Being From Nothingness

Section 3: The Geometry of Constraint Space

3.1 From Configurations to Geometry

Section 2 established that configurations can be represented as 5-vectors C = (C₁, C₂, C₃, C₄, C₅), with viable configurations occupying a bounded region 𝒱. We now develop the geometric structure of this representation.

Recall the caveat from Section 2.5: constraint space is not a pre-existing container but a representational tool.What exists is relational structure; constraint space captures the pattern of that structure. With this understanding, we can fruitfully employ geometric language to analyze relationships between configurations.

The geometry has three aspects:

• Metric structure: How "far apart" are two configurations?

• Potential structure: What organizes and drives change between configurations?

• Curvature structure: What is the local shape of the landscape?

These aspects are not independent. The potential determines the gradient; the gradient and metric together determine geodesics; curvature characterizes how geodesics converge or diverge. We develop each in turn, beginning with the potential—which requires careful derivation from the axiom.

Section 3 of Philosophy Paper

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