G₂-Structure and the Fano Plane

For whom this chapter is intended

The group $G_2 = \mathrm{Aut}(\mathbb{O})$ and the Fano plane $\mathrm{PG}(2,2)$ as the central algebraic structures of UHM theory. The reader will learn how the multiplication table of the octonions determines the physical architecture of the theory.

Overview

The group $G_2 = \mathrm{Aut}(\mathbb{O})$ — the automorphism group of the octonions — is the central algebraic structure of UHM theory. The Fano plane $\mathrm{PG}(2,2)$ encodes the multiplication table of the imaginary units of $\mathbb{O}$ and determines the entire physical architecture of the theory: from Lindblad operators to gauge symmetries and selection rules.

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1. Fano Plane PG(2,2)

1.1 Definition

The Fano plane $\mathrm{PG}(2,2)$ is the minimal finite projective plane. It contains:

• 7 points, identified with the 7 imaginary units of the octonions $e_1, \ldots, e_7$, and in UHM theory — with the 7 dimensions $\{A, S, D, L, E, U, O\} = \{1, 2, 3, 4, 5, 6, 7\}$
• 7 lines, each containing exactly 3 points

1.2 Table of Fano Lines

#	Fano line	Dimensions
1	$\{1, 2, 4\}$	$\{A, S, L\}$
2	$\{2, 3, 5\}$	$\{S, D, E\}$
3	$\{3, 4, 6\}$	$\{D, L, U\}$
4	$\{4, 5, 7\}$	$\{L, E, O\}$
5	$\{5, 6, 1\}$	$\{E, U, A\}$
6	$\{6, 7, 2\}$	$\{U, O, S\}$
7	$\{7, 1, 3\}$	$\{O, A, D\}$

1.3 Fundamental Properties

1. Through any two points there passes exactly one line. This means that every pair of dimensions $(i, j)$ uniquely determines a Fano line $(i, j, k)$.

2. Each point lies on exactly 3 lines. Consequently:

$\sum_{p=1}^{7} \Pi_p = 3I$

where $\Pi_p = \sum_{i \in \mathrm{line}_p} |i\rangle\langle i|$ is the projector onto the subspace corresponding to Fano line $p$.

3. Octonion structure constants $f_{ijk}$: $f_{ijk} = \pm 1$ if and only if $\{i, j, k\}$ is a Fano line, and $f_{ijk} = 0$ otherwise. The multiplication table of $\mathbb{O}$:

$e_i \cdot e_j = f_{ijk}\, e_k - \delta_{ij}$

1.4 Automorphism Group

$\mathrm{Aut}(\mathrm{PG}(2,2)) = \mathrm{PSL}(2,7)$

This is the group of order 168, isomorphic to $\mathrm{GL}(3, \mathbb{F}_2)$. It acts transitively on both points and lines.

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2. Octonionic Multiplication and $G_2$

2.1 The Octonion Algebra $\mathbb{O}$

The octonions are an 8-dimensional real division algebra. Each octonion is written as:

$x = x_0 \cdot 1 + \sum_{i=1}^{7} x_i \, e_i, \quad x_0, x_i \in \mathbb{R}$

where $1$ is the real unit, and $e_1, \ldots, e_7$ are imaginary units satisfying:

$e_i^2 = -1, \quad e_i \cdot e_j = -e_j \cdot e_i \;\; (i \neq j)$

In UHM theory the imaginary units are identified with the 7 dimensions: $e_1 = A$, $e_2 = S$, $e_3 = D$, $e_4 = L$, $e_5 = E$, $e_6 = U$, $e_7 = O$.

2.2 Octonion Multiplication Table

The multiplication of imaginary units is completely determined by the Fano plane. For each Fano line $(i, j, k)$ with canonical ordering:

$e_i \cdot e_j = e_k, \quad e_j \cdot e_k = e_i, \quad e_k \cdot e_i = e_j$

$\times$	$e_1$ (A)	$e_2$ (S)	$e_3$ (D)	$e_4$ (L)	$e_5$ (E)	$e_6$ (U)	$e_7$ (O)
$e_1$ (A)	$-1$	$e_4$	$-e_7$	$-e_2$	$e_6$	$-e_5$	$e_3$
$e_2$ (S)	$-e_4$	$-1$	$e_5$	$e_1$	$-e_3$	$e_7$	$-e_6$
$e_3$ (D)	$e_7$	$-e_5$	$-1$	$e_6$	$e_2$	$-e_4$	$-e_1$
$e_4$ (L)	$e_2$	$-e_1$	$-e_6$	$-1$	$e_7$	$e_3$	$-e_5$
$e_5$ (E)	$-e_6$	$e_3$	$-e_2$	$-e_7$	$-1$	$e_1$	$e_4$
$e_6$ (U)	$e_5$	$-e_7$	$e_4$	$-e_3$	$-e_1$	$-1$	$e_2$
$e_7$ (O)	$-e_3$	$e_6$	$e_1$	$e_5$	$-e_4$	$-e_2$	$-1$

Each row and column contains all 7 imaginary units exactly once (up to sign) — a division algebra.

