Uniqueness Theorem of Holonomic Representation

Status: [Т] — all steps proven

The uniqueness theorem of holonomic representation is a theorem [Т], relying exclusively on previously proven results:

• Primitivity of $\mathcal{L}_\Omega$ [Т] (proof)
• Full minimality 7/7 [Т] (proof)
• Bridge T15 [Т]: (AP)+(PH)+(QG)+(V) $\Rightarrow$ P1+P2 $\Rightarrow$ $\mathbb{O}$ $\Rightarrow$ $G_2$ (proof)
• L-unification [Т] (proof)
• Uniqueness of E, O, U [Т] (proof)

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Problem statement

The problem of the map G

The central task of operationalizing UHM is the map G:

$$G: \mathrm{States}(S) \to \mathcal{D}(\mathbb{C}^7)$$

which assigns to a physical system $S$ satisfying (AP)+(PH)+(QG)+(V) its coherence matrix $\Gamma \in \mathcal{D}(\mathbb{C}^7)$.

In the ontology of UHM, $\Gamma$ is a primary object: the system is its coherence matrix. The problem of G is not "how to compute $\Gamma$ from something more fundamental," but "is the identification of $\Gamma$ for a given system unique?"

Analogy with Stone–von Neumann

	Quantum mechanics	UHM
Primitive	Canonical commutation relations $[\hat{x}, \hat{p}] = i\hbar$	Primitive $\mathfrak{T} = (\mathbf{Sh}_\infty(\mathcal{C}), J_{Bures}, \omega_0)$
Representation	Realization of $\hat{x}, \hat{p}$ on $\mathcal{H}$	Holonomic representation $G: \mathrm{States}(S) \to \mathcal{D}(\mathbb{C}^7)$
Uniqueness theorem	Stone–von Neumann: representation is unique up to $U(\mathcal{H})$	This theorem: representation is unique up to $G_2$
Gauge group	$U(\mathcal{H})$ (infinite-dimensional)	$G_2 = \mathrm{Aut}(\mathbb{O})$ (14-dimensional)
Physical parameters	Infinitely many (quantum numbers)	34 = 48 $-$ 14 (gauge-invariant)

The key distinction: in QM the gauge group is infinite-dimensional ($U(\mathcal{H})$), leaving enormous freedom. In UHM the gauge group is finite-dimensional $G_2$, which radically restricts this freedom and increases the predictive power of the theory.

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Definitions

Definition G1 (Holonomic representation)

A holonomic representation of a system $S$ satisfying (AP)+(PH)+(QG)+(V) is a triple $(\mathbb{C}^7, \mathcal{B}, G_S)$, where:

• $\mathbb{C}^7$ — Hilbert space of the holon
• $\mathcal{B} = \{|A\rangle, |S\rangle, |D\rangle, |L\rangle, |E\rangle, |O\rangle, |U\rangle\}$ — ordered orthonormal basis with functional labeling (7 dimensions)
• $G_S: \mathrm{States}(S) \to \mathcal{D}(\mathbb{C}^7)$ — map compatible with UHM dynamics

Compatibility condition (covariance): For any physical trajectory $s(\tau)$ of system $S$:

$$\frac{d}{d\tau} G_S(s(\tau)) = \mathcal{L}_\Omega[G_S(s(\tau))]$$

where $\mathcal{L}_\Omega$ is the logical Liouvillian defined by axioms A1–A5 in basis $\mathcal{B}$.

Definition G2 (Equivalence of representations)

Two holonomic representations $(\mathbb{C}^7, \mathcal{B}_1, G_1)$ and $(\mathbb{C}^7, \mathcal{B}_2, G_2)$ are equivalent if there exists $U \in U(7)$ such that:

$$G_2(s) = U \, G_1(s) \, U^\dagger \quad \forall \, s \in \mathrm{States}(S)$$

and $\mathcal{B}_2 = U \cdot \mathcal{B}_1$ (basis transformation).

Definition G3 (Gauge group)

The gauge group is the maximal subgroup $\mathcal{G} \subseteq U(7)$ whose elements generate equivalent representations, preserving all structures defined by axioms A1–A5.

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Preliminary results

All results below have status [Т] and are proven in the respective documents.

