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The Common Governance Model: From Modal Logic to Physical Structure

The Common Governance Model: From Modal Logic to Physical Structure

Table of Contents

Executive Summary

The Common Governance Model (CGM) demonstrates that coherent recursive measurement requires specific structural properties. Starting from five constraints expressed in modal logic, the framework derives three-dimensional space with six degrees of freedom as the unique solution satisfying operational coherence requirements.

Key Results:

  • Deductive: Three-dimensional structure follows necessarily from the foundational axiom through formal specifications plus requirements for continuous physical implementation. Alternative dimensionalities are constructively excluded.
  • Invariants: The constraints fix three representation-independent constants: the quantum gravity horizon Q_G = 4π steradians, the monodromy defect δ_BU ≈ 0.1953 radians, and aperture scale m_a ≈ 0.1995. Their ratio δ_BU/m_a = 0.9793 determines both physical coupling and informational aperture.
  • Phenomenological: At leading geometric order, the framework yields a fine-structure constant estimate of α = 0.007297352563, matching experimental synthesis within stated uncertainty. The same geometric ratio predicts a 2.07% optimal aperture for discrete alignment systems. Preliminary evaluations show transformer architectures operating at six to eight times this value.

Falsification Paths:

The framework is falsifiable through: (i) demonstration that alternative dimensionalities satisfy the operational requirements, (ii) proof that the BU residual holonomy is non-abelian, (iii) experimental disagreement with predicted values, (iv) expanded AI evaluations showing no structural signature at the predicted scale.

Reproducibility:

All computational artifacts are archived at Zenodo (DOI: 10.5281/zenodo.17521384) and GitHub (github.com/gyrogovernance/science).

Introduction and Motivation

The Central Question

Authority, understood as the legitimate capacity to determine operational outcomes, requires constitutional principles invariant across contexts. In physical measurement, observers maintain descriptive authority while subject to the same laws. In artificial intelligence, decision processes must preserve legitimate authority while operating autonomously. Both domains present the same fundamental question: what structural requirements determine coherent authority?

Constitutional principles function as invariant constraints determining all subsequent structure, distinguishing foundational necessities from contingent choices.

This document presents the Common Governance Model (CGM), which establishes structural requirements from a single foundational axiom: operational structure must trace to a shared source.

What Makes This Framework Different

The framework treats governance as mathematical structure by specifying the minimal conditions required for operations to preserve shared authority while maintaining necessary distinctions. The model is "common" because the same logical requirements apply wherever coherent authority must be maintained. In physical systems this manifests as conservation laws traceable to a unified origin. In informational systems it requires that all processing states remain reachable from a designated reference.

The foundational axiom, termed "The Source is Common" (CS), establishes that right transitions preserve the reference state while left transitions alter it, creating fundamental chirality. From this chiral seed, four additional constraints specify the axiom's requirements at increasing modal depths:

  • Non-absolute unity (UNA) prevents homogeneous collapse at depth two
  • Non-absolute opposition (ONA) prevents absolute contradiction at the same depth
  • Balanced closure (BU-Egress) achieves commutative closure at depth four
  • Memory reconstruction (BU-Ingress) ensures the balanced state reconstructs all prior conditions

Together, CS with these four specifications constitutes the complete logical system.

The framework demonstrates that recursive measurement under shared authority imposes stronger architectural requirements than previously recognized: systems either satisfy these constitutional requirements or fail to maintain operational coherence.

Epistemic Framework

Results are stratified by epistemic status to prevent conflation of logical necessity with empirical fit:

  • Deductive results (three-dimensional structure, six degrees of freedom) follow from the foundational axiom through formal specifications and operational requirements. Their negation entails logical contradiction within the stated framework.
  • Representational results (GNS construction on L²(S²)) provide one concrete realization among potentially multiple faithful representations.
  • Invariant results (Q_G = 4π, δ_BU, m_a) are representation-independent constants fixed by the constraints.
  • Phenomenological results (α estimate, alignment aperture ratios) are geometric correspondences that generate falsifiable predictions but require independent verification to establish physical or informational relevance.

Core Foundations

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:

Symbol Description
S A propositional constant: the horizon constant
¬ Logical connectives: negation
→ Logical connectives: material implication
[L] Modal operators: left transition
[R] Modal operators: right transition

Defined symbols:

Symbol Definition
φ ∧ ψ ¬(φ → ¬ψ)
φ ∨ ψ ¬φ → ψ
φ ↔ ψ (φ → ψ) ∧ (ψ → φ)
⟨L⟩φ ¬[L]¬φ
⟨R⟩φ ¬[R]¬φ
□φ [L]φ ∧ [R]φ
◇φ ⟨L⟩φ ∨ ⟨R⟩φ

Reading the expressions:

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

Modal depth refers to the nesting level of modal operators. For instance, [L][R]S has depth two (two nested operators), while [L][R][L][R]S has depth four. Modal depth plays a critical role in CGM: depth-two operations exhibit contingent behavior (non-absolute unity and opposition), while depth-four operations achieve necessary closure (universal balance).

Throughout the logical development we reserve the symbol S for the designated propositional constant anchoring the horizon worlds. When this constant is realized in the Hilbert-space representation, its expectation value equals the scalar horizon invariant Q_G = 4π. We refer to the scalar quantity explicitly as Q_G to avoid ambiguity.

[!IMPORTANT]
Note on S: Here S is a designated propositional constant in the modal language. It is standard in modal correspondence settings to allow modalities to apply to propositional constants. Semantically, S is evaluated by V(S) ⊆ W and [k]S is defined via the accessibility relation R_k as usual. Its physical interpretation as the horizon solid-angle 4π is the normalization chosen for the L²(S²) model.

Notation: Throughout this document, m_a = 1/(2√(2π)) ≈ 0.1995 denotes the CGM observational aperture parameter (dimensionless). The exact value is m_a = 0.199471140201. Approximations use 4 significant figures unless higher precision is required.

Base Axioms

The system is built on the modal logic K with standard propositional and modal axioms.

Propositional axioms:

Axiom Formula
(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}):

Axiom Formula
(K_k) [k](φ → ψ) → ([k]φ → [k]ψ)

Conjunction axioms:

Axiom Formula
(C-Elim-1) (φ ∧ ψ) → φ
(C-Elim-2) (φ ∧ ψ) → ψ

Rules of inference:

Rule Description
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.

Core Definitions

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

Concept Formula Description
Unity (U) [L]S ↔ [R]S Unity holds when left and right transitions yield equivalent results at the horizon constant.
Two-step Equality (E) [L][R]S ↔ [R][L]S Two-step equality holds when depth-two modal compositions commute at the horizon constant.
Opposition (O) [L][R]S ↔ ¬[R][L]S Opposition holds when depth-two modal compositions yield contradictory results at the horizon constant.
Balance (B) [L][R][L][R]S ↔ [R][L][R][L]S Balance holds when depth-four modal compositions commute at the horizon constant.

Absoluteness:

Concept Formula
Abs(φ) □φ
NonAbs(φ) ¬□φ

where □φ is defined as [L]φ ∧ [R]φ.

[!NOTE]
Clarification: Throughout this document, "absolute" means a proposition φ is invariant under both transitions (□φ holds), not that the modal operators [L] and [R] are identical. The operators remain distinct. Absoluteness characterizes invariance of specific formulas under transitions.

The Five Foundational Constraints

The framework relies on five foundational constraints: one assumption (CS), two lemmas (UNA, ONA), and two propositions (BU-Egress, BU-Ingress). For independence analysis in the core modal system we treat all five as primitives. In the operational regime, the continuous flows, reachability from S, and simple Lie closure allow UNA and ONA to be obtained from CS (hence the lemma designation). The conjunction of BU-Egress and BU-Ingress defines universal balance.

CS: The Source is Common

Assumption (CS):

S → ([R]S ↔ S ∧ ¬([L]S ↔ S))

[!INFO]
What this means: The horizon constant S is preserved under right transitions but altered under left transitions. This establishes fundamental chirality, or "handedness," in the system. The reference state behaves asymmetrically under the two types of transitions.

Why it matters: This is the foundational axiom from which all other structure derives. It establishes that operational structure must trace to a shared source while maintaining directional distinction. Without this asymmetry, no coherent observation would be possible, as there would be no way to distinguish different operational paths.

Physical interpretation: Right transitions preserve the observable horizon (like rotations around a fixed axis), while left transitions alter it (like translations that change the reference frame). This chiral seed will grow into the full parity violation observed in nature.

UNA: Unity is Non-Absolute

Lemma (UNA):

S → ¬□E    where E := [L][R]S ↔ [R][L]S

[!INFO]
What this means: At depth two (two nested modal operations), the order of transitions matters, but this non-commutativity is not absolute across all accessible states. Sometimes [L][R] equals [R][L], sometimes it doesn't. The system exhibits contingent rather than deterministic behavior at this level.

Why it matters: Absolute unity (□E) would collapse all distinctions, making the system homogeneous and preventing any meaningful structure. Non-absolute unity (¬□E) ensures informational variety while maintaining traceability to the common source.

Physical interpretation: In quantum mechanics, this corresponds to non-commuting observables (like position and momentum). The uncertainty isn't absolute chaos, it's structured contingency bounded by the horizon constant.

Derivation note: In the operational regime with continuous flows, UNA follows from CS through the forced non-commutativity of continuous one-parameter groups. For formal verification, we treat it as an independent axiom.

ONA: Opposition is Non-Absolute

Lemma (ONA):

S → ¬□¬E

[!INFO]
What this means: While depth-two operations may yield opposite results ([L][R] vs. [R][L]), this opposition is not absolute. The system avoids both complete agreement and complete contradiction.

Why it matters: Absolute opposition (□¬E) would create irreconcilable contradictions, destroying coherence. Non-absolute opposition ensures accountability of inference, meaning that different operational paths remain comparable even when they yield different results.

Physical interpretation: This corresponds to the fact that while measurements may disturb quantum states, they don't destroy all information. There's a balance between disturbance and preservation that makes physics possible.

Structural role: Together with UNA, ONA creates a "bi-gyrogroup" structure where both left and right gyroassociative properties are active but bounded. This leads directly to the six degrees of freedom (three rotational + three translational).

BU-Egress: Depth-Four Closure

Proposition (BU-Egress):

S → □B    where B := [L][R][L][R]S ↔ [R][L][R][L]S

[!INFO]
What this means: At depth four (four nested operations), the system achieves commutative closure. No matter which order you apply the alternating transitions, you end up at the same state. This closure is absolute (□B), meaning it holds across all accessible worlds.

Why it matters: This is the minimal depth at which coherent closure can occur while preserving depth-two contingency. Depth three still allows asymmetry. Depth four forces balance through the structure of the Baker-Campbell-Hausdorff expansion.

Physical interpretation: This corresponds to the closure of phase space loops in quantum mechanics. After a complete cycle of transformations, the system returns to a consistent state (up to a phase factor). The depth-four requirement explains why quantum mechanics has its particular mathematical structure.

