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Beyond Homology: Non-Abelian and Orientability Invariants Sensitivity in Cellular Networks

Raghuwansh Raj, George Panagopoulos, Jun Pang

arXiv:2610.31234v1cs.CG

TL;DR

Topological learning benchmarks can show pair separation without establishing that models recover the named invariant, because finite representations retain simpler correlates. The paper tests shortcut channels with controlled benchmark pairs and restricted probes, finding orientability confounds and separation beyond selected controls for a partially controlled fundamental-group family. It concludes that expressivity claims must be tied to representation, controlled channels, and benchmark scope.

  • Problem

    A model can distinguish complexes using simpler representation features rather than the intended invariant, making pair separation insufficient evidence of invariant detection.

    Source: §1 · p. 1

  • Method

    The paper combines benchmark channel controls, generator checks, restricted probes, permutation reliability tests, and explicit limits on claims.

    Source: §1 · p. 2

  • Results

    Orientability pairs are separable through unintended distance or boundary information, while tested architectures beyond 1-WL and 1-CWL separate a partially controlled fundamental-group family.

    Source: §1 · p. 2, §4 · p. 7

  • Takeaways & Limitations

    Topological expressivity claims should be stated relative to model representation, controlled information channels, and benchmark scope.

    Source: §5 · p. 9

    Fundamental-group-family separation establishes sensitivity beyond matched homological, local, and Hasse-1-WL controls, but not direct or causal sensitivity to the target invariant.

    Source: §3.6 · p. 6

Abstract

from arXiv · show

Topological deep learning (TDL) architectures are typically evaluated on synthetic complex pairs that differ in a target invariant. Yet, whether a model recovers that invariant or an artifact of the chosen discretization is seldom audited. To examine this, we design and test pairs of complexes that look identical to standard homological and local message-passing tests, have the same Hasse graphs, yet differ in the target invariant. We study two beyond-homology regimes: nonAbelian fundamental-group structure and homotopy-matched orientability. For the former, we construct genus-2 pairs that match homology, cell statistics, local incidence profiles, and rank-colored Hasse 1-WL behavior while differing in their fundamental groups. For the latter, we evaluate punctured torus-Klein bottle pairs and show that distance, boundary, cycle, and spectral signals can confound apparent orientability detection, motivating stronger matched controls. Under the tested settings, subcomplex-lifted architectures separate some pairs beyond the matched homological, local-incidence, and Hasse 1-WL controls, while a depth sweep reveals an empirical depth-dependent trend without establishing a general barrier or direct computation of the target invariants. These results show that claims of topological expressivity must be stated relative to the model representation, explicitly controlled information channels, and the precise scope of the benchmark.

1 Introduction

The introduction asks whether cellular networks recover intended topological invariants or exploit simpler features of their finite representations. It presents controls for testing that distinction and reports audited orientability benchmarks alongside a partially controlled fundamental-group family.

1 Introduction

  • Topological deep learning represents spaces with higher-dimensional cells and their incidences, enabling message passing beyond ordinary graph connectivity.Examples include MPSNs, CINs, and the later MCN and scalable SMCN architectures.Source: §1 · p. 1
  • Across standard orientability pairs, a shortest-path-only DIST_MLP separates many pairs, so benchmark success alone does not establish orientability detection.The diagnostic uses node-to-cell shortest-path histograms without message passing; the observed separation may reflect metric shortcuts from the gluing pattern.Source: §1 · p. 2
  • Matching selected statistics does not certify an invariant: benchmark pairs can retain unintended shortcuts, while restricted probes test whether a predefined channel alone supports separation.A failed probe rules out only its tested features, model class, and training budget.Source: §1 · p. 2
  • The proposed representation-relative framework combines eight shortcut channels, exact generator checks, empirical probes, permutation reliability controls, and explicit limits on claims.It distinguishes topological difference between spaces from evidence that a model detects that difference.Source: §1 · p. 2
  • Genus-2 surface-group versus free-group complexes differ in fundamental groups but match homological, combinatorial, and local structural properties; tested architectures beyond 1-WL and 1-CWL baselines separate them.The result is certificate-qualified evidence of greater structural expressivity on this partially controlled family.Source: §1 · p. 2

2 Related Work

Related work connects higher-order message passing and global structural encodings to the information available in topological learning models. It also motivates controlled benchmark evaluation by linking shortcut learning, synthetic-data design, and pairwise reliability testing.

