Open Autonomous Intelligence Initiative

This Part develops the formal geometric and category‑theoretic structures that make the Unity–Polarity framework mathematically tractable and suitable for translation into computational, cognitive‑scientific, and physical models. Whereas Part III focused on theorem development in English with light symbolic support, Part IV provides the explicit modeling environment in which those theorems can be expressed, visualized, and generalized. In particular, this section introduces a family of spherical models, axis structures, rotation‑based transformations, and category‑theoretic functoriality that together form the canonical representation of polarity as structured within Unity.

The geometric component focuses on spherical and hyper‑spherical constructions in which Unity is interpreted as the undivided whole, and polarity is represented through axes and structured antipodes. This enables continuous parametrization of differentiation, identity of opposites at π‑rotation, and the embedding of recursive and multi‑axis expressions. Category‑theoretic modeling complements this work by presenting polarity as an involutive structure on objects within a category, with functors preserving polarity structure across conceptual domains—expanding the system’s interpretive and translational power.

These models support three primary goals: (1) to show how the axioms and theorems of Parts II and III can be formally realized in a unified representational space; (2) to provide tools for measuring and comparing contextual expression and harmony; and (3) to ground the computational implementation of Holism and Unity within Open‑SGI.

8.1 Spherical Polarity Models

Spherical polarity models provide a geometric realization of the Unity–Polarity Axiom System (UPA). In this representation, Unity is expressed as a compact, continuous totality—mathematically modeled as a sphere (S^2)—within which generative axes establish structured polar differentiation. Each axis defines a pair of antipodal points representing opposite poles a and σ(a). Their mutual relation arises from involution (reflection through the sphere’s center), ensuring σ(σ(a)) = a.

Under this model, great circles represent axes, and transformation along an axis corresponds to continuous geodesic motion on the sphere. Rotational operators Rθ trace paths between poles, yielding intermediate states that demonstrate continuity (A3b). Novelty (A3c) is geometrically interpreted through the emergence of new axes or new directions not captured by rotation within a preexisting axis. This provides a geometric understanding of lawful transformation and emergent differentiation.

The spherical model supports multi-axis structure (A12). Multiple axes appear as distinct great circles intersecting at the sphere’s center. Cross-axis structure produces composite polarity surfaces, analogous to overlapping coordinate systems, enabling entities to participate in multiple polarity regimes simultaneously.

This framework is particularly useful for understanding phenomena in physics (e.g., spin, polarization), psychology (e.g., multidimensional trait models), and SGI (e.g., latent semantic structures). The sphere also supports hierarchical recursion (A11): sub-axes may be defined locally on sub-manifolds, illustrating nested polarity. Contextual fields (A7) deform the metric of S^2, modulating distances and hence making certain transformations more or less viable.

8.2 Axis / Longitude / Latitude / Level Structure

In the spherical realization of the Unity–Polarity Axiom System (UPA), dimensions provide global and local coordinate structure. Globally, each axis defines antipodal poles a and σ(a), embedded on S^2 as opposite points connected by a great circle. Locally, tangent‑plane (Cartesian) charts support detailed geometric and computational activity. This hybrid representation enables Open‑SGI systems to reason about context both holistically (global location on the sphere) and analytically (local spatial/semantic configuration).

This section introduces the dimensional framework: (1) global spherical placement; (2) local tangent‑plane (Cartesian) structure; (3) hierarchical sub‑axes; (4) contextual deformation of geometry.

8.2.1 Theorem — Spherical–Cartesian Representability

Statement. Given a spherical polarity model with axes defined as great circles and poles as antipodes, there exists, at every non‑singular point of S^2, a local Cartesian coordinate chart supporting representation of shapes, magnitudes, and transformations with respect to spherical position.

Linked Axioms. A2 (Axis), A3b (Continuity), A11 (Recursion), A12 (Multi‑axis).

English Proof Sketch. A2 and A3b ensure that each polarity is expressed along a continuous axis on S^2. Standard differential geometry guarantees that S^2 admits an atlas of tangent‑plane charts in which neighborhoods are locally Euclidean. A11 allows recursive differentiation of structure within local charts, while A12 permits multiple coordinate roles for a single pole. Therefore, local Cartesian encoding naturally complements global spherical position.

