisting of eight vertices v1, . . . , v8 and edges colored 0, 1, 2, 3. 2.3 Concrete example of analyzing a colored graph and its dual pseu￾domanifold Given a (D + 1)-colored graph G, we can construct the pseudomanifold dual to G by determining the simplices corresponding to all bubbles of G. As a concrete example of this procedure, in this subsection we analyze which pseudomanifold i… view at source ↗
Figure 4
Figure 4. Figure 4: Two examples of B ˆ0 v1v2...vn obtained from G. Removing all edges of a given color may cause the graph to decompose into multiple connected components. In this way, in what follows, we attach to each p-simplex the information of the consecutive index numbers of the (p − 1)-simplices that constitute it. Here we explain how a 2-bubble corresponds to a 1-simplex, using B ˆ0ˆ1 1456 = {A1, A3}1(1, 3) as a repr… view at source ↗
Figure 5
Figure 5. Figure 5: A manifold obtained by gluing two octahedra along their boundaries. The colored faces [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Removing all color-0 edges from G yields the direct sum of two connected subgraphs, which correspond to the 3-bubbles B ˆ0 1456 and B ˆ0 2378. As another example, consider the matrix obtained by removing all color-1 edges: M ˆ1 =   2 1 4 3 8 7 6 5 0 0 0 0 0 0 0 0 6 7 8 5 4 1 2 3 4 3 2 1 6 5 8 7   . (3.4) In this case, the only tuple of columns representing a connected graph is again the trivial one… view at source ↗
Figure 7
Figure 7. Figure 7: An example of mutation of a colored graph. [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: An example of crossover of colored graphs. [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Mutation of simplicial complex matrix corresponding to Figure [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Mutation of a simplicial complex matrix corresponding to Figure [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Dual objects in two dimensions shown in one figure: (a) the dual of a vertex with five [PITH_FULL_IMAGE:figures/full_fig_p018_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: the coordinates. to coincide with the vertex A0. Each axis of the coordinate system is chosen so that the coordinates of the vertex A1 become (l01, 0, 0, . . . , 0), and furthermore, for each vertex Ai , its i-th coordinate component is positive and all components from the (i+1)-th onward are zero. Under this definition, the (i + 1)-simplex consisting of the vertices A0, A1, A2, A3, . . . , Ai lies in the… view at source ↗
Figure 13
Figure 13. Figure 13: Structure of the simplicial complex In D dimensions, the output file out_simplex_D.dat generated by generate_simplex_v7.f90 is empty. This is because the corresponding matrix becomes the zero matrix once all D+1 rows are removed. In contrast, the output file out_identify_Ddim.dat produced by identify_simplex_ v9.f90 contains information about the D-simplices. A D-simplex corresponds to a single vertex, an… view at source ↗
read the original abstract

Color graphs and their subgraphs, referred to as bubble graphs, correspond bijectively to the simplicial complexes of pseudomanifolds and their subsimplices, respectively. In this paper, we introduce matrix representations for colored graphs and their associated bubble graphs. By using this correspondence, we define simplicial-complex matrices and subsimplex matrices that encode the simplicial complexes of pseudomanifolds and their subsimplices. Moreover, we formulate mutation and crossover operations on colored graphs. Through the established correspondence among simplicial complexes, colored graphs, and simplicial-complex matrices, we extend these operations to simplicial complexes and simplicial-complex matrices. We further implement an algorithm generating simplicial-complex matrices and a genetic algorithm performing mutation and crossover of them to produce pseudomanifolds exhibiting diverse topologies. In addition, we implement procedures for decomposing the generated simplicial-complex matrices into simplex matrices, reconstructing the simplicial complexes of the associated pseudomanifolds from this information, and computing geometric quantities such as the volume, circumcenter, and dual-simplex volume of each simplex.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper establishes matrix representations for colored graphs and bubble graphs, which bijectively correspond to simplicial complexes of pseudomanifolds and their subsimplices. It formulates mutation and crossover operations on colored graphs, extends them to simplicial-complex matrices via the correspondence, implements a generator for such matrices and a genetic algorithm applying these operations to produce pseudomanifolds of diverse topologies, and provides procedures to decompose matrices into simplex matrices, reconstruct the complexes, and compute geometric quantities including volume, circumcenter, and dual-simplex volume.

