HyperSort: Riemannian Comparison Manifold Sort with GTC

Exact sorting + approximate manifold sort + outlier detection.

HyperSort is a Python package that uses the HyperTensor Geometric Jury framework to provide two sort modes and per-element confidence scoring via a Riemannian Comparison Manifold with Geodesic Trajectory Cache (GTC).

Quick Start

from hypersort import hypersort, ComparisonManifold

# One-shot exact sort
result = hypersort([3.14, 1.41, 2.71, 1.73, 0.57])
print(result.sorted_data)  # [0.57, 1.41, 1.73, 2.71, 3.14]

# Build once, sort many (exact)
manifold = ComparisonManifold()
manifold.build(training_data)  # O(n²), one-time

result = manifold.sort(new_data)         # exact, always correct
result = manifold.sort_manifold(new_data)  # GTC approximate + verdicts
verdicts = manifold.classify(data)        # outlier detection only

Two Sort Modes

Outlier Detection

classify() and sort_manifold() assign each element a verdict tier:

Statistical z-score check (|z| > 3 on sort key) automatically upgrades to EXPLORE.

manifold.build(training_data)
verdicts = manifold.classify(mixed_data)
for v in verdicts:
    if v.verdict_type == 'EXPLORE':
        print(f"Outlier detected! d={v.geodesic_distance:.4f}")

Configuration

API Reference

One-shot

from hypersort import hypersort

result = hypersort(data)                      # exact sort
result = hypersort(data, mode='manifold')     # exact + verdicts + confidence
result = hypersort(data, compute_pairwise=False)  # skip O(n²) pairwise matrix

Reusable Workflow

from hypersort import ComparisonManifold, ManifoldConfig

manifold = ComparisonManifold()               # auto-tune k
# or: ManifoldConfig(intrinsic_dim=16, use_gpu=True)
manifold.build(training_data)                 # O(n²), one-time

# Exact sorting
result = manifold.sort(new_data)              # always correct
result = manifold.sort(new_data, compute_pairwise=True)  # + jury confidence

# GTC approximate sorting
result = manifold.sort_manifold(new_data)     # approximate positions + verdicts

# Outlier detection
verdicts = manifold.classify(data)            # batch
verdict  = manifold.classify_one(element)     # single element (streaming)

Persistence

manifold.save('manifold.npz')
loaded = ComparisonManifold.load('manifold.npz')

Verdict Tiers

from hypersort import VerdictTier

for v in result.verdicts:
    if v.verdict_type == VerdictTier.EXPLORE:
        print(f"Outlier: d={v.geodesic_distance:.4f}, conf={v.confidence:.4f}")

Honest Complexity

How It Works

Build Phase (one-time, O(n²))

Data → [Encoder] → dim 0 = sort key, dims 1+ = manifold features
     → [SVD on dims 1+] → PCA manifold basis [d-1, k]
     → [Normalize] → Unit Sphere S^{k-1} (trajectory cache)
     → [Sort by dim 0] → GTC indexed by ground-truth positions

Sort Phase

• sort(): exact ordering by encoder dim 0 via argsort
• sort_manifold(): project new data onto manifold → query GTC for nearest cached trajectories → approximate positions + verdict tiers
• classify(): project → measure geodesic distance to GTC cache → assign RETRIEVE/AUGMENT/EXPAND/EXPLORE based on proximity and statistical z-score (|z| > 3 → EXPLORE)

Limitations

• String encoder: first 7 chars for positional encoding (exact via Python ints)
• GTC sort is approximate for out-of-distribution data
• Manifold build is O(n²); best with build() once, sort_manifold() many
• GPU acceleration requires CuPy (pip install cupy)

Citation

@software{stewart2026hypersort,
  title     = {{HyperSort}: {Riemannian} Comparison Manifold Sort},
  author    = {Stewart, William Ken Ohara},
  year      = {2026},
  publisher = {GitHub},
  journal   = {GitHub repository},
  url       = {https://github.com/NagusameCS/HyperTensor/tree/main/hypersort}
}

License

MIT — see LICENSE file.

Related Work

• HyperTensor — The parent project
• Papers I–XVIII in ARXIV_SUBMISSIONS/ — Full theoretical foundation
• volume_extended.tex — Master compilation of all papers