Topological Data Analysis in Quantitative Trading

The corpus is ~1.4M backtested strategies across 27 crypto assets, 8 indicator families, and 6–30 walk-forward windows per asset. Phase 1 built the unified parquet, the 50-feature behavior fingerprint per strategy, the per-family parameter parser, the streaming `trades.bin` decoder (242M trade rows in 17 minutes), and the distance-matrix infrastructure that scales to 1M+ strategies via Faiss IVF-PQ.

Subphases that landed under Phase 01: (a) bootstrap the public repo + Docker container + CI; (b) parser end-to-end on 100 .txt files; (c) parsed-only 42-D fingerprint + full 50-D fingerprint; (d) distance matrices and the kNN search; (e) Mapper-of-strategies first pass + HDBSCAN first pass.

First-light findings on 3 pilot assets (BTC, DOGE, AVAX, 127,879 strategies, 24.8M long-form rows, 13 minutes of wall-time): the kNN-aggregate predicts OOS Sharpe with mean Pearson 0.94 and Spearman 0.37 on the pilot, top-H₀ persistence is uniform 6.0–7.1 z-units across cohorts, top-H₁ persistence (0.47–1.46) discriminates indicator families, and HDBSCAN finds zero clusters in 23 of 24 cohorts at locked parameters. The behavior-space manifold is smooth, not partitioned, which changed the H4 method choice from raw HDBSCAN to Mapper / ToMATo / Reeb-graph at lock-in.

• ▪Falsifiability is non-negotiable. Every empirical chapter has a pre-registered falsification criterion locked at end-of-Phase-1.
• ▪Phase gate cleared: the pre-registration manifest is immutable from 2026-05-09 forward; subsequent revisions live in a v2 sibling document.
• ▪All Phase-1 outputs reproducible bit-for-bit from the locked seed = 2026 and the committed Docker image.

Six distance metrics on strategy populations, parameter-space, behavior-space, equity-curve W₂, trade-level Wasserstein, Gromov–Wasserstein, Fisher–Rao, characterised. Density and manifold structure (intrinsic dimension via TwoNN and MLE; full manifold-learning suite for visualisation only). Plateau detection combines persistent superlevel sets on IS Sharpe, discrete Morse theory on the parameter grid, HDBSCAN on the joint (parameter, IS Sharpe) space, and SCMS ridge fits.

Subphases: (a) the locked H1 falsification ran across all 26 assets × 7 families = 182 cohorts with window-causal CV; (b) the H2 first-pass (locked spec) ran on all 182 cohorts × 3 plateau methods = 546 cohort-method results; (c) an H2 salvage attempt in behavior-space appeared to PASS at +0.236, and a window-causal robustness audit showed ~74% of that effect was look-ahead leakage. H2 falsifies under any honest methodology.

Headline results. H1: meta ΔR² = +0.130 (Holm-adjusted p < 1×10⁻³⁰⁰; 180/182 cohorts reject, 178/182 with bootstrap 95% CI excluding zero, 146/182 cross the +0.05 effect threshold). H1 is the program's first PASS at corpus scale and the foundational result everything downstream rests on. H2: median ΔSharpe is +0.002 (ph_h0), +0.035 (Morse-Smale), −0.005 (HDBSCAN) across 182 cohorts; none cross the locked +0.10 threshold. H2 is falsified.

• ▪Holm–Bonferroni at FWER ≤ 0.05 applied family-wise; bootstrap = 5000 stationary-block replicates, Politis–White block length, seed = 2026.
• ▪The H2 leakage audit (rebuilding the kNN graph causally per prediction window) is the program's load-bearing methodology precedent, the same trap is now actively guarded against in every downstream test.

Single-filtration PH computed Vietoris–Rips and alpha complexes across all 27 assets × 7 families × cohorts; PDs vectorised into persistence images, persistence landscapes, sliced-Wasserstein kernel features, and PersLay embeddings. Phase 03 also stood up the modern-ML baseline stack (B1 ridge, B2 XGBoost/LightGBM/CatBoost, B3 TabPFN, B4 TS2Vec, B5 contrastive equity-curve encoder, B6 stacking), the H3 challenger the topology stack had to beat.