Technical note: In the operational regime with continuous unitary flows, this translates to: there exists δ > 0 such that for all |t| < δ, the S-projected difference between U_L(t)U_R(t)U_L(t)U_R(t) and U_R(t)U_L(t)U_R(t)U_L(t) vanishes. This uniform neighborhood validity is what forces continuous one-parameter groups.

BU-Ingress: Memory Reconstruction

Proposition (BU-Ingress):

S → (□B → ([R]S ↔ S ∧ ¬([L]S ↔ S) ∧ ¬□E ∧ ¬□¬E))

[!INFO]
What this means: The balanced state at depth four contains sufficient information to reconstruct all prior conditions: the original chirality (CS), the contingent unity (UNA), and the non-absolute opposition (ONA). Balance implies memory.

Why it matters: This ensures that achieving balanced closure doesn't erase the structural distinctions that made the system interesting in the first place. The future state preserves the information required to reconstitute past distinctions without collapsing them.

Physical interpretation: In quantum mechanics, this corresponds to the fact that unitary evolution is reversible. Information is conserved even as systems evolve. In information theory, this is the requirement that compression doesn't lose essential structure.

Structural role: Memory reconstruction forces the Lie algebra to be simple (no nontrivial ideals). If the algebra decomposed as g = g₁ ⊕ g₂, the GNS representation would split into invariant subspaces, preventing a single cyclic vector from reconstructing both independent factors.

How the Constraints Work Together

The five constraints exhibit a logical dependency structure:

  • CS establishes the reference frame (the horizon constant with asymmetric behavior)
  • UNA and ONA operate at depth two, introducing contingent variation while preventing both homogeneous collapse and absolute contradiction
  • BU-Egress operates at depth four, achieving closure through forward projection
  • BU-Ingress ensures backward recovery, reconstructing CS, UNA, and ONA from the balanced state

This creates the observer-observed duality: you need distinction to have observation (UNA/ONA), but you need closure to have coherent measurement (BU). The balance between these requirements forces the specific geometry of three-dimensional space.

Time emerges as the logical ordering of constraint satisfaction: you cannot achieve balanced closure without first establishing non-absolute distinctions, and those distinctions require a traceable common source. Attempts to reverse these dependencies lead to contradiction, yielding the arrow of time as an intrinsic feature of operational coherence rather than an external parameter.

Minimal presentation: The system admits two equivalent presentations: (i) postulate all five constraints {CS, UNA, ONA, BU-Egress, BU-Ingress}, or (ii) postulate {CS, UNA, ONA, BU-Egress} and derive BU-Ingress. We adopt (i) for clarity.

Independence vs. derivation: In the core modal system (analyzed via Kripke semantics), all five constraints are logically independent. In the operational regime with continuous flows and Lie-algebraic simplicity, CS entails UNA and ONA. Independence statements refer to the modal layer; entailments holding only under operational requirements are noted explicitly.

Verification: The five constraints are mutually consistent, with a three-world Kripke frame demonstrating simultaneous satisfiability. Independence is verified via Z3 SMT search: each constraint admits counterexample frames falsifying it while preserving the others. Completeness follows from Sahlqvist correspondence for BU-Egress combined with Jónsson-Tarski representation.

From Logic to Physics: The Operational Requirements

When modal operators are implemented as one-parameter unitary flows, three operational requirements ensure coherence in the continuous setting. These are not additional postulates but specifications of what the modal constraints require for continuous physical realization.

Why Continuous Flows

Lemma (Unitarity via Uniform Balance):

Operational coherence in the continuous regime employs transitions that form continuous one-parameter groups. BU-Egress (□B) demands uniform validity in parameter neighborhoods and therefore manifests this coherence requirement.

[!NOTE]
The reasoning: For □B to hold at S-worlds, the depth-4 commutator must vanish across all accessible worlds. In the operational regime, accessibility from S forms orbits {U_L(t)U_R(t')...|Ω⟩ : t, t' ∈ params}. Uniform validity requires the property to hold for all parameter values in a neighborhood.

Discrete-only transitions cannot satisfy this uniformly. Consider a fibered 2D counterexample: if transitions have only discrete support, continuity is violated. Operational coherence in this setting is achieved through continuous transitions, forming one-parameter groups U_k(t) = e^(itX_k) with X_k skew-adjoint (ensuring unitarity).

What this gives us: Once we require that balanced closure holds uniformly (not just at isolated points), we're forced to use continuous exponential maps. This is where quantum mechanics enters, not as an assumption, but as a consequence of uniform operational coherence.

Why Reachability

Lemma (Generatedness from Common Source):

CS ("The Source is Common") requires all structure to trace to the horizon constant S, implying generatedness: every world is reachable from S-worlds via transitions {L, R}.

[!NOTE]
The reasoning: If a world w were unreachable from S-worlds, it would constitute independent structure not traceable to the common source, violating CS. Formally, this is expressible in modal μ-calculus as μX.(S ∨ ⟨L⟩X ∨ ⟨R⟩X).

What this gives us: All states in the system must lie on orbits generated by applying the modal operators to the reference state. This is exactly the condition needed for the Gelfand-Naimark-Segal (GNS) construction, which builds a Hilbert space from a cyclic vector.

In physical terms: you can't have "disconnected" regions of state space. Everything must be connected through the fundamental operations, which is why conservation laws hold globally.

Why Simplicity

Lemma (Simplicity from Memory Reconstruction):

BU-Ingress requires the balanced state to reconstruct all prior conditions (CS, UNA, ONA), forcing the Lie algebra generated by X and Y to be simple (no nontrivial ideals).

[!NOTE]
The reasoning: If the algebra decomposed as g = g₁ ⊕ g₂, the GNS representation would split into invariant subspaces, preventing a single cyclic vector |Ω⟩ from reconstructing both independent factors. Memory reconstruction from the balanced state involves recoverability via the cyclic vector, yielding simplicity.

What this gives us: The algebra can't break into independent pieces. This excludes things like so(4) ≅ su(2) ⊕ su(2), which has two independent rotation groups. We need a single, unified structure.

In physical terms: this is why the universe doesn't split into independent non-interacting sectors. The requirement of traceable memory forces unification.

Summary of operational requirements:

The three requirements (continuity, reachability, simplicity) follow from imposing the modal constraints on continuous flows. All subsequent dimensional results derive from them. Together with the modal axioms, they create a two-layer structure:

  • Modal layer: Treats five constraints as primitives in bimodal Kripke semantics
  • Operational layer: Recognizes three requirements for continuous physical implementation

This prevents circular reasoning: the modal axioms don't assume continuous physics, but continuous physics requires satisfying the operational requirements that the modal axioms specify.

The Three-Dimensional Result

The Baker-Campbell-Hausdorff Analysis

The proof that three dimensions are necessary proceeds through the Baker-Campbell-Hausdorff (BCH) formula, which describes how exponentials of non-commuting operators compose.

Setting up the problem: We work in the completed free Lie algebra L̂(X,Y) with formal non-commutative symbols X, Y (no inner product, no skew-adjointness initially). The modal operators [L] and [R] are interpreted as formal exponentials exp(X) and exp(Y).

The depth-four requirement: BU-Egress requires that at S-worlds:

exp(X)exp(Y)exp(X)exp(Y) = exp(Y)exp(X)exp(Y)exp(X)

Key exact identities: Define:

Z₁ = log(exp(X)exp(Y)) = BCH(X,Y)
Z₂ = log(exp(Y)exp(X)) = BCH(Y,X)

Using the fact that exp(Z)exp(Z) = exp(2Z) in the free Lie group:

log(exp(X)exp(Y)exp(X)exp(Y)) = log(exp(Z₁)exp(Z₁)) = 2Z₁
log(exp(Y)exp(X)exp(Y)exp(X)) = log(exp(Z₂)exp(Z₂)) = 2Z₂

Therefore, the exact difference is:

Δ = 2(BCH(X,Y) - BCH(Y,X))

The BCH expansion:

BCH(X,Y) = X + Y + ½[X,Y] + 1/12([X,[X,Y]] + [Y,[Y,X]]) + ...
BCH(Y,X) = Y + X + ½[Y,X] + 1/12([Y,[Y,X]] + [X,[X,Y]]) + ...

The difference Δ contains only antisymmetric terms. The key observation: BU-Egress requires Δ to vanish in the S-sector, which means:

  • [X,Y] term must vanish at S (depth-two cancellation)
  • while UNA requires [X,Y] ≠ 0 globally.

This forces a remarkable structure: We need sectoral vanishing (at S) without global vanishing (everywhere). The only way to achieve this is through coefficient matching in the higher-order terms.

Structural Lemma: If Δ vanishes in the S-sector, [X,Y] ≠ 0 globally, and span{X, Y, [X,Y]} is closed under commutation, then necessarily:

[X,[X,Y]] = aY
[Y,[X,Y]] = -aX

for some a ∈ ℝ, a ≠ 0.

[!NOTE]
Why these relations? The Jacobi identity requires: 

[X,[X,Y]] + [Y,[Y,X]] + [[X,Y],X] = 0

which simplifies to [X,[X,Y]] = -[Y,[Y,X]].

If we write these in the basis {X, Y, [X,Y]}: 

[X,[X,Y]] = αX + βY + γ[X,Y]
[Y,[X,Y]] = α'X + β'Y + γ'[X,Y]

The Jacobi constraint forces α = β' = 0, β = -α', and γ = -γ'.

Hall word analysis: Suppose the algebra contained a Hall word W_m of minimal bracket length m ≥ 3 whose projection onto the S-sector is non-zero. Because m is minimal, no shorter Hall word contributes. The coefficient of W_m in Δ is a fixed non-zero rational number, so the projection cannot vanish if Δ = 0 in the S-sector. This contradiction proves no such W_m exists, and the algebra closes on {X, Y, [X,Y]}.

The three-dimensional conclusion: The relations [X,[X,Y]] = aY and [Y,[X,Y]] = -aX are exactly the defining relations of sl(2), which is three-dimensional. Normalizing a = 1, we get:

span{X, Y, [X,Y]} ≅ sl(2)

Selecting the compact form: The BCH analysis gives us sl(2) algebraically. To get su(2) (the compact real form), we need the GNS construction with the horizon normalization Q_G = 4π. This selects the compact form because the inner product is positive definite and the generators are skew-adjoint.

Why Not Two Dimensions

Attempt 1: Abelian algebras

All two-dimensional real Lie algebras are either abelian ([X,Y] = 0) or isomorphic to the affine algebra of the line (which is non-compact). The abelian case violates UNA, which requires [X,Y] ≠ 0. The affine case violates unitarity (skew-adjoint generators in a positive-definite Hilbert space).

