Scope and Mathematical Architecture

The purpose of this note is to construct an explicit bridge between three classical structures:

$$\sqrt{2}D_4 \hookrightarrow \Lambda_{24} \longrightarrow \text{MOG coordinate data} \subset \mathbb F_4^6.$$

The construction has three logically distinct layers.

1. Lattice layer. An explicit $24\times4$ matrix $V$ embeds a scaled $D_4$ root system into the minimal shell of the Leech lattice.

2. Coding layer. An explicit Golay octad and MOG sextet identify four distinguished duads with four tetrads of the MOG. The ordinary Construction-A reduction and the shell-level reduction are kept conceptually separate.

3. Triality layer. The $24$ roots split into three eight-element sectors with identical metric behavior. These sectors realize the $8_v$, $8_s$, and $8_c$ triality decomposition of $D_4$.

No arithmetic modulus, elliptic curve, complex multiplication condition, or numerical discretization is required for this intrinsic construction.

In particular, the map from the lattice shell to the MOG coordinates is not asserted to be a single linear map on the whole lattice. This distinction is essential and will be made explicit below.

The Leech Coordinate Frame

We work in the standard Conway–Sloane Construction-A coordinate frame $$\Lambda_{24}\subset \frac{1}{\sqrt 8}\mathbb Z^{24}.$$

For integral representatives $$x\in\sqrt8\,\Lambda_{24},$$ the Leech inner product is $$\langle x,y\rangle_{\Lambda} = \frac18 x\cdot y.$$

A minimal vector has squared norm $4$, hence an integral representative satisfies $$\sum_{i=1}^{24}x_i^2=32.$$

The Construction-A description uses the extended binary Golay code $$\mathcal G_{24}\subset\mathbb F_2^{24}.$$ In particular, an integral representative may be characterized by the usual Golay congruence conditions $$x_i\equiv 2c_i\pmod4, \qquad c=(c_1,\ldots,c_{24})\in\mathcal G_{24},$$ together with the corresponding coordinate-sum condition.

We use $0$-indexed coordinates throughout.

Fix the Golay octad $$\mathcal O_0 = \{1,12,13,14,16,17,18,22\}.$$

It will be important that this octad is not subsequently identified with the union of two MOG tetrads. Instead, we choose a sextet for which it has intersection pattern $$(2,2,2,2,0,0).$$

Explicit Embedding of $\sqrt2D_4$

Let $e_i$ denote the standard coordinate vector of $\mathbb R^{24}$. Define four vectors in $\mathbb Z^{24}$ by $$\begin{align} v_1&=-4e_1-4e_{12},\\ v_2&= 2e_1+2e_{12}+2e_{13}+2e_{14} +2e_{16}+2e_{17}+2e_{18}+2e_{22},\\ v_3&=-4e_{13}-4e_{14},\\ v_4&=-4e_{16}-4e_{17}. \end{align}$$

These vectors are understood in the Construction-A coordinate frame, so their Leech norms are computed using the factor $1/8$.

Define the $24\times4$ matrix $$V= \begin{pmatrix} |&|&|&|\\ v_1&v_2&v_3&v_4\\ |&|&|&| \end{pmatrix}.$$

Membership in the Leech Minimal Shell

For $v_1,v_3,v_4$, $$\|v_i\|^2 = \frac18(16+16) = 4.$$

For $v_2$, $$\|v_2\|^2 = \frac18(8\cdot4) = 4.$$

The vectors $v_1,v_3,v_4$ are divisible by $4$, and hence satisfy the zero-codeword parity condition. Their coordinate sums are $$-8\equiv0\pmod8.$$

For $v_2$, $$\frac{v_2}{2}\pmod2 = \mathbf1_{\mathcal O_0},$$ and since $\mathcal O_0$ is a Golay octad, $$\mathbf1_{\mathcal O_0}\in\mathcal G_{24}.$$ Moreover, $$\sum_i(v_2)_i=16\equiv0\pmod8.$$

Thus all four vectors are minimal Leech vectors: $$v_1,v_2,v_3,v_4\in S_{24}.$$

The Exact Gram Matrix

The nonzero pairings are $$\begin{align} \langle v_1,v_2\rangle &= \frac18((-4)(2)+(-4)(2)) =-2,\\ \langle v_2,v_3\rangle &= \frac18((2)(-4)+(2)(-4)) =-2,\\ \langle v_2,v_4\rangle &= \frac18((2)(-4)+(2)(-4)) =-2. \end{align}$$

All other off-diagonal pairings vanish.