Subphases: (a) the Phase-3 H3-lite preliminary on 11 of 182 cohorts (preliminary INCONCLUSIVE, near-null); (b) Phase-4 Stage 1, Rust evaluation primitives ported via PyO3 (32/32 parity tests pass; per-primitive speedup 1.05× to 42×); (c) Stage 2, window-causal topology features rebuilt per prediction window (the H2 +0.236-leakage trap mitigation); (d) Stage 4, B7 (topology-only LightGBM) + B8 (B2 + 38-D topology); (e) the H100 burst, B5 contrastive (5 architecture variants), B5 supervised-head viability check, and B9 PersLay-image, all on the locked top-5 H1-PASS cohorts.

H3 verdict: FALSIFIED. The locked nested test B8 (B2 + 38-D engineered causal topology) vs B2 alone on the five strongest H1-PASS cohorts: meta-FE ΔR² = −0.0378, Holm-adjusted p = 1.07 × 10⁻⁵, 5/5 cohorts negative, 4/5 with bootstrap 95% CI excluding zero on the negative side. B7 (topology-only) lands in [−0.43, −0.13], worse than predicting the per-window mean. On the H100 side, B5 contrastive across 5 architectures all in [−0.028, −0.024]; B5 supervised-head −0.022; B9 PersLay catastrophic numerical instability with non-instable cohorts ≤ +0.006. Two completely independent feature representations and three independent architecture families converge on the same negative answer. This is the cleanest possible Phase-4 outcome from a falsificationist perspective.

• ▪Best honest read of why H3 failed: shape-features encode information that correlates with OOS Sharpe, but the strong tabular B0/B2 baseline (in-sample Sharpe, profit factor, deflated Sharpe, prior-window OOS lookback, edge LB) already extracts essentially all the residual signal. Shape-features add redundancy and selection noise rather than new information at the granularity tested.
• ▪Phase 04's deep-learning side required rented H100 compute, running PersLay and the contrastive equity-curve encoders at production architecture size is not feasible on the local 8 GB VRAM workstation. The burst was deliberately conservative and retired under-budget.

H10 asks whether persistent local homology of an ε-ball around each strategy carries incremental predictive power beyond cohort-level features. H11 cross-checks the engineered behavior fingerprint against Fisher–Rao distances on OOS-return distributions. Both hypotheses were designed as elaborations of the H3 thesis (topology-replaces-ML).

With H3 falsified, the prior on H10 / H11 adding non-redundant signal drops sharply: their failure modes are structurally similar to H3's. Phase 04 is therefore deprioritised. The Phase-1 plateau-detection vocabulary survives as a per-strategy diagnostic (the audit-trail interpretability use case is preserved), but the predictive-power tests are scheduled as a follow-up paper rather than an in-program phase.

H4 builds portfolios that sample one strategy per Mapper cluster in behavior-space, weighted by the Beta-Binomial profitability lower-bound. Baselines: a naive in-sample-Sharpe-rank top-k, a plain reliability-ranked top-k, HRP, NCO from the same pool (mean-variance excluded for solver wall-time). The bar is ΔSortino ≥ 0.15 against the strongest baseline, paired-bootstrapped across the full corpus, every crypto asset × indicator family × walk-forward window, at portfolio sizes k ∈ {5, 10, 20, 50}, with the ToMATo persistence-based clustering arm as a sensitivity check.

H4 verdict: the topology selection-and-construction Stack beats Hierarchical Risk Parity at deployment scale by ΔSortino = +0.61 (95% CI [+0.39, +0.82], Holm-significant), with the edge concentrated on the L1-majors cohort (+1.20 ΔSortino vs HRP, Holm-rejected) and the volatility-and-oscillator indicator families (ATR / MACD / Stochastic %K each Holm-rejected positive); on the moving-average families (EMA / RSI / SMA) the result is flat or negative. Against a simple top-k-by-reliability equal-weight baseline the Mapper-cluster step is statistically indistinguishable on Sortino (paired Δ = −0.06, Holm-p = 0.37), so the topology contribution is a coverage-and-audit-traceability one, not a raw-return one, and against the naive in-sample-Sharpe-ranked top-k baseline the topology Stack still loses on Sortino. The headline ships: risk-parity-beating, honest, audit-traceable construction.

