# Tasks: Phase 5 & Phase 6 (User Stories 3 & 4)

**Feature**: Grand Unified KGC Thesis
**Branch**: `012-grand-unified-kgc-thesis`
**Generated**: 2025-12-16

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## Phase 5: User Story 3 - Demonstrate 4D Temporal Event Sourcing (P2)

**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**:
1. Event Sourcing Calculus appears in Chapter 4 with nanosecond precision timestamps
2. GitBackbone state reconstruction algorithm matches @unrdf/kgc-4d implementation
3. Time-travel query example demonstrates reconstructing state at any past timestamp

**Deliverables**: Chapter 4 sections, Event types, reconstructState pseudocode, 6+ temporal theorems

### Tasks (T051-T060)

- [ ] **T051** [P] [US3] Define temporal event sourcing entities in RDF schema
  - **Path**: `/Users/sac/ggen/specs/012-grand-unified-kgc-thesis/ontology/thesis-schema.ttl`
  - **Desc**: Create temporal event types (CreateEvent, UpdateEvent, DeleteEvent) with nanosecond precision timestamp properties
  - **AC**: CreateEvent class with event:timestamp (xsd:long), UpdateEvent with oldValue/newValue, DeleteEvent with deletedQuad, event:causalPredecessor relationship
  - **Time**: 1h

- [ ] **T052** [P] [US3] Author Chapter 4 introduction and temporal model overview
  - **Path**: `/Users/sac/ggen/specs/012-grand-unified-kgc-thesis/ontology/kgc-unified-content.ttl`
  - **Desc**: Write Chapter 4 Section 1 introducing 4D spacetime coordinates and temporal RDF extensions
  - **AC**: Introduction establishes 4D coordinate system (x,y,z,t), temporal RDF motivation, @unrdf/kgc-4d references, sec:temporal-model with 500+ words
  - **Time**: 2h

- [ ] **T053** [US3] Define Event Sourcing Calculus equations (6+ temporal theorems)
  - **Path**: `/Users/sac/ggen/specs/012-grand-unified-kgc-thesis/ontology/kgc-unified-content.ttl`
  - **Desc**: Create 6 equations: state transition, state reconstruction, causal ordering, snapshot creation, time-travel query, temporal consistency
  - **AC**: δ(G,e)=G∪{τ}, ρ(t)=fold(δ,∅,{e|e.t≤t}), e₁≺e₂⟺e₁.t<e₂.t, freeze(t)=ρ(t), query(Q,t)=SPARQL(Q,ρ(t)), ∀t₁<t₂:ρ(t₁)⊆ρ(t₂), all eq:* labeled
  - **Deps**: T051, T052
  - **Time**: 3h

- [ ] **T054** [US3] Prove Deterministic Reconstruction Theorem
  - **Path**: `/Users/sac/ggen/specs/012-grand-unified-kgc-thesis/ontology/kgc-unified-content.ttl`
  - **Desc**: Formal proof that state reconstruction from event log is deterministic and complete
  - **AC**: Theorem 'For any t, ρ(t) is unique and deterministic', proof uses fold associativity, references eq:causal-order, labeled thm:deterministic-reconstruction, proof ≥300 words
  - **Deps**: T053
  - **Time**: 2h

- [ ] **T055** [US3] Prove Causal Consistency Theorem
  - **Path**: `/Users/sac/ggen/specs/012-grand-unified-kgc-thesis/ontology/kgc-unified-content.ttl`
  - **Desc**: Formal proof that event ordering preserves causal relationships
  - **AC**: Theorem 'If e₁ causes e₂, then e₁.t<e₂.t', proof by contradiction using causalPredecessor, references temporal monotonicity, labeled thm:causal-consistency, includes lemma about transitive closure
  - **Deps**: T053
  - **Time**: 2h

- [ ] **T056** [US3] Prove Event Immutability Theorem
  - **Path**: `/Users/sac/ggen/specs/012-grand-unified-kgc-thesis/ontology/kgc-unified-content.ttl`
  - **Desc**: Formal proof that past events cannot be modified, only compensating events added
  - **AC**: Theorem 'For t₁<t₂, events at t₁ are immutable in ρ(t₂)', proof uses append-only property, references state reconstruction, labeled thm:event-immutability, discusses compensating events
  - **Deps**: T053, T054
  - **Time**: 1.5h

- [ ] **T057** [P] [US3] Document GitBackbone state reconstruction algorithm
  - **Path**: `/Users/sac/ggen/specs/012-grand-unified-kgc-thesis/ontology/kgc-unified-content.ttl`
  - **Desc**: Write Algorithm: reconstructState(t) with pseudocode matching @unrdf/kgc-4d implementation
  - **AC**: Algorithm titled 'State Reconstruction from Event Log', Input: timestamp t (ns), event log E, Output: ρ(t), Steps: filter≤t, sort, fold with δ, labeled alg:reconstruct-state, algorithm2e syntax
  - **Deps**: T053
  - **Time**: 2h

