Statics / Kinematics

Core Objects

The 24+3 Split

The split is intrinsic to the Jordan algebra, not imposed:

The diagonal entries are already algebraically different from the off-diagonal entries. The 24-dimensional off-diagonal sector over integral octonions contains a Leech-isometric sublattice (Baez/Egan). This is the strongest established result in the program.

The Two-Sector Ontology

The Leech lattice being rootless is physically significant: roots generate local reflections. A rootless lattice has no reflection generators — nothing acts on it locally. This makes it a natural candidate for a passive, read-only global state space.

The Leech consistency layer

The Golay code’s minimum weight 8 means the smallest nonzero valid configuration touches 8 coordinates. Geometrically:

Small local defects are not admissible states. This is the geometric analogue of error correction operating on the state space itself.

Non-perturbative physics

• Perturbative physics = local fluctuations = rejected by Leech (no roots)
• Non-perturbative physics = global topology = exactly what Leech admits
• Non-perturbative effects are invisible to perturbation theory because they live on the Leech tier, which perturbative expansion cannot reach by local steps

The Snap Coupling

The natural packaging map

is only bookkeeping. The bridge requires a second map:

specifying how the diagonal entries act on the off-diagonal Leech sector. The natural candidate is the Jordan product $D \circ U$. The test: this map must be equivariant under the residual $SU(3)$ symmetry. Until that is verified, the diagonal bridge is a structural suggestion, not a coupling law.

SU(3) as Color: What is Still Missing

The stabilizer isomorphism is step 1 only:

What is needed:

plus a local gauge connection, coupling law, and anomaly check.

The octonionic structure does produce 3- and 8-dimensional objects naturally under the $u$-selection. The gap is from “present” to “gauged with correct coupling and anomaly structure.”

Honest claim status: candidate color symmetry, not derived color. This may be a permanent structural rather than forced result — some features of the universe may be natural without being uniquely derivable.

Leech Equivariance Check — Results

Part 1: Fixing the Baez/Egan Embedding

Lattice conventions. $E_8$ with min squared norm 2 is self-dual ($\det = 1$). The off-diagonal sector $\mathbb{O}\mathbb{Z}^3$ with the component-sum inner product is $E_8^3$ (min norm 2, det 1, rank 24). The Leech lattice $\Lambda{24}$ in standard normalization has min norm 4, det 1, rank 24.

The correct containment direction. Since both $E_8^3$ and $\Lambda_{24}$ are even unimodular of rank 24 but non-isometric (different min norms), neither can be a proper sublattice of the other in $\mathbb{R}^{24}$ (equal determinants force equal index, hence equality — but they are not isometric). The Baez/Egan embedding uses the scaled Leech lattice $\sqrt{2}\Lambda_{24}$, which has min norm 8 and det $2^{12}$. The index formula

shows that $\sqrt{2}\Lambda_{24}$ is an index-64 sublattice of $E_8^3$. Integrality is satisfied: inner products in $\sqrt{2}\Lambda_{24}$ lie in $2\mathbb{Z} \subset \mathbb{Z}$, so $\sqrt{2}\Lambda_{24}$ is a valid even sublattice of $E_8^3$.

The precise statement of Baez/Egan: $\mathbb{O}\mathbb{Z}^3 \cong E_8^3$ contains $\sqrt{2}\Lambda{24}$ as a sublattice of index 64. The quotient $E_8^3/\sqrt{2}\Lambda_{24}$ has order 64. The embedding uses the octonionic multiplication: the triality structure of $D_4 \subset E_8$ (arising from $\mathbb{H} \subset \mathbb{O}$) interleaves the three $E_8$ components to select the Leech sublattice. The embedding is not unique — there are multiple index-64 sublattices of $E_8^3$ isometric to $\sqrt{2}\Lambda_{24}$, related by $\mathrm{Aut}(E_8^3) = W(E_8)^3 \rtimes S_3$.

Note on previous notation. Prior notes (“off-diagonal sector contains $\Lambda_{24}$”) are imprecise: what is contained is $\sqrt{2}\Lambda_{24}$ (Leech at scale $\sqrt{2}$), an index-64 sublattice of $E_8^3$. In the Jordan trace inner product (which rescales the off-diagonal sector by $\sqrt{2}$ to give $\sqrt{2}E_8^3$ with min norm 4), one instead says $\Lambda_{24} \subset \sqrt{2}E_8^3$ of index 64 — both formulations describe the same geometry.