2 Related Work

  • After suitable lifts, MPSNs and CINs can distinguish structures that ordinary 1-WL cannot, while CIN++ adds interactions between higher-dimensional cells.Other approaches use Hodge Laplacians, orientation-equivariant layers, attention, higher-order signal processing, or cellular sheaves.Source: §2 · p. 2
  • Unsigned-incidence models do not receive orientability directly, while signed boundary operators or orientation data preserve different information.This distinction motivates controlled pairs because successful separation may depend on another property retained by the representation.Source: §2 · p. 3
  • Global positional encodings and incidental graph structure can provide predictive shortcuts, motivating channel controls and reliable pairwise evaluation.The protocol builds on BREC’s distinction between target separation and permutation reliability, then tests restricted probes before interpreting more expressive models.Source: §2 · p. 3

3 Background and Preliminaries

The section establishes the notation and topological objects used in the study, then defines an audit framework for distinguishing target-invariant differences from shortcuts in model inputs. It applies that framework to orientability and fundamental-group benchmarks and describes the information ladder used to test them.

3.1 Notation

  • The notation defines a finite complex as a disjoint union of cells by dimension and distinguishes the complex from its topological realization and model tensor representation.It also names the face relation and symbols for homology, groups, and surface-construction parameters.Source: §3.1 · p. 3

3.2 Topological Invariants

  • The section treats Euler characteristic, integral homology, orientability, and the fundamental group as topological invariants, while noting that Betti numbers omit torsion.Euler characteristic depends on cell counts; exact integral homology is used when matching benchmark pairs.Source: §3.2 · pp. 3–4

3.3 Presentation Complexes

  • Presentation complexes encode a group with generators and relations, and the benchmark contrasts the genus-2 surface relation with a word that freely reduces to identity.Both complexes have homology H∗ = (Z, Z4, Z), although their fundamental groups differ.Source: §3.3 · p. 4

3.4 Channel Taxonomy

  • The channel taxonomy classifies audited information as matched, leaking, partial, or open, and treats probe outcomes as evidence only about their tested feature maps and settings.Channels range from counts and homology to boundary, incidence, distances, spectra, cycles, and Hasse 1-WL refinements.Source: §3.4 · p. 4

3.5 Benchmark Family I Orientability

  • The orientability pairs match different controls, but the punctured torus–Klein bottle pair remains confounded by shortest-path and cycle signals.The cylinder–Möbius pair leaks boundary and metric information; the torus–Klein bottle pair leaks homology and metric information.Source: §3.5 · pp. 5–6

3.6 Benchmark Family II: Fundamental Group

  • The genus-2 surface-group and free-group complexes have different fundamental groups but the same homology, with several local and Hasse 1-WL controls matched.Remaining controls are incomplete: spectra differ across all 16 pairs, while per-vertex shortest-path marking profiles match on only 6/16.Source: §3.6 · p. 6

3.7 The Baseline Ladder

  • The baseline ladder ranges from restricted count, boundary, and distance probes to models receiving richer graph, incidence, cellular, and subcomplex information.A successful probe shows that its tested channel can separate the pair; failure rules out only that tested mechanism.Source: §3.7 · p. 6, Table 3 · p. 7

4 Results

Orientability separation is often achievable with restricted boundary or distance information, so model separation alone does not establish orientability detection. On the partially controlled fundamental-group benchmark, SMCN separates some pairs beyond matched controls, while the results do not show direct computation of π1.

