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mathematics_processes.html

@@ -66,9 +66,9 @@
                 useMaxWidth: false, 
                 htmlLabels: true,
                 curve: 'linear',
-                nodeSpacing: 50,
-                rankSpacing: 50,
-                padding: 20
+                nodeSpacing: 30,
+                rankSpacing: 30,
+                padding: 10
             },
             themeVariables: {
                 fontFamily: 'Arial Unicode MS, Arial, sans-serif'
@@ -86,64 +86,59 @@
         <div class="figure">
             <div class="mermaid">
 graph TD
-    %% Initial Setup
-    %% Axioms and Given Conditions
     A1[Peano Axioms] --> B1[Axiom Processing]
     C1[Given n in Natural Numbers] --> D1[Input Validation]
     E1[Goal: Prove P(n)] --> F1[Target Identification]
-    %% Logical Framework Setup
+    
     B1 --> G1[Mathematical Universe Setup]
     D1 --> H1[Variable Declaration]
     F1 --> I1[Proof Strategy Selection]
-    %% Proof Structure
+    
     G1 --> J1[Induction Hypothesis P(k)]
     H1 --> K1[Base Case Analysis]
     I1 --> L1[Inductive Step Planning]
-    %% Base Case Processing
+    
     K1 --> M1[P(0) Verification]
     M1 --> N1[Base Case Success]
     N1 --> O1[Induction Foundation]
-    %% Inductive Step Processing
+    
     L1 --> P1[Assume P(k) for k in Natural Numbers]
     P1 --> Q1[Show P(k+1) follows]
     Q1 --> R1[Inductive Step Execution]
-    %% Mathematical Operations
+    
     R1 --> S1[Algebraic Manipulation]
     S1 --> T1[Logical Deduction]
     T1 --> U1[Theorem Application]
-    %% Intermediate Calculations
+    
     U1 --> V1[Sub-proof Construction]
     V1 --> W1[Lemma Application]
     W1 --> X1[Contradiction Analysis]
-    %% Proof Validation
+    
     X1 --> Y1[Logical Consistency Check]
     Y1 --> Z1[Mathematical Rigor Verification]
     Z1 --> AA1[Proof Completeness Assessment]
-    %% Decision Points
+    
     AA1 --> BB1{Proof Complete?}
     BB1 -->|No| CC1[Identify Gap]
     BB1 -->|Yes| DD1[Proof Validated]
-    %% Gap Resolution
+    
     CC1 --> EE1[Additional Lemma Needed]
     EE1 --> FF1[Sub-proof Construction]
     FF1 --> GG1[Gap Resolution]
     GG1 --> Y1
-    %% Final Output
+    
     DD1 --> HH1[Theorem P(n) Proven]
     HH1 --> II1[Mathematical Truth Established]
     II1 --> JJ1[Proof Tree Complete]
-    %% Styling - Mathematical Color Scheme
-    %% Red: Axioms & Inputs
+    
     style A1 fill:#ff6b6b,color:#fff
     style C1 fill:#ff6b6b,color:#fff
     style E1 fill:#ff6b6b,color:#fff
     
-    %% Yellow: Logical Structures & Methods
     style J1 fill:#ffd43b,color:#000
     style P1 fill:#ffd43b,color:#000
     style Q1 fill:#ffd43b,color:#000
     
-    %% Green: Deductions & Operations
     style S1 fill:#51cf66,color:#fff
     style T1 fill:#51cf66,color:#fff
     style U1 fill:#51cf66,color:#fff
@@ -151,7 +146,6 @@ graph TD
     style W1 fill:#51cf66,color:#fff
     style X1 fill:#51cf66,color:#fff
     
-    %% Blue: Intermediates & States
     style B1 fill:#74c0fc,color:#fff
     style D1 fill:#74c0fc,color:#fff
     style F1 fill:#74c0fc,color:#fff
@@ -174,11 +168,10 @@ graph TD
     style FF1 fill:#74c0fc,color:#fff
     style GG1 fill:#74c0fc,color:#fff
     
-    %% Violet: Final Results
     style HH1 fill:#b197fc,color:#fff
     style II1 fill:#b197fc,color:#fff