SGI Implications. Open‑SGI agents may use global spherical coordinates to contextualize meaning and local Cartesian coordinates to represent object geometry, feature maps, or abstract semantic subspaces. This hybrid captures both Unity (global) and polarity (local expressions) without sacrificing computational tractability.

8.2.2 Dimensional Anatomy: Axes, Longitudes, and Latitudes

In S^2, an axis is a diameter whose endpoints are antipodal poles. The great circle orthogonal to the axis defines an infinite set of longitudes intersecting both poles. Latitudes represent sets of points equidistant from the equatorial great circle perpendicular to the axis. Thus, axes embody polarity, longitudes encode directional differentiation, and latitudes reflect degrees of similarity. Together, they define a structured manifold supporting continuous transformation.

8.2.3 Levels and Sub‑Axes

Differentiated states along an axis may exhibit internal polarity (A11), generating sub‑axes analogous to local great circles. These structures combine to define levels: graded manifolds encoding hierarchical differentiation. Levels allow Open‑SGI systems to represent nested semantic detail (e.g., trait subdimensions in psychology, subtypes of concepts, or latent embedding subregions).

8.2.4 Contextual Deformation of Dimensions

Context (A7) adjusts the effective metric structure of dimensions. Distances or angles on S^2 may be rescaled to reflect situational relevance, yielding geodesic deformation. This supports dynamic re‑weighting of similarity relations, enabling SGI architectures to adaptively tune geometric neighborhoods, thus preserving viability (A15) under changing circumstances.

8.3 Rotation & Novelty Geometry

Movement on S^2 is more general than rotation. While rotation along a great‑circle axis represents a canonical transformation between opposed poles a and σ(a), many motions—including shifts along latitude or longitude, or traversal across multi‑axis intersections—reflect alternative modes of relational change. This section reconceives Rotation as a special case of Orientation‑preserving movement, emphasizing that movement may proceed in arbitrarily chosen “up–down” or “left–right” directions relative to local frames. These distinctions reflect contextual embeddings rather than inherent absolute directions.

Novelty is not solely tied to rotational progression. In UPA, novelty (A3c) marks the emergence of additional axes and new prime levels of differentiation that cannot be reduced to continuous transformation along pre‑existing axes. While hierarchical expressions of novelty may be treated as sub‑axes (IV.2.3), the appearance of new prime levels is best modeled as the arrival of irreducible dimensions. Importantly, movement toward a pole does not guarantee increased similarity to that pole in terms of systemic harmony (A15); contextual demands and tradeoffs (A10) may cause proximity to increase imbalance (strife), clarifying that geometric closeness is not identical to functional or axiological proximity.

8.3.1 Theorem — Harmony–Distance Decoupling

Statement. Movement toward a pole a on S^2 (i.e., decreasing geodesic distance to a) does not necessarily increase harmony H(a, σ(a) | Ctx); contextual factors and tradeoffs may cause proximity to heighten imbalance (strife).

Linked Axioms. A7 (Context), A10 (Tradeoff), A15 (Harmony/Viability).

English Proof Sketch. A7 asserts context modulates polarity expression; movement toward a may place the system in a context where its characteristics are maladaptive. A10 states poles impose reciprocal constraint; increasing emphasis on a may suppress σ(a) excessively, creating imbalance. A15 defines viability as sufficient harmony; therefore, decreased spatial distance to a does not guarantee increased harmony.

8.3.2 Orientation Frames & Arbitrary Directions

Local tangent‑plane charts define orientation frames; “up–down” and “left–right” are contextual rather than inherent. Movement may proceed along any basis vector in the chart, allowing multi‑directional exploration independent of polar alignment.

8.3.3 Novelty as Emergent Dimensions

Novelty (A3c) corresponds to introduction of new axes not obtainable by intra‑axis rotation. These are prime‑level differentiations, irreducible to prior structure. Novelty may arise from contextual forces (A7) and recursive differentiation (A11), generating new semantic degrees of freedom.

8.4 Initialization of World Levels & Generative Progression

In the Open‑SGI formalism, World Levels represent stratified layers of differentiation emerging from Unity through structured polarity. Initialization describes how a first world‑level (W₀) becomes expressible; generative progression describes how subsequent levels (W₁, W₂, …) emerge. While involution (σ) maps poles within an existing level, the formation of entirely new Unity‑levels requires a distinct mechanism oriented toward synthesis rather than reflection.

