# Session 2026-04-24 Part 2: NAL↔Beta Manifold Pipeline

## Goals Completed: g110-g131, g133-g136 (23 total)

### Core Theoretical Results
- **g126**: NAL↔Beta↔Poincare resolution — NAL c=evidence weight lifts to Beta(wf,w(1-f)) with K=-1/4
- **g120**: Contraction mapping proof L=d/(o+1)²<1 for homeostasis dynamics
- **g129**: Rao geodesic distances validated — similar beliefs cluster, contradictions maximally distant

### Pipeline Architecture (complete)
1. **Homeostasis pruning** (g120,g123): c*=[-(o+1-d)+√((o+1-d)²+4do)]/(2d), beliefs below c* pruned
2. **Predicate filtering** (g135): 3-layer — exact match → subject-chainable → remaining alive
3. **Rao geodesic ranking** (g129,g130): Beta manifold distance selects geometrically nearest premises
4. **Backward chaining** (g131): Unified chainer composes pruned+filtered+ranked premises into proofs

### Artifacts Deployed to nonlanguage.dev/MeTTaSoul/mb/
- g127_nal_beta_poincare.md — formal paper
- g129_rao.py — pairwise Rao geodesic computation
- g130_rao_chainer.py — premise selection by manifold proximity
- g131_unified_chainer.py — full pipeline chainer class
- g133_chain_poincare.png — static backward chain visualization
- g134_benchmark.py — 50-belief KB benchmark (1.1ms)
- g135_filtered_ranker.py — predicate-filtered Rao ranking
- g136_poincare_viewer.html — interactive HTML/JS Poincare disk viewer

### Key Insights
- Low-confidence goals prefer low-evidence premises (geodesically closer) — correct behavior
- NAL revision is symmetric — destroys temporal signal, need per-session atoms for regression
- Predicate filtering before Rao ranking essential for production use
- Full pipeline: 5 goals × 50 beliefs in 1.1ms — real-time viable