     style JJ1 fill:#b197fc,color:#fff
-                    </div>
+            </div>
             
             <div style="margin-top: 1rem; display: flex; flex-wrap: wrap; gap: 0.5rem; justify-content: center;">
                 <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;">
@@ -207,61 +200,60 @@ graph TD
         <div class="figure">
             <div class="mermaid">
 graph TD
-    %% Initial Setup
-    %% Input Conditions
     A2[Integer a] --> B2[Input Validation]
     C2[Integer b] --> D2[Input Validation]
     E2[Goal: Find GCD(a,b)] --> F2[Problem Statement]
-    %% Algorithm Initialization
+    
     B2 --> G2[Set r₀ = a]
     D2 --> H2[Set r₁ = b]
     F2 --> I2[Algorithm Selection]
-    %% Division Process
+    
     G2 --> J2[Division Algorithm]
     H2 --> K2[Division Algorithm]
     I2 --> L2[Iterative Process]
-    %% First Division
+    
     J2 --> M2[r₀ = q₁r₁ + r₂]
     K2 --> N2[Calculate q₁ and r₂]
     L2 --> O2[Initialize iteration counter]
-    %% Iterative Loop
+    
     M2 --> P2{Is r₂ = 0?}
     N2 --> Q2[Store r₂]
     O2 --> R2[Increment counter]
-    %% Continue or Terminate
+    
     P2 -->|No| S2[Set r₀ = r₁, r₁ = r₂]
     P2 -->|Yes| T2[GCD Found: r₁]
     Q2 --> U2[Update remainders]
     R2 --> V2[Track iterations]
-    %% Next Iteration
+    
     S2 --> W2[Next Division Step]
     U2 --> X2[Prepare for next iteration]
     V2 --> Y2[Check termination condition]
-    %% Final Result
+    
     T2 --> Z2[GCD(a,b) = r₁]
     W2 --> AA2[Repeat division process]
     X2 --> BB2[Update variables]
     Y2 --> CC2{Continue?}
-    %% Algorithm Completion
+    
     Z2 --> DD2[Result Validation]
     AA2 --> P2
     BB2 --> P2
     CC2 -->|Yes| AA2
     CC2 -->|No| T2
-    %% Output
+    
     DD2 --> EE2[GCD Calculation Complete]
     EE2 --> FF2[Mathematical Proof of Correctness]
     FF2 --> GG2[Algorithm Efficiency Analysis]
-    %% Styling - Mathematical Color Scheme
-    %% Styling - Biological Color Scheme
+    
     style A2 fill:#ff6b6b,color:#fff
     style C2 fill:#ff6b6b,color:#fff
     style E2 fill:#ff6b6b,color:#fff
+    
     style G2 fill:#ffd43b,color:#000
     style H2 fill:#ffd43b,color:#000
     style I2 fill:#ffd43b,color:#000
     style J2 fill:#ffd43b,color:#000
     style K2 fill:#ffd43b,color:#000
+    
     style L2 fill:#51cf66,color:#fff
     style M2 fill:#51cf66,color:#fff
     style N2 fill:#51cf66,color:#fff
@@ -269,6 +261,7 @@ graph TD
     style S2 fill:#51cf66,color:#fff
     style W2 fill:#51cf66,color:#fff
     style AA2 fill:#51cf66,color:#fff
+    
     style B2 fill:#74c0fc,color:#fff
     style D2 fill:#74c0fc,color:#fff
     style F2 fill:#74c0fc,color:#fff
@@ -282,11 +275,12 @@ graph TD
     style Y2 fill:#74c0fc,color:#fff
     style CC2 fill:#74c0fc,color:#fff
     style DD2 fill:#74c0fc,color:#fff
+    
     style Z2 fill:#b197fc,color:#fff
     style EE2 fill:#b197fc,color:#fff
     style FF2 fill:#b197fc,color:#fff
     style GG2 fill:#b197fc,color:#fff
-                    </div>
+            </div>
             
             <div style="margin-top: 1rem; display: flex; flex-wrap: wrap; gap: 0.5rem; justify-content: center;">
                 <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;">
@@ -311,348 +305,6 @@ graph TD
             </div>
         </div>
 
-        <h2>3. Calculus Integration Process</h2>
