Theorem T3 — Recursive Coherence – Open Autonomous Intelligence Initiative

Associated Axioms: A9 (Mapping), A11 (Recursion), A12 (Multi‑Axis Interaction), A15 (Harmony / Viability)

Symbolic Representation:
σⁿ ⟹ Coherence(n) with H(σ) ≥ θ

Formal Statement:
Coherence across nested levels is achievable iff each level satisfies local harmony and all cross‑level interfaces preserve integrative (mapping‑consistent) constraints.

Interpretation:
Stability in any multi‑level system requires both (a) internal balance within each level, and (b) proper coordination across levels. Recursive coherence fails if either condition collapses.

Domain / Scope: Universal — applies to individuals, groups, worlds, semantic stacks, SGI systems, and any hierarchical or multi‑layer structure.

Function / Role:
Provides a foundation for layered governance, multi‑scale therapeutic integration, and hierarchical SGI architectures in which each layer contributes to — and depends on — the coherence of the whole.

1. Underlying Axioms

A9 — Mapping

Coherence requires that information, constraints, and structures map correctly across levels. If Lᵢ and Lᵢ₊₁ fail to preserve structural correspondences, recursive integrity collapses.

A11 — Recursion

Systems reproduce their organization across nested scales. Recursion ensures that σ (the polarity operation) is structurally similar at every level.

A12 — Multi‑Axis Interaction

Higher‑order coherence requires managing interactions among multiple axes of variation at each level and between levels.

A15 — Harmony (Viability Condition)

Each level must maintain minimum viability (H ≥ θ) to avoid destabilizing the entire stack.

2. Intuitive Explanation

Coherence is not achieved by forcing all levels to be identical. Nor is it achieved by isolating each level from the others.

Instead, recursive coherence requires two simultaneous conditions:

Local Harmony: Each level maintains its own viable balance of opposites (A15).
Cross‑Level Integrity: Interfaces between levels transmit constraints, structure, and feedback reliably (A9 + A11).

If one level becomes chaotic or one interface becomes distorted, coherence breaks down in a patterned, predictable way — either cascading downward (top‑heavy overreach) or upward (local chaos destabilizing global organization).

3. Scope and Applicability

Recursive coherence applies whenever:

Levels are nested (self → parts → subparts).
Levels are functionally coupled (local → regional → global).
Systems require stable feedback across scales.

Examples include:

selves, subselves, and self-coordination
households, communities, cities, nations
layers of SGI (data → models → planners → meta‑controllers)
semantic worlds stacking into multi-world intelligibility fields

4. Role in SGI / Open SGI Architecture

T3 is the primary theorem supporting hierarchical SGI stacks, including:

multi-layer planners
multi-world semantic routing
model‑of‑models introspection
harmonization monitors across levels
safety valves ensuring lower layers remain locally viable

The theorem justifies designing SGI architectures as layered, recursively coherent systems, not monolithic black boxes or fragmented collections of components.

5. Preconditions / Conditions for Satisfaction

1. Defined Interfaces

Each level must have explicit, specification-driven interfaces that:

constrain upward signals,
filter downward commands, and
preserve mapping structure (A9).

2. Feedback Channels

Stable coherence requires bi-directional feedback:

Upward: state reporting, viability signals.
Downward: calibration, constraints, parameterization.

3. Local Viability (H ≥ θ)

No level may fall below viability threshold. Local collapse propagates recursively.

6. Implications

1. Interface Contracts

Every cross-level connection must be treated as a contract: what is transmitted, how, and under what integrity guarantees.

2. Measure Cross‑Level Leakage

Leakage is when constraints meant for level Lᵢ inappropriately alter Lᵢ₊₁ (or vice versa). In SGI, leakage is a measurable expression of violated mappings (A9) or insufficient isolation of axes (A12).

3. Multi‑Scale Monitoring

Systems must monitor:

local harmony metrics (A15) at each level
coherence metrics between levels
recursive propagation effects

7. Failure Modes

1. Siloing

Levels become isolated, losing shared structure; mapping fails and the system fragments.

2. Top‑Down Overreach

Higher levels impose constraints that disrupt local viability, lowering H below θ.

3. Brittle Coupling

Levels are too tightly bound; local perturbations cascade uncontrollably.

These failure modes can be diagnosed using harmony metrics (Appendix F) and mapping/viability constraints (Appendix A).

8. Cross‑Domain Projections

Philosophy — Identity Through Change

The self persists not by static unity but by recursively coherent restructuring through time.

Psychology — Parts ↔ Whole‑Self Integration

Therapeutic change requires both:

healthy parts (subselves), and
healthy integration among them.

Social / Governance — Federal Integration

Federal systems succeed when:

local units remain viable, and
the federal interface preserves integrative constraints.

SGI / Computation — Multi‑Layer Planners

Planning across strategic → tactical → operational layers requires recursive coherence in:

goals,
constraints,
admissible actions,
viability thresholds.

9. Proof Sketch

The proof follows from inductive composition:

Base Case: Level L₀ maintains harmony (A15).
Induction Step: Assume Lᵢ is coherent. If the mapping from Lᵢ → Lᵢ₊₁ preserves structure (A9), and recursive relations hold (A11), then Lᵢ₊₁ inherits coherence provided its own local harmony condition is met.

Since multi-axis interactions are uniformly constrained (A12), the mapping is stable under recursive composition.

Thus, σⁿ yields Coherence(n) whenever each level satisfies viability and each interface preserves constraints.

10. PER / Siggy‑Style Example

A Personal Event Recognition (PER) SGI stack includes:

L₀: raw sensor event layer
L₁: semantic event classifier
L₂: situational plan recognizer
L₃: whole‑home adaptive policy engine

T3 requires:

each layer’s classifier or planner must be locally viable,
upward signals (events, scores, features) must preserve mapping integrity,
downward signals (constraints, comfort parameters, safety locks) must not violate local thresholds.

If L₁ degrades (e.g., ambiguous classifier states), coherence breaks upward (L₂ misreads situations) and downward (L₃ issues misaligned policies). The system stabilizes only when each layer is harmonious and interfaces maintain integrative structure.

11. Summary

The Recursive Coherence Theorem establishes that multi‑level stability emerges only when:

each level is locally harmonious and viable, and
cross‑level mappings and interfaces maintain structural integrity.

This is the foundational theorem behind hierarchical cognition, layered governance, multi-scale psychology, and SGI architectures designed for safety, transparency, and resilience.