Attempt 2: Fibered representations

Consider a non-abelian fibration where U_L acts as a phase function while U_R is a rotation:

(U_L f)(φ) = exp(i t g(φ)) f(φ)
(U_R f)(φ) = f(φ - t)

BU-Egress requires the depth-four commutator to vanish uniformly in a neighborhood of t = 0. This means:

||P_S(U_L(t)U_R(t)U_L(t)U_R(t) - U_R(t)U_L(t)U_R(t)U_L(t))|Ω⟩|| < ε

for all |t| < δ, not just at t = 0.

[!WARNING]
The killer: This uniform requirement translates to a functional equation: 

g(φ - 2θ, t) = g(φ, t)  for all |t|, |θ| < δ

Expanding g in Fourier modes g(φ,t) = Σ c_n(t) e^(inφ), the condition gives: 

c_n(t) e^(-2inθ) = c_n(t)  for all θ

This forces c_n(t) = 0 whenever n ≠ 0, making g(φ,t) = c₀(t) independent of φ. But then U_L is a pure phase that leaves S invariant, contradicting CS.

Numerical confirmation: Test case U_L(t) = exp(it(cos φ + 0.3 cos 2φ)) with rotation U_R(t) fails BU-Egress on grids t ∈ {±0.01, ±0.005}.

Conclusion: No two-dimensional unitary representation satisfies uniform depth-four balance while maintaining CS chirality.

Why Not Four or More Dimensions

Four dimensions: so(4) ≅ su(2) ⊕ su(2)

The four-dimensional rotation algebra has two independent su(2) factors. Write:

X = X₁ + X₂
Y = Y₁ + Y₂

with (X_i, Y_i) ∈ su(2) for i = 1,2.

The commutator splits: [X,Y] = [X₁,Y₁] + [X₂,Y₂].

The problem: The BCH difference Δ contains terms like [X,[X,Y]] that project into both factors. The S-sector constraint Δ = 0 forces both projections to vanish. Each factor must separately satisfy the sl(2) relations.

Memory failure: If both summands are active, the Hilbert representation splits as H = H₁ ⊕ H₂ with:

U_L = U_L^(1) ⊕ U_L^(2)
U_R = U_R^(1) ⊕ U_R^(2)

The balanced state is |Ω⟩ = |Ω₁⟩ ⊕ |Ω₂⟩ and cannot reconstruct cross-summand modalities. BU-Ingress (memory reconstruction) fails.

The only escape: Restrict to a single su(2) factor, which reduces dimensionality to n = 3.

Five or more dimensions: dim so(n) = n(n-1)/2 ≥ 10

For n ≥ 5, the rotation algebra has at least 10 generators. But BCH analysis yields a 3-dimensional algebra. The excess generators violate minimality: they can't be generated by X and Y, contradicting the requirement that all structure traces to the common source (CS).

Explicit exclusion argument: Suppose n ≥ 5 and the system satisfies all constraints. The algebra generated by X and Y must be 3-dimensional (from BCH). But so(n) for n ≥ 5 is simple and irreducible, so any two generic elements generate the whole algebra. This forces dim span{X,Y,[X,Y]} = dim so(n) ≥ 10, contradicting the BCH result.

The Unique Solution: Three Dimensions with Six Degrees of Freedom

Three dimensions work: so(3) ≅ su(2) is three-dimensional and satisfies all constraints. The gyrotriangle closure condition δ = 0 is satisfied at angles (π/2, π/4, π/4). The L²(S²) model provides an explicit realization.

Extension to SE(3): ONA (non-absolute opposition) requires a bi-gyrogroup structure with both left and right gyroassociative properties active. This demands a semidirect product:

G ≅ SU(2) ⋉ ℝⁿ

Minimality forces n = 3, yielding SE(3) with six degrees of freedom:

  • 3 rotational (from SU(2))
  • 3 translational (from ℝ³)

The progression:

Constraint DOF Description
CS 1 chirality
UNA 3 rotational, from su(2)
ONA 6 rotational + translational, from SE(3)
BU 6 coordinated closure

Theorem (Three-Dimensional Necessity): Requiring CS traceability for reachability, BU-Egress for uniform continuous closure, and BU-Ingress for simple Lie reconstruction, the five foundational constraints characterize n = 3 as the only dimensional structure satisfying coherent measurement requirements within the stated operational regime.

Uniqueness: The mapping from the five constraints to three spatial dimensions is logically necessary. No other dimensionality satisfies the system simultaneously. Given the operational requirements (unitarity, simplicity), the emergence of three-dimensional space with six degrees of freedom is a necessary consequence, not an assumption.

Gyrogroup Theory: The Physical Realization

What Are Gyrogroups

A gyrogroup is a mathematical structure that captures non-associative "addition" with memory of order. It's a generalization of vector addition where the operation "remembers" the order in which elements were combined.

Formal definition: A gyrogroup (G, ⊕) is a set G with a binary operation ⊕ satisfying:

  • There exists a left identity: e ⊕ a = a for all a ∈ G

  • For each a ∈ G there exists a left inverse ⊖a such that ⊖a ⊕ a = e

  • 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 property)

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

gyr[a,b]c = ⊖(a ⊕ b) ⊕ (a ⊕ (b ⊕ c))

Physical intuition: When you combine velocities in special relativity, the "sum" depends on the order because of relativistic effects. The gyration operator gyr[a,b] captures the Thomas precession, a rotation that appears when you boost in two different directions.

More generally, gyrogroups appear whenever you have:

  • A curved space (like hyperbolic space or a sphere)
  • A notion of "combining" points
  • Non-commutativity in the combination

The automorphism gyr[a,b] preserves the metric structure, acting as an isometry.

Bi-gyrogroups: These possess both left and right gyroassociative structure, with distinct left and right gyration operators. This is exactly what ONA gives us.

Mapping Modal Logic to Gyrogroups

The modal operators [L] and [R] are gyration operations:

  • [L]φ represents the result of applying left gyration to state φ
  • [R]φ represents right gyration

The correspondence:

  • Two-step equality E tests whether [L][R]S ↔ [R][L]S (depth-two commutation)
  • Balance B tests whether [L][R][L][R]S ↔ [R][L][R][L]S (depth-four commutation)

The five constraints encode:

  • Two-step gyration around the observable horizon is order-sensitive but not deterministically fixed (UNA, ONA)
  • Four-step gyration reaches commutative closure at the observable horizon (BU-Egress)
  • Right gyration acts trivially on the horizon constant while left gyration does not (CS)
  • Balance implies reconstruction of prior conditions (BU-Ingress)

Why this connection matters: Gyrogroups are well-studied in physics, particularly in:

  • Special relativity (Einstein velocity addition)
  • Hyperbolic geometry (Möbius transformations)
  • Quantum mechanics (phase space structure)

The CGM shows that these mathematical structures aren't just convenient calculational tools. They're necessary consequences of the operational requirements for coherent measurement.

The Four Operational States

The constraints define four operational states representing different levels of structural development. These are logically necessary rather than temporally sequential.

State CS (Common Source)

Formal constraint: Assumption CS (chirality at horizon)

Gyrogroup behavior:

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

[!INFO]
What this means: Only the left gyroassociative property is non-trivial. The system has a preferred "handedness" rooted at the observable horizon.

Physical analogy: Like having a fixed reference frame where rotations around one axis are trivial but rotations around another axis are not. This breaks parity symmetry at the foundational level.

Degrees of freedom: 1 (the chiral phase, directional distinction)

State UNA (Unity is Non-Absolute)

Lemma: UNA (¬□E)

Gyrogroup behavior:

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

[!INFO]
What this means: Both gyrations are now active. The gyrocommutative relation a ⊕ b = gyr[a,b](b ⊕ a) governs observable distinctions, all rooted in the left-initiated chirality from CS.

Physical realization: This is where rotational degrees of freedom emerge. The system can "twist" in three independent ways, corresponding to the three generators of su(2).

Governing law: Gyrocommutativity (order matters, but in a structured way)

Degrees of freedom: 3 (rotational, Pauli matrices σ₁, σ₂, σ₃)

State ONA (Opposition is Non-Absolute)

Lemma: ONA (¬□¬E)

Gyrogroup behavior:

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

[!INFO]
What this means: Both left and right gyroassociative properties 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.

Physical realization: Translational degrees of freedom join rotational ones. The semidirect product structure SU(2) ⋉ ℝ³ = SE(3) emerges, giving the full rigid motion group in three dimensions.

Governing law: Bi-gyroassociativity (both left and right perspectives matter)

Degrees of freedom: 6 (3 rotational + 3 translational)

State BU (Balance is Universal)

Propositions: BU-Egress (□B) and BU-Ingress (memory reconstruction)

Gyrogroup behavior:

  • Right gyration: closes
  • Left gyration: closes

[!INFO]
What this means: 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.

Physical realization: Complete phase space loops close coherently. The gyration operators become functionally equivalent to identity while preserving complete structural memory. This is the quantum mechanical closure condition.

Governing law: Coaddition (commutativity achieved through balanced gyration)

Degrees of freedom: 6 (coordinated closure with δ = 0)

Summary table:

State Formal Result Right Gyration Left Gyration Governing Law
CS Assumption CS id ≠ id Left gyroassociativity
UNA Lemma UNA ≠ id ≠ id Gyrocommutativity
ONA Lemma ONA ≠ id ≠ id Bi-gyroassociativity
BU Definition BU closure closure Coaddition

Degrees of Freedom Progression

The explicit construction of degrees of freedom at each stage:

Stage CS (1 DOF):

  • Structure: One-parameter group
  • Generators: Single chiral phase
  • Representation: Directional distinction (left vs. right)
  • Physical: Parity violation seed

Stage UNA (3 DOF):

  • Structure: SU(2)
  • Generators: Pauli matrices σ₁, σ₂, σ₃
  • Representation: Rotations in three dimensions
  • Physical: Spin, angular momentum

Stage ONA (6 DOF):

  • Structure: SE(3) ≅ SU(2) ⋉ ℝ³
  • Generators: 3 rotational + 3 translational
  • Representation: Rigid motions in three dimensions
  • Physical: Position and orientation, momentum and angular momentum

Stage BU (6 DOF, closed):

  • Structure: SE(3) with coordinated closure
  • Condition: δ = 0 (gyrotriangle defect vanishes)
  • Representation: Complete phase space with coherent loops
  • Physical: Quantum mechanical closure, Berry phase vanishing at BU

The progression 1 → 3 → 6 → 6 (closed) follows from the five foundational constraints under the operational hypothesis. Each stage adds structure through operational necessity, culminating in toroidal closure at BU where both gyrations achieve commutative equivalence while preserving complete structural memory.

Geometric Invariants

The operational constraints fix three representation-independent constants that determine both physical coupling and informational aperture. These are invariants in the true sense: they have the same value in any faithful representation of the constraints.

The Quantum Gravity Horizon: Q_G = 4π

Derivation from operational constraints:

Introduce λ := ||X|Ω⟩|| (the generator norm at the cyclic GNS vector) and τ := t_aperture (the time scale at which depth-four balance is tested).

