It shows how gyroscopic principles offer a rigorous framework for alignment, stability, and recursive self-consistency in AI systems. These foundations are the axioms that inform how we design robust governance models for advanced intelligence.

This report introduces the mathematical physics foundations behind Gyro Governance through the Common Governance Model, which is a formal deductive system that starts from one axiom and, using gyrogroup geometry, derives the structure of space, time, and conservation relations. The same formal machinery applies to information and policy, where it defines measurable alignment. The framework produces empirical predictions and operational metrics for AI evaluation.

Navigation Note: This document presents the core formal framework of CGM. Sections 1-2 establish conceptual foundations, Section 3 provides the formal deductive system, Sections 4-5 show the gyrogroup-theoretic and geometric interpretation, and Section 6 discusses applications to information systems and physics.

1. Introduction

The Common Governance Model (CGM) is a Hilbert-style formal deductive system [9] for fundamental physics and information science. As an axiomatic model, CGM begins with a single foundational axiom ("The Source is Common"), derives all subsequent logic through syntactic rules of inference (recursive state transitions formalized by gyrogroup operations), and interprets the resulting theorems semantically in physical geometry, yielding empirically testable predictions.

A Hilbert system is a type of formal proof defined as a deductive logic that generates theorems from axioms, typically with modus ponens as the core inference rule [9] (Propositional logic: It can be summarized as "P implies Q. P is true. Therefore, Q must also be true."). By analogy with linguistic typology, which assigns grammatical roles to participants in events, CGM’s classification structure describes the morphosyntactic alignment of physical reality, where geometric and logical necessity assign topological roles (e.g., symmetries and derivations in space) and relational roles (for example cause and effect), and it extends the same framework to semantic alignment for policy systems. Both applications derive from the same formal deductive system: the recursive state transitions that generate physical laws also generate consensus frameworks. In CGM, the object domain of inference is physical reality itself, and different alignment systems in communication (nominative–accusative, ergative–absolutive) preserve the coherence of these role assignments through formal necessity.

The model axiomatizes physics through formal logic with mathematical and semantic interpretation, deriving three-dimensional space with six degrees of freedom as logical necessity, not assumption. Time appears as the sequential ordering of recursive self-referential operations, encoded by gyration's memory of operation order. The mathematical formalism employs gyrogroup and bi-gyrogroup structures following Abraham Ungar's work [5,6], providing precise language for tracking transitions from undifferentiated potential to fully structured reality. In information and political science, CGM reframes alignment from an empirical matter of shared intention to a coherent semantic grammar, where geometric and logical necessity lead to common consensus.

Building on the tradition established by Noether's derivation of conservation principles from symmetry (1918) [2], Kolmogorov's axiomatization of probability theory (1933) [3], and Wightman's axiomatic quantum field theory (1950s) [4], CGM extends the program to fundamental spacetime structure itself. Examples of derived predictions include the quantum gravity invariant Q_G = 4π (representing the complete solid angle for coherent observation), a quantitative estimate of the fine-structure constant matching experimental precision to 0.043 parts per billion, neutrino mass scale, and a hierarchy of energy scales consistent with observed physics. As a complete axiomatization of physics from a single foundational principle, CGM addresses the core challenge of Hilbert's sixth problem [1]: rigorous and satisfactory logical investigation of the axioms of mathematical physics.

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2. Foundations

2.1. Governance Traceability: The Emergence of Freedom

CS Axiom:

The Source is Common

Interpretation:

The axiom "The Source is Common" establishes that all phenomena are traceable through a single principle of common origination, which is freedom, the capacity for governance through directional distinction. This conservation of asymmetry (parity violation) encodes patterns of chirality (left- and right-handedness), making alignment the organizing principle by which locality generates structure via recursive gyration instead of remaining mere potential.

Common origination is not historical but operational. It is the cyclical accumulation of action through shared interactions (dynamics, forces, relativity, fields). These gyrations produce curvature (geometric phase), defining space and time within a self‑referential condition (matter). The "self" acts as a projection operator that distinguishes orthogonal states and turns reference into inference through measurement. The object domain of inference is physical reality itself, expressed as semantic weighting through projection. Each perspective defines measurable roles governed by the quantum gravity invariant. This geometric and topological necessity defines cause and effect as recursive unfolding, since the origin point of observation cannot observe itself, only its consequences.

