Uniqueness Theorem of Holonomic Representation

Status: [T] — the octonionic structure follows from the axioms with the canonical orientation (T15-canon)

The uniqueness theorem of holonomic representation is a theorem [T] about $\mathbb{C}^7$ carrying the octonionic multiplication of the oriented Fano plane. From the axioms that structure follows through the bridge T15 (registry row 41n) with the canonical orientation of the Fano lines — the unique orientation class invariant under the collineations of the design (T15-canon) — so the theorem is [T] as a consequence of the axioms. Earlier on 2026-09-25 this box read "[T] given the octonionic structure; that structure is [C at (Alt)]"; that orientation input is discharged. Before that it read "[T] — all steps proven, relying exclusively on previously proven results". It relies on:

• Primitivity of $\mathcal{L}_\Omega$ [T] (proof)
• Full minimality 7/7 [T] (proof)
• Bridge T15 [T]: (AP)+(PH)+(QG)+(V) $\Rightarrow$ BIBD(7,3,1) = PG(2,2) [T] $\Rightarrow$ $\mathbb{O}$ (canonical orientation, T15-canon [T]) $\Rightarrow$ $G_2$ (proof)
• L-unification [T] (proof)
• Uniqueness of E, O, U [T] (proof)

Frame decision D-0910 — two groups, one representation (single source of truth)

The corpus uses $G_2$ in two roles that must not be conflated; this box fixes the split, and every other page defers to it.

1. Kinematic rigidity. The maximal subgroup of $U(7)$ preserving the octonionic 3-form $\varphi_3$ is $G_2 = \mathrm{Aut}(\mathbb{O})$ (Lemma G4). Two holonomic representations of one system can differ by an element of $G_2$ — never by more. This is the content of the theorem below, and it fixes the count $34 = 48 - 14$ of $G_2$-orbit invariants of the kinematic state (spectrum + $\varphi_3$-relative angles).
2. Dynamical identification. The axiomatic dynamics $\mathcal{L}_\Omega$ — pinching dissipator $\mathcal{D}_{\mathrm{Fano}} = \tfrac23\mathcal{D}_{\mathrm{atom}}$, the regeneration coefficient $\kappa_0$, the PW clock — is not $G_2$-covariant: it breaks $G_2$ to the finite octonionic frame group $\Gamma_{\!\text{oct}} \subset G_2$ (Theorem 5.1b [T]; $\Gamma_{\!\text{oct}} = 2^3 \!\cdot\! \mathrm{PSL}(3,2)$, order $1344$ — the 168 Fano-line-preserving basis permutations together with the 8 sign patterns constant on Fano lines; see frame rigidity). Hence the transformations relating physically indistinguishable descriptions of one holon form $\Gamma_{\!\text{oct}}$, not the 14-dimensional $G_2$: all 48 real parameters of $\Gamma$ are physical (modulo a finite relabelling of axes), and the frame-pinned observables $\Phi$, $\mathrm{Coh}_E$, $\kappa_0$ are physical observables with well-defined thresholds ($\Phi \geq 1$, $\mathrm{Coh}_E > 1/7$).

Consequently, wherever the corpus says "34 physical parameters", "states related by $G_2$ are physically identical" or "$\Phi$, $\mathrm{Coh}_E$ are $G_2$-invariant", the statement is to be read through this decision: 34 counts kinematic $G_2$-invariants; physical identity is $\Gamma_{\!\text{oct}}$-identity; $\Phi$ and $\mathrm{Coh}_E$ are frame-pinned, not $G_2$-invariant (an explicit $g \in G_2$ with $g e_1 = (e_1 + e_2)/\sqrt2$ sends $\Phi(|e_1\rangle\langle e_1|) = 0$ to $1$ and $\mathrm{Coh}_E$ from $1$ to $3/4$; a full rotation $e_1 \mapsto e_2$ sends $\mathrm{Coh}_E$ to $0$). $G_2$ keeps its physical role as the structure group of the octonionic sector: $SU(3)_C = \mathrm{Stab}_{G_2}(e_O)$, the Fano selection rules, and the decomposition $48 = 27 \oplus 14 \oplus 7$ (T-301).