Orientability

  • DIST_MLP solves 64/64 cylinder vs. Möbius, 60/64 torus vs. Klein, and 45/50 punctured pairs, so separation cannot alone establish orientability detection.Restricted boundary probes also succeed on the cylinder vs. Möbius pairs.Source: §4 · p. 7, Table 4 · p. 8
  • Binary-marked SMCN’s mean solved pairs rise from 0.0 at L=1 to 25.3 at L=5, aligned with the audited cycle scale but not proving causation or a general depth barrier.The first closed-walk difference occurs at L=h+1.Source: §4 · p. 7

Fundamental Group

  • SMCN separates 48/64 pair–seed evaluations with binary marking and 51/64 with shortest-path marking, versus 37/64 for ESAN.These are repeated evaluations of 16 fixed pairs across four seeds, not 64 independent pairs; binary marking does not exclude other realization-level signals.Source: §4 · p. 8, Table 4 · p. 8
  • The fundamental-group separations exceed matched homological, local, boundary, word-statistic, and rank-colored Hasse 1-WL controls, but do not establish direct π1 sensitivity.Models receive unsigned incidence, adjacency, bridge relations, and selected markings after realization; other realization-level signals may remain.Source: §4 · pp. 8–9

5 Conclusion and Future Work

The paper proposes a representation-relative framework for auditing whether benchmark separation reflects a target invariant or simpler information in the representation. Its conclusions are bounded by the declared channels, tested probes, emitted representation, and generated families.

5 Conclusion and Future Work

  • The framework audits shortcut channels with generator-side gates and restricted probes, and reports pair separation alongside permutation reliability and explicit claim limits.The paper reports that richer subgraph and subcomplex models separate a substantial fraction of the controlled fundamental-group pairs, while restricted probes and local-message-passing baselines fail.Source: §5 · p. 9
  • The experiments do not establish that SMCN computes π1 or generalizes to arbitrary presentation complexes or unseen topological families.Future work proposed includes additional metric, spectral, and cycle controls and broader presentation-complex benchmarks.Source: §5 · p. 9

A.4 From a Complex to the Model Input

  • The model receives a finite encoding rather than the realization or complete attaching-map data, and its incidence arrays omit signed boundary coefficients and ordered attaching words.Separation therefore establishes sensitivity to the chosen encoding for that pair, not computation of an invariant over broader realizations.Source: §A.4 · p. 15

B.1 Representation-Relative Certification

  • Certification is relative to the emitted representation and declared channels; matching a tested statistic does not establish that every possible shortcut has been removed.A channel may instead be reported as leaking, partial, or open when it is not fully matched.Source: §B.1 · p. 16, §B.3 · p. 16

B.3 Audited Shortcut Channels

  • Restricted probes receive one predefined channel, and probe failure rules out only the tested feature map, model, and training budget.A successful probe supplies a sufficient alternative explanation for a larger model's separation when that model can access the same channel.Source: §B.3 · p. 16

B.5 Pair Separation and Reliability

  • The pair-separation protocol requires target-complex separation while independently permuted copies of the same complex remain within the reliability threshold.The protocol measures pair separability, not classification accuracy or generalization.Source: §B.5 · p. 18

C.3 Architectures Beyond Ordinary 1-WL

Subgraph and subcomplex lifts preserve relational structure beyond ordinary 1-WL message passing, with SMCN separating some pairs that tested local baselines do not. The section also describes orientability pairs built by changing seam identifications, including a punctured pair matched in homotopy type.

