shcl.mathml.xml

<!--
The file `shcl.mathml` is a MathML document that describes a hierarchical system of shapes and their mathematical properties. Here is a summary of its contents:

- **Title**: Unified Homoiconic Topology
- **Description**: It contains shapes unified by prime numbers (1, 2, 3, 5), with each shape represented at different levels of hierarchy.

### Key Sections:
1. **Level 1: Unity (ex:EvalquoteShape)**
   - Description: \( s_0 = 2^1 \cdot 3^1 \cdot 5^2 = 150 \)
   - Function: \( f_{s_0}(n) = \text{if } (n \text{ has } 2^1, 3^1, 5^2) \text{ return true;} \)

2. **Level 2: Duality (ex:InstanceShape)**
   - Description: \( s_1 = 2^1 \cdot 3^1 \cdot 5^1 \cdot 7^1 = 210 \)
   - Function: \( f_{s_1}(n) = \text{if } (n \text{ has } 3^1, 2^p, 5^q) \text{ return true;} \)

3. **Level 3: Triad (ex:SporeTopologyShape)**
   - Description: \( s_2 = 2^1 \cdot 3^2 \cdot 5^2 \cdot 7^1 = 3150 \)
   - Function: \( f_{s_2}(n) = \text{if } (n \text{ contains } s_1, s_0) \text{ return true} \)

4. **Level 5: Pentad (ex:MetaTopologyShape)**
   - Description: \( s_3 = 2^1 \cdot 3^2 \cdot 5^2 \cdot 7^2 = 22050 \)
   - Function: \( f_{s_3}(n) = \text{if } (n \text{ contains } s_2) \text{ return true} \)

### Recursive Topology:
- Describes a nested structure: \( s_3 ⊃ s_2 ⊃ s_1 ⊃ s_0 \)
- Meta-Hierarchy: Each shape hovers over its instances in a Chord Distributed Hash Table (DHT).

### Annotations:
- Contains descriptions of SHACL shapes, their functions, and the level meanings, linking the math to the system’s semantics.

### Explanation:
- Levels are represented by <mrow> tags with prime factorization.
- Gödel numbers are used to express products of primes with exponents.
- Recursive topology shows the nesting levels of shapes with their respective Gödel numbers.

Would you like further details on any specific aspect or need help with something else in this repository?
-->
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <!-- Title: Unified Homoiconic Topology -->
  <mtext>Unified Homoiconic Shapes with 1, 2, 3, 5</mtext>

  <!-- Level 1: Unity (ex:EvalquoteShape) -->
  <mrow>
    <mn>1</mn>
    <mo>:</mo>
    <mtext>EvalquoteShape</mtext>
    <mo>=</mo>
    <msub>
      <mi>s</mi>
      <mn>0</mn>
    </msub>
    <mo>=</mo>
    <mrow>
      <msup><mn>2</mn><mn>1</mn></msup>
      <mo>⋅</mo>
      <msup><mn>3</mn><mn>1</mn></msup>
      <mo>⋅</mo>
      <msup><mn>5</mn><mn>2</mn></msup>
    </mrow>
    <mo>=</mo>
    <mn>150</mn>
    <semantics>
      <annotation-xml encoding="application/xhtml+xml">
        <div>
          <p>SHACL: ex:EvalquoteShape</p>
          <p>Function: \( f_{s_0}(n) = \text{if } (n \text{ has } 2^1, 3^1, 5^2) \text{ return true; } n' = n \cdot 3^1 \)</p>
          <p>Level: Unity (Seed of Homoiconicity)</p>
        </div>
      </annotation-xml>
    </semantics>
  </mrow>

  <!-- Level 2: Duality (ex:InstanceShape) -->
  <mrow>
    <mn>2</mn>
    <mo>:</mo>
    <mtext>InstanceShape</mtext>
    <mo>=</mo>
    <msub>
      <mi>s</mi>
      <mn>1</mn>
    </msub>
    <mo>=</mo>
    <mrow>
      <msup><mn>2</mn><mn>1</mn></msup>
      <mo>⋅</mo>
      <msup><mn>3</mn><mn>1</mn></msup>
      <mo>⋅</mo>
      <msup><mn>5</mn><mn>1</mn></msup>
      <mo>⋅</mo>
      <msup><mn>7</mn><mn>1</mn></msup>
    </mrow>
    <mo>=</mo>
    <mn>210</mn>
    <semantics>
      <annotation-xml encoding="application/xhtml+xml">
        <div>
          <p>SHACL: ex:InstanceShape</p>
          <p>Function: \( f_{s_1}(n) = \text{if } (n \text{ has } 3^1, 2^p, 5^q) \text{ return true; } n' = n \cdot 11^1 \)</p>
          <p>Level: Duality (Type-Instance)</p>
          <p>Contains: \( s_0 \)</p>
        </div>
      </annotation-xml>
    </semantics>
  </mrow>

  <!-- Level 3: Triad (ex:SporeTopologyShape) -->
  <mrow>
    <mn>3</mn>
    <mo>:</mo>
    <mtext>SporeTopologyShape</mtext>
    <mo>=</mo>
    <msub>
      <mi>s</mi>
      <mn>2</mn>
    </msub>
    <mo>=</mo>
    <mrow>
      <msup><mn>2</mn><mn>1</mn></msup>
      <mo>⋅</mo>
      <msup><mn>3</mn><mn>2</mn></msup>
      <mo>⋅</mo>
      <msup><mn>5</mn><mn>2</mn></msup>
      <mo>⋅</mo>
      <msup><mn>7</mn><mn>1</mn></msup>
    </mrow>
    <mo>=</mo>
    <mn>3150</mn>
    <semantics>
      <annotation-xml encoding="application/xhtml+xml">
        <div>
          <p>SHACL: ex:SporeTopologyShape</p>
          <p>Function: \( f_{s_2}(n) = \text{if } (n \text{ contains } s_1, s_0) \text{ return true} \)</p>
          <p>Level: Triad (Topology of Shapes)</p>
          <p>Contains: \( s_1, s_0 \)</p>
        </div>
      </annotation-xml>
    </semantics>
  </mrow>