- [ ] **T058** [P] [US3] Document freezeUniverse snapshot algorithm
  - **Path**: `/Users/sac/ggen/specs/012-grand-unified-kgc-thesis/ontology/kgc-unified-content.ttl`
  - **Desc**: Write Algorithm: freezeUniverse() for creating immutable snapshots
  - **AC**: Algorithm titled 'Snapshot Creation', Input: event log E, Output: snapshot S with timestamp, Steps: get time, reconstruct, persist, labeled alg:freeze-universe, references reconstructState
  - **Deps**: T057
  - **Time**: 1.5h

- [ ] **T059** [US3] Create time-travel query example demonstrating state reconstruction
  - **Path**: `/Users/sac/ggen/specs/012-grand-unified-kgc-thesis/ontology/kgc-unified-content.ttl`
  - **Desc**: Concrete example showing reconstruction of project state at three timestamps
  - **AC**: Example with CreateEvent at t₁, UpdateEvent at t₂, DeleteEvent at t₃, SPARQL queries reconstructing state at each t, figure showing timeline (fig:time-travel)
  - **Deps**: T057, T058
  - **Time**: 2h

- [ ] **T060** [US3] Add 3 temporal theorems proving completeness guarantees
  - **Path**: `/Users/sac/ggen/specs/012-grand-unified-kgc-thesis/ontology/kgc-unified-content.ttl`
  - **Desc**: Define theorems: Temporal Completeness, Audit Trail Integrity, Snapshot Consistency
  - **AC**: Theorem: every past state reconstructible, event log preserves causality, frozen states never change, each proof ≥200 words, labeled thm:temporal-complete, thm:audit-integrity, thm:snapshot-consistent
  - **Deps**: T054, T055, T056
  - **Time**: 3h

**Phase 5 Summary**: 10 tasks, 20 hours, 3 parallelizable

---

## Phase 6: User Story 4 - Explain Hyperdimensional Information Theory (P2)

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

**Acceptance Criteria**:
1. Hyperdimensional encoding equations with dimensional analysis
2. Shannon entropy applied to ontology distributions with fidelity bounds
3. Mutual information proofs are tight

**Deliverables**: Chapter 3 sections, 15+ equations (Shannon entropy, mutual information, HD encoding, etc.), Semantic fidelity theorem with proof

### Tasks (T061-T075)

- [ ] **T061** [P] [US4] Define hyperdimensional vector space entities in RDF schema
  - **Path**: `/Users/sac/ggen/specs/012-grand-unified-kgc-thesis/ontology/thesis-schema.ttl`
  - **Desc**: Create HD encoding classes: HypervectorSpace, TripleEncoding, SemanticProjection with dimension properties
  - **AC**: HypervectorSpace with hd:dimension=10000, TripleEncoding with subjectVector/predicateVector/objectVector, SemanticProjection with sourceOntology/targetCode/fidelity, hd:bindingOperation for ⊗
  - **Time**: 1h

- [ ] **T062** [P] [US4] Author Chapter 3 introduction to hyperdimensional information theory
  - **Path**: `/Users/sac/ggen/specs/012-grand-unified-kgc-thesis/ontology/kgc-unified-content.ttl`
  - **Desc**: Write Chapter 3 Section 1 introducing HD computing, vector symbolic architectures, info-theoretic foundations
  - **AC**: Introduction establishes 10000D spaces, motivates HD for semantic similarity, references Shannon theory and rate-distortion, sec:hd-intro with 600+ words, cites ACM/IEEE papers
  - **Time**: 2.5h

- [ ] **T063** [US4] Define Shannon entropy equations for ontology distributions (5 equations)
  - **Path**: `/Users/sac/ggen/specs/012-grand-unified-kgc-thesis/ontology/kgc-unified-content.ttl`
  - **Desc**: Create equations: Shannon entropy H(O), conditional entropy H(O|C), joint entropy H(O,C), mutual information I(O;C), KL divergence D(O||C)
  - **AC**: H(O)=-Σp(t)log₂p(t), H(O|C)=-Σp(t,c)log₂p(t|c), H(O,C)=H(O)+H(C|O), I(O;C)=H(O)-H(O|C), D(O||C)=Σp(t)log₂(p(t)/q(t)), labeled eq:entropy through eq:kl-div
  - **Deps**: T061, T062
  - **Time**: 2h