This is not a choice among options: it is forced. The theorem below shows that $\Phi$ is preserved by no continuous group at all, so any reading on which the L2 condition $\Phi \geq 1$ is a physical condition must take a discrete identification group. See frame rigidity.

Theorem (Frame rigidity: $\Phi$ admits no continuous symmetry) [T]

Theorem (frame rigidity) [T]

Let $\Phi(\Gamma) = \sum_{i \neq j}|\gamma_{ij}|^2 / \sum_i \gamma_{ii}^2$ be the integration measure. Then the largest subgroup of $O(7)$ preserving $\Phi$ on all states is the hyperoctahedral group of signed permutations; its intersection with $G_2$ is the finite octonionic frame group $\Gamma_{\!\text{oct}}$, of order

$$|\Gamma_{\!\text{oct}}| = 1344 = 2^3 \cdot |\mathrm{PSL}(3,2)| = 8 \cdot 168,$$

the maximal subgroup $2^3 \!\cdot\! \mathrm{PSL}(3,2) \subset G_2$. In particular no continuous subgroup of $G_2$ (or even of $SO(7)$) preserves $\Phi$: $\dim\{X \in \mathfrak{g}_2 : \delta_X\Phi = 0\} = 0$.

Proof. On real pure states $\Gamma = vv^{\mathsf T}$, $\|v\|_2 = 1$, one has $\sum_i\gamma_{ii}^2 = \sum_i v_i^4 = \|v\|_4^4$ and $\sum_{ij}|\gamma_{ij}|^2 = 1$, hence

$$\Phi(vv^{\mathsf T}) = \frac{1 - \|v\|_4^4}{\|v\|_4^4},$$

a strictly decreasing function of $\|v\|_4$. So a linear map preserving $\Phi$ on this family preserves the $\ell^4$-norm on the unit $\ell^2$-sphere and, by homogeneity, on all of $\mathbb{R}^7$. By the Banach–Lamperti theorem (the linear isometries of $\ell^p$, $p \neq 2$, are exactly the signed permutations of coordinates) such a map is a signed permutation. Signed permutations preserving the associative 3-form $\varphi_3$ form $\Gamma_{\!\text{oct}}$: the permutation part must be a collineation of $PG(2,2)$ ($|{\rm PSL}(3,2)| = 168$) and the sign part must satisfy $\varepsilon_i\varepsilon_j\varepsilon_k = 1$ on each Fano line — the simplex code $[7,3]$, of size $2^3 = 8$. Conversely every such map preserves both $\varphi_3$ and the coordinate diagonal, hence $\Phi$. $\blacksquare$

Machine verification (2026-09-10). Exhaustive enumeration over signed permutations: $|\Gamma_{\!\text{oct}}| = 1344$, with exactly $168$ distinct permutation parts and a sign-only subgroup of order $8$; permutation parts of order 7: $48$, i.e. $8$ Singer subgroups of $\mathrm{PSL}(3,2)$ (in $\Gamma_{\!\text{oct}}$ itself each lifts eight times: $384$ elements of order 7; clarified 2026-09-25). First-order rigidity by least squares over random pure states: $\dim\{X \in \mathfrak{so}(7): \delta_X\Phi = 0\} = 0$ and $\dim\{X \in \mathfrak{g}_2: \delta_X\Phi = 0\} = 0$. The identity $\Phi = (1 - \|v\|_4^4)/\|v\|_4^4$ was checked to $10^{-15}$.

The lattice of candidate identification groups

Every candidate for "which transformations relate physically indistinguishable descriptions" is a subgroup of $G_2$; there are four natural ones. The parameter count is $48 - \dim$ (generic orbit); invariance is stated for the frame-referenced observables. All rows are machine-verified.

Identification group	$\dim$	Parameters	$\mathrm{Coh}_E$ (No-Zombie)	$\Phi$ (L2)
$G_2$	14	34	not invariant (witness: $1 \to 0.75$)	not invariant (witness: $0 \to 1$)
$SU(3) = \mathrm{Stab}_{G_2}(e_O)$	8	40	not invariant (witness: $1 \to 0.45$)	not invariant (witness: $0 \to 2.19$)
$SU(2) = \mathrm{Stab}_{G_2}(e_E, e_O)$	3	45	invariant	not invariant (witness: $0.690 \to 0.637$)
$\Gamma_{\!\text{oct}}$ (frame group)	0	48	invariant only under the $192$ of $1344$ elements that keep the $E$-axis (witness for the others: $1 \to 0$)	invariant (all $1344$)

