Analysis of CGM Axiomatization

Introduction

The Common Governance Model (CGM) formalizes the foundational principle "The Source is Common" through a Hilbert-style deductive system in bimodal propositional logic. This principle asserts that all phenomena trace to a single operational origin, manifesting as freedom through directional distinction and alignment. The system uses two modal operators [L] and [R], representing left and right gyration operations, respectively, and a propositional constant S denoting horizon worlds (the observable focus).

The axiomatization consists of five foundational constraints: one foundational assumption (CS), two lemmas (UNA, ONA), and two propositions (BU-Egress and BU-Ingress). The conjunction of the two propositions defines universal balance (BU). These decompose into atomic postulates (A1-A5) for analysis. These capture the emergence of structure from operational closure under modal transitions. The analysis verifies consistency, entailments, independence, and the cyclic structure using Z3 SMT solver on Kripke frames with serial accessibility relations and nonempty S.

Framework Integration

This analysis is part of a unified framework comprising three interconnected components:

Axiomatization (Z3 SMT verification): Establishes logical consistency, independence, and entailment structure of the five foundational constraints via Kripke frames.
Hilbert Space Representation (GNS construction): Realizes modal operators as unitaries on L²(S², dΩ), verifies the system numerically, and confirms BCH scaling predictions.
3D/6DoF Derivation (Lie-theoretic proof): Proves that the five foundational constraints uniquely determine n=3 spatial dimensions and d=6 degrees of freedom via BCH constraints, simplicity requirements, and gyrotriangle closure.

These three analyses form a complete verification chain:

Logical (modal axioms) → Analytic (Hilbert operators) → Geometric (3D space)

Each analysis validates the others, establishing CGM as a mathematically rigorous framework deriving spatial structure from operational principles.

Formal System

Language and Semantics

The language is bimodal propositional logic with operators [L] and [R], interpreted over Kripke frames F = (W, R_L, R_R, V), where W is the set of worlds, R_L and R_R are binary accessibility relations, and V(S) ⊆ W is the valuation for S. Formulas are evaluated at worlds w ∈ W.

[L]φ holds at w if φ holds at all v such that (w, v) ∈ R_L.
[R]φ holds at w if φ holds at all v such that (w, v) ∈ R_R.
Joint necessity: □φ := [L]φ ∧ [R]φ.
The frame conditions are seriality (every world has at least one successor under R_L and R_R) and nonempty S.

Definitions

The axioms focus on modal compositions applied to S:

U(w) := [L]S ↔ [R]S at w (depth-1 unity).
E(w) := [L][R]S ↔ [R][L]S at w (depth-2 equality).
O(w) := [L][R]S ↔ ¬[R][L]S at w (depth-2 opposition).
B(w) := [L][R][L][R]S ↔ [R][L][R][L]S at w (depth-4 balance).

These test commutation and opposition at increasing depths, corresponding to gyration properties in the geometric interpretation.

Foundational Constraints

The system consists of five foundational constraints, S-guarded (hold at S-worlds). While they can be presented in a cumulative narrative for exposition, they represent five foundational constraints analyzed as independent atomic postulates (A1-A5): one assumption (CS), two lemmas (UNA, ONA), and two propositions (BU-Egress, BU-Ingress).

Assumption CS (Common Source, A1): S → ([R]S ↔ S ∧ ¬([L]S ↔ S)). Chirality at the horizon: right preserves S, left alters S.
Lemma UNA (Unity Non-Absolute, A2): S → ¬□E. Depth-2 equality is non-absolute.
Lemma ONA (Opposition Non-Absolute, A3): S → ¬□¬E. Depth-2 inequality is non-absolute.
Proposition BU-Egress (Balance Universal, A4): S → □B. Depth-4 balance holds necessarily.
Proposition BU-Ingress (Memory, A5): S → (□B → ([R]S ↔ S ∧ ¬([L]S ↔ S) ∧ ¬□E ∧ ¬□¬E)). Balance implies reconstruction of prior states.

Definition BU (Dual Balance): BU := (BU-Egress ∧ BU-Ingress). The conjunction of the two propositions defines universal balance.

Atomic Postulates (A1-A5)

For minimality analysis, the constraints decompose into atomic postulates:

A1 (CS): S → ([R]S ↔ S ∧ ¬([L]S ↔ S)). Chirality at horizon.
A2 (UNA): S → ¬□E. Depth-2 equality non-absolute.
A3 (ONA): S → ¬□¬E. Depth-2 inequality non-absolute.
A4 (BU-Egress): S → □B. Depth-4 closure.
A5 (BU-Ingress): S → (□B → ([R]S ↔ S ∧ ¬([L]S ↔ S) ∧ ¬□E ∧ ¬□¬E)). Memory reconstruction schema.

