Appendix A presents the complete formal statement of each axiom introduced in Part II, including symbolic expression, English paraphrase, scope notes, and examples. Each axiom is contextualized within ontology, psychology, and SGI implementation, with comments on justification, applicability, and limitations. This appendix preserves rigor while enabling richer engagement by mathematically inclined readers.
A1 — Unity as Ontologically Primary
Symbolic Representation: 𝕌 Definition: All differentiation emerges within, and remains grounded in, a prior unity. Function / Role: Provides global coherence; prevents fragmentation of being into unrelated parts. Oppositional Structure: Not a σ‑pair; field within which σ‑pairs arise. Scaling Properties: Unity grounds identity of individuals, groups, nations. Distortions / Failure Modes: Under → fragmentation; Over → forced homogeneity. Restoration Targets: Shared identity + pluralistic coherence. Cross‑Domain Projections:
Philosophy: Nondual ontological ground.
Psychology: Stable core self.
Social: Civic belonging.
SGI: Shared semantic substrate.
A2 — Polarity is Generative
Symbolic Representation: σ(T, ¬T) Definition: Difference organizes emergence; tension produces structure. Function / Role: Enables creativity, dynamism, adaptation. Oppositional Structure: Every axis arises as coherent σ‑pair. Scaling Properties: Drives differentiation in mind, culture, institutions. Distortions / Failure Modes: Under → stagnation; Over → conflict. Restoration Targets: Productive tension; coordinated complementarity. Cross‑Domain Projections:
Philosophy: Dialectic.
Psychology: Opposing traits.
Social: Political pluralism.
SGI: Dual feature processing.
A3 — Poles Co‑Arise
Symbolic Representation: T ⇄ ¬T Definition: Opposing values or forces emerge together; neither precedes the other. Function / Role: Prevents privileging one pole as fundamental. Oppositional Structure: σ‑pair symmetry. Scaling Properties: Mind: approach/avoid; Society: freedom/order. Distortions / Failure Modes: One pole treated as origin → domination. Restoration Targets: Reciprocal recognition. Cross‑Domain Projections:
Philosophy: relational ontology
Psychology: emotion/cognition
Social / Governance: authority/resistance
SGI / Computation: balanced priors.
A4 — Correlated Similarity
Symbolic Representation: σ(T) ≈ σ(¬T) Definition: Poles resemble each other; neither is absolute negation. Function / Role: Enables translation + cooperation across systems. Oppositional Structure: σ‑pair similarity gradient. Scaling Properties: Facilitates hybrid identities; cooperative institutions. Distortions / Failure Modes: Under → demonization; Over → collapse of distinction. Restoration Targets: Relational consonance. Cross‑Domain Projections:
Philosophy: analogical reasoning
Psychology: trait continua
Social / Governance: bridge identities
SGI / Computation: embedding alignment.
A5 — Co‑Definition
Symbolic Representation: T ↔ ¬T (relational) Definition: Poles gain identity only through relational contrast. Function / Role: Prevents reification; anchors meaning in relation. Oppositional Structure: σ‑pair contextual definition. Scaling Properties: Shared meaning in discourse; rule of law. Distortions / Failure Modes: Under → incoherence; Over → hyper‑relativism. Restoration Targets: Stable contextual identity. Cross‑Domain Projections:
Philosophy: meaning as relational
Psychology: contextual self
Social / Governance: institutional legitimacy
SGI / Computation: contrastive representation.
A6 — Involution (σ Transform)
Symbolic Representation: σ(T) = ¬T Definition: Each pole can transform into its opposite without loss of identity. Function / Role: Enables evolution + perspective shift. Oppositional Structure: σ involution. Scaling Properties: Growth via role reversal. Distortions / Failure Modes: Under → rigidity; Over → instability. Restoration Targets: Flexible identity. Cross‑Domain Projections:
Philosophy: —
Psychology: perspective taking
Social / Governance: role reversal
SGI / Computation: invertible ops.