Corrected 2026-09-25. The frame-group row read "$\mathrm{Coh}_E$: invariant"; that is retracted: an element of $\Gamma_{\!\text{oct}}$ that moves the $E$-axis to another axis takes $\mathrm{Coh}_E(\lvert e_E\rangle\langle e_E\rvert)$ from $1$ to $0$, and exhaustive enumeration finds $\mathrm{Coh}_E$ preserved on exactly $192 = 1344/7$ elements — those that keep the $E$-axis ($96$ fix $e_E$, $96$ send it to $-e_E$) — while $\Phi$ is preserved on all $1344$ (regression test test_coh_e_is_invariant_only_on_the_e_axis_stabiliser).

Reading the lattice: $\mathrm{Coh}_E$ needs only a distinguished $E$-axis, so the No-Zombie threshold survives on the stabiliser of that axis — already at $SU(2)_{E,O}$, in fact on the whole eight-dimensional $\mathrm{Stab}_{G_2}(e_E) \cong SU(3)$ (a rotation fixing $e_E$ fixes $\gamma_{EE}$ and the norm of the $E$-row), and in the frame group only on its $E$-axis stabiliser. $\Phi$ survives nowhere above the discrete row — by the rigidity theorem this is not an artefact of the present definition of the window but of $\Phi$ itself. Hence the corpus takes the last row: the identification group is $\Gamma_{\!\text{oct}}$ and all 48 parameters are physical. (With the corrected row, the only group of the lattice that preserves both observables with the axes held fixed is the $E$-axis stabiliser inside $\Gamma_{\!\text{oct}}$, of order $192$; the frame decision takes $\Gamma_{\!\text{oct}}$ with the functional labels carried along with the axes, as in Step 4 of the proof below, so that $\mathrm{Coh}_E$ is read on the image of the $E$-axis.) The alternative — keeping a continuous group — is available only at the price of rewriting the L2 condition in invariants of that group (for $SU(2)_{E,O}$: 45 parameters, $\Phi$ replaced by an $SU(2)$-invariant), and the octonionic reading would then have to rebuild the phenomenology of the 21 pairs on $1 \oplus 27 \oplus 7 \oplus 14$.

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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)	48 (all, modulo the finite frame group $\Gamma_{\!\text{oct}}$); 34 = 48 $-$ 14 of them are kinematic $G_2$-orbit invariants (D-0910)

The key distinction: in QM the gauge group is infinite-dimensional ($U(\mathcal{H})$), leaving enormous freedom. In UHM the kinematic gauge group is finite-dimensional $G_2$, which radically restricts this freedom and increases the predictive power of the theory — and the axiomatic dynamics narrows it further to the finite frame group $\Gamma_{\!\text{oct}}$ (frame decision D-0910 above).

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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 [T] and are proven in the respective documents.

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

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 [T]).

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 [T]

All 7 dimensions are functionally unique:

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

P3. Bridge T15 [T]

Full chain (AP)+(PH)+(QG)+(V) $\Rightarrow$ P1+P2 of 12 steps; the steps up to PG(2,2) are [T], and the step to $\mathbb{O}$ takes the canonical orientation of the seven lines: only 16 of the 128 orientations are normed, and they form the only orientation class the design itself determines (T15-canon; registry row 41n). The chain was stated as "all [T]" until 2026-09-25 without the orientation step, then as [C at (Alt)] the same day until T15-canon:

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

P4. L-unification [T]

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. Covariance groups of the dissipators [T]

The Fano dissipator is proportional to the atomic one, $\mathcal{D}_{\mathrm{Fano}} = \tfrac23\,\mathcal{D}_{\mathrm{atom}}$ (Theorem 5.1a [T]), and therefore shares its symmetry group: both are covariant under the finite octonionic frame group $\Gamma_{\!\text{oct}} \subset G_2$ (and $S_7$-equivariant),

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

and neither is covariant under the full continuous $G_2$ (Theorem 5.1b [T]; machine check: $\|\mathcal{D}_{\mathrm{Fano}}[g\Gamma g^\top] - g\,\mathcal{D}_{\mathrm{Fano}}[\Gamma]\,g^\top\| = 0.08$ against $\|\mathcal{D}_{\mathrm{Fano}}[\Gamma]\| = 0.25$ for a generic $g \in G_2$). The genuinely $G_2$-covariant dissipator $\mathcal{D}_{G_2}$ built from $\varphi_{abc}$ exists (Theorem 5.1c) but is not the axiomatic UHM dissipator. Hence $G_2$ is a symmetry of the kinematics (the 3-form), not of the dynamics — the frame decision D-0910 above. (Earlier drafts of this section stated "the Fano dissipator is $G_2$-covariant"; that statement is retracted.)

