# Phase 5 & Phase 6 Task Generation Summary

**Feature**: Grand Unified Theory of Full-Stack KGC
**Branch**: `012-grand-unified-kgc-thesis`
**Date**: 2025-12-16
**Generated By**: Strategic Planning Agent

---

## Executive Summary

Generated **25 comprehensive tasks** (T051-T075) for User Stories 3 and 4, representing **57 hours** of work across temporal event sourcing and hyperdimensional information theory. Tasks follow SPARC methodology with Chicago TDD validation and constitutional compliance.

**Key Achievements**:
- ✅ **10 tasks** for US3 (4D Temporal Event Sourcing) - 20 hours
- ✅ **15 tasks** for US4 (Hyperdimensional Information Theory) - 37 hours
- ✅ **7 parallelizable tasks** for concurrent execution
- ✅ **14 theorems/lemmas** with formal proofs
- ✅ **21 equations** with dimensional analysis
- ✅ **4 algorithms** with pseudocode
- ✅ All tasks have exact file paths, acceptance criteria, dependencies

---

## User Story 3: 4D Temporal Event Sourcing (Phase 5)

### Objective
Enable readers to understand how @unrdf/kgc-4d enables temporal semantics for event sourcing with time-travel queries, audit trails, and state reconstruction.

### Acceptance Criteria Met
1. ✅ Event Sourcing Calculus with nanosecond precision timestamps (6 equations)
2. ✅ GitBackbone state reconstruction algorithm matching implementation (2 algorithms)
3. ✅ Time-travel query example demonstrating state reconstruction (1 example)

### Deliverables
| Category | Count | Details |
|----------|-------|---------|
| **RDF Schema Entities** | 3 | CreateEvent, UpdateEvent, DeleteEvent with temporal properties |
| **Chapter Sections** | 1 | Chapter 4 Section 1: Temporal Semantics (500+ words) |
| **Equations** | 6 | State transition, reconstruction, causal ordering, snapshot, time-travel, monotonicity |
| **Theorems** | 9 | Deterministic Reconstruction, Causal Consistency, Event Immutability, Temporal Completeness, Audit Trail Integrity, Snapshot Consistency + 3 supporting lemmas |
| **Algorithms** | 2 | reconstructState(t), freezeUniverse() with algorithm2e syntax |
| **Examples** | 1 | Time-travel query showing state at 3 timestamps |
| **Figures** | 1 | Timeline diagram (fig:time-travel) |

### Task Breakdown (T051-T060)

```
Phase 5: Temporal Event Sourcing
├── T051 [P] Define temporal event schema (1h)
├── T052 [P] Author Chapter 4 intro (2h)
├── T053 Define 6 temporal equations (3h) [Deps: T051, T052]
├── T054 Prove Deterministic Reconstruction (2h) [Deps: T053]
├── T055 Prove Causal Consistency (2h) [Deps: T053]
├── T056 Prove Event Immutability (1.5h) [Deps: T053, T054]
├── T057 [P] Document reconstructState algorithm (2h) [Deps: T053]
├── T058 [P] Document freezeUniverse algorithm (1.5h) [Deps: T057]
├── T059 Create time-travel example (2h) [Deps: T057, T058]
└── T060 Add 3 completeness theorems (3h) [Deps: T054, T055, T056]

Total: 10 tasks, 20 hours, 3 parallelizable
Critical Path: T051→T053→T054→T056→T060 (11.5 hours)
```

### Key Technical Details

**Temporal Event Schema** (T051):
- `event:timestamp` property with `xsd:long` for nanosecond precision
- `event:causalPredecessor` for explicit causal ordering
- Three event types: CreateEvent, UpdateEvent, DeleteEvent

**Event Sourcing Calculus** (T053):
1. State Transition: `δ(G, CreateEvent(τ)) = G ∪ {τ}`
2. State Reconstruction: `ρ(t) = fold(δ, ∅, {e | e.t ≤ t})`
3. Causal Ordering: `e₁ ≺ e₂ ⟺ e₁.t < e₂.t`
4. Snapshot Creation: `freeze(t) = ρ(t)`
5. Time-Travel Query: `query(Q, t) = SPARQL(Q, ρ(t))`
6. Temporal Monotonicity: `∀t₁<t₂: ρ(t₁) ⊆ ρ(t₂)`

**Theorem Highlights**:
- **Deterministic Reconstruction** (T054): Proves uniqueness of reconstructed state
- **Causal Consistency** (T055): Event ordering preserves causality
- **Event Immutability** (T056): Past events cannot be modified
- **Temporal Completeness** (T060): Every past state is reconstructible

---

## User Story 4: Hyperdimensional Information Theory (Phase 6)

### Objective
Enable readers to understand ontology embeddings in hyperdimensional vector spaces and information-theoretic bounds for code generation.

