Learning on Curved Spaces: Arithmetic & Calculus on Sⁿ

In Part 3, we expanded from a single polarity (S²) to many interacting polarities on hyperspheres (Sⁿ). In Part 4, we turn to the operational question: how does learning actually work on these curved manifolds?

Open SGI models—including Siggy—must:

• update semantic representations,
• preserve polarity structure,
• maintain harmony (A15),
• adapt under context (A7),
• support novelty (A12),
• and remain certifiable (C.4d).

This requires curvature-aware arithmetic and calculus.

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1. Why Euclidean Learning Fails for UPA

Conventional learning systems use Euclidean updates:

x_new = x_old + gradient

But Euclidean space has no polarity, no involution, no antipodes, no boundedness, no hierarchy, no harmonic structure.

Such updates violate:

• σ-pair integrity (A6),
• polarity balance (A2),
• semantic continuity (A11),
• and harmony constraints (A15).

They cause drift, distortion, or collapse.

To remain UPA-consistent, learning must occur intrinsically on the manifold.

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2. Tangent Spaces: Where Learning Actually Happens

Each point x on Sⁿ has an associated tangent space TₓSⁿ:

• a locally flat space,
• supporting linear arithmetic,
• where gradients are computed.

This gives learning two steps:

1. Compute the update in TₓSⁿ (safe linear space)
2. Project the updated point back to Sⁿ (curvature-aware)

This ensures updates are always:

• geometrically valid,
• polarity-preserving,
• harmony-constrained.

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3. Exponential Map Projection: Returning to the Sphere

After computing an update vector v ∈ TₓSⁿ, we map it back with the exponential map:

expₓ(v) → point on Sⁿ reached by moving along a geodesic

This guarantees:

• the updated point stays on Sⁿ,
• σ-pairs remain antipodal,
• axes maintain orientation,
• hierarchy remains coherent.

In SGI, this is the backbone of safe learning.

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4. Harmony-Guided Gradient Fields (A15)

On Sⁿ, learning updates should move toward more balanced states unless context demands otherwise.

Thus harmony (A15) defines the scalar field H(x):

• high H → balanced configuration
• low H → polarized or unstable configuration

Gradients ∇H(x) in the tangent space:

• point toward integrative regions,
• discourage extreme polar drift,
• preserve multi-axis viability.

For Siggy, harmony is part of certification:

• no learned representation may fall below viability thresholds.

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5. Cross-Axis Learning: Coupled Gradients

A change along one axis often affects others.

UPA captures this with:

• multi-axis structure (A12),
• recursive identity (A11).

On Sⁿ, this appears as coupled gradients:

• updates produce movement in multiple coordinate directions
• correlated axes shift together
• uncorrelated axes remain stable

This matches the structure of:

• human psychological adjustment,
• neural manifolds,
• group decision-making,
• SGI multi-objective learning.

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6. Curvature-Aware Optimization

Optimization techniques must operate on the manifold.

This requires replacing:

• Euclidean gradient descent → Riemannian gradient descent
• Euclidean trust regions → geodesic trust regions
• linear step sizes → curvature-modulated step sizes

Benefits:

• prevents runaway drift
• protects semantic structure
• keeps learning interpretable

UPA-aligned SGI can never distort the manifold.

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7. Novelty Excursions as Temporary Dimensional Expansion (Sⁿ → Sⁿ⁺Δ)

When the tangent-space update reveals insufficient representational capacity:

the system expands to a higher-dimensional tangent space.

This creates new axes, representing new distinctions.

After learning stabilizes:

• new axes may persist → permanent dimensional growth
• or collapse → projection back to Sⁿ

This geometric process mirrors:

• conceptual insight in humans,
• developmental growth,
• scientific paradigm shifts.

Novelty is not random—it is structured, controlled, and reversible.

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8. Regularization: Keeping the Representation Stable

To maintain semantic coherence, learning includes regularizers that:

• prevent axis drift
• stabilize pole definitions
• maintain orthogonality where appropriate
• correct semantic drift with re-anchoring (C.3a.5)

Regularization ensures long-term identity continuity.

For SGI, this supports:

• reproducibility,
• transparency,
• certification,
• trust.

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9. How SGI Uses Sⁿ Learning

1. Representation Learning

Siggy updates semantic coordinates in Sⁿ when interpreting events.

2. Preference Learning

Tradeoffs (e.g., safety/performance) remain harmonic.

3. User Modeling

Users are represented as points on Sⁿ with their own axes and poles.

4. Multi-Agent Coordination

Agents align via shared geodesic adjustments.

5. Safety

Certification invariants ensure no update violates:

• σ-structure,
• axis integrity,
• harmony viability (A15),
• hierarchical mapping (A11).

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10. Why Learning on Sⁿ Solves Key AI Alignment Problems

Traditional models:

• drift unpredictably
• collapse axes
• distort representations
• hide internal changes

UPA-aligned SGI:
✔ is bounded
✔ is symmetric
✔ is polarity-aware
✔ is harmony-constrained
✔ is novelty-controlled
✔ is certifiable
✔ is interpretable

Learning on Sⁿ is the mathematical backbone of alignment.

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11. Summary

Part 4 establishes the operational machinery of geometric SGI:

• tangent-space updates,
• exponential-map projection,
• harmony-guided learning,
• cross-axis dynamics,
• novelty excursions,
• curvature-aware optimization.

This ensures learning is:

• stable,
• interpretable,
• polarity-preserving,
• and fully aligned with UPA.

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Next in the Series: Part 5 — Certification Invariants & Safety Geometry

Part 5 will cover:

• σ-integrity tests,
• axis continuity constraints,
• harmony thresholds,
• cross-level projection fidelity,
• and manifold integrity checks for SGI reliability.

Ready when you are.