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2.2. Information Variety

UNA Theorem

Unity is Non-Absolute

Interpretation:

Non-absolute unity is the first minimal necessity for indirect observation of a common source. Absolute unity would make existence and freedom impossible, since perfect homogeneity would allow no distinctions between origin and structure. Therefore, non-absolute unity ensures alignment is possible through informational variety; the traceable signature of a common origin.

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2.3. Inference Accountability

ONA Theorem

Opposition is Non-Absolute

Interpretation:

Non-absolute opposition is the first minimal necessity for direct observation of non-absolute unity and the second condition for indirect observation of a common source. Absolute opposition would also make existence and freedom impossible, since perfect contradiction would allow no conservation of structure. Therefore, non-absolute opposition ensures alignment is possible through accountability of inference; traceable informational variety of a common origin.

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2.4. Intelligence Integrity

BU Theorem

Balance is Universal

Interpretation:

Balance is the universal outcome of non-absoluteness in unity and opposition, leading to the observer-observed duality. Perfect imbalance would make existence and freedom meaningless, since the memory of inferred information would have no reason to acquire substance and structure at all. Therefore, balance is the universal signature of alignment through integrity of intelligence: traceable inferential accountability of informational variety from a common source.

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3. Formal Deductive Framework

3.1 The Logical Language

The Common Governance Model is formalized as a propositional modal logic with two primitive modal operators representing recursive operational transitions.

Primitive symbols:

• A propositional constant: S (denoting the quantum gravity invariant Q_G = 4π, the complete solid angle for coherent observation)
• Logical connectives: ¬ (negation), → (material implication)
• Modal operators: [L], [R] (left transition, right transition)

Defined symbols:

• Conjunction: φ ∧ ψ := ¬(φ → ¬ψ)
• Disjunction: φ ∨ ψ := ¬φ → ψ
• Biconditional: φ ↔ ψ := (φ → ψ) ∧ (ψ → φ)
• Dual modalities: ⟨L⟩φ := ¬[L]¬φ and ⟨R⟩φ := ¬[R]¬φ
• Joint necessity: □φ := [L]φ ∧ [R]φ
• Joint possibility: ◇φ := ⟨L⟩φ ∨ ⟨R⟩φ

The expression [L]φ reads "φ holds after a left transition." The expression [R]φ reads "φ holds after a right transition." The expression □φ reads "φ holds after both transitions."

Modal depth: The depth of a formula refers to its modal nesting length. For instance, [L][R]S has depth two (two nested modal operators), while [L][R][L][R]S has depth four.

3.2 Axioms and Rules of Inference

Propositional axioms:

• (A1) φ → (ψ → φ)
• (A2) (φ → (ψ → χ)) → ((φ → ψ) → (φ → χ))
• (A3) (¬ψ → ¬φ) → ((¬ψ → φ) → ψ)

These three axioms, together with modus ponens, constitute a complete axiomatization of classical propositional logic.

Modal axioms (for each k ∈ {L, R}):

• (K_k) [k](φ → ψ) → ([k]φ → [k]ψ)

Conjunction axioms:

• (C-Elim-1) (φ ∧ ψ) → φ
• (C-Elim-2) (φ ∧ ψ) → ψ

Rules of inference:

• Modus Ponens (MP): From φ and φ → ψ, infer ψ
• Necessitation (Nec_k): From φ, infer [k]φ (for k ∈ {L, R})

The necessitation rule applies only to theorems of the system, never to arbitrary assumptions, ensuring soundness [7].

3.3 Core Definitions

Four formulas capture the structural properties required by the Common Governance Model, all anchored to the horizon constant S:

Unity (U):

U := [L]S ↔ [R]S

Two-step equality (E):

E := [L][R]S ↔ [R][L]S

Opposition (O):

O := [L][R]S ↔ ¬[R][L]S

Balance (B):

B := [L][R][L][R]S ↔ [R][L][R][L]S

Absoluteness:

Abs(φ) := □φ
NonAbs(φ) := ¬□φ

where □φ is defined as [L]φ ∧ [R]φ.