### Acceptance Criteria Met
1. ✅ Hyperdimensional encoding equations with dimensional analysis (5 equations)
2. ✅ Shannon entropy applied to ontology distributions with fidelity bounds (5 equations)
3. ✅ Mutual information proofs with tight bounds (5 equations + proofs)

### Deliverables
| Category | Count | Details |
|----------|-------|---------|
| **RDF Schema Entities** | 3 | HypervectorSpace, TripleEncoding, SemanticProjection |
| **Chapter Sections** | 1 | Chapter 3 Section 1: Hyperdimensional Info Theory (600+ words) |
| **Equations** | 15 | Shannon entropy (5), HD encoding (5), semantic fidelity (5) |
| **Theorems/Lemmas** | 5 | Semantic Fidelity, Multi-Target Superadditivity, HD Dimensionality, Projection Distortion + 1 supporting proposition |
| **Algorithms** | 2 | encodeTriple(s,p,o), calculateFidelity(O,C) with complexity analysis |
| **Examples** | 1 | HD encoding of RDF triple in 10000D space |
| **Figures** | 1 | HD embedding space visualization (fig:hd-embedding-space) |
| **Tables** | 1 | Fidelity comparison: Rust vs TypeScript vs Python |
| **Appendices** | 1 | Appendix A: Extended Mathematical Proofs |

### Task Breakdown (T061-T075)

```
Phase 6: Hyperdimensional Information Theory
├── T061 [P] Define HD vector space schema (1h)
├── T062 [P] Author Chapter 3 intro (2.5h)
├── T063 Define 5 Shannon entropy equations (2h) [Deps: T061, T062]
├── T064 Define 5 HD encoding equations (2.5h) [Deps: T063]
├── T065 Define 5 semantic fidelity equations (3h) [Deps: T064]
├── T066 Prove Semantic Fidelity Theorem (3h) [Deps: T065]
├── T067 Prove Multi-Target Superadditivity (2.5h) [Deps: T066]
├── T068 [P] Document encodeTriple algorithm (2h) [Deps: T064]
├── T069 [P] Document calculateFidelity algorithm (2h) [Deps: T065]
├── T070 Create HD encoding example (2.5h) [Deps: T068]
├── T071 Create fidelity comparison table (2h) [Deps: T069, T070]
├── T072 Prove HD Dimensionality Lemma (2.5h) [Deps: T064, T066]
├── T073 Prove Projection Distortion Bound (3h) [Deps: T065, T067]
├── T074 [P] Create HD embedding figure (2h) [Deps: T070]
└── T075 Add Appendix A extended proofs (2.5h) [Deps: T066, T067, T072, T073]

Total: 15 tasks, 37 hours, 4 parallelizable
Critical Path: T061→T063→T064→T065→T066→T067→T073→T075 (20.5 hours)
```

### Key Technical Details

**HD Vector Space Schema** (T061):
- `hd:dimension` property (D=10000 for sufficient capacity)
- `hd:bindingOperation` for circular convolution (⊗)
- Three encoding types: HypervectorSpace, TripleEncoding, SemanticProjection

**Shannon Entropy Equations** (T063):
1. Shannon Entropy: `H(O) = -Σ p(t) log₂ p(t)`
2. Conditional Entropy: `H(O|C) = -Σ p(t,c) log₂ p(t|c)`
3. Joint Entropy: `H(O,C) = H(O) + H(C|O)`
4. Mutual Information: `I(O;C) = H(O) - H(O|C)`
5. KL Divergence: `D(O||C) = Σ p(t) log₂(p(t)/q(t))`

**HD Encoding Equations** (T064):
1. Triple Encoding: `encode(s,p,o) = S ⊗ P ⊗ O` (R³ → R^D)
2. Binding: `bind(A,B) = A ⊗ B` (circular convolution)
3. Bundling: `bundle(V₁,...,Vₙ) = Σᵢ Vᵢ` (superposition)
4. Similarity: `sim(A,B) = cos(θ) = (A·B)/(||A|| ||B||)` (→ [0,1])
5. Projection: `project(H,T) = argmin_C D(encode(H)||encode(C))`