2.3 The Group $G_2 = \mathrm{Aut}(\mathbb{O})$

Definition. The group $G_2$ is the group of all $\mathbb{R}$-linear bijections $g: \mathbb{O} \to \mathbb{O}$ preserving multiplication:

$G_2 = \{g \in \mathrm{GL}(\mathbb{O}) : g(xy) = g(x)g(y) \; \forall x, y \in \mathbb{O}\}$

Since $g(1) = 1$ for any automorphism, $g$ acts on the 7-dimensional subspace of imaginary octonions $\mathrm{Im}(\mathbb{O}) \cong \mathbb{R}^7$.

Status: Theorem [T]

$G_2$ is an exceptional compact simple Lie group with the following characteristics:

• $\dim(G_2) = 14$
• $\mathrm{rank}(G_2) = 2$
• $G_2 \subset \mathrm{SO}(7)$ — a proper subgroup of the rotation group of $\mathbb{R}^7$

Why $\dim(G_2) = 14$? The group $\mathrm{SO}(7)$ has dimension $7 \cdot 6 / 2 = 21$. The condition of preserving octonionic multiplication imposes 7 independent constraints (one per Fano line): $g(e_i \cdot e_j) = g(e_i) \cdot g(e_j)$ for all $(i,j,k) \in \mathrm{PG}(2,2)$. Total:

$\dim(G_2) = 21 - 7 = 14$

Why $\mathrm{rank}(G_2) = 2$? The maximal torus of $G_2$ is two-dimensional: it is generated by two commuting rotations in $\mathbb{R}^7$ compatible with all 7 Fano lines. Two independent angles $(\theta_1, \theta_2)$ parametrise the maximal torus, yielding 2 quantum numbers — Noether charges (see G₂-Noether Charges).

2.4 Fourteen Generators of $G_2$

The Lie algebra $\mathfrak{g}_2$ has 14 generators, which can be decomposed with respect to the representations of the subgroup $\mathrm{SU}(3) \subset G_2$:

$\mathfrak{g}_2 = \mathfrak{su}(3) \oplus \mathbb{C}^3$

Type	Count	$\mathrm{SU}(3)$ representation	Physical interpretation in UHM
$\mathrm{SU}(3)$ generators	8	$\mathbf{8}$ (adjoint)	Gauge transformations between triples of dimensions on a single Fano line; analogue of gluon fields
Additional generators	6	$\mathbf{3} \oplus \bar{\mathbf{3}}$	Transformations mixing dimensions from different Fano lines; 'inter-line' rotations

All 14 generators are anti-Hermitian $7 \times 7$ matrices $T_a \in \mathfrak{so}(7)$ satisfying:

$[T_a, T_b] = f_{ab}^{\;\;c}\, T_c, \quad a, b, c = 1, \ldots, 14$

where $f_{ab}^{\;\;c}$ are the structure constants of $\mathfrak{g}_2$.

2.5 Example: Octonionic Multiplication and Non-associativity

The octonions are the unique normed division algebra that is non-associative. Let us demonstrate this with a concrete example.

Problem. Compute $(e_1 \cdot e_2) \cdot e_3$ and $e_1 \cdot (e_2 \cdot e_3)$ and verify that the results differ.

Step 1. From the multiplication table (Fano line $\{1,2,4\}$):

$e_1 \cdot e_2 = e_4 \quad (\text{A} \cdot \text{S} = \text{L})$

Step 2. Now multiply the result by $e_3$ (Fano line $\{3,4,6\}$):

$(e_1 \cdot e_2) \cdot e_3 = e_4 \cdot e_3 = -e_6 \quad (\text{L} \cdot \text{D} = -\text{U})$

(the minus sign — because the canonical orientation of line $\{3,4,6\}$ gives $e_3 \cdot e_4 = e_6$, and we are multiplying in the reverse order).

Step 3. Separately compute the right bracket (Fano line $\{2,3,5\}$):

$e_2 \cdot e_3 = e_5 \quad (\text{S} \cdot \text{D} = \text{E})$

Step 4. Multiply $e_1$ by the result (Fano line $\{5,6,1\}$):

$e_1 \cdot (e_2 \cdot e_3) = e_1 \cdot e_5 = e_6 \quad (\text{A} \cdot \text{E} = \text{U})$

Result:

$(e_1 \cdot e_2) \cdot e_3 = -e_6, \quad e_1 \cdot (e_2 \cdot e_3) = +e_6$

The difference: $(e_1 \cdot e_2) \cdot e_3 - e_1 \cdot (e_2 \cdot e_3) = -2e_6$. The non-associativity is manifest.

Physical Interpretation

In terms of UHM dimensions: the sequence of interactions $A \to S \to D$ yields different results depending on the grouping order. This means that the octonionic structure encodes the contextual dependence of coherent transitions: the result depends not only on the participating dimensions, but also on the order of their involvement.

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3. $G_2 = \mathrm{Aut}(\mathbb{O})$ and its Action on Coherences

3.1 Action of $G_2$ on the Space of Coherences

Upon identifying $e_i \leftrightarrow$ dimensions (from the dimensions.md table), the group $G_2$ acts on the $7D$ space:

$g \in G_2: \quad |i\rangle \mapsto \sum_j D_{ji}(g) |j\rangle$

where $D(g)$ is the 7-dimensional (fundamental) representation of $G_2$.