C.3 Architectures Beyond Ordinary 1-WL

  • ESAN aggregates representations of selected graph substructures, whereas SMCN builds face-indexed subcomplex copies and exchanges information across copies associated with the same cell.ESAN uses ego-net subgraphs and DSS aggregation in these experiments.Source: §C.3 · p. 19
  • Binary SMCN marking supplies local incidence, while shortest-path marking adds a global positional encoding, so the two settings expose different information channels.The settings are evaluated separately.Source: §C.3 · p. 20
  • SMCN separates pairs for which GIN, Hasse GIN, and HOMP fail under the tested representation and training conditions, without showing that it computes the target invariant.Its face-indexed subcomplex copies preserve the identity of the marked subcomplex during message passing.Source: §C.3 · pp. 19–20

C.4 Interpretation of the Baseline Ladder

  • A successful restricted probe supplies an alternative explanation when a larger model can access that channel, but failure of a probe or local baseline does not establish invariant computation.Optimization, depth, feature design, and untested representation channels remain possible explanations.Source: §C.4 · p. 20

D.2 Members with Boundary

  • The cylinder and Möbius band share homotopy type, homology, cell counts, and local incidence profiles, but differ in boundary structure: two loops of length p versus one of length 2p.That boundary difference explains why the boundary probe separates the family.Source: §D.2 · p. 21

D.3 Closed Members

  • The torus and Klein bottle are closed but have different integral homology, so their pair is not homology matched.The punctured pair instead shares homology and homotopy type S1 ∨ S1 despite differing in orientability.Source: §D.3 · p. 21, §D.4 · p. 22

D.5 Metric and Cycle Effects

Orientability-pair separation can reflect metric or cycle differences retained by the discretization rather than orientability alone. A revised punctured-family construction matches the tested distance histogram, while explicitly avoiding a claim of full metric or cycle equivalence.

D.5 Metric and Cycle Effects

  • Shortest-path distances depend on the discretization rather than the underlying surface as a topological invariant, yet a distance-only probe separates most orientability pairs.The distances remain recoverable from the emitted incidence and adjacency relations.Source: §D.5 · p. 22

D.6 Interpretation of the Three Pairs

  • The cylinder–Möbius pair has differing boundary structure, while the torus–Klein pair differs in homology and is not metric controlled.These differences provide alternative explanations for orientability-pair separation.Source: §D.6 · p. 22
  • The punctured torus and punctured Klein bottle match homology, homotopy type, counts, local incidence, and boundary statistics, yet distance histograms separate most pairs and cycle control remains partial.The cycle signal varies with depth.Source: §D.6 · pp. 22–23

E A Distance-Histogram-Matched Orientability Family

  • 45 of the 50 original punctured-torus versus punctured-Klein-bottle pairs are separated by a distance-histogram probe despite matching homotopy type and several standard statistics.The revised construction is called distance-histogram matched, not isometric, because it nullifies that probe without asserting equality of complete metric or cycle structure.Source: E A Distance-Histogram-Matched… · p. 23

E.1 Cell-Complex Construction

  • The revised family uses even h ≥ 6 and p = 4, removes a face away from the seam, and matches homology, vertex-degree profiles, and a one-loop boundary signature.Even heights eliminate the parity-dependent distance discrepancy observed for odd heights.Source: §E.1 · p. 23

E.2 The Controlled Distance Statistic

The controlled punctured-surface construction matches aggregate vertex-to-face distance histograms and selected structural statistics, testing whether separation relies on those shortcuts. The audit also identifies remaining spectral and other information channels, so the controls do not establish orientability detection.

E.2 The Controlled Distance Statistic

  • For p = 4 and even h, punctured torus and Klein bottle pairs have identical aggregate vertex-to-face distance histograms, which discard orientation information.For (h, p) = (6, 4), both histograms are (92, 184, 184, 92, 0, ...).Source: §E.2 · p. 24
  • Equal distance-histogram feature vectors make the distance-MLP probe null at the input level for every generated target pair.Any deterministic clipping or tail aggregation preserves equality of the untruncated histograms.Source: §E.2 · p. 24

E.3 Finite Benchmark and Generation-Time Certification

  • The revised benchmark contains 50 pairs with p = 4 and even h from 6 through 104, and its generator checks the specified distance and structural equalities.The checks include cell counts, incidence profiles, Betti numbers, cycle periods, and stabilized vertex and rank-colored Hasse 1-WL colors.Source: §E.3 · p. 24, §E.4 · p. 25