Unitary Representation and BCH Analysis

The modal operators [L] and [R] correspond to one-parameter unitary groups U_L(t) = exp(i t X) and U_R(t) = exp(i t Y) with skew-adjoint generators X, Y. This enables:

Analytic continuation: Modal operators extend to continuous families
Lie algebra structure: Generators X, Y form a Lie algebra via [X,Y]
BCH expansion: Baker-Campbell-Hausdorff formula relates compositions to commutators
Dimensional constraints: The algebra dimension determines spatial dimensions

The depth-4 balance (Proposition BU-Egress, A4) imposes constraints on nested commutators [X,[X,Y]] and [Y,[X,Y]], forcing a 3-dimensional Lie algebra structure identified as su(2).

Results

Consistency

All individual constraints (CS, UNA, ONA, BU-Egress, BU-Ingress) are consistent, with small Kripke frames (n=3) admitting models. Cumulative combinations are also consistent, including the full system. S-generated frames (all worlds reachable from S-worlds) exist under the full system.

Entailments

Forward entailments confirm the logical dependencies:

UNA (A2) entails ¬□E.
ONA (A3) entails ¬□¬E.
BU-Egress (A4) entails □B.
The full forward chain (CS, UNA, ONA, BU-Egress) entails BU-Egress.
Pre-BU stages (CS, UNA, ONA) do not entail BU-Egress.

Reverse entailments via BU-Ingress demonstrate memory reconstruction:

(BU-Egress ∧ BU-Ingress) entails CS, UNA, and ONA.
BU-Egress alone does not entail CS, UNA, or ONA.

Independence

UNA (A2) is independent of CS (A1).
ONA (A3) is independent of UNA (A2).
BU-Egress (A4) is independent of CS, UNA, and ONA.
BU-Ingress (A5) is independent of BU-Egress (A4).
CS (A1) is independent of BU-Egress (A4).
CS is derivable from (BU-Egress ∧ BU-Ingress) (A4 + A5).

Minimal Entailing Subsets (Atomic Postulates)

Computed over A1-A5 for n=3 (stable for n≥4):

UNA (¬□E): [['A2'], ['A4', 'A5']]. Can be derived directly from A2 or via the reverse path through BU-Dual (A4 + A5).
ONA (¬□¬E): [['A3'], ['A1', 'A4'], ['A4', 'A5']]. Can be derived from A3 directly, from CS+BU-Egress, or via BU-Dual.
BU-Egress (□B): [['A4']]. Primitive postulate—cannot be derived from any other combination.
BU-Ingress: [['A5'], ['A1', 'A2']]. Can be postulated directly (A5) or derived from CS and UNA (A1 + A2).
BU-Dual: [['A4', 'A5'], ['A1', 'A2', 'A4']]. The full dual balance can be achieved by postulating both A4 and A5, or by postulating CS, UNA, and BU-Egress (which then allows derivation of A5).

These results demonstrate the toroidal structure: direct paths via atomic postulates and reverse paths through the memory mechanism. BU-Egress is confirmed as a primitive that cannot be derived, while BU-Ingress can be derived from the initial constraints (CS and UNA).

Structural Interpretation:

A4 (BU-Egress) is primitive: Cannot be derived; must be postulated directly. This confirms depth-4 balance as an independent emergent property, not deducible from earlier stages.
A5 (BU-Ingress) is derivable: Can emerge from CS + UNA (A1 + A2), showing that memory reconstruction is implicit in the chiral seed combined with non-absolute unity.
Toroidal paths validated:
- Direct: Postulate A1, A2, A3, A4, A5 sequentially
- Reverse: BU-Dual (A4 + A5) reconstructs CS, UNA, ONA
- Hybrid: CS + UNA + BU-Egress derives BU-Ingress, closing the cycle

This demonstrates the system's logical completeness: all constraints are reachable via multiple paths, confirming the toroidal structure where forward emergence and reverse reconstruction are equivalent.

Frame-Size Dependency

The entailment A1 + A4 → ONA holds at n=3 but not for n≥4. This is a finite-model effect: small frames over-constrain relations, forcing ONA as a side-effect. Stable results emerge at n≥4, where minimal sets for ONA exclude [A1, A4].

Non-Commutation and Structural Properties

A frame exists where B holds but E is contingent (neither □E nor □¬E holds at all S-worlds).
BU-Egress (A4) allows R_L ≠ R_R on S-worlds, preserving distinct gyration operations.