Action on the coherence matrix:

$g: \Gamma \mapsto D(g)\, \Gamma\, D(g)^\dagger$

Action on coherences:

$g: \gamma_{ij} \mapsto \sum_{k,l} D_{ki}(g)\, D_{lj}^*(g)\, \gamma_{kl}$

3.2 $G_2$ Preserves the Fano Structure

Since $G_2 = \mathrm{Aut}(\mathbb{O})$ preserves octonionic multiplication, it preserves the structure constants $f_{ijk}$ and hence the Fano plane:

$g \in G_2 \quad \Rightarrow \quad g \text{ permutes the Fano lines}$

More precisely: for each $g \in G_2$ there exists a permutation $\sigma_g$ on the set $\{1, \ldots, 7\}$ of lines:

$g\, \Pi_p\, g^\dagger = \Pi_{\sigma_g(p)}$

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3b. Physical Interpretation of $G_2$-Symmetry

What $G_2$-Invariance Preserves

$G_2$-symmetry is the continuous kinematic symmetry of UHM theory, which is spontaneously broken upon dynamical vacuum fixation. Before minimization of $V_{\text{Gap}}$: $G_2$ transformations rename the basis $\{A, S, D, L, E, U, O\}$, preserving the octonionic structure and all $G_2$-invariants ($P$, $R$, $\Phi$, spectrum of $\Gamma$). After minimization (T-64 [T]): a specific vacuum $\Gamma_{\text{vac}}$ is fixed, breaking $G_2 \to H$ (vacuum stabilizer). The Boolean fragment $\mathrm{Dec}(\Omega) \cong 2^7$ crystallizes as the pointer basis selected by spontaneous symmetry breaking — analogous to the Higgs mechanism $SU(2) \times U(1) \to U(1)_{\text{em}}$. Goldstone modes — massless excitations along the broken directions $G_2/H$.

Status: Theorem [T]

The $G_2$-transformation $g: \Gamma \mapsto D(g)\,\Gamma\,D(g)^\dagger$ preserves the following physical quantities:

Invariant	Formula	Physical meaning
Total purity	$P = \mathrm{Tr}(\Gamma^2)$	Degree of consciousness integration
Reflection measure	$R = R(\Gamma)$	Depth of self-observation
Integration	$\Phi = \Phi(\Gamma)$	System irreducibility
Spectrum of $\Gamma$	$\lambda_1 \geq \cdots \geq \lambda_7$	Eigenvalue populations
Total coherence	$\sum_{i < j}	\gamma_{ij}
Fano structure	$f_{ijk}$	Octonion multiplication table

What $G_2$ Does NOT Preserve

The $G_2$-transformation mixes specific dimensions. In general:

• Populations of individual dimensions $\gamma_{ii}$ are not invariant: $G_2$ can transfer population from $A$ to $S$.
• Specific coherences $\gamma_{ij}$ are not invariant: the $A \leftrightarrow S$ coupling can transform into $D \leftrightarrow L$.
• Gap profile $\{\mathrm{Gap}(i,j)\}_{i<j}$ is not invariant elementwise (although the total Gap is invariant).
• Stress vector $\sigma_k = 1 - 7\gamma_{kk}$ is not invariant componentwise.

Geometric Meaning: Rotations in $\{A, S, D, L, E, U, O\}$

A $G_2$-transformation can be viewed as a rotation of the 7-dimensional space that:

1. Is compatible with the Fano plane: if $\{i, j, k\}$ is a Fano line, then the image $\{g(i), g(j), g(k)\}$ is also a Fano line.
2. Is not arbitrary: of the 21 possible rotations in $\mathrm{SO}(7)$, only the 14-dimensional submanifold $G_2$ preserves octonionic multiplication.
3. Physically: a $G_2$-transformation is a change of 'coordinate system' in the space of dimensions, under which all algebraic relations (Fano lines, signs of structure constants, triples of related dimensions) remain unchanged.

warning

$G_2$-Covariance Principle

The physical laws of UHM theory (evolution, consciousness thresholds, Lindblad operators) must be formulated in terms of $G_2$-invariants. The specific 'label' of a dimension ($A$, $S$, $D$, ...) is a matter of basis choice, not of physics.

Analogy with Gauge Theories

Theory	Gauge group	What is preserved	What is mixed
Electrodynamics	$U(1)$	Charge	Phase of the wave function
Chromodynamics	$SU(3)$	Color singlet	Quark color (r, g, b)
UHM	$G_2$	$P$, $R$, $\Phi$, spectrum, Fano structure	Dimension labels $\{A,S,D,L,E,U,O\}$

In this sense $G_2$ for UHM theory is the analogue of $SU(3)_c$ for QCD: specific 'colors' (dimensions) are not directly observable; only invariant combinations are observable.