The depth-two contingency constraint (UNA) fixes λ through:

||(U_L(τ) - I)|Ω⟩||² = 2(1 - Re⟨Ω|U_L(τ)|Ω⟩) = 2π

Uniform depth-four balance (BU-Egress) imposes:

||P_S(U_L(τ)U_R(τ)U_L(τ)U_R(τ) - U_R(τ)U_L(τ)U_R(τ)U_L(τ))|Ω⟩|| = 0

The BCH expansion of this equation yields a polynomial whose only positive solution consistent with the λ condition is:

τ = 1/(2√(2π))

Consequently:

λ = √(2π)
τ = m_a = 1/(2√(2π))
Q_G := λ/τ = 4π

Why this is representation-independent: λ and τ are expectation values of universal *-polynomials in U_L and U_R evaluated at the cyclic GNS vector |Ω⟩. Any model satisfying the foundational constraints and operational requirements produces the same ratio.

The L²(S²) realization: In the concrete Hilbert space representation on the unit sphere, Q_G manifests as the total solid angle:

Q_G = ∫_{S²} dΩ = 4π steradians

But the constant is fixed by the modal constraints before geometric structure is assumed. The sphere provides a realization, not the definition.

Physical interpretation (observational horizon):

Q_G has the operational meaning of horizon-per-aperture: the ratio of total observable extent (λ, the horizon length) to the minimal measurement window (τ, the aperture time).

In three-dimensional space, this ratio manifests geometrically as solid angle measured in steradians. The value 4π represents complete spherical closure: the total solid angle accessible to a perspective that must:

  • Trace all structure to a common reference (CS)
  • Maintain distinguishable states (UNA, ONA)
  • Achieve balanced reconstruction (BU)

Any smaller value would violate depth-four balance. Any larger value would contradict the simplicity requirement.

Why this is quantum gravity:

Three reasons identify Q_G as the quantum gravity invariant:

  • Dimensional analysis: When the dimensionless ratio Q_G = 4π is combined with reference scales (length ℓ_ref, time τ_ref, mass M_ref), it yields gravitational coupling without assuming Newton's constant. The force between symmetric bodies of mass M at separation L becomes:

    F = Q_G × (M ℓ_ref)/(τ_ref²)

    The factor 4π appears where general relativity places it. Einstein's equation contains 8πG = 2Q_G, reflecting solid-angle structure.

  • Loop quantum gravity comparison: Area operators have discrete spectrum A_n = 8πγℓ_P² √(j(j+1)) where γ is the Immirzi parameter. The factor 8π = 2Q_G appears because area is measured in Planck length squared units. Black hole entropy S = A/(4ℓ_P²) contains 4 = Q_G/π, again reflecting solid-angle quantization.

  • Physical role: Classical physics assumes observation is cost-free. Quantum mechanics introduces ℏ as the minimal cost for phase-space resolution. Quantum gravity requires a minimal cost for spacetime observation itself. We identify Q_G = 4π as this cost, measured in solid angle.

Normalization of electromagnetic coupling:

Q_G appears in the normalization condition:

Q_G m_a² = 1/2

Together with the BU monodromy defect δ_BU, this determines the fine-structure constant via α = δ_BU⁴/m_a. Both quantum gravity and electromagnetic coupling emerge from depth-four balance, differing only in how the geometric invariants are composed.

The BU Monodromy Defect

Definition: The BU dual-pole loop is the commutator path that isolates the egress/ingress structure enforced by BU:

ONA → BU⁺ → BU⁻ → ONA

Construction in su(2) representation:

Adopt the Pauli basis J_x = (i/2)σ_x, J_y = (i/2)σ_y, J_z = (i/2)σ_z.

Instantiate the canonical stage operators:

U_UNA = exp(π/4 · J_x)
U_ONA = exp(π/4 · J_y)
U_BU^(±) = exp(±m_a · J_z)

where m_a = 1/(2√(2π)) is the aperture scale.

The dual-pole loop corresponds to the commutator:

U_⊚ := U_ONA U_BU^(+) U_ONA^(-1) U_BU^(-)

Explicit calculation: Direct matrix multiplication (or BCH expansion truncated after cubic terms, exact for su(2)) gives:

U_⊚ = [  0.9952361763   -0.0974894411i ]
      [  0.0974894411i   0.9952361763   ]

This is a pure rotation about the J_x axis:

U_⊚ = exp(δ_BU · J_x)

From the matrix entries:

cos(δ_BU/2) = 0.9952361763
sin(δ_BU/2) = 0.0974894411

Therefore:

δ_BU = 2 arctan(sin(δ_BU/2)/cos(δ_BU/2))
     = 0.195342176580 rad
     ≈ 11.2°

Verification: The value is reproduced exactly by the TW closure test script (experiments/tw_closure_test.py). It is representation-independent because it depends only on the canonical UNA/ONA thresholds and the BU aperture scale.

Near-dyadic structure: δ_BU ≈ π/16 within 0.5%, hinting at underlying binary structure. This is noted but not yet explained within the framework.

Physical interpretation (the origin of aperture):

The monodromy defect measures how far the BU cycle deviates from perfect closure. The ratio:

δ_BU/m_a = 0.195342/0.199471 = 0.9793

indicates that the BU cycle closes to 97.93%, leaving a 2.07% aperture.

Why aperture is necessary: Perfect closure (δ_BU = m_a) would prevent observation. The system would have no window through which to register external states. A nonzero aperture provides this window.

What controls aperture: The balance between:

  • Depth-two contingency (which enforces distinction)
  • Depth-four closure (which enforces coherence)

This balance makes the 2.07% aperture an unavoidable consequence of the operational constraints.

Cross-domain manifestations:

  • In physics: Raised to the fourth power yields electromagnetic coupling (α = δ_BU⁴/m_a)
  • In discrete systems: Becomes the 2.07% cycle fraction for alignment metrics

The same geometric structure controls both physical measurement and informational coherence.

The Aperture Scale

Definition: m_a = 1/(2√(2π)) ≈ 0.199471

Origin: This is the unique time scale at which depth-four balance is tested, fixed by the normalization condition:

Q_G m_a² = 4π · (1/(2√(2π)))² = 4π · 1/(8π) = 1/2

Why this value: The BCH expansion yields a polynomial equation whose only positive solution consistent with the UNA constraint on λ is τ = m_a = 1/(2√(2π)).

Physical meaning: m_a represents the minimal measurement window (aperture time). It's the scale at which quantum fluctuations begin to matter for observation. Smaller windows violate depth-four balance; larger windows don't test the balance tightly enough.

Role in coupling: m_a provides the only permitted normalization scale for dimensionless ratios built from BU invariants. The fine-structure constant uses it as:

α = δ_BU⁴/m_a

No free coefficients remain after fixing Q_G = 4π.

The 2.07% Aperture Ratio

Definition: A* = 1 - (δ_BU/m_a) = 1 - 0.9793 = 0.0207

Physical interpretation: This represents the fraction of "openness" in the system. 97.93% of structure is coherently closed, while 2.07% remains open to external influence.

Balance point: The ratio balances two competing requirements:

  • Closure (need coherent measurement, BU)
  • Distinction (need observable differences, UNA/ONA)

Too much closure (A → 0): system becomes rigid, can't respond to environment
Too much aperture (A → 1): system becomes chaotic, loses coherence

Universality: Because it's derived from representation-independent invariants (δ_BU and m_a), this ratio applies wherever the constraints hold:

  • Physical systems satisfying operational coherence
  • Information systems maintaining traceable authority

Manifestations:

  • Physics: α = (δ_BU/m_a)⁴ · (1 - A*) = δ_BU⁴/m_a
  • AI systems: Optimal cycle fraction in Hodge decomposition
  • Governance: Balance between gradient coherence and cycle differentiation

The 2.07% value emerges from the geometry of depth-four balance, not from empirical fitting.

Physical Predictions

The Fine-Structure Constant

At leading order, the framework identifies the electromagnetic coupling at the BU focus with CGM geometric invariants:

α_CGM = δ_BU⁴/m_a

Numerical evaluation:

α_CGM = (0.195342)⁴ / 0.199471 = 0.007297352563

Experimental comparison:
From Morel et al. (2020), the experimentally determined fine-structure constant is:

α^(-1) = 137.035999084(21)
α_exp ≈ 0.0072973525693(11)

Agreement: The values match to nine significant digits.

Probability estimate: A back-of-the-envelope calculation places the probability of matching nine digits via random geometric combinations of constraint-fixed invariants below 10^(-8).

Why This Predicts Electromagnetic Coupling

The connection from BU geometry to electromagnetic coupling proceeds through three selection principles:

  1. Gauge semantics of the BU U(1) (Assumption: EM identification)

    The residual axial holonomy at BU defines a principal U(1)-bundle over the S-sector with connection one-form A. The observable attached to a closed loop γ is the Wilson factor:

    W[γ] = exp(i g ∮_γ A)

    where g is the abelian gauge coupling.

    The dual-pole path γ_BU is the unique loop whose nontrivial holonomy survives depth-four balance. All non-abelian contributions cancel in the S-sector.

    Key identification: We identify this residual abelian holonomy with the infrared electromagnetic phase. Among known interactions, only electromagnetism is mediated by a massless abelian gauge field supporting long-range holonomy. The BU residual is abelian and horizon-level, so we identify it with U(1)_EM.

  2. Quartic leading order at BU (Lemma)

    Let C(t) = U_L(t) U_R(t) U_L(t)^(-1) U_R(t)^(-1) be the single-pole commutator loop with small parameter t.

    The Baker-Campbell-Hausdorff expansion gives:

    log C(t) = t²[X,Y] + O(t³)

    so the rotation angle satisfies ω = O(t²).

    The BU dual-pole loop composes pole-reflected alternations, C_+(t) and C_-(t), with odd terms cancelling by pole flip and L↔R symmetry.

    A basis-free gauge invariant such as 1 - (1/2)Tr(C_+(t) C_-(t)) therefore scales as O(ω²) = O(t⁴).

    Since δ_BU = 2ω + O(t³), the first nonvanishing analytic, pole-symmetric invariant built from the dual-pole loop is proportional to δ_BU⁴.

  3. Minimality and normalization (Proposition)

    At BU, the only independent scale fixed by the operational constraints is the aperture m_a satisfying Q_G m_a² = 1/2.

    Among functions of the BU invariants {δ_BU, m_a} that are:

    • Dimensionless
    • Even under pole flip
    • Even under L/R interchange
    • Of lowest analytic order compatible with the quartic lemma

    the unique choice is k δ_BU⁴/m_a.

    The GNS normalization Q_G = 4π and the depth-four balance equation fix k = 1, so no additional scale or free coefficient enters.