- [ ] **T064** [US4] Define hyperdimensional encoding equations with dimensional analysis (5 equations)
  - **Path**: `/Users/sac/ggen/specs/012-grand-unified-kgc-thesis/ontology/kgc-unified-content.ttl`
  - **Desc**: Create equations: HD triple encoding, binding, bundling, similarity, projection
  - **AC**: encode(s,p,o)=S⊗P⊗O (R³→R^D), bind(A,B)=A⊗B (circular conv), bundle(V₁,...,Vₙ)=ΣᵢVᵢ, sim(A,B)=cos(θ)=(A·B)/(||A||||B||) (→[0,1]), project(H,T)=argmin_C D(encode(H)||encode(C)), dimensional analysis included
  - **Deps**: T063
  - **Time**: 2.5h

- [ ] **T065** [US4] Define semantic fidelity and rate-distortion equations (5 equations)
  - **Path**: `/Users/sac/ggen/specs/012-grand-unified-kgc-thesis/ontology/kgc-unified-content.ttl`
  - **Desc**: Create equations: semantic fidelity Φ, distortion D, rate-distortion R(D), capacity C, fidelity bounds
  - **AC**: Φ=I(O;C)/H(O) (∈[0,1]), D(o,c)=1-sim(encode(o),encode(c)), R(D)=min I(O;C) s.t. E[D]≤D, C=max I(O;C), Φ≥1-ε for ε-bounded distortion, all eq:* labeled with units
  - **Deps**: T064
  - **Time**: 3h

- [ ] **T066** [US4] Prove Semantic Fidelity Theorem with tight bounds
  - **Path**: `/Users/sac/ggen/specs/012-grand-unified-kgc-thesis/ontology/kgc-unified-content.ttl`
  - **Desc**: Formal proof that Φ is bounded by mutual information ratio with achievable bounds
  - **AC**: Theorem 'Φ∈[0,1] with Φ=1⟺I(O;C)=H(O)', proof shows Φ=1→perfect preservation, Φ=0→independence, constructive proof of achievability, labeled thm:semantic-fidelity, proof ≥400 words
  - **Deps**: T065
  - **Time**: 3h

- [ ] **T067** [US4] Prove Multi-Target Superadditivity Proposition
  - **Path**: `/Users/sac/ggen/specs/012-grand-unified-kgc-thesis/ontology/kgc-unified-content.ttl`
  - **Desc**: Proof that N targets from single ontology has higher total fidelity than N independent specs
  - **AC**: Proposition 'ΣᵢΦ(O→Cᵢ) > ΣᵢΦ(Oᵢ→Cᵢ) for shared O', proof uses data processing inequality, shows cross-module consistency increases info preservation, references Zero-Drift Theorem, labeled prop:multi-target-superadditivity
  - **Deps**: T066
  - **Time**: 2.5h

- [ ] **T068** [P] [US4] Document HD encoding algorithm with dimensional analysis
  - **Path**: `/Users/sac/ggen/specs/012-grand-unified-kgc-thesis/ontology/kgc-unified-content.ttl`
  - **Desc**: Write Algorithm: encodeTriple(s,p,o) showing circular convolution implementation
  - **AC**: Algorithm titled 'Hyperdimensional Triple Encoding', Input: (s,p,o), D=10000, Output: H∈R^D, Steps: generate random vectors, circular conv, normalize, labeled alg:hd-encode, complexity O(D log D) via FFT
  - **Deps**: T064
  - **Time**: 2h

- [ ] **T069** [P] [US4] Document semantic fidelity calculation algorithm
  - **Path**: `/Users/sac/ggen/specs/012-grand-unified-kgc-thesis/ontology/kgc-unified-content.ttl`
  - **Desc**: Write Algorithm: calculateFidelity(O,C) measuring information preservation
  - **AC**: Algorithm titled 'Semantic Fidelity Measurement', Input: ontology O (RDF), code C (AST), Output: Φ∈[0,1], Steps: encode O and C in HD, compute mutual info, calculate Φ, labeled alg:calc-fidelity, references entropy subroutines
  - **Deps**: T065
  - **Time**: 2h

- [ ] **T070** [US4] Create concrete HD encoding example for RDF triple
  - **Path**: `/Users/sac/ggen/specs/012-grand-unified-kgc-thesis/ontology/kgc-unified-content.ttl`
  - **Desc**: Work through complete example encoding (:Task1 rdf:type :ProjectTask) in 10000D
  - **AC**: Example shows random vector generation, circular convolution (FFT), bundling for multiple triples, similarity calculation, figure fig:hd-encoding, numerical results for sim(encode(t1),encode(t2))
  - **Deps**: T068
  - **Time**: 2.5h

- [ ] **T071** [US4] Create table showing fidelity measurements across targets
  - **Path**: `/Users/sac/ggen/specs/012-grand-unified-kgc-thesis/ontology/kgc-unified-content.ttl`
  - **Desc**: Empirical table comparing semantic fidelity for Rust, TypeScript, Python from same ontology
  - **AC**: Table 3 rows (Rust/TS/Python) × 4 cols (H(O), I(O;C), Φ, D), Rust highest fidelity (types preserve info), Python lowest (dynamic typing), values from actual ggen, labeled tab:fidelity-comparison, references Chapter 5 TanStack case study
  - **Deps**: T069, T070
  - **Time**: 2h