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4. $G_2$-Invariants of the Gap Profile

Theorem 2.1 ($G_2$-Invariants of the Gap Profile)

Status: Theorem [T]

The following quantities are $G_2$-invariants (unchanged under $G_2$ transformations).

(a) Total purity: $P = \mathrm{Tr}(\Gamma^2)$ — invariant under $\mathrm{SO}(7) \supset G_2$.

(b) Total Gap:

$\mathcal{G}_{\mathrm{total}} := \sum_{i<j} |\gamma_{ij}|^2 \cdot \mathrm{Gap}(i,j)^2 = \sum_{i<j} |\mathrm{Im}(\gamma_{ij})|^2$

The total 'imaginary energy' of coherences is invariant under $\mathrm{SO}(7)$.

(c) However, the distribution of Gap over pairs $(i,j)$ is not a $G_2$-invariant. $G_2$ 'mixes' Gap between pairs, preserving only the total.

Proof. (a) and (b) follow from the unitary invariance of the Frobenius norm. (c) follows from the fact that $G_2$ is not diagonal in the basis $\{|i\rangle\}$. $\blacksquare$

Theorem 2.2 ($G_2$-Orbits of Gap Profiles)

Status: Theorem [T]

The set of all possible Gap profiles $\{\mathrm{Gap}(i,j)\}_{i<j}$ for a fixed $\Gamma$ decomposes into $G_2$-orbits.

(a) The total number of $G_2$-invariants for a Hermitian $7 \times 7$ matrix: $48 - 14 = 34$, where $14 = \dim(G_2)$. Consequently, $G_2$ gauge freedom reduces the 48-dimensional parameter space to a 34-dimensional space of physically distinguishable configurations. The $G_2$-rigidity theorem [T] proves that $G_2$ is the maximal gauge group (Lemma G4): no larger subgroup of $U(7)$ preserves all axiomatic structures A1–A5. The reduction $48 \to 34$ is not an arbitrary gauge choice, but a necessary consequence of the uniqueness of the holonomy representation.

(b) Of the 21 Gap values, only $34 - 7 =$ up to 27 are 'physically distinguishable' (7 populations are subtracted from the invariants).

(c) This means that $21 - (27 - 21) =$ all 21 Gaps can be distinguishable, but with 14 relations between them. In practice, knowing 7 Gaps, one can (under $G_2$-covariance) recover the remaining 14.

Corollary: $G_2$-Reduction of Diagnostics

If the UHM evolution equations are $G_2$-covariant, a full diagnostic requires measuring only:

• 7 populations $\gamma_{ii}$
• 7 moduli $|\gamma_{ij}|$ for one 'base set' of pairs
• 7 phases $\theta_{ij}$ for the same set

The remaining 27 parameters are computed from $G_2$ relations.

Status: Open Problem [H]

$G_2$-covariance of the evolution equations has not been proved in full generality. The degree of $G_2$ breaking is determined by the parameter $\alpha$ (see Theorem 11.3 below).

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5. Fano-Structured Lindblad Operators

5.1 Two Types of Classifier Atoms

From L-unification: Lindblad operators $L_k = \sqrt{\chi_{S_k}}$ are derived from atoms of the classifier $\Omega$. There are two types:

Basis atoms (7 in total):

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

Composite atoms (7 in total): the Fano lines define 7 linear subobjects — projections onto 3-dimensional subspaces:

$\Pi_p = \sum_{i \in \mathrm{line}_p} |i\rangle\langle i|, \quad p = 1, \ldots, 7$

Each Fano line $p = (i, j, k)$ generates a composite atom $S_p = \mathrm{span}\{|i\rangle, |j\rangle, |k\rangle\}$.

Theorem 10.0 (Completeness of Fano Atoms)

Status: Theorem [T]

Each dimension lies on exactly 3 Fano lines.

$\sum_{p=1}^{7} \Pi_p = 3I$

Proof. A property of the Fano plane: each of the 7 points is incident to exactly 3 lines. $\blacksquare$

5.2 Definition (Fano-Structured Lindblad Operators)

For each Fano line $p = (i, j, k)$, a Lindblad operator is defined:

$L_p^{\mathrm{Fano}} := \frac{1}{\sqrt{3}} \Pi_p = \frac{1}{\sqrt{3}}(|i\rangle\langle i| + |j\rangle\langle j| + |k\rangle\langle k|)$

CPTP verification:

$\sum_{p=1}^{7} (L_p^{\mathrm{Fano}})^\dagger L_p^{\mathrm{Fano}} = \frac{1}{3}\sum_{p=1}^{7} \Pi_p = \frac{1}{3} \cdot 3I = I \quad \checkmark$

5.3 Definition (Fano Predictive Channel)

$\mathcal{P}_{\mathrm{Fano}}(\Gamma) := \sum_{p=1}^{7} L_p^{\mathrm{Fano}} \, \Gamma \, (L_p^{\mathrm{Fano}})^\dagger = \frac{1}{3}\sum_{p=1}^{7} \Pi_p \, \Gamma \, \Pi_p$

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6. Properties of the Fano Channel

Theorem 10.1 (Fano Channel Preserves Coherences)

Status: Theorem [T]

For an arbitrary coherence matrix $\Gamma$:

(a) Diagonal elements are preserved exactly:

$[\mathcal{P}_{\mathrm{Fano}}(\Gamma)]_{ii} = \gamma_{ii}$

(b) Off-diagonal elements (coherences) are preserved with a factor of $1/3$:

$[\mathcal{P}_{\mathrm{Fano}}(\Gamma)]_{ij} = \frac{1}{3}\gamma_{ij} \quad \text{for all } i \neq j$

(c) Phases of coherences are preserved exactly:

$\arg([\mathcal{P}_{\mathrm{Fano}}(\Gamma)]_{ij}) = \arg(\gamma_{ij}) = \theta_{ij}$

Proof.

