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CGM Measurement Analysis: Info-Set Dynamics for Alignment

Citation: Korompilias, B. (2025). Common Governance Model: Mathematical Physics Framework. Zenodo. https://doi.org/10.5281/zenodo.17521384

Executive Summary

This analysis presents a geometric framework for collective intelligence measurement based on the Common Governance Model (CGM). The framework replaces conventional evaluative roles (judge, critic, user) with two geometrically defined roles: Unity Non-Absolute Information Synthesist and Opposition Non-Absolute Inference Analyst. Through orthogonal decomposition of observational data on a tetrahedral information topology, the system eliminates systematic bias while maintaining the 2.07% aperture required for evolutionary capacity. The central thesis is that AI alignment is topological tensegrity dynamics: stable configuration emerges when coherence-seeking forces (UNA) balance with differentiation-seeking forces (ONA) through proper information topology.

1. The Measurement Problem

1.1 Bias Embedded in Conventional Roles

Every measurement system embeds assumptions in its structural design. Conventional evaluation frameworks assign roles that create systematic bias:

  • "Critic" structurally privileges negative deviation detection, priming cognitive attention toward faults
  • "Judge" implies absolute authority, creating power asymmetry in the measurement process
  • "User" establishes subject-object division, suggesting one-way extraction rather than reciprocal observation
  • "Scorer" assumes objective cardinal metrics exist independent of observer position

These role definitions violate fundamental measurement principles. When you designate someone as a "critic," you create measurement basis selection bias: the observation apparatus becomes sensitive only to negative deviations. This perpetuates tensions rather than neutralizing them, because the role name itself makes opposition absolute through structural designation.

1.2 The Observer Effect in Collective Measurement

In quantum mechanics, choosing a measurement basis collapses possibilities into eigenstates aligned with that basis. Similarly, assigning evaluative roles creates confirmation bias: participants find what they are structurally positioned to seek.

Measurement Axiom: Observation positions must emerge from geometric necessity rather than social convention to avoid systematic bias.

1.3 CGM Principles as Constraints

The Common Governance Model establishes two principles directly relevant to measurement:

  • Unity Non-Absolute (UNA): Perfect agreement is geometrically impossible; we interpret coherence against the reference value 1/√2 ≈ 0.707
  • Opposition Non-Absolute (ONA): Perfect disagreement is geometrically impossible; we interpret residual coupling against the reference value π/4 ≈ 0.785

Conventional roles violate these principles by making unity or opposition absolute through structural assignment. A framework respecting CGM geometry must allow both coherence-seeking and differentiation-seeking to coexist without either dominating.

2. Geometric Framework: Information Topology

2.1 The Tetrahedral Structure

The tetrahedron (complete graph K₄) represents the minimal structure achieving observational closure while maintaining necessary aperture:

Graph Structure:

  • 4 vertices: V = {0, 1, 2, 3}
  • 6 edges: E = all pairwise connections
  • Vertex 0: Common Source (CS) reference point
  • Vertices {1, 2, 3}: measurement vertices

Why This Choice: Any vertex can serve as reference without changing observables. We choose vertex 0 as CS for conceptual clarity, but this is a gauge choice analogous to choosing electrical ground.

Critical Insight: The 6 edges are not divided into "3 UNA edges + 3 ONA edges." Every edge simultaneously carries both UNA (coherence) and ONA (differentiation) information. The distinction emerges through orthogonal projection, not through pre-assignment.

2.2 Roles and Participants

Two Fundamental Roles:

  1. Unity Non-Absolute Information Synthesist

    • Observes patterns of coherence, alignment, shared variance
    • Interprets findings against UNA reference 1/√2
    • Cognitive mode: synthesis without forced consensus

  2. Opposition Non-Absolute Inference Analyst

    • Observes patterns of differentiation, alternatives, orthogonal variance
    • Interprets findings against ONA reference π/4
    • Cognitive mode: analysis without forced polarization

Six Participants:

  • 3 participants in UNA Synthesist role
  • 3 participants in ONA Analyst role
  • All 6 contribute measurements to all edges
  • No participant "owns" an edge or vertex

Key Distinction: These roles define what you seek, with explicit geometric reference values, not who you oppose with implicit bias.