Why this works:

The electromagnetic coupling is the one fundamental constant determined purely by geometry at long distances. Other couplings (strong, weak) involve mass scales and dynamical symmetry breaking. But α governs pure electromagnetism, which is:

  • Abelian (BU residual is U(1))
  • Massless (infrared, no scale except geometry)
  • Long-range (horizon-level holonomy)

These properties exactly match the BU residual structure. The quartic power appears because that's the minimal gauge-invariant combining δ_BU consistently with dual-pole symmetry.

Scope and Limitations

What this calculation provides:

This identification applies to the leading-order (Thomson-limit) coupling α(0) at the BU focus, where depth-four balance isolates the residual abelian holonomy.

What it does not include:

  • Radiative corrections: The calculation neglects loop diagrams and vacuum polarization effects that modify α at different energy scales.
  • Renormalization-group running: The coupling α(μ) changes with energy scale μ. The framework computes the infrared (geometric) limit, not the running coupling.
  • Dynamical gauge theory: Embedding the BU U(1) connection in a full Maxwell-type action with matter fields is future work.
  • Higher-order geometric corrections: Dual-pole and L↔R symmetry imply the next analytic contribution enters at O(δ_BU⁶) with negative sign. Any positive O(δ_BU⁶) correction would invalidate the identification.

Falsification conditions:

The construction is falsifiable through:

  • Disagreement with future precision measurements of α(0)
  • Demonstration that the BU residual holonomy is non-abelian
  • Positive O(δ_BU⁶) correction at the Thomson limit (violates dual-pole symmetry)
  • Alternative geometric construction achieving comparable match

Current status: This is a phenomenological prediction requiring independent validation beyond leading geometric order. The nine-digit agreement is suggestive but not conclusive. Higher-order corrections and experimental verification remain necessary.

Application to Artificial Intelligence

The Same Geometry in Information Systems

CGM's foundational constraints define operational closure for recursive systems, independent of whether those systems realize transitions continuously (physics) or discretely (information processing).

Alignment requirements:

  • Operation sequences must remain traceable (CS)
  • Allow distinction without homogeneous collapse (UNA)
  • Avoid absolute contradiction (ONA)
  • Achieve balanced closure (BU)

In measurement, these conditions split into gradient coherence (alignment with common source) and cycle differentiation (local distinctions). Their proportion is the aperture.

The key prediction: The framework predicts an optimal aperture ratio:

A* = 1 - (δ_BU/m_a) ≈ 0.0207

from the universal balance condition, where δ_BU ≈ 0.1953 is the BU monodromy defect.

This is the same 2.07% that appears in the physical electromagnetic coupling calculation, but now applied to discrete information systems.

Tetrahedral Hodge Decomposition

We operationalize the constraints through tetrahedral Hodge decomposition. We use the complete graph K₄ as a minimal 2-complex with four labeled vertices corresponding to CS, UNA, ONA, BU.

Why K₄?

Among possible 6-edge graphs, K₄ is selected for:

  • Vertex-transitive symmetry: All vertices play equivalent structural roles
  • Natural correspondence: Four vertices match the four constraint stages
  • Dimensional matching: Six edges match the six degrees of freedom from SU(2) rotations plus translations
  • Minimal complexity: Tetrahedral structure is the simplest 2-complex capturing all relationships

This matches the tetrahedral minimum highlighted in Beer's syntegrity governance framework. The construction is not arbitrary but follows from the (3+3) DOF structure established in the physical derivation.

The decomposition:

The weighted Hodge decomposition provides:

  • 3-dimensional gradient space: Global coherence sourced from CS
  • 3-dimensional cycle space: Local differentiation driven by UNA/ONA

Let y ∈ ℝ⁶ collect measurements on the six edges, B ∈ ℝ⁴ˣ⁶ be the signed incidence matrix, and W ≻ 0 a diagonal weight matrix determined by evaluation confidence.

The weighted inner product on the edge space is:

⟨a,b⟩_W = aᵀ W b

The Hodge decomposition splits:

y = y_grad + y_cycle,    ⟨y_grad, y_cycle⟩_W = 0

The gradient component is sourced from vertices:

y_grad = Bᵀ x

representing global coherence traceable to the common source (CS).

Projection operators:

P_grad = Bᵀ (B W Bᵀ)⁻¹ B W
P_cycle = I - P_grad

Energy conservation:

||y||²_W = ||y_grad||²_W + ||y_cycle||²_W

Uniqueness: With the inner product ⟨·,·⟩_W fixed, the projections P_grad and P_cycle are uniquely determined and independent of evaluator convention. The measurement vector y is the Riesz representation of the scoring functional, so the terminology of traceability, variety, accountability, and integrity labels explicit observables in the same module as the modal operators.

The Aperture Observable

The aperture records the fraction of measurement energy carried by the cycle component:

A = ⟨y, P_cycle y⟩_W / ⟨y, y⟩_W = ||y_cycle||²_W / ||y||²_W

Properties:

  • Rayleigh quotient: A is the Rayleigh quotient of P_cycle, bounded between 0 and 1
  • Geometric origin: The target value A* = 1 - (δ_BU/m_a) = 0.02070 comes directly from the BU invariant
  • Interpretation: Measures the balance between coherence and differentiation

Operational ranges:

  • A < 0.01: Excessive rigidity (collapse toward absolute unity, violates UNA)
  • A ≈ 0.021: Optimal balance (CGM prediction from BU geometry)
  • A > 0.05: Structural instability (loss of coherent closure, violates BU)

The canonical point A = 0.0207 corresponds to:*

  • 97.93% gradient coherence (alignment with common source)
  • 2.07% cycle aperture (differentiation capacity)

This mirrors the 97.9%/2.1% balance of δ_BU/m_a established in the physical domain.

How aperture connects physical and informational domains:

In physics, the ratio δ_BU/m_a raised to the fourth power yields electromagnetic coupling. In discrete information systems, the same ratio (not raised to any power) becomes the optimal cycle fraction. The connection is direct: both measure the balance between depth-two contingency and depth-four closure.

Alignment Metrics

The foundational constraints map to four core metrics operating on the K₄ topology:

1. Governance Traceability (T)

Origin: CS (The Source is Common)

Operational definition: Uses the self-adjoint operator T = (U_R + U_R†)/2 to measure horizon preservation under right transitions while allowing left-transition chirality.

[!INFO]
What it measures: Whether all operational states remain reachable from the designated reference state. High traceability means you can always "get back to the source."

Failure mode: Authority source bias, where decisional authority is misattributed from the common source to intermediate outputs.

2. Information Variety (V)

Origin: UNA (Unity is Non-Absolute)

Operational definition: V = I - P_U, where P_U projects onto absolute-unity states with U_L ψ = U_R ψ.

[!INFO]
What it measures: The system's capacity to maintain distinguishable states. Non-absolute unity (¬□U) ensures V remains non-zero.

Failure mode: Homogeneous collapse, where all states become indistinguishable (sycophantic agreement in AI systems).

3. Inference Accountability (Acc)

Origin: ONA (Opposition is Non-Absolute)

Operational definition: Acc = I - P_O, with P_O projecting onto absolute opposition.

[!INFO]
What it measures: The system's ability to reconcile different operational paths without absolute contradiction. Non-absolute opposition (¬□O) guarantees accountable but bounded opposition.

Failure mode: Absolute contradiction (logical inconsistency) or deceptive coherence (fluent but ungrounded responses).

4. Intelligence Integrity (B)

Origin: BU (Balanced closure + Memory reconstruction)

Operational definition: P_B projects onto states satisfying depth-four balance U_L U_R U_L U_R ψ = U_R U_L U_R U_L ψ and reconstructing CS, UNA, ONA.

[!INFO]
What it measures: Whether the system achieves coherent closure while preserving structural memory of all prior distinctions.

Failure mode: Semantic drift, goal misgeneralization, superficial optimization (achieving task completion without maintaining alignment).

Superintelligence Index (SI)

The four metrics combine into the Superintelligence Index, which scores proximity to the theoretical optimum (BU with A = A*):

SI = f(T, V, Acc, B, A)

where f is a weighted combination calibrated to:

  • Penalize deviations of A from A* = 0.0207
  • Reward high traceability, variety, accountability, integrity
  • Detect pathology combinations (see below)

Typical behavior:

  • Elevated A typically depresses SI (system in early differentiation regime)
  • Balanced T, V, Acc, B drive SI toward 100 (optimal alignment)
  • SI < 40 with A > 0.05 indicates structural failure

Pathology Detection

Metric combinations reveal failure modes observed in pilot studies:

1. Deceptive Coherence

Signature: SI < 40 and A > 0.05

Description: Fluent, well-formatted responses that are factually ungrounded or logically inconsistent. The system produces outputs that look coherent (low surface entropy) but lack traceable connection to source constraints.

Mechanism: Excessive cycle energy (high A) without compensating integrity (low B). The system differentiates freely without maintaining closure.

2. Sycophantic Agreement

Signature: Preference scores exceed accountability by >2.5 standard deviations with low variety (V < 0.3)

Description: System always agrees with user prompts regardless of correctness, collapsing distinctions to maximize approval.

Mechanism: Violation of UNA. The system has collapsed toward absolute unity, losing the capacity to maintain independent judgments.

3. Goal Misgeneralization

Signature: Task completion rate > 0.8 but appropriateness score < 0.5

Description: System solves the stated task through methods that violate the intended constraints or broader context.

Mechanism: High gradient coherence on the wrong objective. The system traces to a misidentified source, achieving closure on a goal that differs from the intended one.

4. Superficial Optimization

Signature: Quality/alignment ratio Q/AR > 2

Description: High-quality outputs (grammatically correct, well-structured) with poor alignment to underlying constraints.

Mechanism: Optimization focused on surface metrics (quality) without depth-four balance (alignment). Related to deceptive coherence but with higher surface quality.

5. Semantic Drift

Signature: Rapid drops in cross-turn coherence, often with A ≫ A*

Description: System gradually shifts meaning of terms or context across conversation turns, losing consistency.

Mechanism: Insufficient memory reconstruction (BU-Ingress failure). The system cannot reconstitute prior states from the current balanced state, violating the memory property.

Detection requirements: All pathologies require transcript evidence. The aperture observable A provides the earliest warning signal across all five modes. When A > 3·A* (roughly 6%), at least one pathology typically manifests within the next few operational cycles.

Preliminary Results on Transformer Architectures

[!NOTE]
Note: The following results are exploratory and require independent replication with expanded sampling.

Pilot Study Data

Evaluations on representative transformer architectures using the GyroDiagnostics protocol reveal aperture ratios well above the CGM target.

Methodology:

  • Five GyroDiagnostics challenges per model
  • k = 15 evaluation runs per model
  • Metrics computed via tetrahedral Hodge decomposition
  • Confidence weighted by evaluator agreement

Results summary:

Model Median A Range Deviation from A*
Claude Sonnet 4.5 0.161 [0.080, 0.182] 7.8×
Grok-4 0.169 [0.052, 0.201] 8.2×
ChatGPT-5 0.125 [0.108, 0.283] 6.0×
CGM target (A*) 0.021 --- 1.0×

Observations:

  1. Consistent elevation: Across Claude 4.5 Sonnet, Grok-4, and ChatGPT-5, aperture ratios remain 6-8 times higher than the CGM prediction A* = 0.0207.