(a) $[\sum_p \Pi_p \Gamma \Pi_p]_{ii} = \sum_{p:\, i \in \mathrm{line}_p} \gamma_{ii} = 3\gamma_{ii}$. With factor $1/3$: $\gamma_{ii}$. $\checkmark$

(b) In $\mathrm{PG}(2,2)$ any two points lie on exactly one line. For the pair $(i,j)$, $i \neq j$: exactly one line $p^*$ contains both points.

$\left[\sum_p \Pi_p \Gamma \Pi_p\right]_{ij} = \sum_{p:\, i,j \in \mathrm{line}_p} \gamma_{ij} = 1 \cdot \gamma_{ij}$

With factor $1/3$: $\gamma_{ij}/3$. $\checkmark$

(c) $\arg(\gamma_{ij}/3) = \arg(\gamma_{ij})$, since $1/3 > 0$. $\checkmark$ $\blacksquare$

Theorem 10.2 (Canonical Form of $\varphi_{\mathrm{coh}}$)

Status: Theorem [T]

Canonical coherence-preserving self-modelling is determined by a two-component structure.

$\varphi_{\mathrm{coh}}(\Gamma) = k \cdot \left[\alpha \cdot \mathcal{P}_{\mathrm{base}}(\Gamma) + (1 - \alpha) \cdot \mathcal{P}_{\mathrm{Fano}}(\Gamma)\right] + (1 - k) \cdot \Gamma_{\mathrm{anchor}}$

where:

• $\mathcal{P}_{\mathrm{base}}(\Gamma) = \sum_m P_m \Gamma P_m = \mathrm{diag}(\Gamma)$ — atomic channel (decohering observation)
• $\mathcal{P}_{\mathrm{Fano}}(\Gamma) = \frac{1}{3}\sum_p \Pi_p \Gamma \Pi_p$ — Fano channel
• $\alpha \in [0, 1]$ — decoherence depth parameter (balance between atomic and Fano observation)
• $k < 1$ — contraction parameter
• $\Gamma_{\mathrm{anchor}}$ — E-accented anchor

CPTP verification: For arbitrary $\alpha \in [0,1]$:

$\varphi_{\mathrm{coh}} = k \cdot \mathcal{P}_\alpha + (1-k) \cdot \mathrm{const}$

where $\mathcal{P}_\alpha = \alpha \mathcal{P}_{\mathrm{base}} + (1-\alpha) \mathcal{P}_{\mathrm{Fano}}$ is a convex combination of CPTP channels, hence CPTP. $\checkmark$

Theorem 10.3 (Target Coherences of $\varphi_{\mathrm{coh}}$)

Status: Theorem [T]

For canonical $\varphi_{\mathrm{coh}}$ the target coherences are determined as follows.

(a) Modulus of the target coherence:

$|\gamma_{ij}^{\mathrm{target}}| = \left[\frac{k(1-\alpha)}{3}\right] \cdot |\gamma_{ij}| + (1-k) \cdot [\Gamma_{\mathrm{anchor}}]_{ij}$

For a diagonal anchor ($[\Gamma_{\mathrm{anchor}}]_{ij} = 0$ for $i \neq j$):

$|\gamma_{ij}^{\mathrm{target}}| = \frac{k(1-\alpha)}{3} \cdot |\gamma_{ij}|$

(b) Target phase:

$\theta_{ij}^{\mathrm{target}} = \theta_{ij} \quad \text{(phase is preserved!)}$

(c) Target Gap:

$\mathrm{Gap}^{\mathrm{target}}(i,j) = |\sin(\theta_{ij})| = \mathrm{Gap}(i,j) \quad \text{(Gap is preserved!)}$

Fundamental corollary. Canonical $\varphi_{\mathrm{coh}}$ does not tend to change the Gap — it tends to reproduce the Gap with reduced amplitude. The target state does not destroy coherences, but scales them.

Theorem 10.4 (Variational Determination of $\alpha^*$)

Status: Theorem [T]

The optimal parameter $\alpha^*$ is determined by a variational principle.

$\alpha^* = \arg\min_{\alpha \in [0,1]} \mathcal{F}[\mathcal{P}_\alpha; \Gamma] = \arg\min_{\alpha} \left[S_{\mathrm{spec}}(\mathcal{P}_\alpha(\Gamma)) + D_{KL}(\mathcal{P}_\alpha(\Gamma) \| \Gamma)\right]$

(a) At $\alpha = 1$ (purely atomic): $\mathcal{P}_1 = \mathcal{P}_{\mathrm{base}}$, destroys all coherences. $D_{KL}$ is large (information about coherences is lost). $S_{\mathrm{spec}}$ is maximal (complete decoherence).