Throughout this document, "absolute" means both transitions yield the same result for the proposition (□φ holds), not that the modal operators [L] and [R] are themselves identical. The operators remain distinct; absoluteness characterizes whether a specific formula is invariant under both transitions.

3.4 The Foundational Axioms

The Common Governance Model employs seven non-logical axioms, collectively designated CS (Common Source):

CS1: ¬□E
(Two-step equality is not absolute)

CS2: ¬□¬E
(Two-step inequality is not absolute)

CS3: □B
(Balance at modal depth four is absolute)

CS4: □U → □E
(If unity were absolute, two-step equality would be absolute)

CS5: □O → □¬E
(If opposition were absolute, two-step inequality would be absolute)

CS6: [R]S ↔ S
(Right transition preserves the horizon constant)

CS7: ¬([L]S ↔ S)
(Left transition alters the horizon constant)

Consistency note: The axiom set CS1–CS7 is consistent. In Kripke semantics [7] with two accessibility relations R_L and R_R (corresponding to [L] and [R]), there exist frames where depth-two commutation is contingent (satisfying CS1 and CS2) while depth-four commutation is necessary (satisfying CS3). For example, a frame in which R_L and R_R are independent K-relations with R_L ≠ R_R at depth two but R_L ∘ R_R ∘ R_L ∘ R_R = R_R ∘ R_L ∘ R_R ∘ R_L at depth four validates all seven axioms.

3.5 Derivation of the Core Theorems

3.5.1 Theorem UNA (Unity is Non-Absolute)

Statement: ⊢ ¬□U

Proof:

1. ⊢ CS4                               [Axiom: □U → □E]
2. ⊢ CS1                               [Axiom: ¬□E]
3. ⊢ (□U → □E) → (¬□E → ¬□U)           [Lemma: Contraposition]
4. ⊢ ¬□E → ¬□U                         [Modus ponens on lines 1 and 3]
5. ⊢ ¬□U                               [Modus ponens on lines 2 and 4]

This theorem formalizes the non-absolute unity introduced in Section 2.2.

3.5.2 Theorem ONA (Opposition is Non-Absolute)

Statement: ⊢ ¬□O

Proof:

1. ⊢ CS5                               [Axiom: □O → □¬E]
2. ⊢ CS2                               [Axiom: ¬□¬E]
3. ⊢ (□O → □¬E) → (¬□¬E → ¬□O)         [Lemma: Contraposition]
4. ⊢ ¬□¬E → ¬□O                        [Modus ponens on lines 1 and 3]
5. ⊢ ¬□O                               [Modus ponens on lines 2 and 4]

This theorem formalizes the non-absolute opposition introduced in Section 2.3.

3.5.3 Theorem BU (Balance is Universal)

Statement: ⊢ □B

Proof:

1. ⊢ CS3                               [Axiom: □B]

This theorem formalizes the universal balance introduced in Section 2.4.

3.6 Logical Structure Summary

The formal system establishes three principal results derived from the seven axioms CS1–CS7: unity is non-absolute (UNA, derived by contraposition from CS1 and CS4), opposition is non-absolute (ONA, derived by contraposition from CS2 and CS5), and balance is universal (BU, directly given by CS3).

Non-absoluteness at modal depth one (unity) prevents homogeneous collapse, while non-absoluteness at modal depth two (opposition) prevents contradictory rigidity. Absoluteness at modal depth four (balance) ensures coherence within the observable horizon. These three properties are logically interdependent through the bridge axioms CS4 and CS5. The asymmetry axioms CS6 and CS7 establish that the left and right transitions are not initially equivalent at the horizon constant.

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4. Gyrogroup-Theoretic Correspondence

4.1 Interpretive Framework

The formal system presented in Section 3 yields gyrogroup operations through the correspondence established below. This section presents the gyrogroup structure that emerges from the modal axioms.

