This document maps the current public claims in README.md and STATUS.md to code, CI, preregistration records, and result artifacts. It distinguishes three kinds of statements:
Software capability: the implementation exists and is exercised by tests.
Software correctness invariant: CI enforces a stated property such as package coverage.
Empirical claim: a seeded or deterministic study supports a stated scientific conclusion within its declared scope.
Implementation and test coverage do not by themselves establish downstream task benefit. Conversely, a negative or invalidated result for one research architecture does not invalidate unrelated library components.
Methodology
Empirical values are taken from committed result artifacts in notebooks/results/ or from the CI command that produces the invariant.
Pairwise graph experiments use their preregistered decision rules and within-family Benjamini-Hochberg correction.
The former investigation-wide 59-comparison and 76-entry sensitivity analysis is withdrawn as a current claim because it included invalidated H009 comparisons. With H009-R complete the validated comparison set exists, but the retrospective table has not been regenerated and remains withdrawn until it is, with documented inclusion rules. See STATISTICAL_SUMMARY.md.
A non-significant comparison is reported as no significant difference detected, not as proof of equivalence.
Where a preregistered threshold differs from a conventional alpha=0.05 interpretation, the preregistered threshold controls the hypothesis verdict.
Invalidated experiments remain in the public record for provenance but are excluded from current scientific conclusions until corrected and rerun.
Claim 1: The importable topogeoml package is gated at 100% line and 100% branch coverage under full dependencies
The exact test-item count can change as regression tests are added. The maintained invariant is 100% line and 100% branch coverage on the importable topogeoml package in the required full-dependency gate.
Scope
benchmarks/ is research infrastructure outside the package-coverage claim. The default dev-only test matrix can skip torch-dependent paths; the full-dependency gate is authoritative for the package coverage statement.
Claim 2: Hodge-residual has a narrow positive difference from MLP on NCI1
The previous 59-comparison retrospective BH statement is withdrawn because that comparison family included invalidated H009 comparisons.
Scope
NCI1, 30 seeds, 10 epochs, hidden_dim=32, matched-capacity protocol. Later operator controls show that the positive difference is not unique to the Hodge L_0 operator.
Claim 3: No unique L_0 Hodge advantage is supported by the matched operator controls
Field
Evidence
H008c NCI1
gin-residual 0.629 vs Hodge 0.609; median Delta = +0.0195; p_BH = 1.01 x 10^-2
H010 MUTAG
gin-residual 0.789 vs Hodge 0.750; p_BH = 7.44 x 10^-3
H010 PROTEINS
Hodge 0.686 vs gin-residual 0.675; p_BH = 0.292, so no significant operator difference is detected
Artifacts
notebooks/results/h008c_nci1_gin_residual_30seeds.{json,md} and notebooks/results/h010_{mutag,proteins}_operator_30seeds.{json,md}
No tested dataset provides evidence that the matched Hodge L_0 arm is superior to the normalized-adjacency arm. MUTAG and NCI1 favor the adjacency arm at the H010 alpha=0.05 family threshold; PROTEINS detects no significant difference.
Not supported
A universal claim that adjacency is always superior, or that the H010 H44 dataset-level mechanism was identified. H44 remains inconclusive.
Claim 4: The tested external-residual adjacency formulation recovers NCI1 performance after normalization with an internal self path does not
Field
Evidence
H008b
gin-normalised median 0.500 on NCI1
H008c
gin-residual median 0.629; gin-residual vs gin-normalised p_BH = 5.20 x 10^-6
Implementation
benchmarks/hodge/models.py shows gin-normalised uses MLP((1+eps)h + A_norm h), while gin-residual uses act(A_norm h W + b) + h
Supported conclusion
The external-residual formulation is a successful tested architectural change and removes any need to invoke a unique Hodge operator effect in this NCI1 comparison.
Causal limitation
The two GIN formulations differ in placement and parameterization of the self path. The experiment does not prove that residual connections, abstracted from their exact computation form, are the sole causal mechanism in arbitrary models or datasets.
Claim 5: Graph-structural classification signal is detected on the three tested datasets under constant features
See REPRODUCING.md section H006, including the benchmarks.hodge.h006_analysis resolver step that performs the class-prior tests and family correction
Scope
These are Hodge-architecture comparisons against theoretical class-prior controls under constant node features. They demonstrate exploitable graph structure in the three tested datasets, not a unique Hodge mechanism or a universal claim about graph datasets.
Claim 6: H009 is invalidated as evidence for a cellular-sheaf Laplacian
Field
Evidence
Historical artifact
notebooks/results/h009_nci1_sheaf_30seeds.{json,md} retained for provenance only
Implementation defect
The historical sheaf-residual model processed both orientations of each undirected edge independently. The resulting learned matrix was not guaranteed symmetric or positive semidefinite and was not, in general, the claimed delta.T @ delta scalar cellular-sheaf Laplacian.
Capacity defect
On NCI1, Linear(64, 2) contributes 130 learner parameters, producing 2,468 total trainable parameters versus 2,338 for the fixed-operator controls. The difference is about 5.56%, slightly outside the stated 5% tolerance.
Current status
The historical H009 pairwise statistics have no inferential status for a sheaf-Laplacian claim. H39, H40, and H41 are now resolved by the corrective replication H009-R (Claim 6b), not by this artifact.
Repair completed
One undirected edge representation, symmetric PSD construction, identity reduction to L_0, gradient tests, and exact parameter accounting were established at merge b89d196; the fresh 30-seed NCI1 corrective replication was preregistered as H009-R and executed from commit 79329df.