Fixing a canonical embedding. A canonical choice of $\sqrt{2}\Lambda_{24} \subset E_8^3$ is equivalent to fixing a triality frame: a choice of $D_4 \subset E_8$ compatible with the octonionic structure, plus a triality automorphism $\tau$ of order 3 acting on the three $E_8$ components. The $u$-selection fixes a quaternionic plane $\mathbb{H} \subset \mathbb{O}$ (Hurwitz quaternions), which specifies $D_4 \subset E_8$. However, fixing $\mathbb{H}$ does not fix $\tau$ uniquely — a residual $SU(2)$ ambiguity remains from $u^\perp \cap \mathbb{H}$. The triality frame is fully fixed only after the Furey-Hughes step that selects the electroweak $\mathbb{H}$ direction.

Part 2: Equivariance Under $SU(3)$

Continuous equivariance is ill-posed. $SU(3)$ is a connected Lie group acting continuously on $\mathbb{R}^{24}$ via $(X,Y,Z) \mapsto (g(X), g(Y), g(Z))$. The orbit of any point in the discrete lattice $\sqrt{2}\Lambda_{24}$ under this action is a connected submanifold of $\mathbb{R}^{24}$. A non-trivial connected orbit cannot be discrete. Therefore:

• A continuous connected group preserves a discrete set only if it acts trivially on that set.
• $SU(3)$ acts faithfully on $\mathbb{O}$, hence non-trivially on $E_8^3$.
• Continuous $SU(3)$ equivariance of $\sqrt{2}\Lambda_{24}$ is impossible by topology, not merely unverified.

The failure condition in 0 - master.md — “no equivariant embedding under the residual $SU(3)$ can be constructed” — must be read as referring to discrete equivariance.

The correct question. Define: \(G_2(\mathbb{Z}) := G_2 \cap \mathrm{Aut}(E_8^3) \subset SO(24)\) the finite subgroup of $G_2$ that preserves the integral lattice $E_8^3$. The relevant symmetry is $SU(3)_{\mathrm{disc}} := SU(3) \cap G_2(\mathbb{Z})$ — the discrete $u$-stabilizer subgroup compatible with the lattice structure.

Equivariance of the canonical embedding. The Baez/Egan embedding is constructed from the ring structure of $\mathbb{O}\mathbb{Z}$ (via octonionic multiplication and the triality from $\mathbb{H} \subset \mathbb{O}$). Any $g \in G_2(\mathbb{Z})$ is an automorphism of $\mathbb{O}\mathbb{Z}$ as a ring; it therefore maps the Baez/Egan Leech sublattice to another Baez/Egan Leech sublattice (the one defined by the frame $g(\mathbb{H})$). Restricting to $SU(3){\mathrm{disc}}$ (elements that additionally fix $u$): these map $\mathbb{H}$ to another quaternionic plane in $u^\perp$. The specific Leech sublattice $\sqrt{2}\Lambda{24}$ is preserved if and only if the triality frame is also preserved, i.e., if the $SU(2)$ ambiguity in $\mathbb{H}$ is fixed.

Conclusion for Part 2: Discrete equivariance holds in the following sense: the Baez/Egan construction is $G_2(\mathbb{Z})$-natural (it uses only ring structure), so $G_2(\mathbb{Z})$ permutes the set of canonical Leech sublattices. The specific sublattice $\sqrt{2}\Lambda_{24}$ is preserved by those elements of $SU(3)_{\mathrm{disc}}$ that also fix the triality frame. Fixing the frame requires the additional Furey-Hughes $\mathbb{H}$-selection step. Discrete $SU(3)$ equivariance is conditional on a fixed triality frame; with that frame fixed, it holds.

Part 3: Jordan Product $D \circ U$ — Preservation Check

Explicit computation. For diagonal $D = \mathrm{diag}(d_1, d_2, d_3)$ with $d_i \in \mathbb{R}$ and off-diagonal $U = (X,Y,Z) \in \mathbb{O}^3$:

(real scalars commute with octonions, and $D$ is diagonal). Therefore:

The Jordan product maps $(X, Y, Z) \mapsto (\lambda_{12} X,\, \lambda_{13} Y,\, \lambda_{23} Z)$ with scaling factors

Integrality. For $D \circ U \in \mathbb{O}\mathbb{Z}^3$ when $U \in \mathbb{O}\mathbb{Z}^3$: need $\lambda_{ij} \in \mathbb{Z}$, i.e., all $d_i$ the same parity.