  <!-- Level 5: Pentad (ex:MetaTopologyShape) -->
  <mrow>
    <mn>5</mn>
    <mo>:</mo>
    <mtext>MetaTopologyShape</mtext>
    <mo>=</mo>
    <msub>
      <mi>s</mi>
      <mn>3</mn>
    </msub>
    <mo>=</mo>
    <mrow>
      <msup><mn>2</mn><mn>1</mn></msup>
      <mo>⋅</mo>
      <msup><mn>3</mn><mn>2</mn></msup>
      <mo>⋅</mo>
      <msup><mn>5</mn><mn>2</mn></msup>
      <mo>⋅</mo>
      <msup><mn>7</mn><mn>2</mn></msup>
    </mrow>
    <mo>=</mo>
    <mn>22050</mn>
    <semantics>
      <annotation-xml encoding="application/xhtml+xml">
        <div>
          <p>SHACL: ex:MetaTopologyShape</p>
          <p>Function: \( f_{s_3}(n) = \text{if } (n \text{ contains } s_2) \text{ return true} \)</p>
          <p>Level: Pentad (Meta-Topology)</p>
          <p>Contains: \( s_2 \)</p>
        </div>
      </annotation-xml>
    </semantics>
  </mrow>

  <!-- Recursive Topology Annotation -->
  <mtext>Recursive Homoiconic Topology</mtext>
  <mo>:</mo>
  <mrow>
    <msub><mi>s</mi><mn>3</mn></msub>
    <mo>⊃</mo>
    <msub><mi>s</mi><mn>2</mn></msub>
    <mo>⊃</mo>
    <msub><mi>s</mi><mn>1</mn></msub>
    <mo>⊃</mo>
    <msub><mi>s</mi><mn>0</mn></msub>
  </mrow>
  <semantics>
    <annotation-xml encoding="application/xhtml+xml">
      <div>
        <p>Meta-Hierarchy: Each metashape hovers over its metainstances in a Chord DHT.</p>
        <p>Primes \( 1, 2, 3, 5 \) unify the system:</p>
        <ul>
          <li>\( 1 \): Unity (Seed)</li>
          <li>\( 2 \): Duality (Structure)</li>
          <li>\( 3 \): Triad (Nesting)</li>
          <li>\( 5 \): Pentad (Recursion)</li>
        </ul>
      </div>
    </annotation-xml>
  </semantics>
</math>
<!--
Explanation of MathML Structure
Levels (1, 2, 3, 5):
Each level is represented as an <mrow> with the prime number (1, 2, 3, 5), the shape name, its Gödel number variable (
s_i
), and its prime factorization.
The numbers ( 1, 2, 3, 5 ) align with the conceptual unification, with 1 as the implicit unit.
Gödel Numbers:
Expressed as products of primes with exponents (e.g., 
2^1 \cdot 3^1 \cdot 5^2 = 150
) using <msup> for superscripts and <mo>⋅</mo> for multiplication.
Computed values (e.g., 150, 210, 3150, 22050) are shown explicitly.
Annotations:
<semantics> and <annotation-xml> embed SHACL shape descriptions, functions, and level meanings, linking the math to the system’s semantics.
Functions are written in pseudo-code for clarity (e.g., 
f_{s_0}(n)
).
Recursive Topology:
The final <mrow> uses the subset symbol (( ⊃ )) to show the nesting: 
s_3 ⊃ s_2 ⊃ s_1 ⊃ s_0
, annotated with the meta-hierarchy explanation.
Homoiconicity:
The Gödel numbers (data) and their functions (code) are unified in the representation, with primes ( 2, 3, 5 ) recurring across levels, reflecting self-description.
Connection to the Unified Concept
1 (Unity): 
s_0 = 150
 is the seed, with prime 2 initiating homoiconic duality.
2 (Duality): 
s_1 = 210
 introduces type-instance duality, building on prime 2.
3 (Triad): 
s_2 = 3150
 nests shapes, with prime 3 structuring the topology.
5 (Pentad): 
s_3 = 22050
 recurses the system, with prime 5 elevating abstraction.
Chord DHT: Keys 
k_0, k_1, k_2, k_3
 cluster this hierarchy, with each metashape hovering over its metainstances.
Final Response
Here’s a concise summary:
The system is unified in MathML as ( 1, 2, 3, 5 ) with homoiconic shapes: 
s_0 = 2^1 \cdot 3^1 \cdot 5^2 = 150
 (Unity, ex:EvalquoteShape), 
s_1 = 2^1 \cdot 3^1 \cdot 5^1 \cdot 7^1 = 210
 (Duality, ex:InstanceShape), 
s_2 = 2^1 \cdot 3^2 \cdot 5^2 \cdot 7^1 = 3150
 (Triad, ex:SporeTopologyShape), and 
s_3 = 2^1 \cdot 3^2 \cdot 5^2 \cdot 7^2 = 22050
 (Pentad, ex:MetaTopologyShape). Each shape’s Gödel number and function (e.g., 
f_{s_1}(n)
) are homoiconic, stored in a Chord DHT where metashapes hover over metainstances in a recursive topology: 
s_3 ⊃ s_2 ⊃ s_1 ⊃ s_0
, driven by primes ( 1, 2, 3, 5 ).
Would you like me to refine the MathML further, generate a visual rendering, or explore another aspect?
Let me know how to conclude this journey!
-->