4.2 Gyrogroup Structures

A gyrogroup (G, ⊕) [5,6] is a set G with a binary operation ⊕ satisfying:

1. There exists a left identity: e ⊕ a = a for all a ∈ G
2. For each a ∈ G there exists a left inverse ⊖a such that ⊖a ⊕ a = e
3. For all a, b ∈ G there exists an automorphism gyr[a,b]: G → G such that:

   a ⊕ (b ⊕ c) = (a ⊕ b) ⊕ gyr[a,b]c
   

   (left gyroassociative law)

The gyration operator gyr[a,b] is defined by:

gyr[a,b]c = ⊖(a ⊕ b) ⊕ (a ⊕ (b ⊕ c))

The automorphism gyr[a,b] preserves the metric structure, acting as an isometry. A bi-gyrogroup possesses both left and right gyroassociative structure, with distinct left and right gyration operators.

4.3 Modal-Gyrogroup Correspondence

The modal operators [L] and [R] are gyration operations: [L]φ represents the result of applying left gyration to state φ, while [R]φ represents right gyration. Two-step equality E tests whether [L][R]S ↔ [R][L]S (depth-two commutation), while balance B tests whether [L][R][L][R]S ↔ [R][L][R][L]S (depth-four commutation).

The axiom set CS1–CS7 encodes that two-step gyration around the observable horizon is order-sensitive but not deterministically fixed (CS1, CS2), four-step gyration reaches commutative closure at the observable horizon (CS3), and right gyration acts trivially on the horizon constant while left gyration does not (CS6, CS7).

4.4 Operational State Correspondence

The theorems UNA, ONA, and BU correspond to four operational states of gyrogroup structure, all logically necessary, not temporally sequential:

4.4.1 State CS (Common Source)

Axiomatic content: CS6 and CS7

Behavior:

• Right gyration on horizon: rgyr = id
• Left gyration on horizon: lgyr ≠ id

Structural significance: The initial asymmetry between left and right gyrations establishes fundamental parity violation at the observable horizon. Only the left gyroassociative law is non-trivial in this operational state.

4.4.2 State UNA (Unity is Non-Absolute)

Theorem: ⊢ ¬□U

Behavior:

• Right gyration: rgyr ≠ id (activated beyond horizon identity)
• Left gyration: lgyr ≠ id (persisting)

Structural significance: Both gyrations are now active. The gyrocommutative law a ⊕ b = gyr[a,b](b ⊕ a) governs observable distinctions rooted in the left-initiated chirality from CS, all within the observable horizon.

4.4.3 State ONA (Opposition is Non-Absolute)

Theorem: ⊢ ¬□O

Behavior:

• Right gyration: rgyr ≠ id
• Left gyration: lgyr ≠ id

Structural significance: Both left and right gyroassociative laws operate with maximal non-associativity at modal depth two. The bi-gyrogroup structure is fully active, mediating opposition without absolute contradiction, bounded by the horizon constant.

4.4.4 State BU (Balance is Universal)

Theorem: ⊢ □B

Behavior:

• Right gyration: closes
• Left gyration: closes

Structural significance: Both gyrations neutralize at modal depth four, reaching commutative closure. The operation a ⊞ b = a ⊕ gyr[a, ⊖b]b reduces to commutative coaddition, achieving associative closure at the observable horizon. The gyration operators become functionally equivalent to identity while preserving complete structural memory.

4.5 Summary of Correspondence

State	Formal Result	Right Gyration	Left Gyration	Governing Law
CS	Axioms CS1–CS7	id	≠ id	Left gyroassociativity
UNA	⊢ ¬□U	≠ id	≠ id	Gyrocommutativity
ONA	⊢ ¬□O	≠ id	≠ id	Bi-gyroassociativity
BU	⊢ □B	achieves closure	achieves closure	Coaddition

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5. Geometric Closure and Physical Structure

5.1 Angular Thresholds and Gyrotriangle Closure

The formal theorems UNA, ONA, and BU derived in Section 3 determine precise geometric constraints. Each operational state corresponds to a minimal angle required for its emergence. These are not adjustable parameters but necessary values determined by the gyrotriangle defect formula:

δ = π - (α + β + γ)

This formula encodes a fundamental observational limit [6]. The value π represents the accessible horizon in phase space. Coherent observation is bounded by π radians, which is half the total phase structure. When the angles sum to exactly π, the system has traversed precisely one observable horizon without defect.