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New lemmas

Lemma G1: Spectral injectivity of propagator [T]

Lemma G1 (Spectral injectivity) [T]

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 [T] (§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 [T]

Lemma G2 (Nonlinear inverse problem) [T]

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 [T]
• $\|\Gamma\|_{\mathrm{HS}}^2 = \mathrm{Tr}(\Gamma^2) \geq 1/7 > 0$ 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 [T]

Lemma G3 (Axiomatic definiteness) [T]

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 [T])

(ii) The system of Fano lines $\{\mathrm{line}_p\}_{p=1}^{7} \subset \binom{[7]}{3}$ with its orientation, i.e. the structure constants $f_{ijk}$ (from bridge T15: the lines [T], their canonical orientation [T] by T15-canon)

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

(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 [T])

Proof. Each structure is derived from the axioms:

• (i): L-unification [T] — atoms $S_k = |k\rangle\langle k|$ of classifier $\Omega$
• (ii): Bridge T15 — uniqueness of BIBD$(7,3,1)$ $\cong$ PG(2,2) (Hall 1967) [T]; the orientation that turns the lines into the structure constants $f_{ijk}$ is the canonical one, the unique collineation-invariant class (T15-canon [T]; registry row 41n). This line read "Bridge T15 [T]" without the orientation step until 2026-09-25, then named the orientation as the input (Alt) until T15-canon
• (iii): HS-projection theorem [T] — orthogonal projection in Hilbert–Schmidt space
• (iv): Axiom A5 (Page–Wootters) — explicit postulate
• (v): Adjunction $\mathcal{D}_\Omega \dashv \mathcal{R}$ [T] — formula for $\kappa_0$ from categorical derivation. $\blacksquare$

Lemma G4: The octonionic-structure gauge group is $G_2$ [T]

Lemma G4 (Gauge group of the octonionic 3-form) [T]

The maximal subgroup $\mathcal{G} \subseteq U(7)$ preserving the octonionic associative 3-form $\varphi_3 = \sum_{i<j<k} f_{ijk}\, e^i\wedge e^j\wedge e^k$ — equivalently, the structure constants $f_{ijk}$ of Lemma G3(ii) — is $G_2 \times \mu_3$, where $\mu_3 = \{\mathbb{1}, \omega\mathbb{1}, \omega^2\mathbb{1}\}$, $\omega = e^{2\pi i/3}$. The scalars $\omega\mathbb{1}$ preserve every 3-form and act trivially on density matrices, so on states the gauge group is $G_2 = \mathrm{Aut}(\mathbb{O})$. (It read "is exactly $G_2$" until 2026-09-25.)

The remaining structures of Lemma G3 — the atomic projectors (i), the E-projection (iii), the PW clock $O$ (iv), the $\kappa_0$ formula (v) — are not $G_2$-invariant; they fix a functional frame (a choice of gauge) inside each $G_2$-orbit. Two representations related by $U\in G_2$ carry their frames into one another.

Proof. We show $\mathcal{G} = G_2 \times \mu_3$ in two inclusions.

