Our review

Checks the mathematical correctness of topological data analysis (TDA) results against internal consistency checks, null models, and known benchmarks.

Strengths

Structured checklist covering persistence diagrams, total persistence scaling, and Wasserstein-specific tests.
Includes concrete benchmark values from the trajectory_tda/P01 project for direct comparison.
Adds cross-era replication checks and null model plausibility checks.

Limitations

Benchmarks are project-specific (trajectory_tda / P01), limiting general applicability.
Requires result files in an expected format and some TDA background to interpret correctly.
Does not address all possible TDA outputs or validation scenarios.

When to use it

Use it after generating quantitative TDA results (persistence, Wasserstein, nulls) to confirm they are plausible and reproducible.

When not to use it

Do not use it for purely exploratory topology work without numeric outputs or known benchmark references.

Examples

Validate a Wasserstein null result file

/validate-topology trajectory_tda results/trajectory_tda_integration/04_nulls_wasserstein.json

Validate a trajectory TDA persistence result

/validate-topology trajectory_tda results/trajectory_tda_integration/01_persistence_diagrams.json

Check topology results against P01 benchmarks

Compare my trajectory_tda persistence outputs with the known P01 benchmark table using /validate-topology.

/validate-topology — Validate Topological Results

Check mathematical correctness of TDA results against known benchmarks and internal consistency checks.

Usage

/validate-topology [domain] [result-file]

Example: /validate-topology trajectory_tda results/trajectory_tda_integration/04_nulls_wasserstein.json

Validation checklist

Persistence diagram sanity

[ ] All birth values ≥ 0
[ ] All death values > birth (finite features)
[ ] H₀ has exactly one infinite feature (the final connected component)
[ ] Feature counts are plausible for landmark count L: H₀ has L-1 finite features; H₁ count varies

Total persistence scaling

[ ] Total persistence scales approximately linearly with L (±15% across a 2× range)
[ ] Maximum persistence is stable across L values (should vary < 5%)

Null model consistency

[ ] Label/cohort shuffle p-values are non-significant (negative control)
[ ] Markov-2 null generates more total persistence than Markov-1 (higher-order Markov → more structured surrogates)
[ ] Null distribution standard deviations are plausible (not near-zero, not huge)

Wasserstein-specific

[ ] W(obs↔null) and W(null↔null) are of comparable magnitude (within ~3×)
[ ] p-value = proportion of null-null distances ≥ mean(obs-null distances)
[ ] 500 null-null pairs is sufficient for stable p-value at 3 decimal places

Cross-era replication

[ ] BHPS-era order-shuffle H₀ p-value ≈ USoc order-shuffle direction (both significant or both not)
[ ] BHPS-era Markov-1 direction ≈ USoc (both non-significant under total persistence)

Known benchmarks (trajectory_tda / P01)

| Test | Expected | Source | |---|---|---| | USoc order-shuffle H₀ (total persistence, L=5000) | p < 0.005 | P01 v5 Table 2 | | USoc Markov-1 H₀ (total persistence) | p = 1.000 | P01 v5 Table 2 | | USoc Markov-1 H₀ (Wasserstein, L=2000) | p = 0.002 | P01 v5 Table 2b | | BHPS order-shuffle H₀ | p = 0.000 | P01 v5 §4.7 | | BHPS Markov-1 H₀ | p = 1.000 | P01 v5 §4.7 | | GMM bootstrap ARI | 0.646 ± 0.086 | P01 v5 §3.5 |

Validate Topological Results

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