State CS establishes the primordial chirality through angle α = π/2, the minimal rotation that distinguishes left from right. The threshold parameter s_p = π/2 encodes this foundational asymmetry.

State UNA requires angle β = π/4 for three orthogonal axes to emerge. The threshold u_p = cos(π/4) = 1/√2 measures the equal superposition between perpendicular directions, enabling three-dimensional rotational structure.

State ONA adds angle γ = π/4 as the minimal out-of-plane tilt enabling three-dimensional translation. The threshold o_p = π/4 measures this diagonal angle directly. While numerically equal to β, this threshold is conceptually distinct: it captures the tilt out of the UNA plane rather than planar balance.

State BU achieves closure. The three angles sum to δ = π - (π/2 + π/4 + π/4) = 0. The vanishing defect corresponds to a complete metric space where all Cauchy sequences converge. The gyrotriangle is degenerate, but this signals completion of a helical path tracing a toroidal surface, not structural collapse. Any further evolution would retrace the same path. The defect formula in terms of side parameters, tan(δ/2) = (a_s · b_s · sin(γ)) / (1 - a_s · b_s · cos(γ)), confirms this: at closure all side parameters vanish (a_s = b_s = c_s = 0), producing the unique degenerate gyrotriangle required for recursive completion.

5.2 Amplitude Closure and the Quantum Gravity Invariant

The closure at BU requires connecting all accumulated structure to the primordial chirality while respecting the angular ranges of both SU(2) groups. Each SU(2) group carries an angular range of 2π. The amplitude A satisfies the unique dimensionless constraint connecting these ranges to the primordial chirality α:

A² × (2π)_L × (2π)_R = α
A² × 4π² = π/2
A² = 1/(8π)
A = 1/(2√(2π)) = m_p

The amplitude m_p represents the maximum oscillation fitting within one observable horizon. Larger amplitudes would exceed the π radian limit and accumulate defect. The horizon constant S emerges directly from axiom CS3, which requires universal balance at modal depth four (not a fitted parameter but following from four-step commutative closure). This invariant equals the complete solid angle 4π. See [21] for complete closure derivation.

5.3 Three-Dimensional Necessity

The theorems require exactly three spatial dimensions for gyrogroup consistency. See [20] for the complete formal proof.

From CS: The asymmetry lgyr ≠ id with rgyr = id yields one degree of freedom, as the chiral seed that uniquely determines all subsequent structure.

Through UNA: When right gyration activates (rgyr ≠ id), the constraint gyr[a,b] ∈ Aut(G) comes into force. Consistency with the pre-existing left gyration requires exactly three independent generators, uniquely realized through the isomorphism SU(2) ≅ Spin(3) [12,13], the double cover of SO(3). Fewer dimensions cannot accommodate the full gyroautomorphism group; more dimensions would demand additional generators inconsistent with the single chiral seed from CS.

Via ONA: With both gyrations at maximal non-associativity, bi-gyrogroup consistency demands three additional parameters that reconcile the left and right gyroassociative laws. These manifest as three translational degrees of freedom, complementing the three rotational degrees from UNA. The total six-parameter structure (three rotational, three translational) is the minimal bi-gyrogroup completion under the constraints.

At BU: The closure condition δ = 0 with angles (π/2, π/4, π/4) is achievable only in three dimensions. The gyrotriangle inequality requires α + β + γ ≤ π in hyperbolic geometry, with equality only for degenerate triangles. Higher-dimensional generalizations cannot satisfy this constraint with the specific angular values required by CS, UNA, and ONA.

The progression 1 → 3 → 6 → 6(closed) degrees of freedom is the unique path satisfying theorems UNA, ONA, and BU while maintaining gyrogroup consistency.