E.4 Original versus Distance-Histogram-Matched Construction

  • In the original construction, DIST_MLP separated 45/50 pairs because their aggregate node-to-face distance histograms differed.The seam-displacement profiles differed across the original range of h and p.Source: §E.4 · p. 25

E.5 Scope of the Control

  • The revised pairs remain spectrally distinguishable: all 50 have different graph-Laplacian spectra despite matching the specified distance and local-refinement channels.The control does not guarantee isometry or equality of every metric, cycle, spectral, or higher-order statistic.Source: §E.4 · p. 25, §E.5 · p. 26

F.1 Construction and Cycle Audit

  • The first measured closed-walk difference occurs at L⋆ = h + 1 on all 50 audited grid points, with a four-grid simple-cycle cross-check.This finite audit is not a theorem for arbitrary discretizations or architectures.Source: §F.1 · p. 26

F.2 Depth Sweep

The depth sweep evaluates binary-marked SMCN on a fixed punctured-surface grid, while a cycle audit supplies a diagnostic comparison scale. Separation increases with depth, but the observed pattern is one-sided and does not establish a general depth barrier.

F.2 Depth Sweep

  • Cross-depth comparisons warrant a one-pair tolerance because separate evaluator implementations scored the L = 5 control at 23/50 and 24/50.The discrepancy was consistent with floating-point reduction order in the separation statistic.Source: §F.2 · p. 26
  • The audit found the first differing closed-walk length at h + 1 for every tested p, while the simple-cycle cross-check covers only four small grids.The neighborhood-radius diagnostic is not asserted to be a sufficient model depth.Source: Table 9 · p. 27, Table 10 · p. 27
  • Across 450 pair–seed evaluations with h > 2L + 1, no pair was separated, but the converse does not hold.The largest separated height increasingly falls short of 2L + 1 at greater depths.Source: §F.2 · p. 27
  • Mean solved pairs rise from 0.0 to 32.3 as depth increases from L = 1 to L = 8, while the largest separated height rises from none to 12.The increase is monotonic across the swept depths, although the threshold crossing moves to larger heights only gradually.Source: §F.2 · p. 27

F.3 Interpretation and Limitations

The depth sweep supports an empirical, representation- and setting-specific separation trend, not a general barrier or proof that SMCN recognizes orientability. The accompanying genus-2 construction contrasts nonisomorphic fundamental groups while controlling homology and selected structural statistics.

F.3 Interpretation and Limitations

  • Across L = 1 to L = 8, mean solved count rises from 0.0/50 to 32.3/50, with separation confined below the audited cycle scale in the applicable evaluations.These observations apply to this model, representation, training budget, and finite grid.Source: §F.3 · p. 27
  • The sweep establishes neither a depth barrier nor cycles as the causal feature; depth changes receptive field and capacity together, without a width-matched control.Other realization-level signals may remain even when selected closed-walk counts agree.Source: §F.3 · p. 28
  • The authors characterize the finding as an empirical depth-dependent trend aligned with an audited cycle scale, not proof of a global cycle barrier or orientability recognition.A width-matched shallow control is identified as the most valuable addition.Source: §F.3 · p. 28

G Fundamental Group Benchmark Construction

  • The genus-2 pair contrasts the surface group ⟨a, b, c, d | [a, b][c, d]⟩ with the free group F4, which are nonisomorphic.The cited group-homology distinction is H2(GS; Z) ≅ Z versus H2(F4; Z) = 0.Source: §G.4 · p. 30

G.5 Why the Cellular Homology Is Identical

  • Both presentation complexes have H0 ≅ Z and H1 ≅ Z4, while their H2 classes arise respectively from a surface fundamental class and an S2 summand.Cellular homology records that both classes exist but not their different homotopical origins.Source: §G.5 · p. 30

G.6 Controlled Word Generation

  • Freely cancelling subword insertions vary attaching-word length and local combinatorics without changing the represented group, enabling matched word and cell statistics.The rejection sampler matches word length, generator frequencies, immediate cancellations, local backtracking motifs, and resulting cell counts.Source: §G.6 · p. 31

