Clarifying When and Why Polarity Axes Share a Common Origin

In the AIM framework (Axioms of Intelligibility and Mind), polarity axes structure differentiation: each axis represents a tension between two complementary determinations (T ↔ ¬T). A recurring question arises:

When two polarity axes intersect at a common origin, what does this signify? Is a shared origin necessary, characteristic, or a consequence of one axis being a function of another?

This post explains the AIM interpretation of shared origins, functional axes, derived axes, and multi-axis coherence.

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1. Functional Dependence Alone Does Not Require a Shared Origin

In mathematics, a function such as:

• σ₂ = f(σ₁), or
• σ₂ = 2σ₁, σ₂ = σ₁², σ₂ = tanh(σ₁)

does not require both variables to share an origin geometrically.

AIM agrees: pure functional dependence between axes does not automatically imply they share the same neutral point.

But AIM is not merely mathematics—it is an ontology of intelligibility. So the underlying meaning determines when origin-sharing matters.

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2. Derived or Refined Axes Do Require a Shared Origin

AIM distinguishes independent axes from derived (refined) axes.

An axis σ₂ is a refinement of σ₁ when it:

• conditions σ₁,
• elaborates its meaning,
• increases its resolution,
• deepens or specifies one direction of σ₁,
• emerges from the same undifferentiated Unity.

In such cases, AIM requires a shared origin.

This follows from:

• Axiom 1: Unity (all determinations emerge from undifferentiated Unity),
• Axiom 2: Polarity (each axis expresses a differentiation of that Unity),
• Axiom 11: Recursion (refinements maintain structural coherence),
• Axiom 12: Multi-Axis Interaction (combined axes operate from a shared baseline).

A shared origin ensures:

• coherence in meaning,
• coordinated modulation,
• unified salience behavior,
• stable viability under interaction.

Example:

• σ₁ = trust ↔ vigilance
• σ₂ = predictability ↔ uncertainty (a refinement of vigilance)

Both emerge from the same undifferentiated state, so their axes must share the same zero point.

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3. Shared Origins Are a Consequence of Multi-Axis World Coherence

Even when axes are not derivationally related, AIM often requires or encourages a shared origin for system coherence.

A World coordinates meaning across axes via:

• Context modulation (A7),
• Gradient shaping (A14),
• Harmonization constraints (A5),
• Functoriality and mapping (A13),
• Viability boundaries (A15).

These functions work best when all axes share:

• a common baseline,
• a universal neutral state,
• a stable reference point for modulation and integration.

Thus, even independent axes often benefit structurally from sharing an origin, because it:

• simplifies mappings between Worlds,
• supports integrated salience computation,
• prevents runaway distortions,
• ensures coordinated viability evaluation.

Shared origins are not required for independence, but they are frequently the most coherent and AIM-consistent configuration.

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4. Summary: What AIM Tells Us About Axis Origins

1. Not Required by Pure Functional Dependence

A functional relationship alone does not force axes to share a zero point.

2. Required for Derived or Recursive Refinements

If σ₂ is a refinement or specialization of σ₁, they must share an origin because both trace back to the same initial Unity.

3. Convergent Consequence in Multi-Axis Worlds

For modulation, mapping, harmonization, and viability, shared origins are the natural structural consequence of building coherent Worlds.

Final Insight

Sharing a common origin is not a mathematical necessity, but a structural and semantic necessity whenever axes participate in coherent world-formation, recursive derivation, or coordinated modulation.