Solving the system $d_i + d_j = \varepsilon_{ij} \cdot 2$ ($\varepsilon_{ij} \in {\pm 1}$) over all pairs:

For mixed-sign cases, e.g. $(\varepsilon_{12},\varepsilon_{13},\varepsilon_{23}) = (+,-,-)$: solving gives $(d_1,d_2,d_3) = (1,-1,-3)$ and $\lambda_{23} = -2 \neq \pm 1$. Not an isometry.

The only consistent solutions with all $d_i = \pm 1$ are $d_1=d_2=d_3=\pm 1$.

Conclusion for Part 3. The Jordan product $D \circ U$ maps $\sqrt{2}\Lambda_{24}$ to itself only when $D = \pm I$. For any other snap state $(d_1,d_2,d_3) \neq (\pm 1, \pm 1, \pm 1)$ with all entries equal, the map scales the three $E_8$ components by different or non-unit factors, producing a vector in $E_8^3 \setminus \sqrt{2}\Lambda_{24}$. The snap coupling $D \circ U$ generically exits the Leech tier.

Physical Interpretation of the Jordan Product Result

The failure of $D \circ U$ to preserve $\sqrt{2}\Lambda_{24}$ is structurally significant:

Reading A — wrong coupling law. The Jordan product is natural to $J_3(\mathbb{O})$ but not to the Leech tier. The correct coupling $\Gamma: \mathbb{R}^3 \to \mathrm{End}(\mathbb{O}^3)$ requires a Leech-projecting modification, e.g. $\Pi \circ (D \circ U)$ where $\Pi$ is a projection onto $\sqrt{2}\Lambda_{24}$. This would require specifying $\Pi$ non-trivially.

Reading B — tier-transition coupling. The snap coupling is not supposed to preserve the Leech tier; its job is to couple the passive tier to the active sector. The image $D \circ U \in E_8^3$ lies in one of the 64 cosets of $\sqrt{2}\Lambda_{24}$ in $E_8^3$. The index-64 quotient $E_8^3/\sqrt{2}\Lambda_{24} \cong (\mathbb{Z}/2)^6$ encodes the 64 “activated states” above the Leech floor. The snap coordinate $D$ selects which coset class the coupling lands in — a $2^6$-valued observable. Reading B is more compatible with the two-sector ontology: the Leech tier is ground-floor and $D \circ U$ defines a coupling into the quotient. The number 64 may have further structure (e.g., $64 = 2^6$, the weight-8 Golay codewords, the minimum weight of $\mathcal{G}_{24}$).

Reading B is structurally preferable because it does not require a modification of the Jordan product and turns the failure of preservation into a positive feature (the structure of the activated sector is determined by the Leech/E8 geometry).

Part 4: Index-64 Quotient Structure

Theorem. $E_8^3/\sqrt{2}\Lambda_{24} \cong (\mathbb{Z}/2)^6$.

Proof. Since $\sqrt{2}\Lambda_{24} \subset E_8^3$ and both are even unimodular of rank 24, dualizing the inclusion reverses it: $(E_8^3)^* \subset (\sqrt{2}\Lambda_{24})^$, i.e., $E_8^3 \subset \tfrac{1}{\sqrt{2}}\Lambda_{24}$ (using $E_8^{3} = E_8^3$ and $(\sqrt{2}\Lambda_{24})^* = \tfrac{1}{\sqrt{2}}\Lambda_{24}^* = \tfrac{1}{\sqrt{2}}\Lambda_{24}$). Therefore $2v \in \sqrt{2}\Lambda_{24}$ for every $v \in E_8^3$. Every element of the quotient has order $\leq 2$. A group of order $2^6$ killed by 2 is $(\mathbb{Z}/2)^6$. ☐

Chain structure. The duality argument gives the three-step chain \(\sqrt{2}\Lambda_{24} \;\subset\; E_8^3 \;\subset\; \tfrac{1}{\sqrt{2}}\Lambda_{24}\) with $[E_8^3 : \sqrt{2}\Lambda_{24}] = 2^6$ and $[\tfrac{1}{\sqrt{2}}\Lambda_{24} : E_8^3] = 2^6$ (both equal by symmetry of the unimodular case). $E_8^3$ sits precisely at the midpoint of the scale-2 Leech chain. This is the lattice analogue of a self-dual code sitting midway in a glue tower.