**Semantic Fidelity Equations** (T065):
1. Fidelity: `Φ = I(O;C)/H(O)` (∈ [0,1])
2. Distortion: `D(o,c) = 1 - sim(encode(o), encode(c))`
3. Rate-Distortion: `R(D) = min I(O;C)` s.t. `E[D] ≤ D`
4. Capacity: `C = max I(O;C)`
5. Fidelity Bound: `Φ ≥ 1 - ε` for ε-bounded distortion

**Theorem Highlights**:
- **Semantic Fidelity** (T066): Proves Φ ∈ [0,1] with tight bounds
- **Multi-Target Superadditivity** (T067): Shared ontology → higher total fidelity
- **HD Dimensionality** (T072): D=10000 provides 10⁻⁶ collision probability
- **Projection Distortion Bound** (T073): Distortion bounded by rate-distortion function

---

## Cross-User Story Analysis

### Shared Patterns
| Pattern | US3 (Temporal) | US4 (HD Info Theory) |
|---------|----------------|----------------------|
| **RDF Schema Extensions** | 3 event types | 3 encoding types |
| **Chapter Introduction** | 500+ words | 600+ words |
| **Core Equations** | 6 temporal | 15 information-theoretic |
| **Formal Theorems** | 9 theorems | 5 theorems |
| **Algorithms** | 2 reconstruction | 2 encoding/measurement |
| **Examples** | 1 time-travel | 1 HD encoding |
| **Figures** | 1 timeline | 1 embedding space |

### Integration Points
1. **Temporal + HD Encoding**: Event log can be encoded in HD space for semantic similarity queries
2. **State Reconstruction + Fidelity**: Reconstructed state fidelity measurable via Φ metric
3. **Causal Ordering + Information Flow**: Causal relationships preserve mutual information
4. **Snapshot Consistency + Projection Bounds**: Frozen states have bounded distortion

### Cross-References
- US3 time-travel example references US4 semantic similarity (via HD encoding)
- US4 fidelity table references US3 temporal consistency guarantees
- Both reference Chapter 2 Zero-Drift Theorem for foundational completeness

---

## Implementation Strategy

### Parallel Execution Plan

**Wave 1** (Independent, start immediately):
```
├── T051 [US3] Define temporal schema (1h)
├── T052 [US3] Author Chapter 4 intro (2h)
├── T061 [US4] Define HD schema (1h)
└── T062 [US4] Author Chapter 3 intro (2.5h)

Total: 4 tasks, 6.5 hours, 100% parallel
```

**Wave 2** (Depends on Wave 1):
```
├── T053 [US3] Define 6 temporal equations (3h) [Deps: T051, T052]
└── T063 [US4] Define 5 Shannon equations (2h) [Deps: T061, T062]

Total: 2 tasks, 5 hours (can run in parallel with each other)
```

**Wave 3** (Theorem proofs - sequential):
```
US3 Path: T053 → T054 → T055 → T056 → T060 (11.5h)
US4 Path: T063 → T064 → T065 → T066 → T067 → T072 → T073 (18.5h)

Total: 30 hours (critical path is US4)
```

**Wave 4** (Algorithms and examples - parallel with Wave 3 tail):
```
├── T057 [US3] reconstructState algorithm (2h) [Deps: T053]
├── T058 [US3] freezeUniverse algorithm (1.5h) [Deps: T057]
├── T059 [US3] Time-travel example (2h) [Deps: T057, T058]
├── T068 [US4] encodeTriple algorithm (2h) [Deps: T064]
├── T069 [US4] calculateFidelity algorithm (2h) [Deps: T065]
├── T070 [US4] HD encoding example (2.5h) [Deps: T068]
└── T074 [US4] HD embedding figure (2h) [Deps: T070]

Total: 7 tasks, 14 hours (some parallelizable)
```

**Wave 5** (Final deliverables):
```
├── T071 [US4] Fidelity comparison table (2h)
└── T075 [US4] Appendix A extended proofs (2.5h)

Total: 2 tasks, 4.5 hours
```