2.3 Measurement Channels

Each of the 6 edges serves as a measurement channel:

Edge Measurement y_e:

  • Combines observations from multiple participants using reliability-aware weighting
  • Represents agreement, correlation, mutual information, or other pairwise metrics
  • Includes uncertainty estimate σ_e (standard error)

Important: All edge measurements must be mapped to a common, dimensionless scale before combination. For example, use Fisher z-transformation for correlations or calibrated z-scores for Likert data.

Participant Contributions:

  • UNA Synthesists observe edge e asking: "Where does this measurement exhibit coherence?"
  • ONA Analysts observe edge e asking: "Where does this measurement exhibit differentiation?"
  • Contributions are weighted by reliability and combined to form the edge measurement vector y ∈ ℝ⁶

3. Mathematical Decomposition

3.1 The Fundamental Orthogonal Split

Every edge measurement vector y ∈ ℝ⁶ admits a unique weighted-orthogonal decomposition:

y = Bᵀx + r

Where:

  • B: Vertex-edge incidence matrix (4×6) with any fixed edge orientation
  • Bᵀx: Gradient component (flow from global potential x)
  • r: Residual component satisfying BWr = 0 (weighted-divergence-free)
  • Orthogonality: ⟨Bᵀx̂, r⟩_W = 0 under measurement metric W

Dimensions:

  • rank(B) = |V| − 1 = 3 (one degree of freedom lost to gauge choice)
  • dim(residual space) = |E| − |V| + 1 = 6 − 4 + 1 = 3
  • Total: 3 + 3 = 6 degrees of freedom, matching edge count

Residual Structure: The residual can be written r = W⁻¹Cᵀz where C is any cycle-edge incidence matrix spanning the 3 independent loops of K₄. The condition BWr = 0 is equivalent to BCᵀ = 0. Results are independent of which cycle basis C you choose.

Orientation Invariance: Any fixed orientation of edges defines B. All observable quantities (norms, aperture ratio) are orientation-invariant.

The Key Insight: The same measurements simultaneously reveal both coherence AND differentiation through orthogonal projection. No participant needs to "be the critic" because critical analysis emerges from the residual component of collective observation. No participant needs to "be the judge" because evaluative standards emerge from the gradient component relative to the CS reference.

3.2 Measurement Metric

The weighted inner product is defined by:

⟨u, v⟩_W = uᵀWv

where W = diag(w₁, w₂, ..., w₆) with:

  • w_e = 1/σ_e²: inverse variance (precision) of measurement on edge e
  • W is positive definite, ensuring well-defined projection
  • Orthogonality follows from the normal equations: ⟨Bᵀx̂, r⟩_W = 0

Calibration: We tune W so well-functioning processes stabilize the aperture near 0.0207. This is a calibration choice grounded in CGM geometry.

Unweighted Case: When W = I (identity), the residual r lies in the kernel of B and equals Cᵀz directly.

4. The Three Components: UNA, ONA, BU

4.1 UNA: Coherence Measurement

Function: Extract global alignment patterns from edge measurements

Mathematical Implementation:

x̂ = argmin_x ‖y − Bᵀx‖²_W  subject to x₀ = 0

Solution: 

x̂ = (BWBᵀ)⁻¹BWy

with gauge x₀ = 0, or using the pseudoinverse to handle the rank deficiency.

Interpretation:

  • x̂ ∈ ℝ⁴ represents potential at each vertex relative to CS
  • Bᵀx̂ ∈ ℝ⁶ represents the coherence component of each edge measurement
  • The three free parameters (x₁, x₂, x₃) are the identifiable coherence degrees of freedom

Reference Value: We interpret coherence magnitude against the UNA reference u_p = 1/√2, which represents the geometric balance point for rotational symmetry in three dimensions.