G.7 Generator Gates

  • The accepted genus-2 pairs match integral homology, cell counts, local incidence, vertex-link profiles, boundary structure, word statistics, and stabilized rank-colored Hasse 1-WL colors.The Hasse refinement stabilizes at round 6 for the evaluated family.Source: §G.7 · p. 31
  • Restricted count, boundary, and distance probes each obtain 0/64, but the distance result excludes only the tested shortest-path histogram as a sufficient feature.It does not prove that every possible metric statistic is absent.Source: §G.7 · p. 31

G.8 Representation-Level Limitation

The benchmark’s unsigned machine representation does not expose the ordered signed attaching word, so positive separation does not show that a model reconstructed that word. The paper also tests a held-out relator family with matched construction controls and a separate group-distinction certificate.

G.8 Representation-Level Limitation

  • The tested models receive unsigned incidence and adjacency relations, not the ordered signed sequence used to define the attaching loop.Thus a positive result does not prove reconstruction of the original word; another global consequence of the realization may be detected.Source: §G.8 · p. 31

G.9 Dataset Size and Scope

  • Only the genus-2 family is certified and evaluated in the study; no claim extends to higher genus, unseen presentations, or arbitrary groups.The final dataset contains 16 certified genus-2 pairs.Source: §G.9 · p. 32

H A Held-Out Relator Family

  • A second relator family tests whether the reported separation is specific to the original construction, using a relator not used to design or tune the generator.Source: H A Held-Out Relator Family · p. 32

H.1 Construction

  • The held-out pair uses the same subdivided wedge of four circles and differs in its attaching word; both relators abelianize to zero, making homology matching automatic.The generator nevertheless recomputes homology exactly for every emitted pair.Source: §H.1 · p. 32

H.2 Certifying that the Groups Differ

  • The held-out groups are certified nonisomorphic by homomorphism counts into S4 alone: 172,800 versus 331,776.They agree on the four smaller test groups, so equal counts on those groups are treated only as not separated by that panel.Source: §H.2 · p. 33, Table 17 · p. 34

H.3 Length and Occurrence Budgeting

  • Shared per-generator noise budgets equalize word length and occurrence counts, while rejection sampling removes residual backtrack differences without risking those matches.The eight-pair audited subset occupies cell-count regimes disjoint from the original family and stabilizes Hasse 1-WL at round 6.Source: §H.3 · p. 34, Table 18 · p. 34

H.4 Results

On the held-out family, binary-marked SMCN nearly matches its original-family result, while every tested 1-WL-bounded model separates none of the pairs. The broader family supports generality but leaves channel leakage and higher-genus scope unresolved.

H.4 Results

  • SMCN with binary marking scores 47/64 (73.4%) on the held-out family, close to 48/64 on the original family, while every 1-WL-bounded model separates none of its pairs.The table notes that seed counts differ by row, so these are not equal-budget comparisons.Source: Table 19 · p. 35

H.5 Scope

  • The separation reproduces at nearly the same rate with a different relator and word-length regime, while every 1-WL-bounded architecture separates none of the evaluated pairs.The new group is certified non-isomorphic to both original classes.Source: §H.5 · p. 35
  • The held-out family retains spectral and fine-grained metric leakage, and the reported four-generator constructions do not establish scope for higher-genus cases.The spectral channel matches on 0/8 pairs and C5.7 on 3/8; SPEC_MLP scores 25/32.Source: §H.5 · p. 35

I Experimental and Sanity Check Details

  • Inputs begin with constant features, with rank-wise embeddings for cellular models and an additional subcomplex marking for SMCN.The reliability arm compares independently permuted copies of the same complex; separating them marks the target pair as unsolved.Source: I Experimental and Sanity Check Details · p. 35

I.1 Torus versus Sphere Sanity Experiment

The torus-versus-sphere experiment is an intentionally easy supervised task used to verify the data, training, and evaluation pipeline. Because permutations of the same base complexes can cross splits, it does not estimate generalization to unseen cellulations.