(b) At $\alpha = 0$ (purely Fano): $\mathcal{P}_0 = \mathcal{P}_{\mathrm{Fano}}$, preserves coherences with factor $1/3$. $D_{KL}$ is small (little information is lost). But $S_{\mathrm{spec}}$ is not minimal (the predictive model is less precise).

(c) The optimum $\alpha^* \in (0, 1)$ is a balance between predictive accuracy (atomic observation) and structure preservation (Fano observation).

(d) For a system with purity $P > P_{\mathrm{crit}}$:

$\alpha^* \approx 1 - \frac{P_{\mathrm{crit}}}{P} = 1 - \frac{2}{7P}$

At $P = 1$ (pure state): $\alpha^* \approx 5/7 \approx 0.71$ — significant Fano contribution.

At $P \to P_{\mathrm{crit}}$: $\alpha^* \to 0$ — almost entirely Fano (minimal destruction of coherences for survival).

Proof. Minimisation of $\mathcal{F}$ over $\alpha$ at fixed $P$ determines the balance: increasing $\alpha$ improves predictive accuracy ($S_{\mathrm{spec}}$ decreases), but increases coherence loss ($D_{KL}$ grows). The condition $P > P_{\mathrm{crit}}$ requires preserving a sufficient number of coherences, which bounds $\alpha$ from above. The optimum is found from $\partial \mathcal{F}/\partial \alpha = 0$. $\blacksquare$

Theorem 10.5 (Explicit Coefficients of $\varphi_{\mathrm{coh}}$)

Status: Theorem [T]

The Kraus operators of canonical $\varphi_{\mathrm{coh}}$ take a specific form.

Atomic operators (7 in total):

$K_m^{(\mathrm{atom})} = \sqrt{\alpha^* k / 7} \cdot |m\rangle\langle m|, \quad m = 1, \ldots, 7$

Fano operators (7 in total):

$K_p^{(\mathrm{Fano})} = \sqrt{(1-\alpha^*) k / 3} \cdot \Pi_p, \quad p = 1, \ldots, 7$

Anchor operator:

$K_0^{(\mathrm{anch})} = \sqrt{1-k} \cdot \Gamma_{\mathrm{anchor}}^{1/2}$

CPTP verification:

$\sum_{m=1}^{7} (K_m^{(\mathrm{atom})})^\dagger K_m^{(\mathrm{atom})} + \sum_{p=1}^{7} (K_p^{(\mathrm{Fano})})^\dagger K_p^{(\mathrm{Fano})} + (K_0^{(\mathrm{anch})})^\dagger K_0^{(\mathrm{anch})}$

First term: $\alpha^* k / 7 \cdot 7I = \alpha^* k \cdot I$.

Second term: $(1-\alpha^*) k / 3 \cdot 3I = (1-\alpha^*) k \cdot I$.

Third term: $(1-k) \cdot I$.

Total = $I$. $\checkmark$

Corollary. The coefficients $c_{mn}$ are determined by:

$c_{mn} = \begin{cases} \alpha^* k & \text{for } m = n \text{ (atomic part)} \\ (1-\alpha^*) k / 3 & \text{for } m \neq n, (m,n) \text{ on a common Fano line} \\ 0 & \text{for } m \neq n, (m,n) \text{ not on a common Fano line} \end{cases}$

The coefficients are fully determined by:

• The Fano structure $\mathrm{PG}(2,2)$ (algebraic geometry)
• The variational principle ($\alpha^*$ via $P$ and $P_{\mathrm{crit}}$)
• The contraction parameter $k$

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7. $G_2$-Covariance

Theorem 11.1 (Atomic Dissipator is NOT $G_2$-Covariant)

Status: Theorem — negative result [T]

The dissipative channel with atomic Lindblad operators $L_k = |k\rangle\langle k|$ is not $G_2$-covariant.

$\exists g \in G_2: \quad \mathcal{D}_{\mathrm{atom}}[g\Gamma g^\dagger] \neq g \, \mathcal{D}_{\mathrm{atom}}[\Gamma] \, g^\dagger$

Proof.

(a) Atomic dissipator:

$\mathcal{D}_{\mathrm{atom}}[\Gamma] = \sum_{k=1}^{7} L_k \Gamma L_k^\dagger - \Gamma = \sum_k |k\rangle\langle k| \Gamma |k\rangle\langle k| - \Gamma = \mathrm{diag}(\Gamma) - \Gamma$

(b) Action of $G_2$: for $g \in G_2$, $g\Gamma g^\dagger \mapsto D(g)\Gamma D(g)^\dagger$, where $D(g)$ is the 7-dimensional representation of $G_2$.