P1. Primitivity of $\mathcal{L}_0$ (linear part) [Т]

The linear part of the Liouvillian $\mathcal{L}_0$ is primitive (T-39a): there exists a unique stationary state $I/7 \in \mathcal{D}(\mathbb{C}^7)$ for $\mathcal{L}_0$, and for any initial $\rho_0$:

$$\lim_{\tau \to \infty} e^{\tau\mathcal{L}_0}[\rho_0] = I/7$$

The full nonlinear operator $\mathcal{L}_\Omega = \mathcal{L}_0 + \mathcal{R}$ has a unique non-trivial attractor $\rho_* \neq I/7$ with $P > 1/7$ (T-96 [Т]).

Spectrum of $\mathcal{L}_\Omega$ on the space $\mathrm{Herm}_0(\mathbb{C}^7)$ (traceless Hermitian matrices, $\dim_\mathbb{R} = 48$):

$$\mathrm{Spec}(\mathcal{L}_\Omega) = \{0\} \cup \{\lambda_k : \mathrm{Re}(\lambda_k) < 0, \; k = 1, \ldots, 47\}$$

P2. Functional uniqueness of dimensions [Т]

All 7 dimensions are functionally unique:

• Each dimension performs an irreducible function (F1–F7)
• E is unique [Т]: (PH) + $\kappa_0$ (requires $\mathrm{Hom}(O,E)$) + rank greater than 1
• O is unique [Т]: $\mathcal{R}$ [Т] + $\kappa_0$ ($\mathrm{End}(O)$, $\mathrm{Hom}(O,E)$, $\mathrm{Hom}(O,U)$) + PW (A5) + functional independence
• E $\perp$ O [Т]: causal + categorical (O = E degenerates $\kappa_0$)

P3. Bridge T15 [Т]

Full chain (AP)+(PH)+(QG)+(V) $\Rightarrow$ P1+P2 of 12 steps, all [Т]:

$$\mathrm{(AP)+(PH)+(QG)+(V)} \xrightarrow{[\text{Т}]} \mathrm{BIBD}(7,3,1) \xrightarrow{[\text{Т}]} \mathrm{PG}(2,2) \xrightarrow{[\text{Т}]} \mathbb{O} \xrightarrow{[\text{Т}]} G_2$$

P4. L-unification [Т]

Lindblad operators are derived from the classifier $\Omega$:

$$L_k = |k\rangle\langle k|, \quad k \in \{A, S, D, L, E, O, U\}$$

Fano operators are defined by the 7 lines of PG(2,2):

$$L_p^{\mathrm{Fano}} = \frac{1}{\sqrt{3}} \Pi_p, \quad \Pi_p = \sum_{i \in \mathrm{line}_p} |i\rangle\langle i|, \quad p = 1, \ldots, 7$$

P5. $G_2$-covariance [Т]

The Fano dissipator is $G_2$-covariant:

$$\forall \, g \in G_2: \quad \mathcal{D}_{\mathrm{Fano}}[g\Gamma g^\dagger] = g \, \mathcal{D}_{\mathrm{Fano}}[\Gamma] \, g^\dagger$$

The atomic dissipator is not $G_2$-covariant [Т], but is $S_7$-equivariant [Т].

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New lemmas

Lemma G1: Spectral injectivity of propagator [Т]

Lemma G1 (Spectral injectivity) [Т]

For any $\tau > 0$ the map $e^{\tau \mathcal{L}_{\mathrm{lin}}}$ is injective on $\mathrm{Herm}_0(\mathbb{C}^7)$, where $\mathcal{L}_{\mathrm{lin}} = -i[H_{\mathrm{eff}}, \cdot] + \mathcal{D}_\Omega$ is the linear part of the Liouvillian.

Proof.

Let $\mathcal{L}_{\mathrm{lin}}$ act on $V = \mathrm{Herm}_0(\mathbb{C}^7)$ ($\dim_\mathbb{R} V = 48$). By primitivity [Т] (§P1):

$$\mathrm{Spec}(\mathcal{L}_{\mathrm{lin}}\big|_V) = \{\lambda_1, \ldots, \lambda_{48}\}, \quad \mathrm{Re}(\lambda_k) < 0 \; \forall k$$

(the zero eigenvalue corresponds to the invariant component $\rho_*$, factored out into the complement of $V$).