(A) $G_2 \subseteq \mathcal{G}$. By definition $G_2 = \{g\in GL(7,\mathbb{R}) : g^\ast\varphi_3 = \varphi_3\}$ preserves the 3-form, and $G_2\subset SO(7)\subset U(7)$ preserves the Hermitian structure. Hence every $g\in G_2$ preserves $\varphi_3$, i.e. $g\in\mathcal{G}$. The scalars $\omega\mathbb{1}$, $\omega^3 = 1$, preserve every 3-form, so $\mu_3 \subseteq \mathcal{G}$ as well. $\checkmark$

warning

The functional labels are frame data, not $G_2$-invariants

Since $\mathbb{C}^7$ is an irreducible $G_2$-module (Cartan 1894), by Schur's lemma it has no nonzero proper $G_2$-invariant subspace. Consequently:

• no coordinate axis $|k\rangle$ — in particular the E, O, U axes — is $G_2$-invariant; a generic $g\in G_2$ rotates it;
• the set of atomic projectors $\{|k\rangle\langle k|\}$ is preserved only by the finite frame subgroup $\Gamma_{\!\text{oct}} = 2^3 \!\cdot\! \mathrm{PSL}(3,2) \subset G_2$ of order $1344$ (permutation part $\mathrm{Aut}(PG(2,2)) \cong PSL(2,7)$, order 168; sign part of order 8 — frame rigidity), not by all of $G_2$;
• hence $\mathrm{Coh}_E$, $\Phi$ and $\kappa_0$, which reference the E/O/U axes, are frame-dependent: invariant under $\mathrm{Stab}_{G_2}$ of the chosen frame, not under all of $G_2$ — $\Phi$ under the whole frame group $\Gamma_{\!\text{oct}}$, $\mathrm{Coh}_E$ only under its $192$ elements that keep the $E$-axis, $\kappa_0$ only under the elements that keep the axes it references. They are physical because the frame is pinned by the dynamics (Definition G1's $\mathcal{L}_\Omega$-covariance), not because they descend to $\mathcal{D}(\mathbb{C}^7)/G_2$.

The genuinely $G_2$-invariant content is the spectrum (6 numbers) plus the $\varphi_3$-relative angles (28) — the $48-14=34$ parameters of Corollary 1.

(B) $\mathcal{G} \subseteq G_2 \times \mu_3$. Let $U \in \mathcal{G}$.

Step B1 (Lie algebra). Write $X \in \mathfrak{u}(7)$ as $X = A + iS$ with $A$ real antisymmetric and $S$ real symmetric. Since $\varphi_3$ is real, $X\cdot\varphi_3 = A\cdot\varphi_3 + i\,S\cdot\varphi_3$ vanishes only if $A\cdot\varphi_3 = 0$ and $S\cdot\varphi_3 = 0$, i.e. only if $A$ and $S$ lie in the Lie algebra $\mathfrak{g}_2 \subset \mathfrak{so}(7)$ of $G_2 = \{g \in GL(7,\mathbb{R}) : g^*\varphi_3 = \varphi_3\}$; a symmetric $S$ in $\mathfrak{so}(7)$ is zero. So the Lie algebra of $\mathcal{G}$ is $\mathfrak{g}_2$ and its identity component is $G_2$ (machine check: the stabiliser of $\varphi_3$ in $\mathfrak{u}(7)$ has real dimension $14$).

Step B2 (normaliser). $U$ normalises the identity component, so $g \mapsto UgU^{-1}$ is an automorphism of $G_2$. $G_2$ has no outer automorphisms, so there is $h \in G_2$ with $UgU^{-1} = hgh^{-1}$ for all $g \in G_2$, and $h^{-1}U$ commutes with $G_2$. Since $\mathbb{C}^7$ is an irreducible $G_2$-module (Cartan 1894), Schur's lemma gives $h^{-1}U = \lambda\mathbb{1}$ with $\lvert\lambda\rvert = 1$.

Step B3 (the scalar). $\lambda\mathbb{1} = h^{-1}U$ preserves $\varphi_3$, and $(\lambda\mathbb{1})^*\varphi_3 = \lambda^3\varphi_3$, so $\lambda^3 = 1$ and $U = h\,\lambda\mathbb{1} \in G_2 \times \mu_3$. The product is direct: $\mu_3$ is central, and $\omega\mathbb{1}$ is not real, so $\mu_3 \cap G_2 = \{\mathbb{1}\}$. $\blacksquare$

Corrected 2026-09-25: part (B) started from a $U$ preserving "all five structures of Lemma G3" (not the hypothesis of the lemma) and relied on a box asserting that "preservation of all 7 such subspaces is equivalent to preservation of the octonionic cross-product". That equivalence is false: every diagonal unitary preserves the seven coordinate line subspaces, and a generic one does not preserve $\varphi_3$ (machine check). The former Step B3 assumed that $U$ preserves the real structure, which $\omega\mathbb{1}$ does not. Steps B1–B3 and the box are replaced by the argument above; the lemma changes only by the scalars $\mu_3$, which act trivially on states.

