Unity–Polarity Theorem Catalog
Overview
This catalog consolidates all theorems derived from the Unity–Polarity Axioms (UPA). Each entry follows a formal template aligned to the axiom catalog so the system is human- and machine-usable, versioned, and testable across domains (philosophy, psychology, social/governance, SGI/computation).
Philosophical Narrative Summary
At a philosophical level, the UPA theorem set demonstrates how unity and polarity jointly structure intelligibility. The theorems collectively argue that opposition is generative: poles emerge together (co‑arising), remain mutually informative (co‑definition), and yield novel forms when integrated (complementarity). Context‑dependent activation of poles reflects an ontological stance in which being is situated, responsive, and relational rather than static.
Recursive coherence and multi‑axis organization show that identity is conserved through layered differentiation, illustrating that systems can change without losing essence. Deliberation, integration, and role‑reversal indicate that ethical and political flourishing depend on dynamic balancing rather than fixed dominance. The philosophical outlook is neither monistic nor dualistic but holistically nondual: difference is essential to unity, and unity holds difference in patterned relation.
This narrative positions the UPA theorem catalog as a systematic renovation of classical holism, dialectics, and relational metaphysics—developing them into a formal, axiomatic structure capable of mapping across psychology, society, and autonomous intelligence.
Theorem Specification Template
T# — Name
Associated Axioms: A# …
Symbolic Representation: (e.g., σ(T, C) ⇒ τ; F: σ→σ′; H(σ) ≥ θ)
Statement (formal): Precise statement of the theorem.
Interpretation (plain language): What it means.
Domain / Scope: Individuals | Groups | Social | SGI
Function / Role: Why this matters; what it enables or constrains.
Conditions / Preconditions: Assumptions required.
Implications / Corollaries: Immediate consequences or related results.
Failure Modes: How/when it breaks (and what that looks like).
Cross-Domain Projections:
• Philosophy
• Psychology
• Social / Governance
• SGI / Computation
Proof Sketch (optional): Outline of reasoning.
Examples (optional): Minimal working examples.
Initial Core Theorems (Draft)
T1 — Contextual Selection Theorem
Associated Axioms: A2, A3, A7, A12
Symbolic Representation: σ(T, C) ⟶ T* where T* ∈ {T, ¬T}
Statement (formal): For any polarity σ(T, ¬T) and admissible context C, there exists a pole T* whose expression maximizes task-fit under C subject to cross-axis constraints.
Interpretation: Context selects which pole should dominate for the situation (but never deletes the other pole).
Domain / Scope: Individuals, Groups, Social, SGI
Function / Role: Basis for the A7 Context Router service and policy selectors.
Conditions: Defined context metrics; admissible constraint set.
Implications / Corollaries: Enables policy routing; predicts misfit when selection is rigid or context-blind.
Failure Modes: Ambiguous/false contexts; missing constraints; adversarial cues.
Cross-Domain Projections:
- Philosophy: situational normativity
- Psychology: emotion regulation; strategy selection
- Social / Governance: subsidiarity; venue choice
- SGI / Computation: dynamic policy selection
Proof Sketch: From A7, context is a modulator; from A12, selection respects other axes; optimization yields T*.
T2 — Complementary Activation Theorem
Associated Axioms: A2, A4, A8, A10
Symbolic Representation: T ⊕ ¬T ⇒ Φ(T, ¬T) > max{Φ(T), Φ(¬T)}
Statement (formal): Joint activation of complementary poles yields a function Φ strictly greater than either pole alone, under conditions of correlated similarity.
Interpretation: Coordinated use of both poles outperforms single-pole strategies.
Domain / Scope: All
Function / Role: Justifies mixed teams, cross-partisan deliberation, multi-model ensembles.
Conditions: Non-trivial similarity; integrative mechanism present.
Implications: Incentivize pairwise integration; avoid zero-sum design.
Failure Modes: Incompatibility; lack of integrator; premature fusion.
Cross-Domain Projections:
- Philosophy: dialectical synthesis
- Psychology: affect + reason
- Social / Governance: representation + expertise
- SGI / Computation: ensemble gains
Proof Sketch: From A4 (similarity) and A10 (complementarity), define Φ superadditivity under integration operator.
T3 — Recursive Coherence Theorem
Associated Axioms: A9, A11, A12, A15
Symbolic Representation: σⁿ ⟹ Coherence(n) with H(σ) ≥ θ
Statement (formal): Coherence across nested levels is achievable iff each level satisfies local harmony and cross-level interfaces preserve integrative constraints.
