Visual Diagrams

This document contains visual representations of the framework's key concepts using Mermaid diagrams.

1. Equity Bridge Flow Diagram

The equity bridge shows how equity reconciles from one period to the next through various source terms:

flowchart TD
    Start["Opening Equity
E₀
$1,000M"] --> NI["+ Net Income
IAS 1.106
$50M"]
    NI --> OCI1["+ FVOCI Investments
IFRS 9.5.7.1
$28M"]
    OCI1 --> OCI2["+ FX Translation
IAS 21.52
$6M"]
    OCI2 --> OCI3["+ Actuarial Gains
IAS 19.93
$1M"]
    OCI3 --> DIV["- Dividends
IAS 1.107
$10M"]
    DIV --> BUY["- Treasury Stock
IAS 32.33
$25M"]
    BUY --> NCI["± NCI Changes
IFRS 10
$0M"]
    NCI --> End["Closing Equity
E₁
$1,050M"]

    style Start fill:#e1f5ff
    style End fill:#e1f5ff
    style NI fill:#d4edda
    style OCI1 fill:#fff3cd
    style OCI2 fill:#fff3cd
    style OCI3 fill:#fff3cd
    style DIV fill:#f8d7da
    style BUY fill:#f8d7da
    style NCI fill:#d1ecf1

Validation Check:

• Expected: $1,000 + $50 + $35 - $10 - $25 = $1,050 ✓
• If gap exists: Framework identifies missing OCI components or NCI changes

2. M&A Boundary Flux Diagram

This diagram illustrates the Reynolds Transport Theorem concept applied to accounting consolidation:

flowchart LR
    subgraph Before["Before Acquisition (t=0)"]
        direction TB
        A1["Company A
Equity: $500M
Consolidated Group"]
        B1["Company B
Equity: $100M
External Entity"]
    end

    subgraph During["Acquisition Event"]
        direction TB
        Arrow["Boundary Flux
ΦE = $100M
Entity crosses ∂V"]
    end

    subgraph After["After Acquisition (t=1)"]
        direction TB
        A2["Company A (Consolidated)
Parent Equity: $450M
NCI: $50M
Total: $500M"]
        B2["Company B
Now inside boundary
Contributes $50M NCI"]
    end

    Before --> During
    During --> After

    style Before fill:#fff3cd
    style During fill:#d1ecf1
    style After fill:#d4edda
    style Arrow fill:#f8d7da

Mathematical Form:

dE/dt = ∫(sources) dV + ∫(flux across ∂V) dA + ∫(boundary velocity · ρ) dA
        ↑                ↑                      ↑
        operational      static boundary        moving boundary (M&A)

Key Insight: Company B's $100M equity is NOT operational income—it's boundary flux. Framework distinguishes:

• ✅ Operational sources: Net income, OCI
• ✅ Boundary flux: M&A, NCI changes
• ⚠️ Misclassification → Framework detects as reconciliation error

3. Framework Architecture

Technical pipeline from raw XBRL to validated results:

flowchart TD
    subgraph Input["Data Sources"]
        EDGAR["SEC EDGAR
companyfacts API"]
        XBRL["XBRL Filings
.xml format"]
        CSV["Custom Dataset
.csv format"]
    end

    subgraph Parser["Parsing Layer"]
        Extract["XBRL Extractor
scripts/hydrate_companyfacts.py"]
        Map["Concept Mapper
config/concept_aliases.yaml"]
        Taxonomy["Standards Taxonomy
91 IFRS/GAAP mappings"]
    end

    subgraph Validator["Validation Engine"]
        Leverage["Leverage Identity
A = L + E"]
        Bridge["Equity Bridge
ΔE = sources"]
        NCI["NCI Reconciliation
IFRS 10 checks"]
        OCI["OCI Completeness
7 component checks"]
    end

    subgraph Output["Results"]
        Metrics["metrics.json
Pass/fail rates"]
        Report["HTML Reports
index.html"]
        Alerts["Risk Alerts
High-risk flags"]
    end

    Input --> Parser
    Parser --> Validator
    Validator --> Output

    style Input fill:#e1f5ff
    style Parser fill:#fff3cd
    style Validator fill:#d1ecf1
    style Output fill:#d4edda

Key Components:

1. Extractor: Fetches data from SEC EDGAR (rate-limited 10 req/sec)
2. Mapper: Handles XBRL tag variations (us-gaap vs. ifrs-full)
3. Taxonomy: Maps each tag to specific IFRS/GAAP standard
4. Validators: Apply mathematical constraints
5. Metrics: Statistical analysis with Wilson intervals

4. Audit Workflow Integration

How practitioners use the framework in real audit engagements:

flowchart TD
    Start([Audit Planning]) --> Risk["Risk Assessment
ML Model (AUC 0.949)"]
    Risk --> |High Risk| Priority["Prioritize Client"]
    Risk --> |Low Risk| Standard["Standard Procedures"]