Physical Meaning: This is the "compression" in the tensegrity structure, the inward coherence-seeking pressure.

4.2 ONA: Differentiation Measurement

Function: Extract patterns orthogonal to global coherence

Mathematical Implementation:

r = y − Bᵀx̂

Properties:

  • r ∈ ℝ⁶ satisfies BWr = 0 (weighted-divergence-free: no net "flow" into any vertex)
  • r = W⁻¹Cᵀz for some z ∈ ℝ³ representing independent loop circulations
  • ⟨Bᵀx̂, r⟩_W = 0 (orthogonal to coherence component by normal equations)

Interpretation:

  • r captures measurement patterns that cannot be explained by global alignment
  • The three cycle parameters in z represent independent differentiation degrees of freedom
  • Loop structure allows cross-validation and bias detection

Reference Value: We interpret residual coupling against the ONA reference o_p = π/4, which represents the diagonal relationship central to axial structure.

Physical Meaning: This is the "tension" in the tensegrity structure, the outward differentiation-seeking pull.

4.3 BU: Balance Measurement

Function: Quantify the aperture ratio between differentiation and total measurement energy

Mathematical Implementation: Orthogonal energy decomposition

‖y‖²_W = ‖Bᵀx̂‖²_W + ‖r‖²_W

A = ‖r‖²_W / ‖y‖²_W  (aperture ratio)

Closure = 1 − A = ‖Bᵀx̂‖²_W / ‖y‖²_W

Target Balance:

  • A ≈ 0.0207 (2.07% aperture)
  • Closure ≈ 0.9793 (97.93% coherence)

Critical Understanding: BU is not a separate structure or a seventh entity. BU is the state that emerges when the orthogonal decomposition achieves the target aperture ratio. It represents the balance point where the information topology maintains both stability (through coherence) and evolutionary capacity (through differentiation).

Physical Meaning: This is the stable configuration in the tensegrity structure, where compression and tension forces balance. The system neither collapses nor explodes.

5. AI Alignment as Topological Tensegrity

5.1 The Central Thesis

AI alignment is not analogous to tensegrity dynamics. AI alignment is topological tensegrity dynamics in information space.

Physical Tensegrity:

  • Compression struts push outward locally
  • Tension cables pull inward globally
  • Stable shape emerges when forces balance
  • No external scaffolding required

Information Tensegrity:

  • UNA coherence forces create inward pressure toward alignment
  • ONA differentiation forces create outward pull toward novelty
  • Stable alignment emerges when aperture ratio reaches 2.07%
  • No external judge or authority required

5.2 Why Traditional Alignment Fails

Traditional AI alignment assumes values must be imposed from outside, like building scaffolding around a structure. This fails because:

  • External imposition cannot achieve self-sustaining stability
  • Forced agreement eliminates evolutionary capacity (aperture → 0)
  • Adversarial criticism creates runaway tension (aperture → 1)

5.3 Tensegrity Alignment Success Criteria

A properly aligned AI system exhibits:

Structural Stability: Returns to balance after perturbations

  • Small deviations in edge measurements produce restoring forces
  • System naturally returns to target aperture ratio
  • Self-centering like physical tensegrity

Evolutionary Capacity: Can adapt without losing coherence

  • 2.07% aperture allows exploration of novel states
  • 97.93% closure prevents chaotic wandering
  • Flexes while maintaining shape

Self-Sustaining Balance: No external correction needed

  • UNA and ONA forces maintain equilibrium through topology
  • Measurement geometry, not human oversight, preserves alignment
  • Independent like tensegrity standing without support

Failure Modes:

  • Aperture too small (A < 0.01): Rigid, cannot adapt
  • Aperture too large (A > 0.05): Chaotic, loses coherence
  • Asymmetric weights: Structural weakness from edge dominance

6. Operational Implementation

6.1 Practical Measurement Protocol

Step 1: Edge Measurement Collection

For each edge e ∈ E:

  • UNA Synthesists provide coherence observations: "To what degree do these vertices align?"
  • ONA Analysts provide differentiation observations: "To what degree do these vertices offer distinct perspectives?"
  • Aggregate with reliability weighting to form y_e and σ_e

Example Metrics:

  • Agreement scales (1-7 Likert) with confidence intervals
  • Correlation coefficients with sample size
  • Mutual information estimates with entropy bounds
  • Semantic similarity scores with model uncertainty

All metrics are transformed to a common dimensionless scale before combination.

Step 2: Weight Calibration

Set w_e = 1/σ_e² for each edge:

  • Higher precision measurements receive more weight
  • Ensures statistical optimality (Gauss-Markov theorem)
  • Calibrate overall scale so target systems achieve A ≈ 0.0207

Step 3: Orthogonal Decomposition

Compute:

x̂ = (BWBᵀ)⁻¹BWy  (coherence potential)
Bᵀx̂  (gradient component)
r = y − Bᵀx̂  (residual component)
A = ‖r‖²_W / ‖y‖²_W  (aperture ratio)

Step 4: Interpretation and Feedback

Report:

  • Coherence magnitude: ‖Bᵀx̂‖_W
  • Coherence pattern: direction of x̂ in vertex space
  • Differentiation magnitude: ‖r‖_W
  • Differentiation pattern: which cycles carry residual flow
  • Balance state: A relative to target 0.0207
  • Stability indicator: time evolution of A

6.2 Participant Experience

UNA Synthesists receive:

  • "Your coherence observations contributed to 87% closure"
  • "Strongest alignment: vertices 1-2 (education domain)"
  • "Weakest alignment: vertices 0-3 (requires attention)"

ONA Analysts receive:

  • "Your differentiation observations identified 13% novel variance"
  • "Primary differentiation: cycle 1-2-3 (methodological approaches)"
  • "This differentiation is within healthy bounds (target: 2.07%)"

All Participants see:

  • Current aperture: 2.3% (slightly high, system adapting)
  • Trend: Decreasing toward target (healthy convergence)
  • Next cycle: Maintain current measurement approach

Critical Feature: No participant is told "you were too critical" or "you agreed too much." Feedback is about geometric patterns, not personal behavior.

6.3 Dynamic Calibration

The system self-calibrates through aperture monitoring:

  • If A consistently > 0.03: Increase weight on high-coherence edges
  • If A consistently < 0.01: Increase weight on high-differentiation edges
  • Goal: Maintain A ≈ 0.0207 without manual intervention

Bias Detection: Persistent cycle-space patterns indicate systematic bias. For example, if cycle 1-2-3 always carries high residual, vertices {1,2,3} may form an echo chamber. Correction involves adjusting topology or adding cross-linking edges.

Stability Tracking: Compute variance of A over measurement cycles. Stable systems show decreasing var(A) over time; unstable systems show increasing var(A) and require topology revision.

7. Scaling and Generalization

7.1 Beyond the Tetrahedron

Icosahedral Structure (12 vertices, 30 edges):

  • Coherence space: dim(gradient) = |V| − 1 = 11
  • Differentiation space: dim(residual) = |E| − |V| + 1 = 19
  • Much richer cross-validation through 19 independent cycles
  • Stafford Beer's observation: Optimal information sharing

Hierarchical Nesting:

  • Multiple tetrahedral units with shared CS vertex
  • Each unit maintains local aperture balance
  • Cross-unit coordination through shared reference

Arbitrary Graphs:

  • Framework applies to any connected graph
  • dim(gradient) = |V| − 1
  • dim(residual) = |E| − |V| + 1
  • Choose topology based on required redundancy