Hexacode identification (structural). The MOG (Miracle Octad Generator) construction of $\Lambda_{24}$ organizes $\mathbb{R}^{24}$ as a $4\times 6$ array. The hexacode $\mathcal{H}6$ is a $[6,3,4]{\mathbb{F}4}$ linear code with $4^3 = 64$ codewords. As an additive group, $\mathcal{H}_6 \cong (\mathbb{F}_4,+)^3 \cong (\mathbb{Z}/2)^6$. In the MOG construction, a vector $(x,y,z)\in E_8^3$ belongs to $\sqrt{2}\Lambda{24}$ iff its MOG coordinate satisfies the zero hexacodeword condition; the other 63 cosets correspond bijectively to the 63 non-zero elements of $\mathcal{H}_6$. This identifies

as additive groups. Status: the abstract group isomorphism $\cong(\mathbb{Z}/2)^6$ is a theorem (above). The explicit identification of coset labels with hexacode codewords is structurally established via the MOG; writing it as an explicit coset-to-codeword bijection requires fixing the MOG frame, which is precisely the triality frame of §4b.

$S_3$ symmetry. The three $E_8$ factors of $E_8^3$ are permuted by $S_3 \cong \mathrm{Aut}(D_4)/\mathrm{Inn}(D_4)$. The hexacode group $(\mathbb{F}_4)^3$ is permuted by $S_3$ acting on the three $\mathbb{F}_4$ coordinates. These actions are compatible: the $S_3$ symmetry of $E_8^3$ acts on the quotient $(\mathbb{Z}/2)^6 \cong (\mathbb{F}_4)^3$ by permuting the three $\mathbb{F}_4$ factors, each of which contributes exactly 2 bits (one $\mathbb{F}_4 \cong (\mathbb{Z}/2)^2$ per $E_8$ component). This is the “2 bits per $E_8$ factor” structure.

Golay-admissibility. The hexacode has minimum distance 4: every non-zero codeword has weight $\geq 4$, meaning it is supported on $\geq 4$ of the 6 hexacode positions. In the MOG layout, each hexacode position corresponds to one column of 4 rows; weight $\geq 4$ activates $\geq 4$ columns. A weight-4 hexacodeword activates exactly $4 \times 2 = 8$ of the 24 $E_8^3$-coordinate directions — matching the Golay minimum weight 8. Therefore:

• Coset 0 (the Leech ground state) is the unique coset with weight 0.
• All 63 non-zero cosets have hexacode weight $\geq 4$, meaning any deviation from the Leech tier must touch $\geq 8$ coordinates.
• There are no weight-1, 2, or 3 cosets: “small” local deviations from the Leech tier are not admissible coset labels. This is the mathematical formulation of the error-correcting floor — the Golay minimum-weight constraint operates directly on the coset structure.

Snap coupling coset landing. The Jordan product $D \circ U$ for $D = \mathrm{diag}(d_1,d_2,d_3)$ maps $U = (X,Y,Z) \in \sqrt{2}\Lambda_{24}$ to the coset labeled by $(\lambda_{12} \bmod \mathbb{F}4,\, \lambda{13} \bmod \mathbb{F}4,\, \lambda{23} \bmod \mathbb{F}4) \in (\mathbb{F}_4)^3$, where $\lambda{ij} = (d_i+d_j)/2$. The coset is 0 (Leech ground state) iff $\lambda_{12} \equiv \lambda_{13} \equiv \lambda_{23} \equiv 0 \pmod{2}$, which holds iff all $d_i$ have the same parity and $\lambda_{ij} \in 2\mathbb{Z}$ for all pairs — satisfied exactly when $D = \pm I$. For all other snap states, $D \circ U$ lands in a non-zero coset and is Golay-admissible by the minimum-weight property above.