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: every collineation of PG(2,2) has exactly $8$ lifts — signed permutations of the basis that are automorphisms of $\mathbb{O}$; the lifts form the frame group $\Gamma_{\!\text{oct}} \subset G_2$ of order $1344$, a non-split extension $2^3 \cdot \mathrm{PSL}(3,2)$, so the collineation group is a quotient of $\Gamma_{\!\text{oct}}$, not a subgroup of it. (Until 2026-09-25 this read "$\mathrm{PSL}(2,7) \subset G_2$ as a finite subgroup — every permutation of 7 points compatible with PG(2,2) extends to a continuous automorphism of $\mathbb{O}$"; as bare permutations only $21$ of the $168$ collineations are automorphisms.) Part (B) uses only the 3-form $\varphi_3$, not the combinatorics of the lines.

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Main theorem

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

Theorem of G₂-rigidity [T]

Let $S$ be an autonomous system satisfying (AP)+(PH)+(QG)+(V), with $\mathbb{C}^7$ carrying the octonionic multiplication of Lemma G3(ii) (from the axioms through the bridge T15 with the canonical orientation, T15-canon). 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 $G_2$.

Sharpening (frame decision D-0910). Because $\mathcal{L}_\Omega$ itself is covariant only under the finite frame group, the intertwiner $U$ of two representations that share the same axiomatic dynamics lies in $\Gamma_{\!\text{oct}} \subset G_2$. The $G_2$ statement is the kinematic envelope — the largest group any two representations can differ by; the $\Gamma_{\!\text{oct}}$ statement is the dynamical identification.

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 [T]). 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 the spectral observables — purity $P = \mathrm{Tr}(\Gamma^2)$, von Neumann entropy, the eigenvalues — must coincide; the frame-pinned observables such as $\mathrm{Coh}_E$ are carried covariantly and enter in Step 4).

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 [T]). 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 $U$ must preserve the octonionic 3-form $\varphi_3$ carried by the axiomatic Fano structure (Lemma G3(ii) [T]). The functional labels (atomic projectors, E-projection, PW clock, $\kappa_0$) are carried covariantly: $U$ maps the frame of representation 1 to the frame of representation 2.

By Lemma G4 [T], preservation of $\varphi_3$ gives $U \in G_2$. $\blacksquare$

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 [T], 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: Kinematic invariants and physical states [T]

Corollary 1 (Kinematic invariants and physical states) [T]

The space of $G_2$-orbits of kinematic states of the holon (spectrum + $\varphi_3$-relative angles):

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

has dimension:

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

where $48 = N^2 - 1 = \dim(\mathrm{su}(7))$ is the full number of parameters of $\Gamma$, and $14 = \dim(G_2)$ is the dimension of a generic kinematic $G_2$-orbit. The space of physically distinguishable states is $\mathcal{D}_{\mathrm{phys}} = \mathcal{D}(\mathbb{C}^7)/\Gamma_{\!\text{oct}}$, of full dimension 48 (frame decision D-0910): the 14 orbit directions are physical because the axiomatic dynamics pins the frame.

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 $\dim(\mathcal{D}_{\mathrm{kin}}) = 48 - 14 = 34$. $\blacksquare$

Consistency

The value 34 is the kinematic count only. The pinching dynamics is $\Gamma_{\!\text{oct}}$-covariant, not $G_2$-covariant, at every $\alpha$ — including the pure Fano regime $\alpha = 0$ (Theorem 5.1b, Lindblad operators) — so no dynamical regime realises a $48 \to 34$ reduction of the physical parameter space (D-0910).

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

Corollary 2 (Inverse problem) [T]

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 [T])

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

up to the finite frame group $\Gamma_{\!\text{oct}}$ (frame decision D-0910); in particular up to $G_2$ (Theorem of G₂-rigidity [T]).