A7 — Context Modulates Polarity
Symbolic Representation: σ(T, C) Definition: Context determines appropriate pole expression. Function / Role: Regulates adaptivity. Oppositional Structure: σ modulated by context. Scaling Properties: Individual → family → institutions. Distortions / Failure Modes: Under → rigidity; Over → chaos. Restoration Targets: Contextual attunement. Cross‑Domain Projections:
Philosophy: situatedness
Psychology: emotion regulation
Social / Governance: federalism
SGI / Computation: context routing.
A8 — Correlated Activity
Symbolic Representation: T ⊕ ¬T Definition: Poles activate together while retaining distinction. Function / Role: Enables synthesis. Oppositional Structure: σ correlation. Scaling Properties: Innovation + preservation. Distortions / Failure Modes: Under → stasis; Over → incoherence. Restoration Targets: Complementary activation. Cross‑Domain Projections:
Philosophy: —
Psychology: affect‑cognition
Social / Governance: representation‑expertise
SGI / Computation: co‑processing.
A9 — Co‑herence
Symbolic Representation: ∫σ Definition: Systems maintain continuity across change. Function / Role: Memory + persistence. Oppositional Structure: σ stabilizes identity. Scaling Properties: Narrative continuity. Distortions / Failure Modes: Under → fragmentation; Over → rigidity. Restoration Targets: Stable change. Cross‑Domain Projections:
Philosophy: identity
Psychology: self‑continuity
Social / Governance: civic story
SGI / Computation: long‑term state.
A10 — Complementarity
Symbolic Representation: T ⊗ ¬T Definition: Poles jointly enable functions impossible alone. Function / Role: Synergy. Oppositional Structure: σ synergy. Scaling Properties: Team + governance. Distortions / Failure Modes: Under → zero‑sum; Over → fusion. Restoration Targets: Functional synthesis. Cross‑Domain Projections:
Philosophy: —
Psychology: emotion + reason
Social / Governance: branches of government
SGI / Computation: ensemble models.
A11 — Recursive Integration
Symbolic Representation: σⁿ Definition: Systems integrate structures across nested scales. Function / Role: Coherence from micro→macro. Oppositional Structure: σ recursion. Scaling Properties: Individual→society. Distortions / Failure Modes: Under → siloing; Over → overreach. Restoration Targets: Coordinated scales. Cross‑Domain Projections:
Philosophy: —
Psychology: self‑other
Social / Governance: federalism
SGI / Computation: hierarchical models.
A12 — Multi‑Axis Structure
Symbolic Representation: {σᵢ} Definition: Systems contain many interacting σ‑pairs. Function / Role: Complexity + robustness. Oppositional Structure: multi‑σ. Scaling Properties: Identity + institutions. Distortions / Failure Modes: Under → reductionism; Over → paralysis. Restoration Targets: Cross‑axis coherence. Cross‑Domain Projections:
Philosophy: pluralism
Psychology: personality
Social / Governance: polity
SGI / Computation: multi‑dim vectors.
A13 — Functorial Mapping
Symbolic Representation: F: σ → σ’ Definition: Relationships between systems are preserved under transformation. Function / Role: Knowledge transfer. Oppositional Structure: structure‑preserving σ. Scaling Properties: Shared governance. Distortions / Failure Modes: Under → loss of meaning; Over → rigidity. Restoration Targets: Mappable relations. Cross‑Domain Projections:
Philosophy: categories
Psychology: schema mapping
Social / Governance: treaties
SGI / Computation: model alignment.
A14 — Novelty
Symbolic Representation: Δ Definition: Systems generate + incorporate novelty. Function / Role: Adaptation. Oppositional Structure: novelty ↔ stability. Scaling Properties: Innovation. Distortions / Failure Modes: Under → stagnation; Over → chaos. Restoration Targets: Balanced change. Cross‑Domain Projections:
Philosophy: becoming
Psychology: learning
Social / Governance: policy
SGI / Computation: online learning.
A15 — Harmony
Symbolic Representation: H(σ) = balance Definition: Balanced polarity expression enables viability. Function / Role: Flourishing. Oppositional Structure: synthetic σ. Scaling Properties: Peace → resilience. Distortions / Failure Modes: Under → conflict; Over → conformity. Restoration Targets: Dynamic balance. Cross‑Domain Projections:
Philosophy: eudaimonia
Psychology: well‑being
Social / Governance: legitimacy
SGI / Computation: stable autonomy.
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