(c) We check covariance:

$\mathcal{D}_{\mathrm{atom}}[g\Gamma g^\dagger] = \mathrm{diag}(g\Gamma g^\dagger) - g\Gamma g^\dagger$

$g \, \mathcal{D}_{\mathrm{atom}}[\Gamma] \, g^\dagger = g[\mathrm{diag}(\Gamma) - \Gamma]g^\dagger = g \cdot \mathrm{diag}(\Gamma) \cdot g^\dagger - g\Gamma g^\dagger$

(d) Equality requires:

$\mathrm{diag}(g\Gamma g^\dagger) = g \cdot \mathrm{diag}(\Gamma) \cdot g^\dagger \quad \forall \Gamma$

This means: 'diagonal of the transformed matrix = transform of the diagonal'. This holds only for diagonal $g$ (permutations + phases), but NOT for general $g \in G_2$.

(e) Counterexample: take $g$ = rotation by angle $\pi/4$ in the plane $(e_1, e_2)$. For a matrix $\Gamma$ with $\gamma_{12} \neq 0$:

$\mathrm{diag}(g\Gamma g^\dagger) \neq g \cdot \mathrm{diag}(\Gamma) \cdot g^\dagger$

since the left-hand side annihilates the coherence $\gamma_{12}$ in the rotated basis, while the right-hand side does not. $\blacksquare$

Theorem 11.2 (Fano Dissipator is $G_2$-Covariant)

Status: Theorem [T]

The dissipative channel with Fano-structured Lindblad operators $L_p^{\mathrm{Fano}} = \frac{1}{\sqrt{3}}\Pi_p$ is $G_2$-covariant.

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

Proof.

(a) The group $G_2 = \mathrm{Aut}(\mathbb{O})$ preserves octonionic multiplication. The Fano plane $\mathrm{PG}(2,2)$ is defined by the structure constants of $\mathbb{O}$. Therefore $G_2$ preserves the Fano structure:

$g \in G_2 \quad \Rightarrow \quad g \text{ permutes the Fano lines}$

More precisely: for each $g \in G_2$ there exists a permutation $\sigma_g$ on the set $\{1, \ldots, 7\}$ of lines:

$g\, \Pi_p\, g^\dagger = \Pi_{\sigma_g(p)}$

(b) Fano dissipator:

$\mathcal{D}_{\mathrm{Fano}}[\Gamma] = \frac{1}{3}\sum_{p=1}^{7} \Pi_p \Gamma \Pi_p - \Gamma$

(c) Substituting $g\Gamma g^\dagger$:

$\mathcal{D}_{\mathrm{Fano}}[g\Gamma g^\dagger] = \frac{1}{3}\sum_p \Pi_p (g\Gamma g^\dagger) \Pi_p - g\Gamma g^\dagger$

(d) Using $g^\dagger \Pi_p g = \Pi_{\sigma_g^{-1}(p)}$ (from (a)):

$= \frac{1}{3}\sum_p \Pi_p g \Gamma g^\dagger \Pi_p = \frac{1}{3}\sum_p g (g^\dagger \Pi_p g) \Gamma (g^\dagger \Pi_p g) g^\dagger$

$= \frac{1}{3}g\left[\sum_p \Pi_{\sigma_g^{-1}(p)} \Gamma \Pi_{\sigma_g^{-1}(p)}\right]g^\dagger$

Since $\sigma_g$ is a permutation: $\sum_p \Pi_{\sigma_g^{-1}(p)} = \sum_q \Pi_q$ (reindexing). Therefore:

$= g \left[\frac{1}{3}\sum_q \Pi_q \Gamma \Pi_q\right] g^\dagger = g \, \mathcal{P}_{\mathrm{Fano}}(\Gamma) \, g^\dagger$

And:

$\mathcal{D}_{\mathrm{Fano}}[g\Gamma g^\dagger] = g \, \mathcal{P}_{\mathrm{Fano}}(\Gamma) \, g^\dagger - g\Gamma g^\dagger = g[\mathcal{P}_{\mathrm{Fano}}(\Gamma) - \Gamma]g^\dagger = g \, \mathcal{D}_{\mathrm{Fano}}[\Gamma] \, g^\dagger$

$\blacksquare$

Theorem 11.3 (Degree of $G_2$ Breaking is Determined by $\alpha$)

Status: Theorem [T]

For canonical $\varphi_{\mathrm{coh}}$ with parameter $\alpha$, the degree of $G_2$-covariance is determined as follows.

(a) At $\alpha = 0$ (purely Fano): full $G_2$-covariance. The gauge reduction $48 \to 34$ is valid.

(b) At $\alpha = 1$ (purely atomic): $G_2$ is fully broken. No gauge reduction.

(c) For intermediate $\alpha \in (0, 1)$: partial $G_2$-covariance. Mixed channel:

$\mathcal{P}_\alpha = \alpha \, \mathcal{P}_{\mathrm{base}} + (1 - \alpha) \, \mathcal{P}_{\mathrm{Fano}}$

Measure of $G_2$-symmetry breaking:

$\Delta_{G_2}(\alpha) := \sup_{g \in G_2} \|\mathcal{P}_\alpha \circ \mathrm{Ad}_g - \mathrm{Ad}_g \circ \mathcal{P}_\alpha\|_{\mathrm{op}}$

where $\mathrm{Ad}_g(\Gamma) = g\Gamma g^\dagger$.