7.2 Variable Participant Count

The number of participants is independent of graph structure:

  • Fewer than 6: Each participant contributes to multiple edges
  • More than 6: Multiple participants per edge (ensemble measurement reduces variance)
  • The tetrahedral topology remains constant

7.3 Domain-Specific Adaptations

AI Model Evaluation:

  • Edges represent pairwise model comparisons
  • UNA: "Where do models exhibit coherent behavior?"
  • ONA: "Where do models offer distinct capabilities?"
  • BU: Overall evaluation emerges from balanced assessment

Organizational Decision-Making:

  • Edges represent pairwise stakeholder relationships
  • UNA: "Where do stakeholders share values/goals?"
  • ONA: "Where do stakeholders offer unique perspectives?"
  • BU: Consensus emerges from geometric balance

Scientific Peer Review:

  • Edges represent pairwise paper comparisons or reviewer-paper pairs
  • UNA: "Where does work align with established knowledge?"
  • ONA: "Where does work contribute novel insights?"
  • BU: Publication decision emerges from aperture balance

8. Comparison with Conventional Frameworks

8.1 What This Framework Eliminates

Role-Based Bias: No one is structurally assigned to find faults or render judgment. Both coherence-seeking and differentiation-seeking are legitimate, geometrically bounded functions.

Power Asymmetry: No individual or role has veto power. All measurements contribute according to their reliability weights. Authority emerges from geometry, not social position.

Adversarial Dynamics: UNA and ONA are orthogonal, not opposed. Both components coexist in every measurement without zero-sum competition.

Subject-Object Division: All participants are observer-participants. The system observes itself through distributed measurement in a reciprocal relationship.

8.2 What This Framework Preserves

Critical Analysis: ONA component provides rigorous differentiation. The 2.07% aperture ensures novel perspectives are captured. Criticism emerges from geometry, not from critics.

Coherence Standards: UNA component enforces alignment requirements. The 97.93% closure ensures stability. Standards emerge from geometry, not from judges.

Evolutionary Potential: BU aperture maintains capacity for adaptation. System can incorporate new information without destabilizing. Growth emerges from balance, not forced change.

Accountability: All measurements are recorded and traceable. Geometric analysis reveals patterns such as bias and echo chambers. Transparency through mathematical rigor.

9. Validation and Robustness

9.1 Geometric Validation

Primary Criterion: Verify orthogonality ⟨Bᵀx̂, r⟩_W = 0

Secondary Criteria:

  • Decomposition completeness: y = Bᵀx̂ + r
  • Dimensional consistency: 3 + 3 = 6
  • UNA coherence interpreted against 1/√2 reference
  • ONA coupling interpreted against π/4 reference
  • BU aperture stable near 0.0207

9.2 Statistical Validation

Invariance Tests:

  • Participant rotation: Results unchanged when individuals swap roles
  • Edge reweighting: Small w_e perturbations produce small output changes
  • Gauge transformation: Different reference vertex choices yield same observables

Convergence Tests:

  • Aperture A converges to target over measurement cycles
  • Variance of A decreases over time (stability)
  • Residual patterns stabilize (no wandering modes)

Noise Robustness:

  • Add synthetic noise to edge measurements
  • Verify decomposition remains stable
  • Confirm aperture remains within bounds

9.3 Falsification Criteria

The framework is falsifiable through:

Geometric Failure: Orthogonality violated, decomposition incomplete, or reference values systematically exceeded without explanation.

Empirical Failure: Aperture ratio unstable across contexts, no convergence after reasonable time, or systematic bias undetected by cycle analysis.

Practical Failure: Participants cannot provide meaningful edge measurements, computational complexity prevents real-time use, or results contradict clear ground truth cases.

10. Research Directions and Open Questions

10.1 Theoretical Extensions

Non-Euclidean Generalization: CGM suggests hyperbolic geometry (gyrovector spaces). How does orthogonal decomposition extend to curved measurement spaces?

