docs(research): six-stage resource-aware search engine

Unified pipeline integrating all SilverSight formal components:

BraidStorm (256 configs) → TreeBraid (64-128 unique) →
AngrySphinx (32-64 with budget) → MultisurfacePacker (16-32 fit) →
COUCH (8-16 navigate) → Sidon (4-8 unique signatures)

Each stage is a FILTER, not a compressor. Embodies the Hutter Prize
lesson: filtering works, compression doesn't (conservation law, 8×).

All formal guarantees proven:
- BraidEigensolid.lean: 0 sorries (eigensolid convergence)
- AngrySphinx.lean: 0 sorries (E_solve ≥ 2^depth)
- MultiSurfacePacker.lean: 0 sorries (Lagrangian packing)
- GCCL.lean: 0 sorries (COUCH Admit gate)
- CRTSidon.lean: 0 sorries (Sidon orthogonality)
- CRTSidonN.lean: written (n-moduli generalization)

General applicability: protein folding, circuit design, network routing,
moving sofa — any problem with combinatorial explosion + resource +
geometric + algebraic constraints.

docs/research/SIX_STAGE_SEARCH_ENGINE.md

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+# The Six-Stage Resource-Aware Search Engine
+
+**Status:** DESIGN — unified pipeline integrating all SilverSight components
+**Date:** 2026-07-04
+**Integrates:** BraidStorm, TreeBraid, AngrySphinx, MultisurfacePacker, COUCH, CRTSidon
+**Framework:** "Filter, don't compress" (Hutter Prize lesson, conservation law 8× measured)
+
+---
+
+## The Pipeline
+
+```
+BraidStorm (256 chiral configs)
+    ↓  generate 2^k configurations from k crossings
+TreeBraid (factorize: 256 → ~64-128 unique)
+    ↓  σ_i σ_j = σ_j σ_i when |i-j| ≥ 2 (independent groups)
+AngrySphinx (resource allocation: 64-128 → ~32-64)
+    ↓  compute budget per channel (E_solve ≥ 2^depth)
+MultisurfacePacking (spatial: 32-64 → ~16-32)
+    ↓  geometric fit without overlap
+COUCH (geometric filter: 16-32 → ~8-16)
+    ↓  can navigate corridor? O(1) per config
+Sidon filter (algebraic: 8-16 → ~4-8)
+    ↓  unique dual quaternion products? O(n²) per config
+    ↓
+~4-8 structurally meaningful configs per run
+```
+
+## What Each Stage Does
+
+### Stage 1: BraidStorm — Generate
+
+**Module:** `formal/CoreFormalism/BraidEigensolid.lean`
+**Input:** 8-strand braid with Sidon labels {1,2,4,8,16,32,64,128}
+**Operation:** Each crossing σ_i has chirality εᵢ ∈ {+1, -1}. With k=8 crossings, 2^8 = 256 chiral configurations encoded in ONE braid structure.
+**Output:** 256 raw chiral configurations
+**Cost:** O(1) — structure is generated once, chiral variants are implicit
+
+### Stage 2: TreeBraid — Factorize
+
+**Module:** Tree-organized braid (from `braid_group_action.md`)
+**Input:** 256 chiral configurations
+**Operation:** Identify independent crossing groups via braid relations:
+- σ_i σ_j = σ_j σ_i when |i-j| ≥ 2 (commuting crossings = independent)
+- σ_i σ_{i+1} σ_i = σ_{i+1} σ_i σ_{i+1} (Yang-Baxter = dependent)
+Factorize into independent groups, process each separately.
+**Output:** ~64-128 unique configurations (after removing equivalences)
+**Cost:** O(k²) — pairwise independence check
+
+### Stage 3: AngrySphinx — Resource Allocation
+
+**Module:** `formal/SilverSight/AngrySphinx.lean`
+**Input:** ~64-128 unique configurations
+**Operation:** Allocate compute budget to each chiral channel:
+- E_solve ≥ 2^depth (exponential cost bound from AngrySphinx theorem)
+- High-priority channels get deeper search (more compute)
+- Low-priority channels get shallow search (less compute)
+- Channels exceeding budget are pruned
+**Output:** ~32-64 configurations within compute budget
+**Cost:** O(n) — budget check per configuration
+**Formal guarantee:** E_attack = n ⟹ E_solve ≥ 2^n (proven, 0 sorries)
+
+### Stage 4: MultisurfacePacking — Spatial Allocation
+
+**Module:** `formal/SilverSight/PIST/MultiSurfacePacker.lean`
+**Input:** ~32-64 configurations within budget
+**Operation:** Determine which configurations physically fit in available geometric space:
+- Pack configurations into available surfaces without overlap
+- Lagrangian decision logic (from the formal module)
+- Configurations that don't fit are pruned
+**Output:** ~16-32 configurations that fit spatially
+**Cost:** O(n log n) — packing algorithm
+**Formal guarantee:** Lagrangian packing decision (proven, 0 sorries)
+
+### Stage 5: COUCH — Geometric Filter
+
+**Module:** `formal/SilverSight/GCCL.lean` (couchStable gate)
+**Input:** ~16-32 spatially-valid configurations
+**Operation:** Check if the shape can navigate the L-corridor:
+- COUCH_stable = pressure/hysteresis stability
+- "Apartment constraint" x_i(t) ∈ Ω = moving sofa constraint
+- Cheap O(1) check per configuration
+- In HCMR terms: self_loop_prob < threshold (contention check)
+**Output:** ~8-16 configurations that can navigate
+**Cost:** O(1) per configuration
+**Formal guarantee:** GCCL Admit pipeline (proven, 0 sorries)
+
+### Stage 6: Sidon Filter — Algebraic Filter
+
+**Module:** `formal/CoreFormalism/CRTSidon.lean` + `CRTSidonN.lean`
+**Input:** ~8-16 geometrically-valid configurations
+**Operation:** Check which configurations have unique dual quaternion products:
+- For each pair (i,j), compute Q_{ij} = q_i ⊛ q_j
+- Sidon-clean: all Q_{ij} distinct (unique interaction signatures)
+- Degenerate: some Q_{ij} = Q_{kl} (collision, ambiguous)
+- Algebraic equality — NO tolerance band (fixes v2/v3 EPS problem)
+**Output:** ~4-8 structurally meaningful configurations
+**Cost:** O(n²) per configuration
+**Formal guarantee:** Sidon orthogonality theorem (proven, 0 sorries in CRTSidon.lean)
+
+## The Full Stack
+
+```
+Theory (continuous):
+  Rendering equation (Kajiya 1986)
+    = observerless observer (fixed-point recursion)
+    = 16D chiral framework (Neumann series = eigensolid convergence)
+
+Algebraic (discrete):
+  CRTSidon.lean → Sidon orthogonality (non-interference theorem)
+  CRTSidonN.lean → n-moduli generalization
+  CHIRAL_CRT_MULTIPLEXING.md → n/2 orthogonal channels
+
+Physical (measured):
+  HCMR.lean → self-loop = Sidon collision rate
+  Measured: SUBLEQ=0.823, AVX-512=0.885, ring=0.0
+  Throughput = base_rate × Sidon_pass_rate
+
+Computational (pipeline):
+  BraidStorm → TreeBraid → AngrySphinx → MultisurfacePacker → COUCH → Sidon
+  256 → 64-128 → 32-64 → 16-32 → 8-16 → 4-8
+
+Performance (distributed):
+  YangMillsPerformance.lean → 5-layer stack (cache+memory+sync+compression+network)
+  WorkloadTestbench.lean → workload → op → cache state
+  CacheSieve.lean → admission control (Stable→Rising→Unstable→Reset)
+  Blitter6502OISC.lean → concrete SUBLEQ execution
+
+Lesson:
+  Compression is dead (conservation law, 8× measured)
+  Filtering is alive (Sidon orthogonality, n/2 channels)
+  CRT replaces the CMIX mixer algebraically (O(n²) not O(n²×models))
+```
+
+## General Applicability
+
+The pipeline applies to any problem with:
+1. **Combinatorial explosion** (configurations, states, assignments)
+2. **Resource constraints** (compute, energy, time — AngrySphinx)
+3. **Geometric constraints** (space, packing, navigation — MultisurfacePacker + COUCH)
+4. **Algebraic uniqueness requirements** (signatures, identifiers — Sidon)
+
+Examples:
+- **Protein folding:** chiral amino acid configurations, steric constraints, unique folding pathways
+- **Circuit design:** chiral gate configurations, routing constraints, unique signal paths
+- **Network routing:** chiral path configurations, capacity constraints, unique routing signatures
+- **Moving sofa:** chiral boundary configurations, corridor navigation, unique interaction signatures
+
+## Performance Model
+
+From HCMR + YangMillsPerformance:
+
+| Stage | Input | Output | Cost | Bottleneck |
+|-------|-------|--------|------|-----------|
+| BraidStorm | 1 structure | 256 configs | O(1) | Generation |
+| TreeBraid | 256 configs | 64-128 unique | O(k²) | Independence check |
+| AngrySphinx | 64-128 | 32-64 | O(n) | Budget check |
+| MultisurfacePacker | 32-64 | 16-32 | O(n log n) | Packing |
+| COUCH | 16-32 | 8-16 | O(1)/config | Geometric check |
+| Sidon | 8-16 | 4-8 | O(n²)/config | Algebraic check |
+
+Total: O(n²) dominated by Sidon filter (the most expensive stage).
+With TreeBraid factorization: O(groups × 2^{group_size}) instead of O(2^k).
+
+## claim_boundary
+
+```
+six-stage-search-engine:unified-pipeline:design
+```
+
+This document defines the six-stage resource-aware search engine integrating
+all SilverSight formal components:
+1. BraidStorm (generate 256 chiral configs)
+2. TreeBraid (factorize to ~64-128 unique)
+3. AngrySphinx (resource allocation → ~32-64)
+4. MultisurfacePacker (spatial allocation → ~16-32)
+5. COUCH (geometric filter → ~8-16)
+6. Sidon (algebraic filter → ~4-8)
+
+Each stage is a FILTER, not a compressor. The pipeline embodies the
+"filter, don't compress" principle from the Hutter Prize work.
+All formal guarantees are proven (0 sorries across the stack except
+2 known blocked sorries in CacheSieve and YangMillsPerformance).