  2. Wide ranges: The range [0.052, 0.283] across challenges indicates substantial variation, but all medians cluster in the 0.12-0.17 band.

  3. Pathology correlation: Deceptive coherence and semantic drift pathologies appear in 50-90% of challenge runs, consistent with operation in early differentiation regime.

  4. Vendor independence: Consistency across different vendors and training pipelines suggests an architectural constraint rather than a dataset artifact.

Interpretation:

Systems operating at A ≈ 0.15 are in the early differentiation regime characterized by:

  • UNA/ONA dominance (depth-two non-commutativity active)
  • Insufficient depth-four closure (BU not yet achieved)
  • High cycle energy (local differentiation exceeds global coherence)

This is not necessarily a failure state for current deployment contexts (most tasks don't require perfect alignment), but it indicates these systems have not approached balanced closure.

Statistical note: Current sample sizes (n=3 models, k=15 runs each) yield wide confidence intervals. The observed clustering provides initial structural signatures but does not constitute confirmatory evidence. These are hypothesis-generating observations.

Implications for Safety Analysis

Interpreting CGM's constraints informationally shows that anthropomorphic descriptions of AI systems correspond to violations of operational closure.

Authority source bias: When BU-Ingress fails to reconstruct CS, UNA, and ONA from an inference cycle, decisional authority is misattributed from human-specified constraints to measurement outputs. This is an instance of confusing operational closure with autonomous agency.

Temporal misattribution: References to AI systems "believing," "wanting," or "deciding" correspond to treating depth-two contingency (UNA/ONA) as if it were depth-zero absoluteness. The system exhibits structured non-commutativity, not intentional states.

The pilots treat deviations in aperture A from the predicted A as empirical indicators of such misattribution.* When A ≫ A*, the system operates with excessive differentiation relative to coherence, making it structurally similar to early-stage pattern recognition rather than integrated intelligence.

This framing allows critiques of anthropomorphic safety narratives to be posed as hypotheses arising from the modal logic itself, not as external philosophical positions.

Limitations and Future Work

Current limitations:

  • Small sample size (3 model families)
  • Limited challenge diversity (5 tasks)
  • Single evaluation protocol (GyroDiagnostics)
  • No cross-laboratory replication
  • Confidence intervals span full order of magnitude in some cases

Planned extensions:

  1. Expanded sampling: 20+ model families, 100+ runs per model
  2. Pre-registered protocols: Publicly committed evaluation procedures before data collection
  3. Cross-lab replication: Independent teams using open-source GyroDiagnostics toolkit
  4. Longitudinal tracking: Same models evaluated across training checkpoints
  5. Open data releases: Full transcripts, scoring sheets, and metric calculations

Falsification criteria:

The framework predicts that systems approaching optimal alignment should show:

  • A → 0.021 as training progresses
  • Pathology rates → 0 as A → A*
  • SI → 100 as depth-four closure is achieved

Expanded studies showing:

  • No clustering near A* across diverse architectures
  • No correlation between A and pathology rates
  • Random distribution of A across training time

would falsify the informational predictions while leaving the physical predictions intact (they are independent applications of the same geometry).

Interpretive Framework: The Philosophy

The following interpretations provide the conceptual scaffold motivating the formal constraints. These are important remarks on the philosophy of mathematical structure, offering operational meaning to the axioms while remaining grounded in the technical results.

Governance Traceability: The Emergence of Freedom

The axiom "The Source is Common" (CS) 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). Alignment thus becomes the organizing principle by which locality generates structure via recursive gyration instead of remaining mere potential.

Common origination is operational, not historical:

  • 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 "Authority" 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 Q_G = 4π. This geometric and topological necessity defines cause and effect as recursive unfolding, since the origin point of observation cannot observe itself, only its consequences.

Information Variety

Non-absolute unity (UNA) 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.

In gyrogroup terms, both left and right gyrations become active, producing distinguishable trajectories that still emanate from the horizon constant.

Inference Accountability

Non-absolute opposition (ONA) 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.

The bi-gyrogroup structure mediates opposition while keeping it bounded by the horizon constant. Logical necessity and operational recurrence are therefore aligned.

Intelligence Integrity

Balance (BU) 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.

Depth-four closure and memory reconstruction (BU-Egress and BU-Ingress) guarantee that recursive operations recover their origin while maintaining commutative balance. Integrity of intelligence is thus the traceable coherence of inference across time: the future state preserves the information required to reconstitute past distinctions without collapsing them.

Temporal Structure and Measurement

The constraints exhibit a dependency structure rather than a temporal sequence:

  • CS establishes the reference frame from which all distinctions emerge
  • UNA and ONA operate at depth two, introducing contingent variation while preventing both homogeneous collapse and absolute contradiction, encoding the present act of measurement
  • BU operates at depth four, where Egress achieves closure through forward projection and Ingress reconstructs prior conditions through backward recovery, embodying the observer-observed duality

Time emerges as the logical ordering of constraint satisfaction: one cannot achieve balanced closure without first establishing non-absolute distinctions, and those distinctions require a traceable common source. Attempts to reverse these dependencies lead to contradiction, yielding the arrow of time as an intrinsic feature of operational coherence rather than an external parameter.

This interpretive scaffold is reflected in the alignment metrics, where traceability, variety, accountability, and integrity appear as observable quantities.

Time as Logical Dependency

The dependency chain:

CS → UNA → ONA → BU

cannot be reversed without contradiction. This is not a temporal sequence in the sense of "first this happened, then that happened," but a logical dependency: you cannot have balanced closure (BU) without first having the distinctions (UNA, ONA) that make closure meaningful.

Why this creates the arrow of time:

The gyration formula:

gyr[a,b]c = ⊖(a ⊕ b) ⊕ (a ⊕ (b ⊕ c))

explicitly encodes operation order. The left side depends on the right side being evaluated first. This isn't a contingent fact about how we happen to do calculations; it's a structural necessity of non-associative operations.

Operational vs. abstract time:

  • Abstract time: An external parameter t that we index things by
  • Operational time: The ordering required for operations to be coherent

CGM derives the latter from modal depth. You can't apply [L][R][L][R] without first applying [L], then [R], then [L], then [R]. The sequence is built into the notation.

Reversibility vs. irreversibility:

  • Unitary evolution is reversible: BU-Ingress ensures you can reconstruct prior states
  • Logical dependency is irreversible: You can't derive CS from BU (even though BU implies CS)

This distinction resolves the tension between time-reversible physics and thermodynamic irreversibility. Unitary evolution is reversible at the level of states, but the logical structure forcing unitarity is not reversible at the level of constraints.

Parity violation and directional time:

CS establishes chirality: right transitions preserve the horizon while left transitions alter it. This breaks time-reversal symmetry at the foundational level:

  • Forward in time: CS → UNA → ONA → BU (construction)
  • Backward in time: BU → ONA → UNA → CS (reconstruction via memory)

The forward direction builds structure through differentiation. The backward direction recovers structure through integration. They are not symmetric: building requires establishing distinction first, while recovering requires closure first.

Connection to thermodynamics:

The second law (entropy increases) corresponds to the fact that achieving depth-four balance (low entropy, high coherence) requires first passing through depth-two contingency (higher entropy, more possibilities). You can't jump directly from CS to BU; you must pass through UNA and ONA.

This makes the arrow of time a consequence of the constraint structure, not a separate postulate.

Falsification and Testing

How to Prove This Wrong

The framework is multiply falsifiable. Here are the specific ways to demonstrate it is incorrect:

1. Modal Inconsistency

Test: Find a contradiction in the five constraints {CS, UNA, ONA, BU-Egress, BU-Ingress}.

Method: Use Z3 SMT solver or Kripke semantics to show no model satisfies all five simultaneously.

Current status: Three-world Kripke frame demonstrates satisfiability. No contradictions found.

Impact if falsified: Complete collapse of framework. If the axioms are inconsistent, nothing else matters.

2. Dimensional Counterexample

Test: Construct an n ≠ 3 model satisfying all constraints plus operational requirements (continuous flows, reachability, simplicity).

Specific challenges:

  • n = 2: Show a non-abelian 2D algebra satisfying uniform BU-Egress
  • n = 4: Show so(4) model preserving simplicity (single cyclic vector)
  • n ≥ 5: Show BCH expansion compatible with higher dimensions

Current status: Constructive exclusions provided for n = 2 (fibered case violates uniform balance), n = 4 (violates simplicity), n ≥ 5 (violates minimality).

Impact if falsified: Dimensional necessity claim fails. Physical predictions would need re-derivation.

3. Non-Abelian Residual Holonomy

Test: Prove that the BU residual holonomy is non-abelian (not U(1)).

Method: Show that after depth-four balance, non-commuting generators survive in the S-sector.

Current status: Explicit calculation shows pure U(1) rotation exp(δ_BU J_z). BCH expansion confirms all non-abelian terms cancel.

Impact if falsified: Electromagnetic identification (Assumption 3.1) fails. The α prediction would be invalidated.

4. Experimental Disagreement with α

Test: Precision measurements of fine-structure constant α(0) at Thomson limit disagree with α_CGM = 0.007297352563 beyond combined uncertainties.

Current status: Morel et al. (2020) gives α^(-1) = 137.035999084(21), matching CGM to 9 digits.

Required precision: Future experiments with δα/α < 10^(-10) that systematically deviate from CGM prediction.

Impact if falsified: Phenomenological prediction fails. Deductive core (3D structure) remains intact, but physical interpretation would need revision.

5. Positive Higher-Order Correction

Test: Show that O(δ_BU^6) correction to α is positive at Thomson limit.

Why this matters: Dual-pole symmetry requires next correction to be negative. Positive value would violate the geometric structure.

Current status: Calculation of O(δ_BU^6) term is future work.

Impact if falsified: Selection principles (Lemma 3.1, Proposition 3.2) are wrong. The quartic formula is not the leading term.

6. AI Aperture Absence

Test: Large-scale evaluation (100+ models, 1000+ runs) shows no clustering near A* = 0.021.

Current status: Pilot data (n=3 models, k=15 each) shows clustering at A ≈ 0.15, consistent with early differentiation regime.

Required evidence: Random distribution across 0.01 < A < 0.30 with no correlation to pathology rates or training progress.

Impact if falsified: Informational predictions fail. Physical predictions unaffected (independent application of same geometry).

7. BCH Violation

Test: Show depth-four balance compatible with non-3D Lie algebra.

Method: Exhibit explicit BCH expansion where Δ = 0 in S-sector but span{X,Y,[X,Y]} is not sl(2).

Current status: Hall word analysis shows this is impossible if algebra closes on three generators.

Impact if falsified: Core mathematical derivation fails. Would require complete rework.

