Theorem T4 — Multi-Axis Tradeoff – Open Autonomous Intelligence Initiative

Associated Axioms: A2 (Polarity), A12 (Multi-Axis Interaction), A15 (Harmony / Viability)

Symbolic Representation:
Optimize Σ wᵢ · σᵢ subject to H(σ) ≥ θ

Formal Statement:
Any improvement along one axis that drives the harmony metric H(σ) below its viability threshold θ is non-viable. All admissible optima lie on a harmony‑constrained Pareto front, rather than on unconstrained extrema.

Interpretation:
Optimizing one axis (e.g., speed, autonomy, efficiency) at the expense of others (e.g., safety, robustness, ethical guardrails) is only legitimate if overall system harmony remains above the viability threshold. Otherwise, the optimization is structurally self-defeating.

Domain / Scope: Universal — applies to individuals, organizations, semantic worlds, policy systems, SGI architectures, and any multi-axis optimization environment.

Function / Role:
Defines the guardrails for tuning systems with multiple dimensions. Ensures that optimization is viability-aware, transparent, and resistant to collapse from over‑optimization on any single axis.

1. Underlying Axioms

A2 — Polarity

Each axis constitutes a structured polarity (T / ¬T) or more generally a continuum of opposed pressures.

A12 — Multi-Axis Interaction

Systems exhibit interactions among axes; pushing one dimension typically alters the others. Tradeoffs are structural, not incidental.

A15 — Harmony (Viability Condition)

A system must maintain a minimum harmony value H ≥ θ in order to remain viable. Optimization must therefore be viability-constrained.

2. Intuitive Explanation

A system is rarely judged by a single metric. Instead, many competing axes shape its performance:

speed ↔ safety
autonomy ↔ oversight
precision ↔ compute cost
liberty ↔ order
novelty ↔ stability

The Multi-Axis Tradeoff Theorem states:

It is impossible to maximize one axis in isolation without affecting the others.
Pushing one dimension too far causes overall viability (H) to fall below the threshold θ.
Therefore, the true optimum lies on a Pareto front constrained by viability, not the theoretical unconstrained maximum.

The theorem explains why “just optimize X” never works: optimization is entangled.

3. Scope and Applicability

T4 applies wherever:

multiple meaningful dimensions exist,
axes are coupled (A12),
viability constraints matter (A15),
tradeoffs are inherent rather than optional.

This includes:

personal psychological balance,
interpersonal relationships,
organizational decision-making,
governance and policy design,
SGI architectures and multi-objective planners.

4. Role in SGI / Open SGI Architecture

The theorem is central for safe and transparent SGI tuning, ensuring:

planners respect viability constraints,
optimization surfaces are published and interpretable,
safety–latency–quality tradeoffs are explicit,
models cannot silently degrade harmony values,
hyperparameter tuning includes harmonics, not just performance.

In Open SGI, T4 is a structural guardrail for both:

inner-loop optimization (per-layer tuning), and
outer-loop meta-optimization (coordination across layers).

5. Preconditions / Conditions for Satisfaction

1. Harmony Metric θ

There must exist a clearly defined viability threshold. Below it, performance is unstable or unsafe.

2. Weight Vector wᵢ

Relative priorities among axes must be specified, disclosed, or learned under oversight.

3. Structured Axes (A2, A12)

Each axis must be defined, measurable, and interpretable. Hidden axes distort the optimization.

4. Constraints Are Active

The viability constraint H ≥ θ cannot be symbolic; it must actively shape the feasible region.

6. Implications

1. Use Constrained Optimization, Not Unconstrained Tuning

Algorithms must enforce viability. Squeezing out “a little more performance” at the cost of falling below θ is forbidden.

2. Publish Tradeoff Surfaces

SGI systems, governments, and organizations should publish:

admissible regions,
Pareto fronts,
forbidden regions (H < θ),
sensitivity analysis.

3. Avoid Metric Gaming

If one axis is optimized by proxy while others degrade, the harmony constraint detects and blocks the failure.

4. Detect Missing Axes

If H drops unexpectedly, it indicates that optimization ignored a relevant dimension.

7. Failure Modes

1. Metric Gaming

Optimizing one metric gives the illusion of improvement while true viability collapses.

2. Missing Axes

A system fails when a relevant dimension is omitted — e.g., safety, ethics, or long-term sustainability.

3. Over-Optimization

Hyper-fixation on a single axis (e.g., latency) produces brittleness or catastrophic safety failure.

8. Cross-Domain Projections

Philosophy — Practical Reason Under Constraints

Reasonable action is always constrained by viability — one cannot maximize a single virtue at the cost of moral collapse.

Psychology — Values Balancing

Healthy functioning requires balancing autonomy, intimacy, achievement, play, coherence, and rest.

Social / Governance — Policy Tradeoffs

Societies must balance liberty and order, growth and sustainability, innovation and stability.

SGI / Computation — Latency/Quality/Safety Tuning

The theorem directly models the tension among:

speed (latency),
output quality,
risk/safety constraints.

SGI systems must formalize and expose these surfaces.

9. Proof Sketch

From A12, systems operate along multiple interacting axes. From A15, system viability requires H ≥ θ. Therefore:

Define the feasible region F = {σ | H(σ) ≥ θ}.
Since axes interact (A12), altering σᵢ alters H.
Optimization is valid only within F.
Since unconstrained extrema typically lie outside F, the true optima must sit on the boundary of F.
This boundary is precisely the harmony-constrained Pareto front.

Thus, any improvement along one axis that destroys viability is inadmissible, and all realizable optima lie on the constrained front.

10. PER / Siggy-Style Example

Consider a home PER system balancing:

latency (speed of detection),
accuracy, and
safety thresholds.

If latency is aggressively optimized (e.g., ultra-fast but noisy models), accuracy may drop and false alarms increase — reducing harmony.

If accuracy is optimized without regard for compute limits, latency becomes too high to be useful, reducing harmony.

The viable region is where:

detection is fast enough,
classification is accurate enough,
alerts are safe enough.

Optimization finds the Pareto surface that balances all three while maintaining H ≥ θ.

11. Summary

The Multi-Axis Tradeoff Theorem states that:

optimization is multi-dimensional,
viability constraints cannot be violated,
feasible optima lie on a harmony-constrained Pareto frontier.

This result anchors practical reasoning, healthy psychological balance, responsible governance, and safe SGI architecture tuning.