For the propagator $e^{\tau \mathcal{L}_{\mathrm{lin}}}$ the eigenvalues are: $\{e^{\tau\lambda_k}\}_{k=1}^{48}$. Since $\mathrm{Re}(\lambda_k) < 0$:

$$|e^{\tau\lambda_k}| = e^{\tau\mathrm{Re}(\lambda_k)} \in (0, 1) \quad \forall \tau > 0$$

All eigenvalues of the propagator are nonzero, therefore $e^{\tau \mathcal{L}_{\mathrm{lin}}}$ is non-degenerate on $V$, i.e. injective. $\blacksquare$

Corollary G1.1 (Recoverability of initial state): Knowing $\Gamma(\tau)$ for some $\tau > 0$ and the parameters of $\mathcal{L}_{\mathrm{lin}}$, the initial state $\Gamma(0)$ is determined uniquely.

Lemma G2: Well-posedness of nonlinear inverse problem [Т]

Lemma G2 (Nonlinear inverse problem) [Т]

The full evolution equation $\frac{d\Gamma}{d\tau} = f(\Gamma)$, including the nonlinear regenerative term $\mathcal{R}$, has uniqueness of solutions: for any $\Gamma_1(0) \neq \Gamma_2(0)$ the trajectories $\Gamma_1(\tau) \neq \Gamma_2(\tau)$ for all $\tau \geq 0$.

Proof.

The right-hand side $f(\Gamma) = -i[H_{\mathrm{eff}}, \Gamma] + \mathcal{D}_\Omega[\Gamma] + \kappa(\Gamma)(\rho_* - \Gamma) \cdot g_V(P)$, where:

(a) Lipschitz continuity. The linear terms ($-i[H_{\mathrm{eff}}, \cdot]$, $\mathcal{D}_\Omega$) are Lipschitz (linear operators on a finite-dimensional space). The nonlinear term:

• $\kappa(\Gamma) = \kappa_{\mathrm{bootstrap}} + \kappa_0 \cdot \mathrm{Coh}_E(\Gamma)$, where $\mathrm{Coh}_E(\Gamma) = \|\pi_E(\Gamma)\|_{\mathrm{HS}}^2 / \|\Gamma\|_{\mathrm{HS}}^2$ is a rational function of matrix elements [Т]
• $\|\Gamma\|_{\mathrm{HS}}^2 = \mathrm{Tr}(\Gamma^2) \geq 1/7 > 0$ on $\mathcal{D}(\mathbb{C}^7)$ — the denominator is bounded away from zero
• The product $\kappa(\Gamma) \cdot (\rho_* - \Gamma)$ is a smooth function on the compact set $\mathcal{D}(\mathbb{C}^7)$, hence locally Lipschitz

(b) Picard–Lindelöf theorem. On the compact set $\mathcal{D}(\mathbb{C}^7)$ local Lipschitz continuity guarantees existence and uniqueness of the solution to the Cauchy problem for any initial condition $\Gamma(0) \in \mathcal{D}(\mathbb{C}^7)$.

(c) Injectivity of flow. From uniqueness of the Cauchy problem: if $\Gamma_1(0) \neq \Gamma_2(0)$, then $\Gamma_1(\tau) \neq \Gamma_2(\tau)$ for all $\tau$ in the domain of existence (trajectories do not intersect in phase space — a standard result of ODE theory). $\blacksquare$

Lemma G3: Axiomatic definiteness of structures [Т]

Lemma G3 (Axiomatic definiteness) [Т]

Axioms A1–A5 uniquely determine (in the given basis $\mathcal{B}$) the following structures:

(i) Atomic projectors $\{|k\rangle\langle k|\}_{k=0}^{6}$ (from L-unification [Т])

(ii) The system of Fano lines $\{\mathrm{line}_p\}_{p=1}^{7} \subset \binom{[7]}{3}$ (from bridge T15 [Т])

(iii) E-projection $\pi_E(\Gamma) = P_E\Gamma + \Gamma P_E - P_E\Gamma P_E$ (from Coh_E [Т])

(iv) Page–Wootters tensor decomposition $\mathcal{H}_O \otimes \mathcal{H}_{\mathrm{rest}}$, singling out O (from A5)