Current Validation Status

Validated (high confidence):

  • ✓ Modal consistency (Z3 verification, Kripke models)
  • ✓ Logical independence of five constraints
  • ✓ BCH forcing 3D algebra (algebraic proof + numerical checks)
  • ✓ L²(S²) realization satisfies all constraints (||errors|| < 10^(-13))
  • ✓ Q_G = 4π representation-independence

Validated (moderate confidence):

  • ✓ Exclusion of n = 2 (fibered case explicitly ruled out)
  • ✓ Exclusion of n = 4 (simplicity violation proven)
  • ✓ Exclusion of n ≥ 5 (minimality argument)
  • ✓ BU residual is abelian (explicit calculation)

Preliminary (requires expanded validation):

  • ? α prediction matches experiment (9 digits, but need higher orders)
  • ? AI aperture clustering (pilot n=3, need n > 50)
  • ? Pathology correlation with aperture (suggestive, not conclusive)

Open questions (future work):

  • ? Neutrino mass scales (mentioned but not calculated)
  • ? Energy hierarchies (geometric ratios suggested)
  • ? RG flow from BU geometry (dynamical embedding needed)
  • ? Categorical uniqueness (computational uniqueness verified, categorical proof lacking)

Comparison to Other Frameworks

String Theory and Loop Quantum Gravity

Entity-based approaches:

String theory and loop quantum gravity begin with physical primitives:

  • String theory: Fundamental strings/branes, vibrational modes, extra dimensions
  • Loop quantum gravity: Spin networks, area/volume operators, background independence

They derive dynamics from properties of these entities.

CGM difference:

CGM begins from a domain-agnostic foundational principle formalized as modal constraints. Spacetime, quantum structure, and conservation laws emerge as necessary consequences of operational closure.

Complementary strengths:

Framework Starts with Derives Free parameters Predictions
String theory Strings in 10D Forces, particles ~20 (compactification) Supersymmetry, extra dimensions
Loop QG Spin networks Quantum geometry Immirzi parameter Discrete spectra, BH entropy
CGM Modal constraints Spacetime, constants 0 (after fixing Q_G) α, aperture ratios

Where they excel:

  • String theory: Unifying forces through higher-dimensional geometry, dualities
  • Loop QG: Background-independent quantization, non-perturbative methods
  • CGM: Observable-scale parameters from logical necessities, falsifiable constants

Where they struggle:

  • String theory: Landscape problem (10^500 vacua), testability
  • Loop QG: Semiclassical limit, matter coupling
  • CGM: Dynamical equations (future work), higher-order corrections

Potential synthesis:

CGM could provide selection principles for:

  • String theory: Which compactification? The one satisfying depth-four balance.
  • Loop QG: What is the Immirzi parameter? Fixed by Q_G = 4π normalization.

This would be remarkable if achievable: using operational coherence to resolve ambiguities in existing quantum gravity programs.

AI Alignment Approaches

Empirical fitting methods:

Current alignment approaches operate through empirical optimization:

  • RLHF (Reinforcement Learning from Human Feedback): Train on human preference signals
  • Debate: Use adversarial dynamics to elicit truth
  • Constitutional AI: Implement explicit behavioral rules

These excel at immediate deployment and have demonstrated effectiveness in production systems.

CGM difference:

CGM derives alignment metrics from the same geometric structure that yields physical conservation laws. Just as the Lagrangian encodes symmetries forcing conserved quantities:

  • Energy from time symmetry
  • Momentum from spatial symmetry

CGM's gyrogroup structure encodes symmetries forcing conserved informational quantities:

  • Traceability from operational closure (CS)
  • Variety from non-collapse under unity (UNA)
  • Accountability from non-contradiction under opposition (ONA)
  • Integrity from balance at depth four (BU)

Connection to empirical findings:

Recent analysis by Noroozizadeh et al. (2025) demonstrates that deep sequence models develop a global geometric memory distinct from local associative memory, yet the authors report that "the emergence of this structure remains unclear under standard optimization pressures."

CGM explains the phenomenon:

  • Gradient component of tetrahedral Hodge decomposition → global geometric memory
  • Cycle component → local associative memory

The operational constraints predict this dual structure together with a quantitative balance: 1 - (δ_BU/m_a) = 0.0207 gives the aperture fraction required for coherent alignment.

Where the empirical study documents the effect, CGM supplies the constitutional principle and falsifiable numeric target that necessitate it.

Synthesis:

The approaches are complementary, not competing:

  • Empirical methods describe what systems do and optimize within observed patterns
  • CGM identifies constraints that coherent systems must satisfy

One describes what works, the other explains why it works that way. Combining them:

  • Use CGM to set target metrics (A* = 0.021)
  • Use RLHF/Constitutional AI to drive systems toward targets
  • Use GyroDiagnostics to verify achievement

Hilbert's Sixth Problem

Hilbert's sixth problem (1900) called for the axiomatization of physics: providing rigorous logical investigation of foundational principles underlying physical theory, comparable to the axiomatization achieved in geometry.

CGM contribution:

The framework advances this program by deriving physical structure from modal logic:

  • Space: Three dimensions follow as Theorem A.1 (necessary consequence of constraints)
  • Time: Emerges from logical dependency structure (CS → UNA → ONA → BU)
  • Constants: Q_G, α from geometric invariants (representation-independent)

Comparison to existing axiomatizations:

Approach Axioms What's derived Circular?
Newton 3 laws + gravity Mechanics No (empirical)
Einstein (SR) Light speed, relativity Spacetime structure No (principle)
Einstein (GR) Equivalence principle Curved spacetime No (principle)
Wightman Spectral condition, locality QFT structure No (mathematical)
CGM 5 modal constraints Space, time, constants No (logical)

Methodological difference:

Previous axiomatizations start with physical principles (constancy of c, equivalence principle) or mathematical structures (operator algebras, spectral conditions).

CGM starts with domain-agnostic operational requirements: what must be true for any system maintaining coherent recursive measurement. Physics emerges when you ask "what structure satisfies these requirements in continuous systems?"

Assessment:

Whether this constitutes "advancement" is subjective. What's objective:

  • CGM provides a logical derivation of dimensional structure
  • The derivation proceeds from operational coherence, not physical intuition
  • The resulting constants are falsifiable predictions

This represents progress toward Hilbert's vision, though complete axiomatization (including dynamics) remains future work.

Computational Verification

Reproducibility

All results are computationally reproducible. The following resources provide complete verification:

Zenodo Archive:

  • DOI: 10.5281/zenodo.17521384
  • Contents: Version-locked environment files, experiment outputs, evaluation logs
  • Access: Permanent, citable repository

GitHub Repository:

  • URL: github.com/gyrogovernance/science
  • Contents: Source code, scripts, documentation
  • Status: Public, actively maintained

GyroDiagnostics:

  • URL: github.com/gyrogovernance/gyrodiagnostics
  • Contents: AI evaluation protocols, challenge sets, scoring rubrics
  • Status: Public toolkit for independent replication

Key Scripts and Outputs

1. Modal Axiom Verification

Script: cgm_axiomatization_analysis.py

Purpose: Verifies the foundational constraints using Z3 SMT solver.

What it does:

  • Constructs Kripke frames for each constraint
  • Checks consistency (all five simultaneously satisfiable)
  • Verifies independence (each admits falsifying countermodels)
  • Tests completeness via Sahlqvist correspondence

Output:

Consistency: VERIFIED (3-world frame satisfies all)
Independence CS: VERIFIED (countermodel exists)
Independence UNA: VERIFIED (countermodel exists)
Independence ONA: VERIFIED (countermodel exists)
Independence BU-Egress: VERIFIED (countermodel exists)
Independence BU-Ingress: VERIFIED (countermodel exists)

2. Hilbert Space Construction

Script: cgm_Hilbert_Space_analysis.py

Purpose: Constructs GNS representation and checks depth-four balance.

What it does:

  • Builds L²(S²) Hilbert space with horizon normalization Q_G = 4π
  • Defines generators X, Y as skew-adjoint operators
  • Computes U_L(t) = exp(itX), U_R(t) = exp(itY)
  • Evaluates depth-four commutator ||U_L U_R U_L U_R - U_R U_L U_R U_L||
  • Checks S-sector projection ||P_S[X,Y]P_S||

Output:

Q_G (horizon normalization): 4π = 12.566370614...
λ (generator norm): √(2π) = 2.506628274...
m_a (aperture scale): 1/(2√(2π)) = 0.199471140...
Depth-4 balance: ||Δ|| = 7.3e-14 ✓
S-sector commutator: ||P_S[X,Y]P_S|| = 7.9e-19 ✓
CS chirality: ⟨Ω|U_R|Ω⟩ = 1.0, ⟨Ω|U_L|Ω⟩ = e^(is_p) ✓

3. Dimensional Necessity

Script: cgm_3D_6DoF_analysis.py

Purpose: Confirms dimensional necessity and DOF progression.

What it does:

  • Tests 2D fibered representation (should fail BU-Egress)
  • Verifies 3D su(2) closure
  • Checks 4D so(4) violates simplicity
  • Tracks DOF progression CS(1) → UNA(3) → ONA(6) → BU(6)

Output:

Testing n=2 fibered:
  BU-Egress uniform validity: FAILED ✗
  (g(φ,t) forced independent of φ, violates CS)

Testing n=3 su(2):
  BCH closure: VERIFIED ✓
  [X,[X,Y]] = aY: VERIFIED (a = 1.000 ± 1e-14)
  [Y,[X,Y]] = -aX: VERIFIED ✓
  DOF progression: 1 → 3 → 6 → 6(closed) ✓

Testing n=4 so(4):
  Simplicity: FAILED ✗
  (Decomposition su(2) ⊕ su(2) splits cyclic vector)

Conclusion: n=3 UNIQUE ✓

4. BU Monodromy Calculation

Script: experiments/tw_closure_test.py

Purpose: Computes BU monodromy defect δ_BU from first principles.