Hexacode weight distribution. The hexacode $\mathcal{H}6$ as a $[6,3,4]{\mathbb{F}_4}$ code (equivalently a 3-dimensional $\mathbb{F}_4$-vector space) has weight enumerator $A_0 = 1$, $A_4 = 45$, $A_6 = 18$ (total: 64 ✓). Among the 63 non-zero cosets: 45 are weight-4 (octad-type, 71%) and 18 are weight-6 (hexad-type, 29%). This can be verified from the systematic generator matrix: messages with exactly one or two non-zero components always give weight-4 codewords (36 of them); messages with all three components non-zero split 9 weight-4 and 18 weight-6.

What the snap state $D$ determines. The Jordan product $D \circ U$ with $D = \mathrm{diag}(d_1, d_2, d_3)$ maps the coset label in $(\mathbb{Z}/2)^6 \cong (\mathbb{F}4)^3$ to $(\lambda{12} \bmod \mathbb{F}4, \lambda{13} \bmod \mathbb{F}4, \lambda{23} \bmod \mathbb{F}_4)$. However, the hexacode weight is a property of the 6-dimensional codeword $c \in \mathcal{H}_6 \subset \mathbb{F}_4^6$, not the 3-dimensional information vector $(m_1, m_2, m_3) \in \mathbb{F}_4^3$. The weight depends on the generator matrix $G$, which encodes the MOG frame (= the triality frame from §4b).

What is and is not determined by $D$ alone. The snap state $D$ contributes the “inter-block” coset data: the 3 parities $(\lambda_{12} \bmod 2, \lambda_{13} \bmod 2, \lambda_{23} \bmod 2) \in {0,1}^3$, which are a function of $d_i \bmod 4$. The remaining 3 bits (intra-block, one per $E_8$ factor) come from the specific element $U \in \sqrt{2}\Lambda_{24}$ and depend on how $U$ is positioned within its $E_8$ block. Therefore the hexacode weight of the coset is NOT determined by $D$ alone — it also depends on $U$.

Consequence for the snap coupling. For a given snap state $D \neq \pm I$, the coset type (octad vs hexad) varies across different $U \in \sqrt{2}\Lambda_{24}$. The snap coupling $D \circ U$ samples different cosets as $U$ varies over the Leech tier. Whether the octad (weight-4) or hexad (weight-6) type is preferred depends on which $U$-values are dynamically selected — a question about the measure on the Leech tier, not on the algebra alone.

Conclusion for item 7. The hexacode weight cannot be determined from $D$ alone without the explicit triality frame (item 4b) fixing the MOG generator matrix. With that frame fixed: the full 6-bit coset label is computable, and for any given $(D, U)$ pair the weight-4 or weight-6 classification follows directly. The a priori probability of weight-4 vs weight-6 (assuming uniform measure on $U$) is 45:18 ≈ 71%:29%. Open: explicit triality frame needed to give specific snap states a definite weight classification.

Conclusion for item 5. The structure of the index-64 quotient is now determined:

1. $E_8^3/\sqrt{2}\Lambda_{24} \cong (\mathbb{Z}/2)^6$ — theorem (duality proof).
2. The 64 cosets biject with the hexacode $\mathcal{H}_6$ — structurally established (MOG; explicit bijection requires triality frame from §4b).
3. Golay-admissibility is automatic for all 63 non-zero cosets, with minimum activation size 8 — established from hexacode minimum distance 4.
4. The snap coupling $D \circ U$ lands on a non-trivial (Golay-admissible) coset for all $D \neq \pm I$ — established from Part 3.

The remaining open question: does the specific coset landed on by a given snap state $D$ correspond to a weight-4 (octad) or weight-6 hexacodeword? Weight-4 codewords (octads) are the most constrained activated states. Whether generic snap states produce octad-type or hexad-type coset labels is not yet determined.

Status

Triality Frame from Furey-Hughes: Results

The key structural fact: $G_2 = SO(8)^\tau$

The triality automorphism $\tau$ of $D_4 \cong \mathfrak{so}(8)$ is an outer automorphism of order 3. The classical theorem (Élie Cartan; see also Adams) is:

$G_2$ is precisely the fixed-point subgroup of the $D_4$ triality in $SO(8)$. This is why $G_2$ is the automorphism group of $\mathbb{O}$: the octonionic multiplication is the structure preserved by $\tau$-fixed isometries.

This fact has an immediate consequence for the Leech embedding: any construction of $\sqrt{2}\Lambda_{24} \subset E_8^3$ that is built from $\tau$-covariant conditions is automatically covariant under the diagonal $G_2$ action at the level of the ambient algebraic data. This does not by itself show that the full continuous group acts as a symmetry of the discrete integral lattice; that stronger claim is ruled out by the topology argument above.