    Priority --> Hydrate["Hydrate Client Data
scripts/hydrate_companyfacts.py"]
    Standard --> Hydrate

    Hydrate --> Validate["Run Validation
scripts/demo.py --ticker CLIENT"]

    Validate --> Results{Results?}

    Results --> |All Pass| Green["✓ Green Flag
Document & Continue"]
    Results --> |Failures| Red["⚠ Red Flag
Investigate"]

    Red --> Diagnose["Diagnose Gap
scripts/diagnose_equity_gap.py"]

    Diagnose --> Root{Root Cause?}

    Root --> |XBRL Gap| Request["Request OCI Rollforward
from Client"]
    Root --> |Disclosure Gap| Adjust["Adjust Expectations
Document Limitation"]
    Root --> |Potential Error| Escalate["Escalate to Partner
Consider Adjustment"]

    Request --> Revalidate["Re-run Validation"]
    Adjust --> Document["Document in Workpapers"]
    Escalate --> Document

    Revalidate --> Final{Resolved?}
    Final --> |Yes| Document
    Final --> |No| Escalate

    Green --> Final_Report([Issue Audit Opinion])
    Document --> Final_Report

    style Start fill:#e1f5ff
    style Risk fill:#fff3cd
    style Validate fill:#d1ecf1
    style Green fill:#d4edda
    style Red fill:#f8d7da
    style Final_Report fill:#e1f5ff

Time Savings:

• Traditional: 8-12 hours manual tracing per client
• Framework: 2 minutes automated validation + targeted follow-up
• Efficiency Gain: ~95% time reduction on reconciliation procedures

5. Validation Test Hierarchy

Relationship between different validation tests:

graph TD
    Root["Financial Statement Validation"]

    Root --> L1_Leverage["Leverage Identity
A = L + E"]
    Root --> L1_Equity["Equity Conservation
ΔE = sources + flux"]

    L1_Leverage --> L2_Balance["Balance Sheet Integrity
Assets = Liabilities + Equity"]

    L1_Equity --> L2_Bridge["Equity Bridge Closure
E₁ = E₀ + ΔE"]
    L1_Equity --> L2_NCI["NCI Reconciliation
IFRS 10 compliance"]

    L2_Bridge --> L3_Sources["Source Decomposition"]

    L3_Sources --> L4_NI["Net Income
IAS 1.106"]
    L3_Sources --> L4_OCI["OCI Components
7 items"]
    L3_Sources --> L4_Owner["Owner Transactions
Div, Buybacks"]

    L4_OCI --> L5_FVOCI["FVOCI
IFRS 9.5.7.1"]
    L4_OCI --> L5_FX["FX Translation
IAS 21.52"]
    L4_OCI --> L5_Pension["Actuarial
IAS 19.93"]
    L4_OCI --> L5_Hedge["Hedging
IFRS 9 hedge"]

    style Root fill:#e1f5ff
    style L1_Leverage fill:#d1ecf1
    style L1_Equity fill:#d1ecf1
    style L2_Balance fill:#fff3cd
    style L2_Bridge fill:#fff3cd
    style L2_NCI fill:#fff3cd
    style L3_Sources fill:#ffeaa7
    style L4_NI fill:#d4edda
    style L4_OCI fill:#d4edda
    style L4_Owner fill:#d4edda
    style L5_FVOCI fill:#dfe6e9
    style L5_FX fill:#dfe6e9
    style L5_Pension fill:#dfe6e9
    style L5_Hedge fill:#dfe6e9

Test Strategy:

1. Level 1: Fundamental identities (must pass)
2. Level 2: Component reconciliation (high-level)
3. Level 3: Source term decomposition
4. Level 4: Standard-specific validation
5. Level 5: Granular tag-level checks

6. Empirical Results Summary

Visual representation of 500-company validation results:

%%{init: {'theme':'base', 'themeVariables': { 'fontSize':'16px'}}}%%
graph LR
    subgraph Results["N=500 S&P 500 Companies (2,000 Filings)"]
        direction TB

        L["Leverage Identity
72.9% pass
[70.8%, 74.9%]"]
        E["Equity Bridge
54.9% pass
[52.6%, 57.2%]"]
        N["NCI Reconciliation
68.3% pass
[66.1%, 70.4%]"]
        O["OCI Completeness
43.2% pass
[41.0%, 45.4%]"]
    end

    subgraph Interpretation["Interpretation"]
        direction TB

        I1["Not accounting fraud
XBRL data quality issues"]
        I2["Framework = diagnostic tool
for disclosure gaps"]
        I3["ML audit risk model
AUC 0.949"]
    end