8.4.1 Initialization of W₀

The base world‑level W₀ arises when Unity (U) differentiates minimally along one or more axes (A2), generating primary σ‑pairs. W₀ provides a coherent polarity space sufficient to support movement (8.3) and local semantic integration (8.2).

8.4.2 The τ‑Operator (“T‑operator”): Formal Definition

Let Str(Wₙ) denote the structured objects and relations resident at level Wₙ (axes, σ‑pairs, contexts, harmony metrics, mappings). Define a generative synthesis operator

τₙ : Str(Wₙ) → Wₙ₊₁

that constructs a new world‑level Wₙ₊₁ by synthesizing selected structure from Wₙ into a new quasi‑Unity that supports new axes and induced σ‑pairs at Wₙ₊₁.

Core properties.

1. Non‑involutive: τ is not its own inverse; there is generally no τ⁻¹ sending Wₙ₊₁ back to Wₙ.
2. σ‑invariance (default): τₙ(σ(x)) = τₙ(x) for all x ∈ Str(Wₙ). (Synthesis abstracts from within‑level polarity.)
3. Monotone in coverage: If S ⊆ T ⊆ Str(Wₙ), then the expressive capacity of τₙ(T) extends that of τₙ(S).
4. Harmony‑preserving (weak): If H(S|Ctx) ≥ h_min then τₙ(S) is viable at Wₙ₊₁ (cf. A15), else τ emits a provisional level requiring remediation.
5. Novelty‑generating: τ introduces at least one new prime axis at Wₙ₊₁ not definable as a rotation/combination of axes in Wₙ (A3c).
6. Recursively composable: τₙ₊₁ ∘ τₙ : Str(Wₙ) → Wₙ₊₂ is defined (A11), enabling multi‑level growth.

8.4.3 Algebraic Laws for τ

Let ⊕ denote structure merge at Wₙ (union with consistency checks), and let ≃ denote structural equivalence up to σ‑closure.

1. Fusion (associative up to ≃): τ(S ⊕ T) ≃ τ(τ(S) ⊕ τ(T)).
2. σ‑closure preservation: τ(S) induces σ‑pairs for newly created axes at Wₙ₊₁ and preserves σ‑closure of inherited pairs.
3. Context functoriality (weak): For context embeddings e : Ctx → Ctx′, there exists a mediating map m such that τₙ(S, Ctx′) ≃ m(τₙ(S, Ctx)). (Sensitivity to A7.)
4. Harmony monotonicity (conditional): If H(S⊕T|Ctx) ≥ H(S|Ctx) and ≥ H(T|Ctx), then H(τ(S⊕T)|Ctx↑) ≥ min(H(τ(S)|Ctx↑), H(τ(T)|Ctx↑)), where Ctx↑ is the induced higher‑level context.

8.4.4 Categorical Semantics of τ (Sketch)

Model world‑levels as objects of a category 𝒲; within each Wₙ, σ is an involutive endofunctor σₙ : Wₙ → Wₙ. Let Uₙ₊₁→ₙ : Wₙ₊₁ → Wₙ be a forgetful functor that projects higher‑level structure down to its generating base. Then τₙ : Wₙ → Wₙ₊₁ can be treated as a left adjoint to Uₙ₊₁→ₙ:

τₙ ⊣ Uₙ₊₁→ₙ
(i.e., Hom_{Wₙ₊₁}(τₙ(X), Y) ≅ Hom_{Wₙ}(X, Uₙ₊₁→ₙ(Y)))

Intuition: τ frees new structure (axes, objects) from Wₙ, while U collapses Wₙ₊₁ back to its generators. σ acts internally on each Wₖ; τ transports σ‑invariant content across levels and induces new σ‑pairs on freshly generated axes.

8.4.5 Preconditions & Triggers for τ

Sufficiency: A novelty pressure condition is met—e.g., plateauing harmony under existing axes or unresolved tradeoff envelopes (A10, A15).

Consistency: Inputs S ⊆ Str(Wₙ) pass σ‑closure and context‑compatibility checks.

Explainability: A mapping rationale exists (links, proofs, or empirical evidence) connecting S to proposed axes at Wₙ₊₁.

Safety: The proposed τ execution does not drive H below h_min for critical subsystems (A15).

