Theorem T1 — Contextual Selection Theorem

theorems

✴︎

11.27.2025

Expanded Statement, Detailed Explanation, and PER/Siggy Implementation Example

1. Formal Theorem Statement

Symbolic Representation

σ(T, ¬T) ⟶₍C₎ T* where T* ∈ {T, ¬T}

Formal Statement

For any polarity axis σ(T, ¬T) and any admissible context C, there exists a pole T* such that:

T* maximizes task-fit under context C,
subject to cross-axis constraints defined by the World’s polarity system (Π),
and yields a viable (𝒱-preserving) expression of the polarity.

The non-selected pole is never deleted; it remains latent and potentially active under a different context C′.

2. Underlying Axioms

This theorem directly draws from four UPA axioms:

A2 — Polarity: Every dimension of meaning is structured as a σ-pair (T, ¬T).
A3 — Continuity: Polarity expressions shift continuously, not discretely.
A7 — Context: Context modulates meaning, salience, and expression.
A12 — Multi-Axis Interaction: No axis expresses independently of others; selection preserves cross-axis consistency.

The theorem is essentially the interaction of these axioms.

3. Intuitive Explanation

The theorem states that:

Context determines which pole of a polarity should dominate behavior, interpretation, or evaluation—without erasing or invalidating the opposite pole.

In practice:

The polarity always exists.
But only one pole is contextually expressed as the “best fit” at a given moment.
The choice of pole depends on:
- environmental conditions,
- situational goals,
- active contexts (C),
- cross-axis considerations,
- and harmony/viability constraints.

Example of what this means cognitively or computationally:

You may need to be cautious (¬T) when walking on ice, even if you are generally confident (T).
Siggy may need to emphasize routine adherence (T) in one context but flexibility (¬T) when the user is fatigued.

The pole expressed is never absolute—it is context-fitted.

4. Theorem Scope

This theorem applies across all UPA-relevant domains:

Individuals: personal decisions, emotional responses.
Groups: strategy selection, conflict response.
Social & Governance: venue selection, institutional subsidiarity.
SGI / Open SGI: dynamic policy routing and contextual inference.

It is one of the most operationally important theorems for SGI.

5. Functional Role in Open SGI

T1 is the conceptual basis for:

The Context Router Service (A7 implementation)
Dynamic policy selection systems
Behavioral modulation based on situational cues
Salience-based decision making (A14 integration)
Adaptive event interpretation

Open SGI uses T1 internally to determine:

which behaviors to activate,
which rules to apply,
which axes should shift,
which outputs are appropriate for the user or caregiver.

This theorem is core infrastructure.

6. Preconditions / Conditions

The theorem requires:

6.1 Context Definition

A clear schema for representing context (C):

temporal,
spatial,
physiological,
behavioral,
environmental,
social,
emotional (user-inferred).

6.2 Admissibility Conditions

Context must be:

well-formed,
interpretable,
allowed within the World,
not contradictory with strong constraints.

6.3 Cross-Axis Constraints (from A12)

You cannot select a pole in one axis that:

destabilizes viability (A15),
violates structural mappings (A13),
contradicts higher-level axes (A11 recursion).

7. Implications & Corollaries

7.1 Policy Routing

Contexts determine:

which policy applies,
which decision tree Siggy should use,
whether to escalate or down-regulate alerts.

7.2 Misfit Prediction

If a system selects a pole without considering context, the following emerge:

false positives,
false alarms,
non-adaptive responses,
user frustration,
caregiver fatigue.

7.3 Explanatory Transparency

Siggy can explain:

“Based on current context (evening, low light, reduced mobility), I have shifted to the cautious pole on the stability axis.”

7.4 Robustness Against Context Blindness

A system that cannot adapt to context will:

overfit
misinterpret events
or become brittle.

T1 enforces contextual adaptability.

8. Failure Modes

There are three primary failure modes:

8.1 Ambiguous Context

Context is unclear, e.g.:

conflicting signals,
partial sensor readings,
missing metadata.

8.2 Missing Constraints

Context selects a pole that violates higher-level axes.

8.3 Adversarial Context

Incorrect environmental cues cause mis-selection.
(For PER: heat reflections, shadows, unusual lighting, sensor misfires.)

9. Cross-Domain Projections

Philosophy — Situational Normativity

Context determines the appropriate expression of a value or trait.

Psychology — Emotion Regulation & Strategy Selection

Approach vs. avoidance depends on context.

Social / Governance — Subsidiarity & Venue Choice

Different contexts require different decision venues:

local vs. national,
executive vs. legislative,
caregiver vs. professional.

SGI / Computation — Dynamic Policy Selection

Context determines which operational mode or decision-policy dominates.

10. Proof Sketch

From A2 (Polarity): Every dimension has a σ-structure.
From A3 (Continuity): Polarity expression is graded, not binary.
From A7 (Context): Context modulates polarity weights.
From A12 (Multi-Axis): Selection must respect interactions across axes.

Therefore, for any σ(T, ¬T) and any admissible context C, an optimization over:

contextual weights,
cross-axis constraints,
viability constraints

will produce a maximizing pole T*.

QED (structurally).

11. PER/Siggy Example

Below is a concrete, fully implemented Siggy scenario.

11.1 Polarity Axis

σ(stability, mobility)

T = stability (cautious movement)
¬T = mobility (freer movement)

11.2 Context C

Late evening
Low lighting
Prior near-fall event today
Slight gait instability detected

11.3 Theorem Application

Given this context, Siggy evaluates:

increased fall-risk (cross-axis constraint from SafetyWorld),
lower lighting (context modifier),
fatigue markers (daily routine deviations),
recent novelty (A10/A14/A16 integration).

Result:

T* = stability, i.e. the cautious pole should dominate.

11.4 System Behavior (Open SGI implementation)

Mobility encouragement decreases.
Step stability detection thresholds tighten.
Siggy recommends: “Use available lighting when moving.”
Caregiver World receives a mild advisory notification.
Viability score recalculated for next interval.

11.5 Pole Not Deleted

When context changes tomorrow morning:

better lighting,
rested state,
normal gait,

the mobility pole (¬T) may become dominant again.

12. Summary

The Contextual Selection Theorem explains how Open SGI systems—especially Siggy in PER applications—select the appropriate expression of any polarity based on context while preserving cross-axis integrity and global viability.

It is one of the essential bridges between:

UPA theory (A2, A3, A7, A12),
SGI architecture (context routing, policy selection),
and practical functionality (fall detection, behavior modulation, safety alerts).