    Results --> Interpretation

    style L fill:#d4edda
    style E fill:#fff3cd
    style N fill:#d4edda
    style O fill:#f8d7da
    style I1 fill:#e1f5ff
    style I2 fill:#e1f5ff
    style I3 fill:#d1ecf1

Key Findings:

• 72.9% leverage pass rate: Most companies have balance sheet integrity
• 54.9% equity bridge: Significant XBRL/disclosure gaps (especially OCI)
• 43.2% OCI completeness: Weakest area—most companies don't itemize OCI components
• AUC 0.949: ML model successfully predicts high-risk filings

7. Standards Taxonomy Structure

Organization of 91 IFRS/GAAP standards mapped to equity source terms:

graph TD
    Root["Equity Change Sources
ΔE = ΣSource Terms"]

    Root --> Cat1["Profit/Loss
IAS 1"]
    Root --> Cat2["Other Comprehensive Income
IAS 1.7"]
    Root --> Cat3["Owner Transactions
IAS 32/IFRS 2"]
    Root --> Cat4["Consolidation
IFRS 10/IFRS 3"]
    Root --> Cat5["Measurement
IFRS 13/IAS 8"]

    Cat2 --> OCI1["Financial Instruments
IFRS 9"]
    Cat2 --> OCI2["Foreign Currency
IAS 21"]
    Cat2 --> OCI3["Employee Benefits
IAS 19"]
    Cat2 --> OCI4["Revaluation
IAS 16/IAS 38"]

    Cat3 --> Owner1["Dividends
IAS 1.107"]
    Cat3 --> Owner2["Share Buybacks
IAS 32.33"]
    Cat3 --> Owner3["Share-Based Payment
IFRS 2"]
    Cat3 --> Owner4["Capital Contributions
IAS 1.108"]

    Cat4 --> Consol1["Acquisitions
IFRS 3"]
    Cat4 --> Consol2["NCI Changes
IFRS 10.23"]
    Cat4 --> Consol3["Loss of Control
IFRS 10.25"]

    OCI1 --> Tag1["us-gaap:OCI_FVOCI"]
    OCI2 --> Tag2["us-gaap:OCI_ForeignCurrency"]
    OCI3 --> Tag3["us-gaap:OCI_PensionPlans"]

    style Root fill:#e1f5ff
    style Cat1 fill:#d4edda
    style Cat2 fill:#fff3cd
    style Cat3 fill:#f8d7da
    style Cat4 fill:#d1ecf1
    style Cat5 fill:#ffeaa7
    style Tag1 fill:#dfe6e9
    style Tag2 fill:#dfe6e9
    style Tag3 fill:#dfe6e9

Complete Taxonomy: See docs/standards/STANDARDS_CROSSWALK.md

8. Cross-Domain Conservation Analogy

Structural similarity across different domains (conceptual only):

graph TD
    subgraph Physics["Physics: Mass Conservation"]
        P1["∂ρ/∂t + ∇·J = s
Density change = flow + source"]
        P2["Example: Fluid in control volume"]
    end

    subgraph Accounting["Accounting: Equity Conservation"]
        A1["ΔE = flows + sources
Equity change = transactions + income"]
        A2["Example: Company balance sheet"]
    end

    subgraph Math["Mathematical Structure"]
        M1["Graph Theory
Nodes + Directed Edges"]
        M2["Discrete Continuity
Balance = In - Out + Source"]
    end

    Physics --> Math
    Accounting --> Math

    style Physics fill:#e1f5ff
    style Accounting fill:#d4edda
    style Math fill:#fff3cd

Important Qualifier: This is a structural analogy, not a claim that "accounting IS physics." The mathematical form appears similar, but:

• Physics: Empirical physical law (mass/energy cannot be created/destroyed)
• Accounting: Conventional system designed by humans (double-entry is a definition)
• Framework: Uses similar mathematical tools to formalize accounting rules

Usage in Documentation

These diagrams are embedded in:

• README.md: Simplified equity bridge flow
• FOR_AUDITORS.md: Audit workflow integration
• FOR_RESEARCHERS.md: Mathematical structure diagrams
• index.html: Consider adding interactive versions

Generating Custom Diagrams

To create custom diagrams for specific companies:

# Generate equity bridge diagram for specific ticker
python scripts/make_equity_bridge_diagram.py --ticker AAPL --quarter 2024Q3

# Output: docs/figures/equity_bridge_AAPL_2024Q3.svg

Note: All diagrams use Mermaid syntax and render natively in GitHub, GitLab, and most modern markdown viewers. For PDF exports, use mermaid-cli:

npm install -g @mermaid-js/mermaid-cli
mmdc -i docs/DIAGRAMS.md -o docs/DIAGRAMS.pdf

Last Updated: 2025-01-05