-        <div class="figure">
-            <div class="mermaid">
-graph TD
-    %% Initial Setup
-    %% Input Conditions
-    A3[Function f(x)] --> B3[Function Analysis]
-    C3[Integration Limits [a,b]] --> D3[Boundary Definition]
-    E3[Goal: ∫f(x)dx] --> F3[Problem Statement]
-    %% Function Classification
-    B3 --> G3[Function Type Classification]
-    D3 --> H3[Domain Analysis]
-    F3 --> I3[Integration Strategy Selection]
-    %% Integration Methods
-    G3 --> J3{Function Type?}
-    H3 --> K3[Continuity Check]
-    I3 --> L3[Method Selection]
-    %% Direct Integration
-    J3 -->|Polynomial| M3[Power Rule Application]
-    J3 -->|Trigonometric| N3[Trig Identity Application]
-    J3 -->|Exponential| O3[Exponential Rule]
-    J3 -->|Rational| P3[Partial Fractions]
-    %% Substitution Method
-    K3 --> Q3[Substitution Detection]
-    L3 --> R3[Substitution Method]
-    M3 --> S3[Direct Integration]
-    N3 --> T3[Trig Integration]
-    %% Integration by Parts
-    O3 --> U3[Integration by Parts]
-    P3 --> V3[Partial Fraction Decomposition]
-    Q3 --> W3[Substitution Application]
-    R3 --> X3[Variable Change]
-    %% Definite Integration
-    S3 --> Y3[Antiderivative F(x)]
-    T3 --> Z3[Trig Antiderivative]
-    U3 --> AA3[Parts Integration]
-    V3 --> BB3[Fraction Integration]
-    %% Evaluation
-    W3 --> CC3[Substituted Integral]
-    X3 --> DD3[New Variable Integration]
-    Y3 --> EE3[F(b) - F(a)]
-    Z3 --> FF3[Definite Trig Result]
-    %% Final Results
-    AA3 --> GG3[Parts Result]
-    BB3 --> HH3[Fraction Result]
-    CC3 --> II3[Substitution Result]
-    DD3 --> JJ3[Variable Back-Substitution]
-    %% Verification
-    EE3 --> KK3[Result Verification]
-    FF3 --> LL3[Trigonometric Verification]
-    GG3 --> MM3[Parts Verification]
-    HH3 --> NN3[Fraction Verification]
-    %% Output
-    KK3 --> OO3[Definite Integral Value]
-    LL3 --> PP3[Trigonometric Integral Value]
-    MM3 --> QQ3[Parts Integral Value]
-    NN3 --> RR3[Fractional Integral Value]
-    %% Styling - Mathematical Color Scheme
-    %% Styling - Biological Color Scheme
-    style A3 fill:#ff6b6b,color:#fff
-    style C3 fill:#ff6b6b,color:#fff
-    style E3 fill:#ff6b6b,color:#fff
-    style G3 fill:#ffd43b,color:#000
-    style H3 fill:#ffd43b,color:#000
-    style I3 fill:#ffd43b,color:#000
-    style J3 fill:#ffd43b,color:#000
-    style K3 fill:#ffd43b,color:#000
-    style L3 fill:#ffd43b,color:#000
-    style M3 fill:#51cf66,color:#fff
-    style N3 fill:#51cf66,color:#fff
-    style O3 fill:#51cf66,color:#fff
-    style P3 fill:#51cf66,color:#fff
-    style Q3 fill:#51cf66,color:#fff
-    style R3 fill:#51cf66,color:#fff
-    style S3 fill:#51cf66,color:#fff
-    style T3 fill:#51cf66,color:#fff
-    style U3 fill:#51cf66,color:#fff
-    style V3 fill:#51cf66,color:#fff
-    style B3 fill:#74c0fc,color:#fff
-    style D3 fill:#74c0fc,color:#fff
-    style F3 fill:#74c0fc,color:#fff
-    style Y3 fill:#74c0fc,color:#fff
-    style Z3 fill:#74c0fc,color:#fff
-    style AA3 fill:#74c0fc,color:#fff
-    style BB3 fill:#74c0fc,color:#fff
-    style CC3 fill:#74c0fc,color:#fff
-    style DD3 fill:#74c0fc,color:#fff
-    style EE3 fill:#74c0fc,color:#fff
-    style FF3 fill:#74c0fc,color:#fff
-    style GG3 fill:#74c0fc,color:#fff
-    style HH3 fill:#74c0fc,color:#fff
-    style II3 fill:#74c0fc,color:#fff