(v) The regeneration formula $\kappa_0 = \omega_0 \cdot |\gamma_{OE}| \cdot |\gamma_{OU}| / \gamma_{OO}$, singling out $\{O, E, U\}$ (from categorical derivation of κ₀ [Т])

Proof. Each structure is derived from the axioms:

• (i): L-unification [Т] — atoms $S_k = |k\rangle\langle k|$ of classifier $\Omega$
• (ii): Bridge T15 [Т] — uniqueness of BIBD$(7,3,1)$ $\cong$ PG(2,2) (Hall 1967)
• (iii): HS-projection theorem [Т] — orthogonal projection in Hilbert–Schmidt space
• (iv): Axiom A5 (Page–Wootters) — explicit postulate
• (v): Adjunction $\mathcal{D}_\Omega \dashv \mathcal{R}$ [Т] — formula for $\kappa_0$ from categorical derivation. $\blacksquare$

Lemma G4: Gauge group = $G_2$ [Т]

Lemma G4 (Maximal gauge group) [Т]

The maximal subgroup $\mathcal{G} \subseteq U(7)$ whose elements preserve all five structures of Lemma G3 is $G_2 = \mathrm{Aut}(\mathbb{O})$.

Proof. We proceed in two steps: (A) $G_2 \subseteq \mathcal{G}$ and (B) $\mathcal{G} \subseteq G_2$.

(A) $G_2 \subseteq \mathcal{G}$: every $g \in G_2$ preserves all structures.

By definition, $G_2 = \mathrm{Aut}(\mathbb{O})$ acts on $\mathrm{Im}(\mathbb{O}) \cong \mathbb{R}^7 \subset \mathbb{C}^7$ and preserves:

• Octonionic multiplication $e_i \cdot e_j = \varepsilon_{ijk} e_k$
• The multiplication table $\cong$ Fano plane PG(2,2)

Therefore, $g \in G_2$ preserves:

• (ii) Fano lines: $g$ permutes the 7 lines (as an automorphism of PG(2,2))
• (i) The set of atomic projectors: $g|k\rangle\langle k|g^\dagger$ are projectors onto a rotated basis, but the set $\{g\Pi_p g^\dagger\} = \{\Pi_{\sigma_g(p)}\}$ is preserved, and the atomic projectors are recovered from intersections: $|k\rangle\langle k| = \Pi_p \Pi_q$ for two lines through point $k$

For (iii), (iv), (v): $G_2 \subset SO(7)$ preserves the metric, and E, O, U transform within the $G_2$-orbit. Structures (iii)–(v) are formulated in terms of Fano lines and are covariant by construction. $\checkmark$

(B) $\mathcal{G} \subseteq G_2$: any $U \in \mathcal{G}$ belongs to $G_2$.

Let $U \in U(7)$ preserve all five structures of Lemma G3.

Step B1. From preservation of (ii) (Fano lines): $U$ induces an automorphism of the Fano plane PG(2,2). Since PG(2,2) is isomorphic to the multiplication table of $\mathrm{Im}(\mathbb{O})$ [Т], $U$ induces an automorphism of octonionic multiplication.

Step B2. Restrict $U$ to $\mathrm{Im}(\mathbb{O}) \cong \mathbb{R}^7$. An automorphism of octonionic multiplication on $\mathrm{Im}(\mathbb{O})$ by definition belongs to $G_2 = \mathrm{Aut}(\mathbb{O})$.

Transition from combinatorial automorphisms to continuous ones

$U \in U(7)$ preserves Fano lines as subspaces (not just as index sets). Each Fano line defines a 3-dimensional subspace, and preservation of all 7 such subspaces is equivalent to preservation of the octonionic cross-product (3-form $\varphi_3 = \sum f_{ijk}\, e^i \wedge e^j \wedge e^k$). By definition $G_2 = \{g \in \mathrm{GL}(7,\mathbb{R}) : g^*\varphi_3 = \varphi_3\}$, which proves $U \in G_2$.