Source document
docs/hypotheses/HYPOTHESIS-009-sheaf-laplacian.md contains the full invalidation and rerun requirements.
Claim 6b: The H009-R corrective replication resolves H39-H41 for the repaired scalar sheaf operator
Field
Evidence
Artifact
notebooks/results/h009r_nci1_sheaf_v2_30seeds.{json,md}, committed unmodified from GitHub Actions Run Experiment #8 (run ID 31419047248). SHA-256 of the uploaded artifact archive as recorded by GitHub: 3d595aa27a1c95ab258051ac1a152d4e4fc4a4d8b04012e3dcf605a6f93dc920; per-file digests of the committed files are in the H009-R hypothesis document section 9.1
Preregistration
docs/hypotheses/HYPOTHESIS-009R-sheaf-corrective-replication.md, committed and merged before execution; decision rules for H39-H41 fixed in advance
Model
sheaf-residual 2.0.0: one coboundary row per undirected edge, L_F = delta.T @ delta symmetric PSD by construction, invariant-tested; 2,403 parameters versus 2,338 for the controls (+2.78%, inside the 5% tolerance)
H39 (sheaf vs MLP)
Supported. Median 0.604 versus 0.523; median Delta = +0.0809; p_BH = 4.73 x 10^-3 < 0.05; sheaf above MLP on 20/30 seeds
H40 (sheaf vs Hodge)
Not supported. Median 0.604 versus 0.605; p_BH = 0.428. No significant difference detected; not an equivalence claim
H41 (sheaf vs gin-residual)
Falsification condition not met. Median 0.604 versus 0.630; median Delta = -0.0262; p_BH = 0.0342, which does not cross the preregistered 0.01 threshold. Conventionally significant at 0.05, so the comparison is inconclusive under the stricter preregistered rule; the observed direction (sheaf below gin on 22/30 seeds) is reported without an equivalence or non-inferiority interpretation
Supported conclusion
Under the fixed matched-capacity, one-layer, 10-epoch NCI1 protocol: the repaired construction is above the matched MLP (H39); no significant sheaf-Hodge difference is detected at this power (not an equivalence finding); and it is directionally below the matched normalized-adjacency arm without crossing the strict falsification threshold. No learned-sheaf advantage over either fixed operator was detected.
Not supported
Any learned-sheaf advantage over fixed operators; sheaf-Hodge equivalence; generalization beyond this dataset, capacity, depth, and training budget; validation of the invalidated historical H009 numbers.
Claim 7: The NCI1 L_1 experiment does not establish a higher-order advantage
Field
Evidence
Artifact
notebooks/results/h011_nci1_l1_30seeds.{json,md}
L_1 vs MLP
median Delta = +0.0669; p_BH = 0.0957. H47 refuted.
L_1 vs L_0 Hodge
median Delta = -0.0195; p_BH = 0.0787. H48 refuted under its stated rule.
L_1 vs gin-residual
median Delta = -0.0389; p_BH = 0.00676. H49 refuted under its preregistered 0.01 rule.
Structural audit
3961/4110 NCI1 graphs contain no triangles, so the B_2 B_2^T up-Laplacian term is absent for 96% of graphs.
Supported conclusion
The tested NCI1 L_1 arm does not outperform the node-level controls under the preregistered rules. NCI1 is also a poor test of the intended triangle-rich higher-order mechanism.
Frontier
H011b on COLLAB is the correct triangle-rich follow-up. Its one-seed smoke result is directional only and the preregistered 30-seed statistical run remains incomplete.
Claim 8: The public library surface is implemented and tested, but downstream benefit is component-specific and often unproven
Capability
Implementation
Test evidence
Claim boundary
Persistent-homology feature pipeline
topogeoml/pipelines/feature_pipeline.py
tests/test_feature_pipeline.py, full package gate
Implemented scikit-learn transformer; task-specific predictive benefit depends on data and model
Differentiable Vietoris-Rips primitives and TopologyRegularizer
topogeoml/nn/diff_ph.py
tests/test_diff_ph.py, full package gate, path-conditional diff-PH benchmark workflow
Gradient path is tested in exercised regimes; not a claim of globally smooth persistence or universal training improvement
Differentiable cubical persistence and CubicalTopologyLoss
topogeoml/nn/cubical_diff_ph.py
tests/test_cubical_diff_ph.py, full package gate
Implemented and gradient-tested; no powered end-to-end segmentation improvement claim yet
Simplicial/Hodge algebra and fixed-complex neural primitive
14 documents. HYPOTHESIS-009R is a corrective replication protocol for the existing H39-H41 sub-predictions, not a 15th hypothesis, and adds no sub-prediction IDs
Git history for each hypothesis file supplies the preregistration timestamp; experiment artifacts were added later
Sequential-design limitation
Hypothesis selection was sequential and informed by earlier results. The former investigation-wide 59/76 sensitivity analysis remains withdrawn until the retrospective table is regenerated from the validated comparison set, which H009-R (Claim 6b) now completes.
Current unresolved evidence
Investigation-wide multiplicity sensitivity table: the withdrawn 59/76 retrospective analysis has not been regenerated from the validated comparison set now that H009-R is complete; no current conclusion depends on it.
H011b COLLAB L_1: one-seed, one-epoch directional smoke result only; no statistical claim licensed.
CubicalTopologyLoss downstream segmentation benefit: implementation exists, but the powered end-to-end study is not complete.
Topology-divergence callback: exploratory, floor-limited, no non-overfitting negative control.
Independent external reproduction: the validated empirical record has not yet been reproduced by an independent team.