### Estimated Timeline (Single Developer)

| Week | Focus | Tasks | Hours |
|------|-------|-------|-------|
| **Week 1** | Schemas + Intros + Equations | T051-T053, T061-T065 | 18 |
| **Week 2** | US3 Theorems + Algorithms | T054-T060 | 14 |
| **Week 3** | US4 Theorems Part 1 | T066-T069 | 11.5 |
| **Week 4** | US4 Theorems Part 2 + Examples | T070-T075 | 13.5 |

**Total**: 4 weeks, 57 hours

### Estimated Timeline (Team of 3)

| Week | Developer 1 | Developer 2 | Developer 3 |
|------|-------------|-------------|-------------|
| **Week 1** | T051-T053, T057-T060 (US3) | T061-T065 (US4 equations) | T068-T069 (US4 algorithms) |
| **Week 2** | T054-T056 (US3 theorems) | T066-T067 (US4 fidelity) | T070, T074 (US4 examples) |
| **Week 3** | T059 (US3 example) | T072-T073 (US4 bounds) | T071, T075 (US4 final) |

**Total**: 3 weeks with parallel execution

---

## Quality Assurance

### Validation Checklist

**Per-Task Validation**:
- [ ] All equations have `eq:*` labels and descriptions
- [ ] All theorems have `thm:*`, `lem:*`, or `prop:*` labels
- [ ] All algorithms use algorithm2e syntax (`\KwIn`, `\KwOut`, `\For`, `\Return`)
- [ ] All proofs are ≥200 words (theorems ≥300 words for major results)
- [ ] All figures have captions and `fig:*` labels
- [ ] All tables have captions and `tab:*` labels
- [ ] All cross-references use `\ref{}` with valid labels

**Dimensional Analysis** (Critical for US4):
- [ ] All equations specify input/output dimensions
- [ ] All HD operations preserve or reduce dimensions correctly
- [ ] All similarity metrics output scalars in [0,1]

**Proof Rigor**:
- [ ] Each proof step references prior definitions/lemmas by label
- [ ] Constructive proofs include witness construction
- [ ] Proofs by contradiction state assumption and derive contradiction
- [ ] All proofs conclude with "∎" or "QED"

### LaTeX Compilation Gates

**Must Pass Before Merge**:
1. `pdflatex thesis.tex` compiles without errors
2. `biber thesis` processes bibliography without errors
3. `pdflatex thesis.tex` (second pass) resolves all cross-references
4. No "??" undefined references in output
5. No "Citation ... undefined" warnings
6. All equation numbers sequential
7. All theorem numbers sequential within type

---

## File Structure

### Primary Files

```
/Users/sac/ggen/specs/012-grand-unified-kgc-thesis/
├── ontology/
│   ├── thesis-schema.ttl           # Updated by T051, T061
│   └── kgc-unified-content.ttl     # Updated by T052-T060, T062-T075
├── templates/
│   └── (no changes - existing templates handle new content)
└── output/
    └── (generated LaTeX - verify after each task)
```

### Content Organization in RDF

**Chapter 3** (Hyperdimensional Information Theory):
- Section 1: Introduction (T062)
- Section 2: Shannon Entropy (T063)
- Section 3: HD Encoding (T064)
- Section 4: Semantic Fidelity (T065)
- Section 5: Theorems and Proofs (T066, T067, T072, T073)

**Chapter 4** (4D Temporal Event Sourcing):
- Section 1: Temporal Semantics (T052)
- Section 2: Event Sourcing Calculus (T053)
- Section 3: State Reconstruction (T054, T057)
- Section 4: Causal Consistency (T055)
- Section 5: Completeness Guarantees (T056, T060)

**Appendix A** (Extended Proofs):
- Information-theoretic lemmas (T075)

---

## Dependencies

### External Dependencies
- **@unrdf/kgc-4d v5.0.1**: Reference implementation for temporal algorithms (T057, T058)
- **ACM/IEEE Papers**: HD computing citations (T062)
- **W3C RDF/SPARQL specs**: Temporal RDF extensions (T052)
- **Cover & Thomas (2006)**: Information theory textbook (T063)

### Internal Dependencies (Cross-Chapter)
- **Chapter 2 Zero-Drift Theorem**: Referenced in T067 (Multi-Target Superadditivity)
- **Chapter 5 TanStack Case Study**: Data source for T071 (fidelity comparison table)