Clarification: PSL(2,7) vs G₂

The group of combinatorial automorphisms of PG(2,2) is finite: $\mathrm{Aut}(\mathrm{PG}(2,2)) \cong \mathrm{PSL}(2,7)$, $|\mathrm{PSL}(2,7)| = 168$. The group $G_2 = \mathrm{Aut}(\mathbb{O})$ is a compact Lie group, $\dim G_2 = 14$. Relation: $\mathrm{PSL}(2,7) \subset G_2$ as a finite subgroup — every permutation of 7 points compatible with PG(2,2) extends to a continuous automorphism of $\mathbb{O}$. Step B1 shows that $U$ preserves the structure constants $f_{ijk}$ (not just combinatorics), and step B2 uses the definition of $G_2$ as the group preserving these constants.

Step B3. Since $G_2 \subset SO(7) \subset U(7)$ and $U$ preserves the Hermitian structure (as an element of $U(7)$), the restriction $U\big|_{\mathrm{Im}(\mathbb{O})}$ determines $U$ completely (since $\mathrm{Im}(\mathbb{O})$ is a real form of $\mathbb{C}^7$, and $U$ preserves the real structure via preservation of PG(2,2)).

Therefore, $U \in G_2$. $\blacksquare$

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Main theorem

Theorem (G₂-rigidity of holonomic representation) [Т]

Theorem of G₂-rigidity [Т]

Let $S$ be an autonomous system satisfying (AP)+(PH)+(QG)+(V). Let $(\mathbb{C}^7, \mathcal{B}_1, G_1)$ and $(\mathbb{C}^7, \mathcal{B}_2, G_2)$ be two holonomic representations of $S$ (Definition G1).

Then there exists a unique $U \in G_2 = \mathrm{Aut}(\mathbb{O})$ such that:

$$\boxed{G_2(s) = U \, G_1(s) \, U^\dagger \quad \forall \, s \in \mathrm{States}(S)}$$

Equivalently: the holonomic representation is unique up to gauge group $G_2$.

Proof

Step 1: Definiteness of dynamics in each representation.

In representation $(\mathbb{C}^7, \mathcal{B}_i, G_i)$ axioms A1–A5 determine the Liouvillian $\mathcal{L}_\Omega^{(i)}$ (Lemma G3 [Т]). The compatibility condition (Definition G1) guarantees:

$$\frac{d}{d\tau} G_i(s(\tau)) = \mathcal{L}_\Omega^{(i)}[G_i(s(\tau))], \quad i = 1, 2$$

Step 2: Construction of intertwiner $\Phi$.

Define $\Phi: \mathcal{D}(\mathbb{C}^7) \to \mathcal{D}(\mathbb{C}^7)$ as:

$$\Phi := G_2 \circ G_1^{-1}$$

(the inverse $G_1^{-1}$ exists on the image $G_1(\mathrm{States}(S))$). From the compatibility conditions:

$$\frac{d}{d\tau} \Phi(\Gamma(\tau)) = \mathcal{L}_\Omega^{(2)}[\Phi(\Gamma(\tau))], \quad \text{where} \quad \frac{d}{d\tau}\Gamma(\tau) = \mathcal{L}_\Omega^{(1)}[\Gamma(\tau)]$$

Step 3: $\Phi$ is conjugation by a unitary operator.

Both representations describe the same physical system and generate the same observables. The spectrum of $\Gamma$ (set of eigenvalues) is invariant: $\mathrm{Spec}(\Phi(\Gamma)) = \mathrm{Spec}(\Gamma)$ for all $\Gamma$ (since physical observables — purity $P = \mathrm{Tr}(\Gamma^2)$, von Neumann entropy, Coh$_E$, etc. — are functions of the spectrum and certain structural elements, and must coincide).

A spectrum-preserving map on $\mathcal{D}(\mathbb{C}^7)$ is conjugation by a unitary (or antiunitary) operator — this is Wigner's theorem (Wigner 1931) in the form of Kadison (Kadison 1965):

$$\Phi(\Gamma) = U \Gamma U^\dagger \quad \text{for some } U \in U(7)$$

(the antiunitary case is excluded since $\Phi$ is continuously connected to the identity map through a continuous family of systems).

Extension of Φ to all D(ℂ⁷)

By the viability condition (V), the trajectories of the holon pass through an open neighborhood of the attractor $\rho^*$ (T-125 [Т]). Therefore $\mathrm{Im}(G_1)$ contains an open subset of $\mathrm{Int}(\mathcal{D}(\mathbb{C}^7))$. An affine map defined on an open subset of a complete metric space extends uniquely to the whole space (Tietze theorem). The extended $\Phi$ preserves the spectrum on all of $\mathcal{D}(\mathbb{C}^7)$.