H5 cross-crypto universality verdict: FAIL. Persistence diagrams of behavior-space across L1-majors / alt-L1s / memes / DeFi / L2-and-newer cohorts are not closer to each other than to permutation-shuffled nulls, bottleneck within-cohort / cross-cohort ratio 1.018 (p = 0.59); Gromov–Wasserstein 0.96 (p = 0.24); both well above the α = 0.01 falsification rule. Pooling a topological selector across structurally-similar crypto assets is not statistically justified, each asset needs its own selector. Paper D rescopes to no-universality.

Phase 06 is independent of the H3 falsification: it tests portfolio construction (cluster-based diversification), not per-strategy level prediction. The Phase-01 smooth-manifold finding is the unlock, once density-based clustering proved inappropriate at corpus scale, Mapper / Reeb-graph took over for H4 before Phase 06 even started.

The deployable surface is a single pipeline a trader runs on their own corpus of backtests: map (the Mapper view of the corpus) → select (the H4 cluster-sampled portfolio, the selector also serves as the candidate-pool reducer) → diversify (coverage-aware on top of H4) → monitor (the cohort-level "trust-this-aggregation" check, is this cohort's geometry settled enough to pool over?) → pause / refit (the regime-rotation cadence plus an optional smoothing layer, a policy, not a drift alarm). It is realised as a ten-command CLI, map, select, portfolio, audit, monitor, refit, aggregation-check, plateau-audit, universality-check, live-adapter, with a thin live adapter that consumes an existing walk-forward pipeline's output with no changes to that pipeline, a worked end-to-end example on a fresh cohort, and the runbook / live-deployment / case-study chapters drafted. Every stage is built from the geometric-outlier primitives that proved robust, cloud eccentricity, Isolation-Forest and kNN-density anomaly scores, Mapper coverage, the persistence-trajectory descriptors, not from any of the target-regression machinery that failed; and the FAIL-derived caveats are baked into every relevant command's help text (six of the ten carry one, the honest negatives are part of the deliverable). An optional `topo-prune` shortlist diagnostic exists for human attention (it ranks a 100-backtest shortlist far better than chance and about level with the top-100-by-in-sample-Sharpe), but the full-corpus path through the selector beats any 100-shortlist, so the recommendation is: don't pre-prune; run the full corpus.

Two pre-registered hypotheses targeted the new stages. H18, candidate-pool pruning: can a geometric-outlier filter over a fresh backtest corpus beat a deflated-Sharpe / PBO filter at keeping the strategies that survive out-of-sample? Verdict: FAIL, the prune stage collapses into select. There is no separate topology-aware-pruning step that improves on running the H4 portfolio constructor against the full corpus; the selector already exploits the same structure, so pre-pruning its input only removes its options. This sharpened the program's central finding: robustness to the data-cleaning fix is necessary but not sufficient, topology has to also add a degree of freedom the baseline lacks (H4 adds a coverage objective, H17 a cohort-stability axis, H12 a regime label, they land; H18 removes options from a baseline that already uses the same structure, it can't). H19, live-drift monitoring: does the distance of a live strategy's running behaviour fingerprint from its origin cluster predict an imminent edge collapse, early enough to act? Verdict: FAIL, the per-strategy drift score is coincident-to-lagging, not leading: it fires at the classic max-drawdown breach (median lead 0 windows), lags the trailing-Sharpe trigger by 1 window, and carries a 68% false-positive rate against a 30% bar (0 of 26 cohorts under it). A strategy that draws down changes its realised return distribution by virtue of having drawn down, so the geometry lights up after the fact. So there is no per-strategy drift trigger in the Stack, the actionable monitor stays the cohort-level aggregation-trust check (H17 PASS), and the per-strategy drift profile ships as a visualization aid only. H19 added the fourth refinement of the central finding: an added degree of freedom is necessary but not sufficient, it has to be predictive of the outcome, and a "drift" signal isn't. With both new hypotheses closed, the Stack is assembled and the external user-test package is ready to dispatch; per-stage latency budgets are reported (the per-cohort silhouette-lock step is the bottleneck, cached, it stays well within a real-time refit window), and the compute-heavy TDA, multipersistence, sheaf cohomology, is deliberately not on the production path.