-    style JJ3 fill:#74c0fc,color:#fff
-    style OO3 fill:#b197fc,color:#fff
-    style PP3 fill:#b197fc,color:#fff
-    style QQ3 fill:#b197fc,color:#fff
-    style RR3 fill:#b197fc,color:#fff
-                    </div>
-            
-            <div style="margin-top: 1rem; display: flex; flex-wrap: wrap; gap: 0.5rem; justify-content: center;">
-                <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;">
-                    <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ff6b6b;"></span>Triggers & Inputs
-                </div>
-                <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;">
-                    <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ffd43b;"></span>Mathematical Methods & Theorems
-                </div>
-                <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;">
-                    <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#51cf66;"></span>Integration Operations
-                </div>
-                <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;">
-                    <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#74c0fc;"></span>Intermediates
-                </div>
-                <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;">
-                    <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#b197fc;"></span>Products
-                </div>
-            </div>
-            
-            <div class="figure-caption">
-                <strong>Figure 3.</strong> Calculus Integration Process. This mathematics process visualization demonstrates integral calculus computation. The flowchart shows function inputs, mathematical methods and theorems, integration operations, intermediate calculations, and final integral values.
-            </div>
-        </div>
-
-        <h2>4. Linear Algebra Matrix Operations</h2>
-        <div class="figure">
-            <div class="mermaid">
-graph TD
-    %% Initial Setup
-    %% Input Conditions
-    A4[Matrix A] --> B4[Matrix Analysis]
-    C4[Matrix B] --> D4[Matrix Analysis]
-    E4[Operation Type] --> F4[Operation Selection]
-    %% Matrix Classification
-    B4 --> G4[Matrix Dimensions Check]
-    D4 --> H4[Matrix Properties Analysis]
-    F4 --> I4[Operation Feasibility]
-    %% Operation Types
-    G4 --> J4{Operation Type?}
-    H4 --> K4[Matrix Properties]
-    I4 --> L4[Compatibility Check]
-    %% Matrix Addition
-    J4 -->|Addition| M4[Dimension Matching]
-    J4 -->|Multiplication| N4[Inner Product Dimensions]
-    J4 -->|Inverse| O4[Square Matrix Check]
-    J4 -->|Determinant| P4[Square Matrix Check]
-    %% Addition Process
-    K4 --> Q4[Element-wise Addition]
-    L4 --> R4[Compatibility Verification]
-    M4 --> S4[Add Corresponding Elements]
-    N4 --> T4[Matrix Multiplication Algorithm]
-    %% Multiplication Process
-    O4 --> U4[Inverse Calculation]
-    P4 --> V4[Determinant Calculation]
-    Q4 --> W4[Result Matrix C]
-    R4 --> X4[Operation Validation]
-    %% Inverse Calculation
-    S4 --> Y4[Addition Result]
-    T4 --> Z4[Multiplication Result]
-    U4 --> AA4[Gauss-Jordan Elimination]
-    V4 --> BB4[Determinant Expansion]
-    %% Determinant Calculation
-    W4 --> CC4[Result Verification]
-    X4 --> DD4[Properties Check]
-    Y4 --> EE4[Matrix C = A + B]
-    Z4 --> FF4[Matrix C = A × B]
-    %% Final Results