What $\mathbb{H}$-selection determines

Step 1 ($u$-selection → $D_4$). Fixing $u \in \mathrm{Im}(\mathbb{O})$ singles out the subalgebra $\mathrm{span}(1, u) \cong \mathbb{C} \subset \mathbb{O}$. The stabilizer $\mathrm{Stab}(u) = SU(3) \subset G_2$ and $u^\perp \cong \mathbb{C}^3$.

This does not yet determine $\mathbb{H}$. There is a residual $SU(2)$ freedom: the set of quaternionic subalgebras $\mathbb{H} \ni u$ forms a 2-sphere $S^2 \cong SU(3)/U(2)$ within $u^\perp$.

Step 2 ($\mathbb{H}$-selection → $D_4 \subset E_8$). Fixing $\mathbb{H} \subset \mathbb{O}$ (with $u \in \mathbb{H}$) determines the Cayley-Dickson decomposition $\mathbb{O} = \mathbb{H} \oplus \mathbb{H}\ell$ for a unit $\ell \perp \mathbb{H}$. The Hurwitz quaternion sublattice $\mathbb{H}\mathbb{Z} \subset \mathbb{O}\mathbb{Z}$ is then a specific rank-4 sublattice of $E_8$, isometric to $D_4^+$ (the $F_4$ root lattice, or equivalently $D_4$ with minimum norm 1 before rescaling to match $E_8$’s normalization).

The $D_4$ sublattice of $E_8$ is fixed by $\mathbb{H}$, and the $D_4$ Dynkin diagram carries a canonical order-3 symmetry (the triality $\tau$) that extends uniquely (up to inversion) to an automorphism of $E_8$.

Step 3 (octonionic orientation → uniqueness of $\tau$). The extension of $D_4$-triality to $E_8$-triality has a residual $\mathbb{Z}/2$ ambiguity: $\tau$ vs $\tau^{-1} = \tau^2$. These correspond to swapping two of the three $\tau$-orbits of $D_4$ representations, equivalently swapping the two spinor representations $S^+, S^-$ of $\mathfrak{so}(8)$ while fixing the vector $V$.

This ambiguity is resolved by the orientation of $\mathbb{O}$. The octonionic multiplication table specifies a particular orientation of $\mathrm{Im}(\mathbb{O}) \cong \mathbb{R}^7$ (via the associative 3-form $\phi(x,y,z) = \langle x, yz \rangle$). This orientation, together with the choice of $u$ and $\mathbb{H}$, gives an ordered frame $(u, v, w, uv, uw, vw, uvw)$ in $\mathrm{Im}(\mathbb{O})$ that distinguishes $\tau$ from $\tau^{-1}$.

Conclusion for item 4b: The two-step cascade $(u\text{-selection}) + (\mathbb{H}\text{-selection})$, with the octonionic orientation fixed by the multiplication table, fully determines the triality frame and hence the canonical Leech sublattice $\sqrt{2}\Lambda_{24} \subset E_8^3$. No further free choices are required.

The cascade encodes two structures simultaneously

The physical interpretation: the same cascade that forces the SM Higgs doublet also fixes the Leech ground state.

The program’s cascade is therefore doubly canonical: each step has a SM interpretation and a Leech/global-state-space interpretation, both forced by the same algebraic choice.

Clarification: the triality-based argument here is about covariance of the ambient algebraic construction. It should not be read as overturning Part 2’s topology result. A connected continuous group still cannot act nontrivially as a literal symmetry of a discrete integral lattice; the actual lattice symmetry remains the discrete subgroup compatible with the fixed triality frame.

Equivariance - clarified statement

Using $G_2 = SO(8)^\tau$ and the fact that the Leech sublattice is defined by $\tau$-covariant conditions, the most defensible conclusion is this: the canonical Leech embedding is $G_2$-covariant at the level of the ambient algebraic data. That explains why the discrete subgroup $G_2(\mathbb{Z})$ acts naturally on the integral construction once the triality frame is fixed. It should not be read as a literal continuous symmetry of the discrete lattice itself; at the lattice level, the actual symmetry remains the discrete subgroup compatible with the fixed triality frame.