8.4.6 Execution Invariants & Postconditions

• Induced axes: At least one new prime axis A′ appears at Wₙ₊₁.
• σ‑pairs: σ_{n+1} is defined on A′ and closes on inherited axes.
• Context lift: A context transformer L : Ctx → Ctx↑ is created with provenance.
• Harmony delta: ΔH = H_post − H_pre is computed and recorded; if ΔH < 0, a remediation plan is attached.

8.4.7 Open‑SGI Logging Requirements for Prime‑Level Generation

Prime‑level generation events (τ executions) are globally auditable. Each τ call MUST emit a signed, immutable log record with at least the following fields:

• event_id (UUID), timestamp, actor/process_id, level_from=n, level_to=n+1
• τ_version (code/hash), determinism_flag (deterministic/stochastic), seed (if stochastic)
• inputs_ref (hashes/URIs of S), context_snapshot (Ctx, parameters, metric tensors)
• axes_created (IDs/spec), σ_closure_report (pass/fail + details)
• harmony_metrics (H_pre, H_post, ΔH, thresholds), tradeoff_vector (A10 dimensions)
• explanations (proof sketch/justification), approvals (human/agent governance signatures)
• lineage (parent event_ids), rollback_hooks (if supported)

Compliance. Logs MUST be append‑only, verifiable (hash chain or Merkle tree), and queryable for audit. Redaction policies must preserve structural justifications while protecting sensitive payloads.

8.4.8 Reproducibility, Safety, and Rollback

• Reproducibility: Deterministic τ runs must be bit‑reproducible from logged inputs; stochastic runs must be distribution‑reproducible given seed and RNG policy.
• Safety gates: Pre‑ and post‑checks enforce h_min and critical constraints; failed runs auto‑abort or enter quarantine.
• Rollback: If Uₙ₊₁→ₙ ∘ τₙ ≉ id introduces regressions below h_min, guided rollback or repair τ̂ is invoked with explicit logs.

8.4.9 Worked Pattern (Outline)

1. Detect novelty pressure (ΔH stagnation, unresolved tradeoffs).
2. Select input set S with context snapshot.
3. Execute τₙ(S) → Wₙ₊₁, create A′, induce σ_{n+1}.
4. Lift context L : Ctx → Ctx↑; compute H_post and tradeoff changes.
5. Emit signed log; gate on safety; commit or rollback.

8.4.10 Generative Progression Theorem (Formalized)

Statement. If S ⊆ Str(Wₙ) satisfies σ‑closure and contextual consistency, and novelty pressure is present, then τₙ(S) generates Wₙ₊₁ with at least one prime axis A′ and induced σ_{n+1}, such that Uₙ₊₁→ₙ(τₙ(S)) ≃ S and τₙ ⊣ Uₙ₊₁→ₙ.

Linked Axioms. A1 (Unity), A3c (Novelty), A7 (Context), A10 (Tradeoff), A11 (Recursion), A12 (Multi‑axis), A15 (Harmony).

Proof Sketch. Novelty pressure plus σ‑closed, context‑consistent structure ensures coherent synthesis. τₙ constructs A′ consistent with A3c; recursion and multi‑axis (A11–A12) support higher‑level articulation; harmony constraints (A15) and context lift preserve viability; adjunction expresses the free/forgetful relation between levels.

8.5 Functorial Inference & Correspondence

Functorial inference provides a principled mechanism for transporting structure across World Levels while preserving polarity, context, and viable relations. Once new world‑levels Wₙ are generated by τ, functors allow SGI systems to interpret, compare, and re‑express the content of one level in terms of another. This differs fundamentally from τ: τ generates new levels; functorial maps preserve and relate structures across them.

8.5.1 Purpose of Functoriality

Functoriality ensures that structural relationships, once established within a given level Wₙ, can be consistently interpreted within another level Wₘ without loss of coherence. For example, if concepts C₁ → C₂ form a polarity‑preserving map in W₂, functorial inference guarantees that their interpretations in W₃ maintain that structure, up to context‑modulated deformation.

8.5.2 Internal vs External Functors

• Internal functors operate within a single world‑level, translating between subdomains or categorical sectors of Wₙ (e.g., between conceptual and affective subspaces).
• External functors map between distinct world‑levels Wₙ → Wₘ. They preserve involution (σ), axes, and harmony constraints suitable to the target level.