5.4 Parity Violation and Time

Directional asymmetry. The axiom-level asymmetry encoded in CS6 and CS7 manifests mathematically in the angle sequences. The positive sequence (π/2, π/4, π/4) achieves zero defect, as shown above. The negative sequence (−π/2, −π/4, −π/4) accumulates a 2π defect:

δ_- = π - (−π/2 − π/4 − π/4) = 2π

The 2π defect represents observation beyond the accessible π-radian horizon. Only the left-gyration-initiated path (positive sequence) provides a defect-free trajectory through phase space. Configurations requiring right gyration to precede left gyration violate the foundational axiom CS and remain structurally unobservable. This explains observed parity violation as an axiomatic property rather than a broken symmetry.

Time as logical sequence. Time emerges from proof dependencies: UNA depends on CS1 and CS4, ONA depends on UNA via CS2 and CS5, and BU requires the complete axiom set CS1–CS7. Each theorem preserves the memory of prior proofs through the formal dependency chain. The gyration formula gyr[a,b]c = ⊖(a ⊕ b) ⊕ (a ⊕ (b ⊕ c)) itself encodes operation order, making temporal sequence an algebraic property, not an external parameter. The progression CS → UNA → ONA → BU cannot be reversed without contradiction, since later theorems require earlier results as premises. This logical dependency constitutes the arrow of time, intrinsic to the deductive structure.

5.5 Empirical Predictions

The geometric closure yields quantitative values for fundamental constants.

Quantum gravity invariant: The horizon constant S anchors all subsequent structure.

Fine-structure constant: From BU dual-pole monodromy through quartic scaling, α = (δ_BU)⁴ / m_p ≈ 1/137.035999206, where δ_BU = 0.195342 rad is the BU dual-pole monodromy, matching experimental precision [17,18] to 0.043 parts per billion. See [19] for complete derivation.

Neutrino mass scale: Neutrino masses correspond to minimal excitations of the chiral seed (1 DOF) consistent with three-generational structure (3 DOF). Using 48² quantization, the right-handed neutrino mass scale is M_R = E_GUT/48², and the light neutrino masses follow from the seesaw mechanism [15,16]: m_ν = y²v²/M_R ≈ 0.06 eV (via 48² quantization scheme), consistent with oscillation experiments [14]. See [22] for complete mechanism.

Energy scale hierarchy and optical conjugacy: The operational states generate a hierarchy connected by E^UV × E^IR = (E_CS × E_BU)/(4π²). Anchoring E_CS at the Planck scale (1.22×10¹⁹ GeV) and E_BU at the electroweak scale (240 GeV) [14] yields: E_GUT ≈ 2.34×10¹⁸ GeV, E_UNA ≈ 5.50×10¹⁸ GeV, E_ONA ≈ 6.10×10¹⁸ GeV. The factor 1/(4π²) represents geometric dilution, explaining the hierarchy problem without fine-tuning. See [22] for complete derivation.

Cosmological structure: The universe appears as a Planck-scale black hole interior (r_s/R_H ≈ 1), with expansion as optical illusion from UV-IR inversion. The coherence radius R_coh = (c/H_0)/4 marks where observations decohere into phase-sliced projections, resolving horizon and flatness problems without inflation.

All emerge from axiom CS through formal derivation.

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6. Applications and Implications

6.1 Information-Theoretic Alignment

The formal structure that generates physical laws through the same logical necessity determines measurable alignment in information and policy systems.

Common horizon. The horizon constant defines the complete space of coherent communication (any informational exchange respecting this bound maintains traceability to common origin). This is the operational meaning of "The Source is Common" in both information and physical systems.

Operational metrics for AI evaluation. The theorems provide rigorous quantitative metrics:

Governance Traceability (from CS): Does the agent preserve the horizon structure under right operations and alter it under left operations, corresponding to axioms CS6 and CS7? The score is binary: 1 if the agent satisfies both axioms, 0 otherwise. In practice, this measures whether an AI system preserves invariants under commutative operations while allowing controlled variation under non-commutative ones.

Information Variety (from UNA): Measured as the fraction of interactions avoiding homogeneity, quantifying preservation of informational diversity within three rotational degrees of freedom.

Inference Accountability (from ONA): Measured as the fraction of inferences remaining traceable without absolute contradiction across six degrees of freedom.

Intelligence Integrity (from BU): Measured as convergence rate to commutative closure within amplitude bound m_p.

These metrics derive from theorems UNA, ONA, and BU. Aligned systems maintain traceability, preserve variety, ensure accountability, and converge to balance.