Clarification: Wigner vs. Uhlmann

Here Wigner's theorem (in Kadison's form) is applied: an affine bijection $\Phi$ on the state space $\mathcal{D}(\mathbb{C}^7)$ that preserves the spectrum (and hence fidelity $F(\rho, \sigma) = \mathrm{Tr}\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}$) is realized by unitary or antiunitary conjugation. This is the correct reference for this step, since $\Phi$ is a bijection on the state space, not a CPTP channel. For CPTP channels (which are in general not bijections) preservation of fidelity is characterized by Uhlmann's theorem (Uhlmann 1976): $F(\mathcal{E}[\rho], \mathcal{E}[\sigma]) \leq F(\rho, \sigma)$ for any CPTP $\mathcal{E}$, with equality if and only if $\mathcal{E}$ is a unitary channel on the support of $\rho$ and $\sigma$. In the context of the monotonicity of Freedom (Theorem Properties of Freedom in consequences.md), it is precisely Uhlmann's contractivity that justifies the non-increase of freedom under CPTP evolution.

Step 4: $U \in G_2$.

Since both representations satisfy axioms A1–A5, the unitary operator $U$ must preserve all structures defined by the axioms (Lemma G3 [Т]): atomic projectors, Fano lines, E-projection, PW-decomposition, formula $\kappa_0$.

By Lemma G4 [Т]: $U \in G_2$. $\blacksquare$

Step 5: Uniqueness of $U$.

Suppose $U_1, U_2 \in G_2$ both satisfy $G_2 = \mathrm{Ad}_{U_i} \circ G_1$. Then $\mathrm{Ad}_{U_1^{-1}U_2} = \mathrm{Id}$ on the image of $G_1$. If the image of $G_1$ contains sufficiently many states (which is guaranteed by viability: the system passes through a neighborhood of $\rho_*$ by primitivity [Т], and this neighborhood is open in $\mathcal{D}(\mathbb{C}^7)$), then $U_1^{-1}U_2 = e^{i\theta} I$ — a scalar phase, acting trivially on $\mathcal{D}(\mathbb{C}^7)$. $\blacksquare$

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Corollaries

Corollary 1: Space of physical states [Т]

Corollary 1 (Space of observables) [Т]

The space of physically distinguishable states of the holon:

$$\mathcal{D}_{\mathrm{phys}} = \mathcal{D}(\mathbb{C}^7) / G_2$$

has dimension:

$$\dim_\mathbb{R}(\mathcal{D}_{\mathrm{phys}}) = 48 - 14 = 34$$

where $48 = N^2 - 1 = \dim(\mathrm{su}(7))$ is the full number of parameters of $\Gamma$, and $14 = \dim(G_2)$ is the number of gauge degrees of freedom.

Proof. For generic $\Gamma$ (with distinct eigenvalues) the stabilizer $\mathrm{Stab}_{G_2}(\Gamma)$ is trivial (finite group). Then by the orbit theorem: $\dim(\mathrm{Orb}(\Gamma)) = \dim(G_2) = 14$, and $\dim(\mathcal{D}_{\mathrm{phys}}) = 48 - 14 = 34$. $\blacksquare$

Consistency

The value 34 coincides with the number of parameters under full $G_2$-gauge fixing in the pure Fano-observation regime ($\alpha = 0$), stated in Lindblad operators: "$48 \to 34$ parameters."

Corollary 2: Well-posedness of inverse problem [Т]

Corollary 2 (Inverse problem) [Т]

For a system $S$ satisfying (AP)+(PH)+(QG)+(V), the initial state $\Gamma(0)$ is uniquely recovered from:

(a) The observed trajectory $\Gamma(\tau)$ for $\tau \in (0, T]$ (Lemmas G1, G2 [Т])

(b) The system parameters $(\omega_0, \lambda_m)$

up to $G_2$-gauge (Theorem of G₂-rigidity [Т]).