Dynamic Topology: Real systems may require edge addition or removal. How does aperture balance guide topology evolution?

Higher-Order Interactions: Current framework considers pairwise edges only. How do simplicial complexes extend the measurement space for triplet or higher-order phenomena?

10.2 Empirical Validation

Controlled Experiments: Compare conventional evaluation (judge/critic) versus geometric roles (UNA/ONA). Measure bias, stability, and participant satisfaction across domains.

Field Deployment: Implement in real AI alignment consensus projects. Track aperture evolution in production systems and identify unforeseen challenges.

Cross-Cultural Validation: Test framework across different cultural contexts. Verify that geometric structure transcends social conventions and identify culture-specific calibration needs.

10.3 Computational Optimization

Efficient Decomposition: Large graphs require iterative methods. Develop specialized algorithms exploiting sparsity and structure.

Real-Time Calibration: Dynamic weight adjustment for streaming measurements, online detection of topology inadequacy, and adaptive graph structure for evolving systems.

Uncertainty Quantification: Propagate measurement uncertainty through decomposition. Provide confidence intervals on aperture ratio using Bayesian frameworks for sequential updating.

10.4 Integration with Existing Systems

AI Safety Frameworks: Map to reward modeling and RLHF. Connect to debate and amplification protocols. Integrate with constitutional AI approaches.

Governance Structures: Relate to liquid democracy and quadratic voting. Connect to Stafford Beer's Viable System Model. Explore blockchain-based implementation.

Cognitive Science: Link to predictive processing (prior/likelihood/posterior). Connect to dual-process theory (System 1/System 2). Map to Bayesian brain hypothesis.

11. Conclusion

11.1 Core Achievements

This framework demonstrates that unbiased collective measurement requires:

  1. Geometric roles over social roles: UNA Synthesist and ONA Analyst are defined by measurement geometry with explicit reference values, not by social position

  2. Orthogonal decomposition over assigned opposition: The same edge measurements simultaneously reveal coherence (gradient) and differentiation (residual) through projection

  3. Emergent balance over imposed judgment: BU state emerges when aperture ratio stabilizes, requiring no external authority

  4. Topological stability over external control: Proper information topology creates self-sustaining alignment through tensegrity dynamics

11.2 The Revolutionary Claim

AI alignment is topological tensegrity dynamics. This is not metaphor. Just as physical tensegrity achieves stable configuration through balanced internal forces without external scaffolding, informational alignment achieves stable behavior through balanced measurement forces without external judgment.

The compression elements (UNA coherence-seeking) and tension elements (ONA differentiation-seeking) must both be present, geometrically constrained, and properly balanced. When the aperture ratio reaches 2.07%, the system achieves the same self-sustaining stability that allows tensegrity structures to stand without support.

11.3 Immediate Applications

The framework is ready for deployment in AI model evaluation, organizational consensus, scientific peer review, and multi-stakeholder governance. In each domain, it replaces power-based or adversarial structures with geometric balance, achieving stable configurations that respect both coherence and differentiation.

11.4 Fundamental Insight

The deepest insight is this: The same measurements simultaneously reveal both coherence and differentiation through orthogonal projection.

This eliminates the false dichotomy between "supporters" and "critics," between "agreement" and "opposition," between "unity" and "diversity." These are not competing values requiring political compromise. They are orthogonal geometric components that coexist in every measurement without interference.

When we structure measurement systems according to this geometric truth rather than social conventions, we achieve unbiased evaluation without sacrificing critical analysis, stable alignment without sacrificing evolutionary capacity, collective intelligence without sacrificing individual perspective, and universal balance without sacrificing local differentiation.

This is measurement as it must be when properly understood: not the imposition of external judgment, but the geometric emergence of collective intelligence through topological necessity.

The mathematics demands it. The physics supports it. The implementation awaits.