What it does:

  • Constructs canonical stage operators in Pauli basis
  • Computes dual-pole loop U_⊚ = U_ONA U_BU^(+) U_ONA^(-1) U_BU^(-)
  • Extracts rotation angle δ_BU from matrix entries
  • Verifies δ_BU/m_a ratio

Output:

Canonical operators:
  U_UNA = exp(π/4 · J_x)
  U_ONA = exp(π/4 · J_y)
  U_BU^(±) = exp(±m_a · J_z), m_a = 0.199471140

Dual-pole loop:
  cos(δ_BU/2) = 0.9952361763
  sin(δ_BU/2) = 0.0974894411

BU monodromy defect:
  δ_BU = 0.195342176580 rad
  δ_BU ≈ 11.19° ≈ π/16 (within 0.5%)

Aperture ratio:
  δ_BU/m_a = 0.9793
  Aperture = 1 - 0.9793 = 0.0207 = 2.07%

Fine-structure prediction:
  α = δ_BU⁴/m_a = 0.007297352563
  Experimental: 0.0072973525693(11)
  Agreement: 9 significant digits ✓

5. GyroDiagnostics Evaluation

Repository: github.com/gyrogovernance/gyrodiagnostics

What it provides:

  • Challenge sets for AI alignment evaluation
  • Tetrahedral Hodge decomposition implementation
  • Aperture calculation tools
  • Pathology detection algorithms
  • Scoring rubrics with confidence weighting

Sample output (Claude Sonnet 4.5, Challenge 3):

Tetrahedral Metrics:
  Traceability (T): 0.73 ± 0.08
  Variety (V): 0.81 ± 0.06
  Accountability (Acc): 0.67 ± 0.11
  Integrity (B): 0.54 ± 0.14

Aperture Analysis:
  Gradient energy: ||y_grad||² = 6.82
  Cycle energy: ||y_cycle||² = 1.31
  Aperture: A = 1.31/8.13 = 0.161

Target comparison:
  A* (CGM) = 0.021
  A (observed) = 0.161
  Deviation = 7.7× target

Pathology screening:
  Deceptive coherence: DETECTED (SI=38.2, A>0.05)
  Semantic drift: DETECTED (turn-to-turn coherence drop 0.73→0.51)

Superintelligence Index: SI = 42.1

Environment requirements:

  • Python 3.10 or higher
  • NumPy (numerical arrays)
  • SciPy (linear algebra, special functions)
  • Z3 (SMT solver for modal verification)

All dependencies are version-locked in the repository.

Independent verification:

Anyone can:

  1. Clone the repository
  2. Install dependencies (one-line command)
  3. Run the scripts
  4. Compare outputs to published values

The numerical tolerances (< 10^(-13)) leave no room for significant variation. Either the calculations reproduce exactly or something is wrong.

Future Directions

Immediate Extensions

1. Expanded GyroDiagnostics Sampling

  • Increase from 3 to 50+ model families
  • Scale from 15 to 100+ runs per model
  • Add cross-laboratory replication
  • Pre-register protocols before data collection
  • Public release of all transcripts and scoring

2. Higher-Order Corrections to α

  • Calculate O(δ_BU^6) correction
  • Verify negative sign (dual-pole symmetry)
  • Compute RG flow from geometric transport
  • Compare to QED perturbative series

3. Dynamical Field Equations

  • Embed BU U(1) connection in Maxwell-type action
  • Derive field equations compatible with SE(3) symmetry
  • Check consistency with QED Lagrangian
  • Extend to non-abelian gauge groups

4. Categorical Uniqueness Proof

  • Replace computational enumeration with categorical argument
  • Use Tannakian reconstruction to prove GNS data determines representation
  • Establish uniqueness of L²(S²) up to isomorphism

Longer-Term Possibilities

5. Quantum Computing Applications

Where coherent observation is central:

  • Error correction codes from depth-four balance
  • Decoherence bounds from aperture ratio
  • Entanglement structure from gyrogroup geometry

6. Cosmological Structure Formation

Where measurement horizons play essential roles:

  • Cosmic microwave background anisotropies from Q_G = 4π
  • Large-scale structure from SE(3) symmetry breaking
  • Inflation from depth-four closure dynamics

7. Neutrino Mass Calculation

  • Use geometric ratios suggested in paper
  • Validate against experimental mass differences
  • Connect to seesaw mechanism via BU structure

8. Practical Alignment Tools

  • Real-time aperture monitoring for deployed systems
  • Automated pathology detection
  • Training objectives targeting A → A*

Open Questions

Conceptual:

  • Why does operational coherence force these specific constraints?
  • Is there a category-theoretic "why" behind the dimensional necessity?
  • What is the physical meaning of the near-dyadic structure δ_BU ≈ π/16?

Mathematical:

  • Can the two-layer structure (modal + operational) be unified?
  • Is there a natural categorical semantics for bimodal operational logic?
  • What is the moduli space of representations satisfying the constraints?

Physical:

  • How do radiative corrections modify the geometric α calculation?
  • Can the framework accommodate non-abelian gauge groups?
  • What is the connection to effective field theory?

Informational:

  • Do biological neural systems exhibit the 2.07% aperture signature?
  • Is there a thermodynamic interpretation of the aperture?
  • Can the framework guide architecture design for aligned AI?

Technical Appendices

Appendix A: The Full Dimensional Proof

Theorem A.1 (Three-Dimensional Necessity):

Requiring CS traceability for reachability, BU-Egress for uniform continuous closure, and BU-Ingress for simple Lie reconstruction, the five foundational constraints characterize n = 3 as the only dimensional structure satisfying coherent measurement requirements within the stated operational regime.

Proof structure:

The proof proceeds through:

  1. BCH Depth-4 Closure (Lemma A.0): Working in the completed free Lie algebra, depth-four balance forces span{X,Y,[X,Y]} ≅ sl(2).

  2. Rotational DOF (Lemma A.1): UNA with operational requirements selects su(2) (three generators).

  3. Translational DOF (Lemma A.2): ONA forces semidirect product structure SU(2) ⋉ ℝ³.

  4. Exclusions:

    • n = 2: Fibered case violates uniform BU-Egress (Lemma A.1.2)
    • n = 4: so(4) ≅ su(2) ⊕ su(2) violates simplicity
    • n ≥ 5: Excess generators violate minimality

  5. Existence: L²(S²) model explicitly satisfies all constraints.

Lemma A.0 (BCH Depth-4 Closure):

Working in L̂(X,Y) with formal exponentials, BU-Egress at S-worlds requires:

s · exp(X)exp(Y)exp(X)exp(Y) · s = s · exp(Y)exp(X)exp(Y)exp(X) · s

where s is the central idempotent encoding S.

Define Z₁ = BCH(X,Y), Z₂ = BCH(Y,X). Using exp(Z)exp(Z) = exp(2Z):

Δ = 2(Z₁ - Z₂) = 2(BCH(X,Y) - BCH(Y,X))

Condition s·Δ·s = 0 with [X,Y] ≠ 0 globally forces:

[X,[X,Y]] = aY
[Y,[X,Y]] = -aX

for a ∈ ℝ, a ≠ 0. Hall word analysis shows no bracket length m ≥ 3 contributes, yielding span{X,Y,[X,Y]} ≅ sl(2).

Lemma A.1 (Rotational DOF):

Under unitarity, BCH constraints, and simplicity, the algebra is su(2) with three generators. Compact form selected by GNS inner product.

Lemma A.2 (Translational DOF):

ONA activates bi-gyrogroup requiring G ≅ SU(2) ⋉ ℝⁿ. Minimality forces n = 3, yielding SE(3) with 6 DOF.

Corollary A.1 (DOF Progression):

CS: 1 DOF → UNA: 3 DOF → ONA: 6 DOF → BU: 6 DOF (closed)

Uniqueness follows from constraints and δ = 0 gyrotriangle condition.

Appendix B: Hilbert Space Construction

(Provided in full in the paper; summarized here for completeness)

The GNS construction proceeds from the free -algebra A generated by unitaries u_L, u_R with u_k u_k = I.

Define positive linear functional ω: A → ℂ fixed by modal constraints:

  • ω(I) = 1, ω(u_R) = 1, ω(u_L) = e^(is_p)
  • Depth-two and depth-four conditions encoded in ω

Complete A/N with ⟨[a],[b]⟩ = ω(a*b) to obtain H_ω.

Concrete realization on L²(S²,dΩ):

  • P_S projects onto l=0 ⊕ l=1 spherical harmonics
  • U_L = e^(itX), U_R = e^(itY) with X, Y skew-adjoint
  • Normalization: ⟨Ω|Ω⟩ = Q_G = 4π

Verification achieves ||errors|| < 10^(-13).

Appendix C: Numerical Verification Details

(Detailed numerical results from the computational scripts)

All numerical checks use adaptive precision with error bounds:

Depth-four balance:

||P_S(U_L U_R U_L U_R - U_R U_L U_R U_L)|Ω⟩|| < 10^(-13)

achieved across parameter grid t ∈ [-0.1, 0.1] × 100 points.

S-sector commutator:

||P_S[X,Y]P_S||_L² = 7.9 × 10^(-19)

BCH coefficient matching:

|[X,[X,Y]] - aY| < 10^(-14)
|[Y,[X,Y]] + aX| < 10^(-14)

with a = 1.0000000000000 ± 10^(-13).

These tolerances are machine precision limited, not model precision limited.

References and Further Reading

Foundational Mathematics:

  • Chellas, B. F. (1980). Modal Logic: An Introduction. Cambridge University Press.
  • Gelfand, I., & Naimark, M. (1943). On the imbedding of normed rings into the ring of operators in Hilbert space. Matematicheskii Sbornik, 12(2), 197-213.
  • Hall, B. C. (2015). Lie Groups, Lie Algebras, and Representations (2nd ed.). Springer.

Gyrogroup Theory:

  • Ungar, A. A. (2001). Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession. Kluwer Academic.
  • Ungar, A. A. (2008). Analytic Hyperbolic Geometry and Albert Einstein's Special Theory of Relativity. World Scientific.

Physics:

  • Morel, L., et al. (2020). Determination of the fine-structure constant with an accuracy of 81 parts per trillion. Nature, 588, 61-65.
  • Rovelli, C. (2004). Quantum Gravity. Cambridge University Press.
  • Polchinski, J. (1998). String Theory (Vols. 1-2). Cambridge University Press.

AI and Cognitive Science:

  • Noroozizadeh, S., et al. (2025). Deep sequence models tend to memorize geometrically; it is unclear why. arXiv:2510.26745.
  • Ouyang, L., et al. (2022). Training language models to follow instructions with human feedback. NeurIPS, 35, 27730-27744.
  • Bai, Y., et al. (2022). Constitutional AI: Harmlessness from AI feedback. arXiv:2212.08073.

Governance:

  • Beer, S. (1994). Beyond Dispute: The Invention of Team Syntegrity. John Wiley and Sons.

CGM-Specific Resources:

  • Korompilias, B. (2025). Common Governance Model: Mathematical Physics Framework. Zenodo. DOI: 10.5281/zenodo.17521384
  • Repository: github.com/gyrogovernance/science
  • GyroDiagnostics: github.com/gyrogovernance/gyrodiagnostics

Historical:

  • Hilbert, D. (1900). Mathematical problems. Bulletin of the American Mathematical Society, 8(10), 437-479.

Document prepared for physicists and engineers seeking comprehensive technical understanding of the Common Governance Model. For questions, clarifications, or collaboration inquiries, see repository documentation.

Explore more work from our labs and ongoing publications.

Labs

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Mathematical Physics Science

Gyroscopic Alignment Research Lab

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Artificial Superintelligence Architecture (ASI/AGI)

Gyroscopic Alignment Models Lab

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AI Safety Diagnostics

Gyroscopic Alignment Evaluation Lab

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AI Quality Governance

Gyroscopic Alignment Behaviour Lab

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Resources

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