- [ ] **T072** [US4] Prove HD Dimensionality Lemma (capacity vs. collision)
  - **Path**: `/Users/sac/ggen/specs/012-grand-unified-kgc-thesis/ontology/kgc-unified-content.ttl`
  - **Desc**: Lemma proving D=10000 provides sufficient capacity with negligible collision probability
  - **AC**: Lemma 'For D≥10000, P(sim(A,B)>0.9)<10⁻⁶ for random A,B', proof uses birthday paradox, shows C≥log₂(2^D) bits, labeled lem:hd-dimensionality, references Johnson-Lindenstrauss
  - **Deps**: T064, T066
  - **Time**: 2.5h

- [ ] **T073** [US4] Prove Projection Distortion Bound Theorem
  - **Path**: `/Users/sac/ggen/specs/012-grand-unified-kgc-thesis/ontology/kgc-unified-content.ttl`
  - **Desc**: Theorem proving distortion in projection from ontology to code is bounded by rate-distortion
  - **AC**: Theorem 'For any L, E[D(O,C)]≥R⁻¹(H(O))', proof uses rate-distortion theory, shows bound is tight (achievable), labeled thm:projection-distortion, discusses multi-target implications
  - **Deps**: T065, T067
  - **Time**: 3h

- [ ] **T074** [P] [US4] Create figure showing HD embedding space visualization
  - **Path**: `/Users/sac/ggen/specs/012-grand-unified-kgc-thesis/ontology/kgc-unified-content.ttl`
  - **Desc**: Design 3D projection of 10000D space showing ontology concepts as clusters
  - **AC**: Figure shows t-SNE/PCA to 3D, clusters for related concepts (all :Task instances), distance=semantic similarity, labeled fig:hd-embedding-space, caption explains dimensionality reduction, TikZ or external image
  - **Deps**: T070
  - **Time**: 2h

- [ ] **T075** [US4] Add appendix with extended information theory proofs
  - **Path**: `/Users/sac/ggen/specs/012-grand-unified-kgc-thesis/ontology/kgc-unified-content.ttl`
  - **Desc**: Create Appendix A with detailed proofs of info-theoretic lemmas from Chapter 3
  - **AC**: Appendix A 'Extended Mathematical Proofs', contains data processing inequality proof, Fano's inequality application, channel coding theorem relevance, labeled app:extended-proofs, cross-references Chapter 3
  - **Deps**: T066, T067, T072, T073
  - **Time**: 2.5h

**Phase 6 Summary**: 15 tasks, 37 hours, 4 parallelizable

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## Overall Summary

| Metric | Phase 5 (US3) | Phase 6 (US4) | Total |
|--------|---------------|---------------|-------|
| **Tasks** | 10 | 15 | 25 |
| **Estimated Hours** | 20 | 37 | 57 |
| **Parallelizable** | 3 | 4 | 7 |
| **Theorems/Lemmas** | 9 | 5 | 14 |
| **Equations** | 6 | 15 | 21 |
| **Algorithms** | 2 | 2 | 4 |
| **Examples** | 1 | 1 | 2 |
| **Figures** | 1 | 1 | 2 |
| **Tables** | 0 | 1 | 1 |
| **Appendices** | 0 | 1 | 1 |

**Critical Path**: T051→T053→T054→T056→T060 (Phase 5), T061→T063→T064→T065→T066→T067→T073→T075 (Phase 6)

**Parallel Opportunities**:
- Phase 5: T051, T052, T057, T058 can start independently
- Phase 6: T061, T062, T068, T069, T074 can start independently

**Key Dependencies**:
- All US3 tasks require temporal event schema (T051) and calculus equations (T053)
- All US4 tasks require HD vector space schema (T061) and entropy equations (T063)
- Theorems build on equations (must complete equations first)
- Examples and algorithms can proceed in parallel with some theorem proofs

**Quality Gates**:
- All equations must have dimensional analysis and eq:* labels
- All theorems must have proofs ≥200 words with step-by-step arguments
- All algorithms must use algorithm2e syntax (\\KwIn, \\KwOut, \\For, \\Return)
- All figures must have captions and labeled references
- Cross-references must be valid (no "??" in LaTeX compilation)

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**Legend**:
- `[P]` = Parallelizable (can start without waiting for prior tasks)
- `[US3]` = User Story 3 (4D Temporal Event Sourcing)
- `[US4]` = User Story 4 (Hyperdimensional Information Theory)
- **Deps** = Task dependencies (must complete before starting)
- **AC** = Acceptance criteria (verification requirements)
- **Time** = Estimated time to completion