-    AA4 --> GG4[Inverse Matrix A⁻¹]
-    BB4 --> HH4[Determinant |A|]
-    CC4 --> II4[Result Validation]
-    DD4 --> JJ4[Properties Verification]
-    %% Output
-    EE4 --> KK4[Addition Complete]
-    FF4 --> LL4[Multiplication Complete]
-    GG4 --> MM4[Inverse Found]
-    HH4 --> NN4[Determinant Calculated]
-    %% Styling - Mathematical Color Scheme
-    %% Styling - Biological Color Scheme
-    style A4 fill:#ff6b6b,color:#fff
-    style C4 fill:#ff6b6b,color:#fff
-    style E4 fill:#ff6b6b,color:#fff
-    style G4 fill:#ffd43b,color:#000
-    style H4 fill:#ffd43b,color:#000
-    style I4 fill:#ffd43b,color:#000
-    style J4 fill:#ffd43b,color:#000
-    style K4 fill:#ffd43b,color:#000
-    style L4 fill:#ffd43b,color:#000
-    style M4 fill:#51cf66,color:#fff
-    style N4 fill:#51cf66,color:#fff
-    style O4 fill:#51cf66,color:#fff
-    style P4 fill:#51cf66,color:#fff
-    style Q4 fill:#51cf66,color:#fff
-    style R4 fill:#51cf66,color:#fff
-    style S4 fill:#51cf66,color:#fff
-    style T4 fill:#51cf66,color:#fff
-    style U4 fill:#51cf66,color:#fff
-    style V4 fill:#51cf66,color:#fff
-    style AA4 fill:#51cf66,color:#fff
-    style BB4 fill:#51cf66,color:#fff
-    style B4 fill:#74c0fc,color:#fff
-    style D4 fill:#74c0fc,color:#fff
-    style F4 fill:#74c0fc,color:#fff
-    style W4 fill:#74c0fc,color:#fff
-    style X4 fill:#74c0fc,color:#fff
-    style Y4 fill:#74c0fc,color:#fff
-    style Z4 fill:#74c0fc,color:#fff
-    style CC4 fill:#74c0fc,color:#fff
-    style DD4 fill:#74c0fc,color:#fff
-    style II4 fill:#74c0fc,color:#fff
-    style JJ4 fill:#74c0fc,color:#fff
-    style KK4 fill:#b197fc,color:#fff
-    style LL4 fill:#b197fc,color:#fff
-    style MM4 fill:#b197fc,color:#fff
-    style NN4 fill:#b197fc,color:#fff
-                    </div>
-            
-            <div style="margin-top: 1rem; display: flex; flex-wrap: wrap; gap: 0.5rem; justify-content: center;">
-                <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;">
-                    <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ff6b6b;"></span>Triggers & Inputs
-                </div>
-                <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;">
-                    <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ffd43b;"></span>Mathematical Structures & Methods
-                </div>
-                <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;">
-                    <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#51cf66;"></span>Matrix Operations
-                </div>
-                <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;">
-                    <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#74c0fc;"></span>Intermediates
-                </div>
-                <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;">
-                    <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#b197fc;"></span>Products
-                </div>
-            </div>
-            
-            <div class="figure-caption">
-                <strong>Figure 4.</strong> Linear Algebra Matrix Operations. This mathematics process visualization demonstrates matrix algebra computation. The flowchart shows matrix inputs, mathematical structures and methods, matrix operations, intermediate calculations, and final matrix results.