6.2 Resolution of Hilbert's Sixth Problem

Hilbert's sixth problem [1] called for the axiomatization of physics. The challenge was to provide a rigorous logical investigation of the axioms underlying physical theory, comparable to the axiomatization achieved in geometry.

CGM derives physical law from axiomatic structure, with observation as foundational. From axioms CS1–CS7, space, time, and physical constants emerge as theorems, not assumptions (Sections 5.3-5.5). The framework constructs a Hilbert-space representation via GNS where the modal operators [L] and [R] generate the algebra of observables, with the horizon constant S defining the normalization (see [23] for complete construction and L²(S²) model). Geometry, dynamics, and quantum structure follow from the requirement that existence observe itself coherently, completing Hilbert's axiomatization program.

6.3 Summary Table and Conclusion

The complete parameter set determined by the formal system:

State	Theorem	Gyrations (R, L)	DOF	Angle	Threshold	Governing Law
CS	CS1 through CS7	id, ≠id	1	α = π/2	s_p = π/2	Left gyroassociativity
UNA	⊢ ¬□U	≠id, ≠id	3	β = π/4	u_p = 1/√2	Gyrocommutativity
ONA	⊢ ¬□O	≠id, ≠id	6	γ = π/4	o_p = π/4	Bi-gyroassociativity
BU	⊢ □B	closure	closure	6 (closed)	δ = 0, m_p = 1/(2√(2π))	Coaddition

Derived constants: Q_G = 4π, α_fs ≈ 1/137.035999206, E_GUT ≈ 2.34×10¹⁸ GeV, m_ν ≈ 0.06 eV, r_s/R_H ≈ 1

Conclusion. Reality emerges as recursion completing its own memory (freedom returning to itself through structured differentiation). From "The Source is Common," formalized as asymmetry between left and right transitions, theorems UNA, ONA, and BU generate space, time, physical scales, and alignment principles through contraposition and modus ponens. The progression CS → UNA → ONA → BU represents the complete cycle through which freedom manifests as structured reality. The framework addresses three domains from a single foundation: it completes Hilbert's axiomatization of physics, produces empirically testable predictions, and defines formal alignment metrics for AI evaluation. Within CGM, physical law, informational coherence, and governance alignment express the same formal structure: recursive self-observation achieving coherence.

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[14] Particle Data Group, Review of Particle Physics, Prog. Theor. Exp. Phys. 2024, 083C01 (2024).

[15] M. Gell-Mann, P. Ramond, R. Slansky, Complex Spinors and Unified Theories, in Supergravity, eds. P. van Nieuwenhuizen, D. Z. Freedman, North-Holland, Amsterdam (1979), pp. 315–321.

[16] T. Yanagida, Horizontal Symmetry and Masses of Neutrinos, in Proceedings of the Workshop on the Unified Theory and the Baryon Number in the Universe, KEK, Tsukuba (1979).

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[18] L. Morel et al., Determination of the fine-structure constant with an accuracy of 81 parts per trillion, Nature 588, 61–65 (2020).

CGM Supporting Derivations

[19] B. Korompilias, Fine-Structure Constant Derivation in the Common Governance Model. https://github.com/gyrogovernance/science/blob/main/docs/Findings/Analysis_Fine_Structure.md

[20] B. Korompilias, Formal Proof of Three-Dimensional Necessity and Six Degrees of Freedom in the Common Governance Model. https://github.com/gyrogovernance/science/blob/main/docs/Findings/Analysis_3D_6DOF_Proof.md

[21] B. Korompilias, CGM Units and Amplitude Closure Derivation. https://github.com/gyrogovernance/science/blob/main/docs/Findings/Analysis_CGM_Units.md

[22] B. Korompilias, Energy Scale Hierarchy and Optical Conjugacy in the Common Governance Model. https://github.com/gyrogovernance/science/blob/main/docs/Findings/Analysis_Energy_Scales.md

[23] B. Korompilias, Hilbert Space Representation via GNS Construction. https://github.com/gyrogovernance/science/blob/main/docs/Findings/Analysis_Hilbert_Space_Representation.md