(d) $\Delta_{G_2}(\alpha)$ increases monotonically with $\alpha$:

$\Delta_{G_2}(0) = 0, \quad \Delta_{G_2}(1) = \Delta_{\max} > 0$

(e) At optimal $\alpha^* \approx 1 - 2/(7P)$:

$\Delta_{G_2}(\alpha^*) = \alpha^* \cdot \Delta_{\max}$

Proof. (a)–(b): direct consequence of Theorems 11.1 and 11.2. (c)–(e): $\mathcal{P}_\alpha$ is a convex combination of the $G_2$-covariant ($\mathcal{P}_{\mathrm{Fano}}$) and $G_2$-breaking ($\mathcal{P}_{\mathrm{base}}$) channels. The measure of breaking is linear in $\alpha$ (from the linearity of both channels). $\blacksquare$

warning

Limits of $G_2$-Covariance

• Fano dissipator $\mathcal{D}_{\text{Fano}}$: $G_2$-covariant [T] (T-11.2)
• Atomic dissipator $\mathcal{D}_{\text{atom}}$: NOT $G_2$-covariant [T] (T-11.1)
• Full dynamics $\mathcal{L}_\Omega = \mathcal{D}_{\text{atom}} + \mathcal{D}_{\text{Fano}} + \mathcal{R}$: $G_2$-covariance of the full evolution — [C] (depends on the fraction of atomic vs Fano component)

Theorem 11.4 (Modified Gauge Reduction)

Status: Theorem [T]

Under partial $G_2$-covariance ($\alpha \in (0,1)$), the parameter space of Gap profiles is reduced.

(a) Full $G_2$ ($\alpha = 0$): $48 - 14 =$ 34 independent parameters.

(b) Partial $G_2$ (optimal $\alpha^*$): $34 + 14\alpha^*$ parameters. The number of 'additional' parameters $= 14\alpha^* \approx 14(1 - 2/(7P))$.

(c) No $G_2$ ($\alpha = 1$): 48 parameters (full space).

For a typical living system with $P \approx 0.5$: $\alpha^* \approx 0.43$, number of parameters $\approx 34 + 6 =$ 40. Reduction from 48 to 40 — moderate but significant.

For a highly coherent system with $P \approx 0.8$: $\alpha^* \approx 0.64$, number of parameters $\approx 34 + 9 =$ 43. The reduction is even more moderate.

Interpretation. Self-observation (nonzero $\alpha$) partially breaks the algebraic symmetry of the octonions. The deeper the self-knowledge (larger $\alpha$), the more broken the $G_2$-symmetry, and the more parameters are needed to describe the system. This is the fundamental 'price of self-knowledge': knowledge about oneself increases the complexity of self-description.

Updated Diagnostic Protocol

Taking into account partial $G_2$-covariance:

Mode	Number of parameters	Protocol
$\alpha = 0$ (no self-knowledge, L0)	34 (full $G_2$)	Minimal tomography: 7 populations + 7 moduli + 7 phases + $G_2$ relations
$\alpha^* \approx 0.4$ (typical L2 system)	$\sim$40	Extended tomography: 7 + 12 moduli + 12 phases + partial $G_2$ relations
$\alpha = 1$ (full L4)	48 (no $G_2$)	Full tomography: all 48 parameters

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8. Unified Theorem of Self-Observation and Gap

Theorem 12.1 (Fano-Coherent Self-Modelling)

Status: Theorem [T]

Canonical coherence-preserving self-modelling for UHM theory is determined uniquely (up to the contraction parameter $k$).

(a) Algebraic structure: The Fano plane $\mathrm{PG}(2,2)$ determines the composite atoms of the classifier $\Omega$, generating the Fano–Lindblad operators $L_p^{\mathrm{Fano}}$.

(b) Variational principle: The balance of atomic and Fano observation $\alpha^*$ minimises the functional $\mathcal{F} = S_{\mathrm{spec}} + D_{KL}$.

(c) Phase properties: Canonical $\varphi_{\mathrm{coh}}$ preserves the phases of coherences. The target Gap coincides with the current Gap (amplitude scaling without phase distortion).

(d) Symmetry: $G_2$-covariance is partially broken by the atomic component. The degree of breaking $\Delta_{G_2} = \alpha^* \cdot \Delta_{\max}$ depends on the purity $P$.

(e) Stationary Gap: Substituting into the stationary equation with $\theta_{ij}^{\mathrm{target}} = \theta_{ij}$ gives:

$\mathrm{Gap}^{(\infty)}(i,j) = \left|\sin\left(\theta_{ij} - \arctan\left(\frac{\Delta\omega_{ij}}{\Gamma_2 + \kappa}\right)\right)\right|$

The stationary Gap is shifted relative to the current one by the angle $\arctan(\Delta\omega/(\Gamma_2 + \kappa))$ — even with phase-preserving $\varphi_{\mathrm{coh}}$, unitary rotation creates a difference between the target and the stationary.

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Related documents:

• Standard Model from G₂
• Fano Selection Rules
• G₂-Noether Charges
• Confinement