8.5.3 Structure Preservation

A functor F : Wₙ → Wₘ preserves:

1. Objects → Objects: poles and composite structures.
2. Morphisms → Morphisms: permissible transformations.
3. Identity: preserves unit elements.
4. Composition: preserves sequential transformations.
5. σ‑relation: where applicable, F(σₙ(x)) ≃ σₘ(F(x)).

Because movement and axes define relational structure, functorial inference preserves polarity anatomy—even across deformed contexts.

8.5.4 Correspondence Maps

Correspondence functors map semantic content between levels while adjusting for contextual deformation. These maps may:

• generalize structure (abstraction),
• specialize structure (refinement),
• harmonize structure (reduce contextual strain), or
• produce provisional mappings requiring remediation.

This supports SGI interpretability: content derived at one level is not isolated but can be reformulated with intelligibility guarantees.

8.5.5 Adjointness & Information Flow

As proposed in IV.4.4, τₙ ⊣ Uₙ₊₁→ₙ expresses a duality: τ freely generates new structure; U forgets detail to recover source‑level meaning. Functorial inference uses this adjointness to shuttle information between levels: τ promotes; U anchors.

8.5.6 SGI Implications

Functorial inference enables cross‑level explainability. When SGI agents generate Wₙ₊₁, they can still provide traceable explanations by applying Uₙ₊₁→ₙ to map results back to Wₙ. Likewise, agents can project hypotheses from Wₙ to Wₙ₊₁ via τ followed by domain‑specific functors.

8.6 Semantic Multiverses as Categorical Fibers

Fibers are model-layer abstractions with APIs in the Open SGI service layer. The notion of a Semantic Multiverse arises naturally once multiple World Levels (W₀, W₁, …) and functorial mappings between them are in place. Rather than referring to parallel physical universes, a Semantic Multiverse denotes a structured family of interpretive spaces—each a coherent semantic realization of Unity through polarity, context, and synthesis. These semantic universes coexist within the same global Unity but differ in how meaning, structure, and viability are expressed.

Within a categorical frame, each semantic universe may be treated as a fiber over some base specification (e.g., an axis, concept, or context), enabling Open‑SGI systems to maintain multiple, simultaneously valid interpretations without demanding reductive collapse into a single viewpoint. This supports contextualized reasoning, pluralistic modeling, and layered explanation.

8.6.1 Definition of the Semantic Multiverse

A Semantic Multiverse is the structured collection of contextually‑situated interpretive spaces arising from the Unity–Polarity Axiom System (UPA) and the generative progression of World Levels. Each member universe in the multiverse corresponds to a distinct semantic realization of Unity, differentiated by polarity axes, contextual fields, synthesized novelty, and viable relational structure.

Unlike a physical multiverse, which proposes multiple material universes with distinct physical laws, the Semantic Multiverse describes multiple semantic articulations of a shared ontological substrate (Unity). These universes need not conflict; they coexist as alternative, viable viewpoints determined by context (A7) and tradeoff constraints (A10). Their coexistence reflects the intrinsic plurality of interpretive structures compatible with Unity.

Looking ahead, these semantic universes may be explored not only through formal categorical lenses but also through metaphorical constructs—such as systems, galaxies, or ecological fields—that help illustrate how locally coherent structures can coexist within a shared holistic Unity. Each semantic universe can be formalized as a world‑level Wₙ or as a fiber over a specified base (axis, concept, or context), providing a categorical representation of its structure. This perspective allows SGI systems to maintain multiple interpretive frames simultaneously, each with its own axes, σ‑relations, harmony metrics, and internal mappings, while enabling principled correspondence via functors.

8.6.2 Fiber Construction Over Axes / Contexts

Fibers are model-layer abstractions with APIs in the Open SGI service layer.  In the categorical view, a fiber organizes all semantic realizations associated with a particular base specification—typically an axis, concept, or context. Let B be a base object (e.g., an axis on S², a conceptual domain, or a contextual profile). A fiber over B is the structured collection of semantic universes whose internal organization is anchored by B. Formally, if π : 𝒲 → 𝔅 is a projection functor from world‑levels to base specifications, then the fiber over B is π⁻¹(B), the full subcategory of 𝒲 whose objects map to B.