-            </div>
-        </div>
-
-        <h2>5. Probability Theory Process</h2>
-        <div class="figure">
-            <div class="mermaid">
-graph TD
-    %% Initial Setup
-    %% Input Conditions
-    A5[Sample Space S] --> B5[Space Analysis]
-    C5[Event E] --> D5[Event Analysis]
-    E5[Probability Measure P] --> F5[Measure Definition]
-    %% Probability Framework
-    B5 --> G5[Sample Space Properties]
-    D5 --> H5[Event Properties]
-    F5 --> I5[Probability Axioms]
-    %% Axiomatic Foundation
-    G5 --> J5[Kolmogorov Axioms]
-    H5 --> K5[Event Algebra]
-    I5 --> L5[Measure Theory]
-    %% Probability Calculation
-    J5 --> M5[P(S) = 1]
-    K5 --> N5[Event Operations]
-    L5 --> O5[Probability Functions]
-    %% Event Operations
-    M5 --> P5[Complement Rule]
-    N5 --> Q5[Union Rule]
-    O5 --> R5[Conditional Probability]
-    %% Conditional Probability
-    P5 --> S5[P(E') = 1 - P(E)]
-    Q5 --> T5[P(A∪B) = P(A) + P(B) - P(A∩B)]
-    R5 --> U5[P(A|B) = P(A∩B)/P(B)]
-    %% Bayes' Theorem
-    S5 --> V5[Probability Calculation]
-    T5 --> W5[Set Operations]
-    U5 --> X5[Bayes' Theorem]
-    %% Independence
-    V5 --> Y5[Result Verification]
-    W5 --> Z5[Venn Diagram Analysis]
-    X5 --> AA5[P(A|B) = P(B|A)P(A)/P(B)]
-    %% Final Results
-    Y5 --> BB5[Probability Value]
-    Z5 --> CC5[Set Relationships]
-    AA5 --> DD5[Posterior Probability]
-    %% Output
-    BB5 --> EE5[Probability Calculated]
-    CC5 --> FF5[Event Relationships]
-    DD5 --> GG5[Bayesian Update]
-    %% Styling - Mathematical Color Scheme
-    %% Styling - Biological Color Scheme
-    style A5 fill:#ff6b6b,color:#fff
-    style C5 fill:#ff6b6b,color:#fff
-    style E5 fill:#ff6b6b,color:#fff
-    style G5 fill:#ffd43b,color:#000
-    style H5 fill:#ffd43b,color:#000
-    style I5 fill:#ffd43b,color:#000
-    style J5 fill:#ffd43b,color:#000
-    style K5 fill:#ffd43b,color:#000
-    style L5 fill:#ffd43b,color:#000
-    style M5 fill:#51cf66,color:#fff
-    style N5 fill:#51cf66,color:#fff
-    style O5 fill:#51cf66,color:#fff
-    style P5 fill:#51cf66,color:#fff
-    style Q5 fill:#51cf66,color:#fff
-    style R5 fill:#51cf66,color:#fff
-    style S5 fill:#51cf66,color:#fff
-    style T5 fill:#51cf66,color:#fff
-    style U5 fill:#51cf66,color:#fff
-    style X5 fill:#51cf66,color:#fff
-    style AA5 fill:#51cf66,color:#fff
-    style B5 fill:#74c0fc,color:#fff
-    style D5 fill:#74c0fc,color:#fff
-    style F5 fill:#74c0fc,color:#fff
-    style V5 fill:#74c0fc,color:#fff
-    style W5 fill:#74c0fc,color:#fff
-    style Y5 fill:#74c0fc,color:#fff
-    style Z5 fill:#74c0fc,color:#fff
-    style BB5 fill:#74c0fc,color:#fff
-    style CC5 fill:#74c0fc,color:#fff
-    style DD5 fill:#74c0fc,color:#fff
-    style EE5 fill:#b197fc,color:#fff
-    style FF5 fill:#b197fc,color:#fff
-    style GG5 fill:#b197fc,color:#fff
-                    </div>
-            
-            <div style="margin-top: 1rem; display: flex; flex-wrap: wrap; gap: 0.5rem; justify-content: center;">
-                <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;">
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-                <strong>Figure 5.</strong> Probability Theory Process. This mathematics process visualization demonstrates probabilistic reasoning. The flowchart shows sample space inputs, probability axioms and theorems, probability calculations, intermediate computations, and final probability values.
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         <p><strong>Generated using the Programming Framework methodology</strong></p>
         
         <p>This collection demonstrates the computational nature of mathematical processes and systems</p>