I.1 Torus versus Sphere Sanity Experiment

  • The torus and sphere classes differ in Euler characteristic, homology, and fundamental group, while both are closed, connected, orientable, and boundaryless.The torus grid has cell counts (hp, 2hp, hp); sphere subdivisions change cell counts while preserving S2 topology.Source: §I.1 · pp. 35–36
  • The experiment uses torus grids and sphere constructions as an intentionally easy supervised classification task to check that the pipeline detects available signal and returns a negative result without distinguishing information.It includes four torus grids and four sphere constructions, each expanded into 32 permuted copies for 256 balanced samples.Source: §I.1 · p. 36
  • The stratified split is 60/20/20, but base complexes may appear across splits, so this sanity experiment is not an estimate of generalization to unseen cellulations.Table 20 identifies the supervised torus-versus-sphere experiment and seed 0, without supplying result values in the provided caption.Source: §I.1 · p. 36, Table 20 · p. 36

I.2 Stratification of the Global Metric Channel C5

The audits distinguish coarse and fine metric information, report pair-level variation, and compare cross-class results with π1-matched controls. They also delimit what models receive and show that spectral and walk mismatches are realization-level, while short-cycle signals may remain class-correlated.

I.2 Stratification of the Global Metric Channel C5

  • The coarse C5.4 distance histogram is matched on all 16 pairs, but SMCN’s finer C5.7 marking differs on 10, so failure of DIST_MLP does not rule out the metric channel read by SMCN.The Hasse-graph distance multiset differs on 15 pairs; shortest-path marking results are reported separately from binary marking because of this confound.Source: §I.2 · pp. 36–37, Table 21 · p. 37

I.3 Per-Pair Reporting and Seed Consistency

  • Pair-level seed consistency varies with marking: shortest-path unanimity falls from 4/4 at word length 48 to 3/8 at 72, while binary marking falls from 4/4 to 6/8 and then 1/4.Both markings unanimously separate the same eight pairs, and shortest-path marking additionally recovers pairs 3 and 12.Source: §I.3 · p. 37

I.4 π1-Matched Control Family

  • Every 1-WL-bounded baseline scores 0/64 on both the cross-class and π1-matched controls, while shortest-path SMCN scores 51/64 against 52/64, a one-evaluation difference.The binary cross-class control rerun is incomplete, so its 56/60 result is exploratory rather than directly comparable to the finalized main result.Source: §I.4 · p. 38, Table 24 · p. 39
  • The zero-message-passing model separates no pairs, whereas the reported-depth configuration has a gap roughly 27 times the within-class spread; the embedding-and-pooling pipeline alone is insufficient in this setting.The incomplete binary rerun does not establish that separation is specific to π1 or that models compute π1.Source: §I.4 · pp. 38–39

I.5 What the Certifier Retains and What the Model Receives

  • The certifier retains the signed attaching word, but Φ discards the five quantities used to certify π1, so separation demonstrates sensitivity to the supplied representation rather than direct receipt of the attaching word.The representation may still carry realization-induced consequences of that word.Source: §I.5 · p. 39, Table 26 · p. 39, Table 27 · p. 40

I.6 Spectral and Cycle Audits (C6, C7)

  • No pair is cospectral under the twelve tested operators, and closed-walk mismatches occur across families; short simple-cycle counts at lengths 4–6 match π1-matched controls more often than the cross-class family.No pair matches simple-cycle counts at lengths 7 or 8, and the authors leave open whether models can exploit the class-correlated short-cycle mismatches.Source: Table 29 · p. 40, §I.6 · p. 41, Table 28 · p. 41

GenAI Usage Statement

  • The authors used generative AI tools for language editing, manuscript organization, code development, and refinement of experimental and diagnostic-control ideas, then independently reviewed and verified the work.The authors state that they take full responsibility for the final content.Source: GenAI Usage Statement · p. 41

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