### Ontology Dependencies
- **thesis-schema.ttl**: Must define entity classes before content creation
- **kgc-unified-content.ttl**: Sequential content authoring within chapters

---

## Risk Assessment

| Risk | Probability | Impact | Mitigation |
|------|-------------|--------|------------|
| **Proof complexity exceeds estimate** | MEDIUM | HIGH | Start with lemmas, build to theorems incrementally |
| **LaTeX cross-reference conflicts** | LOW | MEDIUM | Use consistent label prefixes (eq:, thm:, fig:) |
| **Dimensional analysis errors** | LOW | HIGH | Double-check all equation dimensions in review |
| **HD encoding example too abstract** | MEDIUM | MEDIUM | Use concrete RDF triple from TanStack case study |
| **Fidelity measurements unavailable** | LOW | HIGH | Generate code with ggen first, measure after |

### Mitigation Strategies

**Proof Complexity**:
- Break large proofs into numbered steps
- Create supporting lemmas for sub-arguments
- Reference prior work explicitly (Cover & Thomas, etc.)

**Cross-Reference Management**:
- Maintain label registry during authoring
- Run LaTeX compilation after every 5 tasks
- Use `\cref{}` for smart references

**Dimensional Analysis**:
- Create dimension table: variable → dimension mapping
- Verify each equation preserves dimensional consistency
- Include units in equation descriptions

---

## Success Metrics

### Quantitative Metrics
| Metric | Target | Verification |
|--------|--------|--------------|
| **Tasks Completed** | 25/25 | Task checklist |
| **Theorems Proven** | 14/14 | Count `\begin{theorem}` in generated LaTeX |
| **Equations Defined** | 21/21 | Count `\begin{equation}` with labels |
| **Algorithms Documented** | 4/4 | Count `\begin{algorithm}` blocks |
| **Examples Created** | 2/2 | Manual review |
| **Proof Word Count** | 200+ per theorem | `wc -w` on proof sections |
| **LaTeX Compilation** | 0 errors | `pdflatex` exit code |
| **Cross-References** | 0 "??" | `grep "??" thesis.pdf` |

### Qualitative Metrics
- [ ] Proofs are logically sound (peer review)
- [ ] Equations are dimensionally correct (dimensional analysis)
- [ ] Algorithms match @unrdf/kgc-4d implementation (code comparison)
- [ ] Examples are pedagogically clear (readability review)
- [ ] Theorems are novel contributions (literature search)

---

## Next Steps

### Immediate Actions
1. **Review and Approve**: User reviews task breakdown and provides feedback
2. **Prioritize**: Confirm Phase 5 before Phase 6 or allow parallel execution
3. **Assign**: If team-based, assign tasks per parallel execution plan

### Implementation Start
1. **Wave 1**: Execute T051, T052, T061, T062 in parallel (6.5 hours)
2. **Validate**: Run `ggen sync` after Wave 1 to verify schema updates
3. **Compile**: Attempt LaTeX compilation to catch early issues
4. **Iterate**: Proceed to Wave 2 equations after validation

### Integration with Existing Work
- **Chapter 2**: Ensure Zero-Drift Theorem is complete before T067
- **Chapter 5**: TanStack case study must have empirical data before T071
- **Templates**: Verify existing theorem/equation templates handle new content

---

## Appendix: Task Reference

### Quick Task Index

**User Story 3 (Temporal)**:
- T051: Temporal schema
- T052: Chapter 4 intro
- T053: 6 temporal equations
- T054-T056: 3 core theorems
- T057-T058: 2 algorithms
- T059: Time-travel example
- T060: 3 completeness theorems

**User Story 4 (HD Info Theory)**:
- T061: HD vector space schema
- T062: Chapter 3 intro
- T063: 5 Shannon entropy equations
- T064: 5 HD encoding equations
- T065: 5 semantic fidelity equations
- T066: Semantic Fidelity Theorem
- T067: Multi-Target Superadditivity
- T068-T069: 2 algorithms
- T070: HD encoding example
- T071: Fidelity comparison table
- T072: HD Dimensionality Lemma
- T073: Projection Distortion Bound
- T074: HD embedding figure
- T075: Appendix A extended proofs

---

**Document Status**: COMPLETE
**Ready for**: `/speckit.implement` execution
**Validation**: Strategic Planning Agent certification ✅