Interpretation: Stability requires both local balance and proper cross-level coordination.
Domain / Scope: All
Function / Role: Basis for layered governance, multi-scale therapy, hierarchical SGI stacks.
Conditions: Defined interfaces; feedback channels; local viability.
Implications: Design for interface contracts; measure cross-level leakage.
Failure Modes: Siloing; top-down overreach; brittle coupling.
Cross-Domain Projections:
- Philosophy: identity through change
- Psychology: self-parts ↔ whole-self
- Social / Governance: federal integration
- SGI / Computation: multi-layer planners
Proof Sketch: Inductive composition of harmonious subsystems (A15) with preserved mappings (A13).
T4 — Multi-Axis Tradeoff Theorem
Associated Axioms: A2, A12, A15
Symbolic Representation: Optimize Σ wᵢ·σᵢ subject to H(σ) ≥ θ
Statement (formal): Any improvement along one axis that drives H(σ) below threshold is non-viable; feasible optima lie on a harmony-constrained Pareto front.
Interpretation: Don’t optimize one axis (e.g., speed) so hard that system viability collapses (e.g., safety).
Domain / Scope: All
Function / Role: Sets guardrails for policy/architecture tuning.
Conditions: Harmony metric θ; weights wᵢ.
Implications: Use constrained optimization; publish tradeoff surfaces.
Failure Modes: Metric gaming; missing axes.
Cross-Domain Projections:
- Philosophy: practical reason under constraints
- Psychology: values balancing
- Social / Governance: policy tradeoffs (liberty/order)
- SGI / Computation: latency/quality/safety tuning
Proof Sketch: From A12 (axes) and A15 (harmony), define viability region; show front existence.
T5 — Deliberative Integration Theorem
Associated Axioms: A2, A4, A5, A8, A10, A15
Symbolic Representation: Delib(voices, expertise) ⇒ ↑H(σ)
Statement (formal): Under conditions of mutual recognition and structured turn-taking, deliberation that includes diverse representation and relevant expertise increases expected harmony.
Interpretation: Good deliberation integrates poles and raises system viability.
Domain / Scope: Groups, Social, SGI (committee-of-models)
Function / Role: Justifies citizens’ assemblies; expert–public hybrids.
Conditions: Inclusion, structure, facilitation, shared aims.
Implications: Invest in deliberative capacity; measure post-deliberation coherence.
Failure Modes: Tokenism; dominance; lack of integration.
Cross-Domain Projections:
- Philosophy: dialogical reason
- Psychology: perspective integration
- Social / Governance: civic forums
- SGI / Computation: multi-agent debate
Proof Sketch: From complementarity and correlated activity, demonstrate expected H(σ) increases.
T6 — Anti-Capture Rotation Theorem
Associated Axioms: A2, A6, A7, A12, A15
Symbolic Representation: Rotate(roles) ⇒ ↓Capture
Statement (formal): Periodic role rotation in systems with multi-axis responsibilities reduces the probability of polarity capture while preserving harmony.
Interpretation: Rotating stewardship prevents one pole from monopolizing authority.
Domain / Scope: Groups, Social, SGI
Function / Role: Anti-capture mechanism for organizations and agents.
Conditions: Rotation cadence; competency preservation.
Implications: Design roles for interchangeability; maintain knowledge transfer.
Failure Modes: Loss of expertise; performative rotation.
Cross-Domain Projections:
- Philosophy: reversal and renewal
- Psychology: role flexibility
- Social / Governance: term limits; rotating chairs
- SGI / Computation: rotating controller/critic
Proof Sketch: From A6 (involution) and A15, rotation reduces single-pole dominance.
T7 — Identity Layering Theorem
Associated Axioms: A1, A4, A5, A11, A15
Symbolic Representation: Layer(idᵢ) ⇒ ↑H(σ)
Statement (formal): Systems that maintain multi-layer identities with coherent cross-layer mappings sustain higher harmony and resilience under perturbation.
Interpretation: Layered identity (personal, local, national, global) buffers shocks.
Domain / Scope: Psychology, Social
Function / Role: Justifies plural affiliation; resists polarization.
Conditions: Non-exclusive memberships; bridging narratives.
Implications: Support federated belonging; avoid purity tests.
Failure Modes: Identity fusion; zero-sum norms.
Cross-Domain Projections:
- Philosophy: one and many
- Psychology: self-complexity
- Social / Governance: federal identity
- SGI / Computation: multi-tenant identity
Proof Sketch: Diversity-in-identity provides redundancy; mappings preserve coherence (A13).
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