Significance. If the operations are shown to preserve validity, the work supplies a concrete computational framework for generating and mutating pseudomanifolds in tensor-model or group-field-theory contexts, together with explicit reconstruction and geometric-measurement code; the implementation of the generator and GA is a tangible strength that could be reused for exploring topological ensembles.

major comments (2)
  1. [section on extension of operations to matrices] The description of the mutation and crossover operations on simplicial-complex matrices (in the section extending the graph operations) does not include an explicit argument or check that the resulting matrices remain in the image of the bijection, i.e., that they continue to encode properly edge-colored graphs whose gluings produce pseudomanifolds. This is load-bearing for the central claim that the GA produces valid pseudomanifolds exhibiting diverse topologies.
  2. [section describing the genetic algorithm] The account of the genetic algorithm implementation supplies no validation step, error check, or sample output demonstrating that post-mutation/crossover matrices decode to objects satisfying the bubble-graph subsimplex property or the pseudomanifold condition; without such verification the assertion that the generated structures are pseudomanifolds remains unsupported.
minor comments (2)
  1. [section introducing matrix representations] Notation for the simplicial-complex matrix entries and the precise mapping from matrix entries to colored-graph edges could be clarified with an explicit small example.
  2. The abstract states that geometric quantities are computed but does not indicate whether these computations are performed on the reconstructed complex or directly from the matrix; a sentence clarifying the data flow would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed review and valuable feedback on our manuscript. We address each major comment below and plan to incorporate revisions to address the concerns regarding the preservation of validity under the operations and the validation of the generated pseudomanifolds.

read point-by-point responses

  1. Referee: The description of the mutation and crossover operations on simplicial-complex matrices (in the section extending the graph operations) does not include an explicit argument or check that the resulting matrices remain in the image of the bijection, i.e., that they continue to encode properly edge-colored graphs whose gluings produce pseudomanifolds. This is load-bearing for the central claim that the GA produces valid pseudomanifolds exhibiting diverse topologies.

    Authors: We acknowledge that the current manuscript does not provide an explicit verification that the extended mutation and crossover operations on the matrices preserve the bijection to valid colored graphs and pseudomanifolds. Although the operations are formulated by transporting the graph-level operations through the established correspondence, a direct check ensuring the resulting matrices correspond to properly edge-colored graphs is absent. In the revised version, we will add an explicit argument or lemma demonstrating that these operations maintain the necessary properties for the gluings to yield pseudomanifolds. revision: yes

  2. Referee: The account of the genetic algorithm implementation supplies no validation step, error check, or sample output demonstrating that post-mutation/crossover matrices decode to objects satisfying the bubble-graph subsimplex property or the pseudomanifold condition; without such verification the assertion that the generated structures are pseudomanifolds remains unsupported.

    Authors: We agree that the manuscript lacks a description of any validation procedure or sample outputs confirming that the matrices after mutation and crossover satisfy the required conditions. To address this, we will include in the revised manuscript a validation step, along with example outputs and checks showing that the decoded structures meet the bubble-graph and pseudomanifold criteria. revision: yes

Circularity Check

0 steps flagged

No circularity; operations defined via external correspondence

full rationale

The paper's chain begins from an established bijective correspondence between colored graphs and simplicial complexes of pseudomanifolds, then defines matrix representations and extends mutation/crossover operations through that correspondence. No step reduces a claimed result to a fitted parameter, self-definition, or load-bearing self-citation whose content is itself unverified within the paper. The generation algorithm and genetic algorithm are presented as implementations that produce outputs asserted to be pseudomanifolds, without any equation or definition that is tautological by construction. The derivation therefore remains self-contained against the stated external correspondence.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claim rests on the bijective correspondence between colored graphs and simplicial complexes as the foundation for all subsequent matrix definitions and operations; the genetic algorithm introduces control parameters whose specific values are not stated in the abstract.

free parameters (2)
  • mutation probability
    Control parameter required by any genetic algorithm performing mutation, though its value is not given in the abstract.
  • crossover probability
    Control parameter required by any genetic algorithm performing crossover, though its value is not given in the abstract.
axioms (1)
  • domain assumption Color graphs correspond bijectively to simplicial complexes of pseudomanifolds and bubble graphs correspond bijectively to their subsimplices.
    This correspondence is invoked at the outset to justify the matrix representations and the extension of mutation and crossover operations.

pith-pipeline@v0.9.1-grok · 5729 in / 1439 out tokens · 39053 ms · 2026-06-26T20:09:31.585390+00:00 · methodology

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