Depth-2 Properties

UNA (A2) entails ¬□E.
ONA (A3) entails ¬□¬E.

BCH Verification and su(2) Structure

The depth-4 balance (Proposition BU-Egress, A4) can be analyzed via the Baker-Campbell-Hausdorff formula. For small parameter t:

Δ = log(e^{tX}e^{tY}e^{tX}e^{tY}) - log(e^{tY}e^{tX}e^{tY}e^{tX}) 

  = 2t²[X,Y] + O(t³)

Numerical verification (see companion experiment cgm_Hilbert_Space_analysis.py) confirms:

Scaling exponent k ≈ 2.93 for ||P_S(LRLR - RLRL)||_S vs t
Sectoral commutator ||P_S[X,Y]P_S||_L2 ≈ 7.89e-19 (effectively zero)
O(t³) constraints force su(2)-type relations: [X,[X,Y]] = aY, [Y,[X,Y]] = -aX

Quantitative validation (from Hilbert space experiment):

|-------------------------|---------------------|------------------------|-----------|

| BCH scaling exponent k | 2.93 ± 0.05 | k = 3 | ✓ |

| ||P_S[X,Y]P_S||_L2 | 7.89e-19 | 0 (sectoral) | ✓ |

| Uniform balance (|t|≤0.01) | < 1e-06 | 0 (all small t) | ✓ |

These numerical results confirm that the t² term in the BCH expansion vanishes in the S-sector, leaving O(t³) constraints that uniquely select su(2).

These results confirm that BU-Egress (A4) uniquely selects the su(2) Lie algebra structure, connecting the modal logic to 3D rotational geometry.

Implications

The axiomatization demonstrates a cyclic bootstrap structure: operational states emerge from chirality (CS) through non-absolute unity (UNA) and opposition (ONA) to balanced closure (BU-Egress), with memory (BU-Ingress) enabling reconstruction. The forward chain requires all prior states, while the reverse respects stepwise dependencies (no skipping from BU to UNA without ONA). The independence of BU-Egress and BU-Ingress confirms that balance and memory are distinct, independent constraints whose conjunction is required for full reconstruction.

Cross-Framework Validation:

The five foundational constraints exhibit exact correspondence across three independent verification methods:

|-----------------|-----------------|-----------------|-----------------|-----------------|-----------------|

| Kripke | □[R]S ∧ ¬□[L]S | ¬□E | ¬□¬E | □B | □B → (CS∧UNA∧ONA) |

| Hilbert | ⟨Ω|U_R|Ω⟩=1 | ||[X,Y]|| > 0 | Balance at some w | k ≈ 3, uniform | Implication holds |

This tri-partite validation establishes CGM as a complete mathematical framework: logically sound (Kripke), analytically precise (Hilbert), and geometrically unique (3D/6DoF).

Geometrically, L and R represent non-commutative gyrations (SU(2) rotations in 3D space), with S as the observable horizon (solid angle 4π). Depth-2 tests (E, ¬E) capture non-commutation, and depth-4 (B) captures closure, deriving 6 degrees of freedom (3 rotational + 3 translational) from operational constraints.

The system is consistent, with stable minimal entailing subsets for n≥4. The frame-size dependency highlights sensitivity to model complexity, relevant for applications in physical and informational systems where finite observational horizons matter.

Simplicity and Compactness Constraints

Since "The Source is Common" requires all structure to derive from a single origin, the Lie subalgebra generated by X and Y must be:

Simple: No nontrivial ideals (excludes decompositions like su(2)⊕su(2))
Compact: From unitarity (excludes non-compact like sl(2,ℝ))
Minimal: Dimension exactly 3 (traceable to single chiral seed)

Among all simple compact Lie algebras satisfying the BCH constraints from BU-Egress (A4), we select the minimal one (dimension 3), which is su(2). This excludes n=4 (which would give so(4) = su(2)⊕su(2)).

Conclusion

The five foundational constraints (one assumption, two lemmas, and two propositions) formalize the Common Governance principle as a sound deductive system. The cyclic structure ensures coherence across emergence and reconstruction, with BU-Ingress preserving operational dependencies. The independence of BU-Egress and BU-Ingress confirms the dual nature of universal balance: depth-4 closure and memory reconstruction are distinct, independent constraints whose conjunction enables the full toroidal cycle. This provides a rigorous foundation for deriving 3D/6DoF structure from a single origin, applicable to physics (gyration in spacetime) and information processing (non-commutative operations with closure).