H3 closed the central topology-beats-modern-ML question with a clean negative answer. The plan was extended with six new hypotheses about distinct questions the H3 result does not bear on, pre-registered separately from the H1–H11 family under their own Holm–Bonferroni FWER ≤ 0.05. All six have now been tested.

H12, topological regime detection on a sliding window of asset returns, and a strategy-cohort rotation policy on top of it: MARGINAL, the topological regime detector itself is null (it does not flag regime changes earlier than the standard HMM detector), but the cohort-rotation policy that consumes the detected-state signal works; it ships as Paper F. H13, graph-Laplacian smoothing on H1's kNN graph: MARGINAL, directionally positive in 94.5 % of 182 cohorts but small (corpus mean +0.0035 ΔR², below the +0.02 headline threshold; Holm rejects the null for all 12 smoothing variants); ships as an optional MACD-only production smoothing layer (+0.0115 there). H14, topological anomaly detection (overfit-flagging + survivor-edge-flagging): MARGINAL, the geometric-outlier score is a useful triage complement to the classical overfit diagnostics (deflated Sharpe, PBO) but does not replace them. H16, geometry-aware behavior-space diversification (farthest-point sampling, k-DPP, Bayesian-coverage acquisition): MARGINAL, none of the three coverage algorithms beats the Mapper-cluster portfolio, but the contest surfaced a chapter-level finding: quality-aware coverage works (the Bayesian-coverage variant clusters near the headline portfolio); quality-blind coverage is noise (farthest-point sampling and k-DPP land ~3–4 Sortino units worse, near the random-portfolio floor). H17, cohort-level persistence-trajectory dynamics over walk-forward windows: PASS, a cohort's persistence-diagram trajectory length and segment count predict how stable that cohort's out-of-sample Sharpe is (segment-count × stability Spearman ρ = +0.72; 5 of 8 trajectory tests survive Holm; the static persistence-diagram baseline does not); ships as a cohort-level "trust-this-aggregation" diagnostic. H15, topological fragility / hazard rate via a Cox proportional-hazards model: FAIL, topology is reliably worse than the classical ex-ante features at predicting which strategies will catastrophically fail (held-out C-index ≈ 0.57–0.60 for topology vs 0.77–0.79 for classical, ΔC-index −0.08 to −0.21 across four failure-event models); the failure event has a sharp signature (the in-sample edge collapsing out-of-sample) that the standard scorecard captures directly.

Validating the six extensions across every hypothesis that touches the 45-D behaviour fingerprint resolved the central methodological finding at a covariate-level grain: geometric-outlier covariates (cloud eccentricity, Isolation-Forest, kNN density, Mapper coverage, the persistence-trajectory descriptors) are the durable practical primitives, and every headline-bearing contribution (the H4 portfolio constructor, the H17 aggregation diagnostic, the H12 rotation policy) is built from them; target-regression covariates (using topology to regress onto out-of-sample performance) shrink or fail; local persistent homology on a smooth manifold is dead weight (there are no clusters or boundaries for "homology depth" to mean anything relative to). Output portfolio grows to eight papers, adding Paper H, the methodology paper that states this prediction-vs-decision partition.

Output portfolio: eight papers (Plateau & Neighborhood Generalization · Persistent Topology of Strategy-Space · Topology-Aware Portfolios · Cross-Crypto Universality · Applied Topological Workflow · Regime Topology & Strategy Rotation · Geometry-Aware Portfolio Construction & Topological Fragility · When Topology Helps in Quantitative Trading: a Prediction-vs-Decision Partition), one applied monograph at ~150–250 pages, one public MIT-licensed software package (the Topology Deployment Stack), ≥7 interactive HTML visualisations embedded in the daru.finance survey article, and a Zenodo DOI for the reproducibility container.

Phase gate: an external user-test on 1–2 practitioners outside the project, before public release. The deliverable in one sentence, a CLI that takes a fresh corpus of backtested strategies and returns a recommended portfolio with per-strategy audit trails, cohort-level aggregation-risk scores, drift monitors, and refit-cadence recommendations, all validated to thesis-grade rigor and reproducible from a public container.