Claude-Code-Workflow/.codex/skill_lib/numerical-analysis-workflow/specs/phase-topology.md

Phase Topology — Diamond Deep Tree

Wave coordination patterns for the Numerical Analysis Diamond Workflow (NADW).

Purpose

Phase	Usage
Phase 1 (Decomposition)	Reference when assigning waves and dependencies
Phase 2 (Execution)	Context flow between waves
Phase 3 (Aggregation)	Structure the final report by topology

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1. Topology Overview

The NADW uses a Staged Diamond topology — six sequential waves, each with 3 parallel tracks. Context flows cumulatively from earlier waves to later ones.

         Wave 1: [T1.1] [T1.2] [T1.3]     Global Survey (3 parallel)
              ↓ Context P1
         Wave 2: [T2.1] [T2.2] [T2.3]     Theory (3 parallel)
              ↓ Context P1+P2
         Wave 3: [T3.1] [T3.2] [T3.3]     Algorithm (3 parallel)
              ↓ Context P1+P2+P3
         Wave 4: [T4.1] [T4.2] [T4.3]     Module (3 parallel)
              ↓ Context P1-P4
         Wave 5: [T5.1] [T5.2] [T5.3]     Local (3 parallel)
              ↓ Context P1-P5
         Wave 6: [T6.1] [T6.2] [T6.3]     Integration (3 parallel)

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2. Wave Definitions

Wave 1: Global Survey

Property	Value
Phase Name	Global Survey
Track Count	3
Dependencies	None (entry wave)
Context Input	Project codebase only
Context Output	Governing equations, architecture map, validation KPIs
Max Parallelism	3

Tracks:

ID	Role	Dimension	Scope
T1.1	Problem_Domain_Analyst	domain_modeling	Full project
T1.2	Software_Architect	architecture_analysis	Full project
T1.3	Validation_Strategist	validation_design	Full project

Wave 2: Theoretical Foundations

Property	Value
Phase Name	Theoretical Foundations
Track Count	3
Dependencies	Wave 1
Context Input	Context Package P1
Context Output	LaTeX formulas, convergence theorems, complexity bounds
Max Parallelism	3

Tracks:

ID	Role	Dimension	Deps	context_from
T2.1	Mathematician	formula_derivation	T1.1	T1.1
T2.2	Convergence_Analyst	convergence_analysis	T1.1	T1.1
T2.3	Complexity_Analyst	complexity_analysis	T1.1;T1.2	T1.1;T1.2

Wave 3: Algorithm Design & Stability

Property	Value
Phase Name	Algorithm Design
Track Count	3
Dependencies	Wave 2
Context Input	Context Package P1+P2
Context Output	Pseudocode, stability conditions, performance model
Max Parallelism	3

Tracks:

ID	Role	Dimension	Deps	context_from
T3.1	Algorithm_Designer	method_selection	T2.1	T2.1;T2.2;T2.3
T3.2	Stability_Analyst	stability_analysis	T2.1;T2.2	T2.1;T2.2
T3.3	Performance_Modeler	performance_modeling	T2.3	T1.2;T2.3

Wave 4: Module Implementation

Property	Value
Phase Name	Module Implementation
Track Count	3
Dependencies	Wave 3
Context Input	Context Package P1-P3
Context Output	Code-algorithm mapping, data structure decisions, API contracts
Max Parallelism	3

Tracks:

ID	Role	Dimension	Deps	context_from
T4.1	Module_Implementer	implementation_analysis	T3.1	T1.2;T3.1
T4.2	Data_Structure_Designer	data_structure_review	T3.1;T3.3	T2.3;T3.1;T3.3
T4.3	Interface_Analyst	interface_analysis	T3.1	T1.2;T3.1

Wave 5: Local Function-Level

Property	Value
Phase Name	Local Function-Level
Track Count	3
Dependencies	Wave 4
Context Input	Context Package P1-P4
Context Output	Optimization recommendations, edge case catalog, precision risk matrix
Max Parallelism	3

Tracks:

ID	Role	Dimension	Deps	context_from
T5.1	Code_Optimizer	optimization	T4.1	T3.3;T4.1
T5.2	Edge_Case_Analyst	edge_case_analysis	T4.1	T1.1;T3.2;T4.1
T5.3	Precision_Auditor	precision_audit	T4.1;T4.2	T3.2;T4.1;T4.2

Wave 6: Integration & QA

Property	Value
Phase Name	Integration & QA
Track Count	3
Dependencies	Wave 5
Context Input	Context Package P1-P5 (ALL cumulative)
Context Output	Final test plan, benchmark report, QA assessment
Max Parallelism	3

Tracks:

ID	Role	Dimension	Deps	context_from
T6.1	Integration_Tester	integration_testing	T5.1;T5.2	T1.3;T5.1;T5.2
T6.2	Benchmark_Engineer	benchmarking	T5.1	T1.3;T3.3;T5.1
T6.3	QA_Auditor	quality_assurance	T5.1;T5.2;T5.3	T1.1;T2.1;T3.1;T4.1;T5.1;T5.2;T5.3

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3. Context Flow Map

Directed Context (prev_context column)

T1.1 ──► T2.1, T2.2, T2.3
T1.2 ──► T2.3, T3.3, T4.1, T4.3
T1.3 ──► T6.1, T6.2
T2.1 ──► T3.1, T3.2
T2.2 ──► T3.1, T3.2
T2.3 ──► T3.1, T3.3, T4.2
T3.1 ──► T4.1, T4.2, T4.3
T3.2 ──► T5.2, T5.3
T3.3 ──► T4.2, T5.1, T6.2
T4.1 ──► T5.1, T5.2, T5.3
T4.2 ──► T5.3
T5.1 ──► T6.1, T6.2
T5.2 ──► T6.1
T5.3 ──► T6.3

Broadcast Context (discoveries.ndjson)

All agents read/append to the same discoveries.ndjson. Key discovery types flow across waves:

Wave 1: governing_equation, architecture_pattern ──► all subsequent waves
Wave 2: convergence_property ──► Wave 3-6
Wave 3: stability_issue, numerical_method ──► Wave 4-6
Wave 4: (implementation findings) ──► Wave 5-6
Wave 5: precision_risk, performance_bottleneck ──► Wave 6

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4. Perspective Reuse Matrix

How each wave's output is reused by later waves:

Source	P2 Reuse	P3 Reuse	P4 Reuse	P5 Reuse	P6 Reuse
P1 Equations	Formalize → LaTeX	Constrain methods	Code-eq mapping	Singularity sources	Correctness baseline
P1 Architecture	Constrain discretization	Parallel strategy	Module boundaries	Hotspot location	Integration scope
P1 Validation	-	Benchmark selection	Test criteria	Edge case sources	Final validation
P2 Formulas	-	Parameter constraints	Loop termination	Precision requirements	Theory verification
P2 Convergence	-	Mesh refinement	Iteration control	Error tolerance	Rate verification
P2 Complexity	-	Performance baseline	Data structure choice	Optimization targets	Perf comparison
P3 Pseudocode	-	-	Impl reference	Line-by-line audit	Regression baseline
P3 Stability	-	-	Precision selection	Cancellation detection	Numerical verification
P3 Performance	-	-	Memory layout	Vectorization targets	Benchmark targets
P4 Modules	-	-	-	Function-level focus	Module test plan
P5 Optimization	-	-	-	-	Performance tests
P5 Edge Cases	-	-	-	-	Regression tests
P5 Precision	-	-	-	-	Numerical tests

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5. Diamond Properties

Property	Value
Total Waves	6
Total Tasks	18 (3 per wave)
Max Parallelism per Wave	3
Widest Context Fan-in	T6.3 (receives from 7 tasks)
Deepest Dependency Chain	T1.1 → T2.1 → T3.1 → T4.1 → T5.1 → T6.1 (depth 6)
Context Accumulation	Cumulative (each wave adds to previous context)
Topology Type	Staged Parallel with Diamond convergence at Wave 6