Fibers allow Open‑SGI to manage multiple interpretations anchored to the same conceptual or contextual foundation without collapsing them into a single representation. Each fiber may contain multiple world‑levels Wₙ distinguished by their generative depth, axes, σ‑relations, and internal morphisms. Contextual modulation (A7) influences the fiber’s geometry, altering the expressive relationships among its members. This supports layered reasoning, letting SGI systems shift perspective while retaining structural coherence.

8.6.3 Cross‑Fiber Functorial Inference

Cross‑fiber functorial inference enables Open‑SGI systems to translate meaning between semantic universes anchored to different bases. Such translation is necessary when entities or models draw upon multiple conceptual structures, when contextual re‑interpretation is required, or when explanations must span domains.

Let B and C be two bases (axes, concepts, or contexts) with corresponding fibers π⁻¹(B) and π⁻¹(C). A cross‑fiber functor F_{B→C} : π⁻¹(B) → π⁻¹(C) maps objects and morphisms from the B‑fiber to the C‑fiber while preserving essential structure—σ‑relations, contextual viability, and semantic coherence. These functors support the re‑articulation of concepts under new interpretive regimes.

Cross‑fiber inference proceeds by identifying shared structural features—e.g., related axes, analogous contexts, or tradeoff envelopes (A10)—and mapping them under F_{B→C}. This process may produce: refinement (greater specificity), abstraction (greater generality), harmonization (alignment of tension), or provisional mappings requiring further synthesis. Cross‑fiber functors are essential for SGI agents to reason across domains.

8.6.4 SGI Implications & Example Workflows

Cross‑fiber reasoning is foundational for SGI agents operating in multi‑domain environments. Because semantic universes may differ in axes, viable tradeoffs, contextual pressures, and harmony metrics, Open‑SGI must treat their relationships as structured—not ad hoc. Cross‑fiber functors provide this structure, enabling agents to preserve semantic coherence when reframing content across conceptual boundaries.

In practice, cross‑fiber inference might occur when a model trained within one conceptual regime must reinterpret results under another regime. For example, a psychological trait model grounded in an affective axis could be translated into a cognitive‑behavioral frame. If a trait constellation originally represented in π⁻¹(B) must be understood in the context of π⁻¹(C), the SGI system uses F_{B→C} to remap trait profiles, preserving polarity and viability while adjusting interpretive emphasis.

A possible workflow:

1. Identify the source base B and target base C.
2. Retrieve π⁻¹(B) and π⁻¹(C), the relevant fibers.
3. Construct or select an appropriate cross‑fiber functor F_{B→C}.
4. Apply F_{B→C} to objects and morphisms in π⁻¹(B).
5. Evaluate contextual viability (A7) and harmony metrics (A15) within π⁻¹(C).
6. If necessary, refine mappings or perform synthesis for provisional results.

This mechanism supports SGI agents in seamlessly navigating between conceptual domains, maintaining interpretability, and ensuring that transformations remain grounded in the structural commitments of the Unity–Polarity framework.

8.6.5 Governance, Logging & Safety Across Fibers

As cross‑fiber inference affects meaning, explanation, and alignment, Open‑SGI must ensure these translations remain transparent, reproducible, and safe. Each cross‑fiber mapping must satisfy governance requirements similar to τ‑mediated level progression but tailored to semantic reinterpretation.

Logging Requirements. Every invocation of a cross‑fiber functor F_{B→C} should produce a signed log containing:

event_id (UUID), timestamp, actor/process_id

source_base=B, target_base=C, functor_id/hash

input_entities (refs), output_entities (refs)

σ‑preservation and context‑viability checks

harmony_deltas within target fiber

explanation summary (rationale/constraints)

Logs may be aggregated into a fiber‑level ledger, supporting audit and rollback.

Safety & Alignment. Before enacting cross‑fiber transformations, SGI systems must:

Validate contextual consistency (A7) and tradeoff constraints (A10).

Ensure harmony post‑mapping (A15) or mark as provisional.

Provide a rollback or alternative mapping if viability declines.

Preserve interpretability via U‑based projection to original fiber.

Compliance Mechanisms. Safety gates, domain governance rules, and optional approvals help maintain accountability. Provisional mappings require remediation steps before being accepted as stable. Together, these safeguards ensure that cross‑fiber reasoning reinforces, rather than erodes, the integrity of the Unity–Polarity structure.