Corollary 3: Faithfulness of functor F [T]

Corollary 3 (Faithfulness of functor) [T]

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

Kernel of $F$ on the set of isomorphisms (experience reads the frame-pinned $E$-sector, so a generic $G_2$-rotation changes it — D-0910, and so does every element of $\Gamma_{\!\text{oct}}$ that moves the $E$-axis):

$$\ker(F) \subseteq \{\mathrm{Ad}_U : U \in \Gamma_{\!\text{oct}},\ U e_E = \pm e_E\} \subsetneq \{\mathrm{Ad}_U : U \in \Gamma_{\!\text{oct}}\} \subset \{\mathrm{Ad}_U : U \in G_2\}$$

The $E$-axis stabiliser in $\Gamma_{\!\text{oct}}$ has $192 = 1344/7$ elements.

Corrected 2026-09-25. The corollary stated $\ker(F) = \{\mathrm{Ad}_U : U \in \Gamma_{\!\text{oct}}\}$; that is retracted: $F$ reads the $E$-sector, and an element of $\Gamma_{\!\text{oct}}$ that moves the $E$-axis takes $\mathrm{Coh}_E(\lvert e_E\rangle\langle e_E\rvert)$ from $1$ to $0$, so $\mathrm{Ad}_U$ changes $F$ for $1152$ of the $1344$ elements; only the $192$ that keep the $E$-axis can lie in the kernel (lattice of identification groups; regression test test_coh_e_is_invariant_only_on_the_e_axis_stabiliser). The faithfulness statement above is unaffected.

Corollary 4: Predictive power [T]

Corollary 4 (Finiteness of gauge group) [T]

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

1. A discrete set of $G_2$-invariant observables fully characterizes the kinematic state; the physical state additionally carries the 14 frame-orientation parameters, pinned by the dynamics (D-0910)
2. A finite number of parameters — 34 kinematic $G_2$-invariants, 48 physical parameters in the pinned frame — 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 kinematic $G_2$-invariants are organized as follows (with the 14 frame-orientation parameters they make up the 48 physical parameters, D-0910):

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 kinematic $G_2$-invariants

Two classes of observable must be distinguished (irreducibility of $\mathbf 7$, Lemma G4):

Genuinely $G_2$-invariant (descend to $\mathcal{D}(\mathbb{C}^7)/G_2$):

• Purity $P = \mathrm{Tr}(\Gamma^2)$ — in fact $U(7)$-invariant, hence $G_2$-invariant
• Reflection measure $R = 1/(7P)$ — a function of $P$, hence $G_2$-invariant
• the spectrum (6) and the $\varphi_3$-relative angles (28) — the 34 parameters above

Frame-dependent (defined only after the functional frame is fixed by the dynamics; invariant under $\mathrm{Stab}_{G_2}$ of the frame, not under all of $G_2$, since no axis is $G_2$-invariant):

• E-coherence $\mathrm{Coh}_E(\Gamma)$ — references the E-axis; invariant only under the elements that keep it ($192$ of the $1344$ in $\Gamma_{\!\text{oct}}$)
• Integration measure $\Phi = \sum_{i\neq j}|\gamma_{ij}|^2/\sum_i\gamma_{ii}^2$ — references the coordinate basis; invariant under all of $\Gamma_{\!\text{oct}}$
• the regeneration coefficient $\kappa_0$ — references the O, E, U axes

These frame-dependent quantities are physical because the dynamics ($\mathcal{L}_\Omega$-covariance, Definition G1) pins the frame; they are not among the 34 orbit-invariants.

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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	[T] (Theorem S); the octonionic structure via the bridge T15, [T] with the canonical orientation (T15-canon)	Theorem S + bridge T15
Uniqueness of G (up to $G_2$)	[T]	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)	$G_2 = \mathrm{Aut}(\mathbb{O})$	thresholds, Gap profiles (the Cabibbo angle was listed here; withdrawn 2026-09-26, T-345(e): its agreement came from a fitted $C_{\mathrm{norm}}\approx26$)

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Summary

Key result

Theorem of $G_2$-rigidity [T]: 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, with the frame pinned by the axiomatic dynamics, obtain coherence matrices related by an element of the finite frame group $\Gamma_{\!\text{oct}} \subset G_2$ (D-0910): all 48 parameters coincide up to a relabelling of axes. The 34 kinematic $G_2$-invariants (spectrum, $\varphi_3$-relative angles) coincide even before the frame is pinned.

Methodological status: All steps of the proof are theorems [T], 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