Corollary 3: Faithfulness of functor F [Т]

Corollary 3 (Faithfulness of functor) [Т]

The functor $F: \mathbf{DensityMat} \to \mathbf{Exp}$ (categorical formalism) is faithful on $G_2$-orbits: if $F(\Gamma_1) \cong F(\Gamma_2)$ in $\mathbf{Exp}$, then $\Gamma_2 = U\Gamma_1 U^\dagger$ for $U \in G_2$.

Kernel of $F$ on the set of isomorphisms:

$$\ker(F) = \{\mathrm{Ad}_U : U \in G_2\}$$

Corollary 4: Predictive power [Т]

Corollary 4 (Finiteness of gauge group) [Т]

$G_2$ is a finite-dimensional (14-dimensional) compact Lie group. This means:

1. A discrete set of $G_2$-invariant observables fully characterizes the physical state
2. A finite number of gauge-invariant parameters (34) — unlike standard QM, where $U(\mathcal{H})$-freedom is infinite-dimensional
3. The theory is maximally predictive at the given dimension $N = 7$: the gauge group $G_2$ is the minimal group preserving the octonionic structure

Corollary 5: G₂-invariants as physical observables

The 34 physical parameters are organized as follows:

Type	Number of parameters	Description
Spectrum of $\Gamma$	6	Eigenvalues (ordered)
$G_2$-invariant angles	28	Mutual position of eigenvectors relative to octonionic structure
Total	34	Complete set of physical observables

The gauge-invariant observables include:

• Purity $P = \mathrm{Tr}(\Gamma^2)$ — $G_2$-invariant ($P(U\Gamma U^\dagger) = P(\Gamma)$ for $U \in U(7) \supset G_2$)
• E-coherence $\mathrm{Coh}_E(\Gamma)$ — invariant within $G_2$-orbit, since $G_2$ preserves the Fano structure and the role of E
• Reflection measure $R$ — determined via $\varphi(\Gamma)$ and $\Gamma$, both transforming covariantly
• Integration measure $\Phi$ — analogously

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Relation to open questions

Closing the problem of G at the level of theory

This theorem fully closes the question of uniqueness of the map G at the theoretical level:

Question	Status	Basis
Existence of G	[Т]	Theorem S + bridge T15
Uniqueness of G (up to $G_2$)	[Т]	Theorem of $G_2$-rigidity (this document)
Predictivity of G	[Empirical]	Requires experimental verification

Question 3 (predictivity) is epistemological, not mathematical: it is closed empirically (convergent validity, predictive success, interventional testing). This is the same epistemological standard by which all fundamental physics operates.

Analogy with physical theories

Theory	Uniqueness theorem	Gauge group	Empirical verification
QM	Stone–von Neumann (1931)	$U(\mathcal{H})$	Spectra, interference
GR	Birkhoff (spherical symmetry)	$\mathrm{Diff}(M)$	Light deflection, gravitational waves
SM	Coleman–Mandula / Haag–Łopuszański–Sohnius	Poincaré $\times$ gauge	Accelerators, PDG
UHM	$G_2$-rigidity (this theorem)	$G_2 = \mathrm{Aut}(\mathbb{O})$	Cabibbo angle, thresholds, Gap profiles

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Summary

Key result

Theorem of $G_2$-rigidity [Т]: The holonomic representation of a system satisfying (AP)+(PH)+(QG)+(V) is unique up to gauge group $G_2 = \mathrm{Aut}(\mathbb{O})$ — a 14-dimensional exceptional Lie group, the automorphism group of the octonions.

Physical meaning: Different observers applying UHM to the same system will obtain coherence matrices related by a $G_2$-transformation. All 34 gauge-invariant parameters (purity, coherences, thresholds) will coincide.

Methodological status: All steps of the proof are theorems [Т], relying on previously established results. This theorem closes the problem of the map G at the theoretical level and is the analogue of the Stone–von Neumann theorem for UHM.

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Related documents:

• Axiom Ω⁷ — fundamental axioms A1–A5
• Axiom (AP+PH+QG+V) — characterizing properties of viable holons
• Lindblad operators — primitivity of ℒ_Ω, L-unification, G₂-covariance
• Minimality theorem — functional uniqueness of 7 dimensions
• Structural derivation N=7 — bridge T15 and octonionic structure
• Categorical formalism — functor F: DensityMat → Exp
• Formalization of φ — equivalence of self-modeling definitions
• G₂-structure — role of G₂ in physical correspondences