Therefore $$V^TV = \begin{pmatrix} 4&-2&0&0\\ -2&4&-2&-2\\ 0&-2&4&0\\ 0&-2&0&4 \end{pmatrix}.$$

With the standard $D_4$ Cartan/Gram matrix $$G_{D_4} = \begin{pmatrix} 2&-1&0&0\\ -1&2&-1&-1\\ 0&-1&2&0\\ 0&-1&0&2 \end{pmatrix},$$ we obtain the exact identity $$\boxed{V^TV=2G_{D_4}}.$$

Hence $V$ realizes an isometric copy of $\sqrt2D_4$ inside $\Lambda_{24}$.

The Four Distinguished Duads

Define $$\begin{align} D_1&=\{1,12\},\\ D_2&=\{13,14\},\\ D_3&=\{16,17\},\\ D_4'&=\{18,22\}. \end{align}$$

The octad is the disjoint union $$\mathcal O_0 = D_1\sqcup D_2\sqcup D_3\sqcup D_4'.$$

For convenience write $$e_{D_k} = \sum_{i\in D_k}e_i.$$

Then $$\begin{align} v_1&=-4e_{D_1},\\ v_2&=2(e_{D_1}+e_{D_2}+e_{D_3}+e_{D_4'}),\\ v_3&=-4e_{D_2},\\ v_4&=-4e_{D_3}. \end{align}$$

The fourth Type-2 root generated by the $D_4$ root system is $$r_4=v_1+2v_2+v_3+v_4.$$

Substitution gives $$\begin{align} r_4 &= -4e_{D_1} +4(e_{D_1}+e_{D_2}+e_{D_3}+e_{D_4'}) -4e_{D_2} -4e_{D_3}\\ &= 4e_{D_4'}. \end{align}$$

Thus the four Type-2 root directions are precisely the four duad directions: $$\pm4e_{D_1},\quad \pm4e_{D_2},\quad \pm4e_{D_3},\quad \pm4e_{D_4'}.$$

The MOG Sextet

Choose the following six tetrads: $$\begin{align} B_1&=\{0,1,12,21\},\\ B_2&=\{4,11,13,14\},\\ B_3&=\{5,6,16,17\},\\ B_4&=\{8,9,18,22\},\\ B_5&=\{15,19,20,23\},\\ B_6&=\{2,3,7,10\}. \end{align}$$

They partition the $24$ coordinates: $$\{0,\ldots,23\} = B_1\sqcup\cdots\sqcup B_6.$$

Their intersections with $\mathcal O_0$ are $$\begin{align} B_1\cap\mathcal O_0&=\{1,12\}=D_1,\\ B_2\cap\mathcal O_0&=\{13,14\}=D_2,\\ B_3\cap\mathcal O_0&=\{16,17\}=D_3,\\ B_4\cap\mathcal O_0&=\{18,22\}=D_4',\\ B_5\cap\mathcal O_0&=\varnothing,\\ B_6\cap\mathcal O_0&=\varnothing. \end{align}$$

Thus the intersection pattern is $$\boxed{(2,2,2,2,0,0)}.$$

The six tetrads form a Golay sextet: every pairwise union $$B_i\cup B_j$$ is an octad of $\mathcal G_{24}$.

Why the Union-of-Tetrads Normalization Fails

The choice $$\mathcal O_0=B_1\sqcup B_2$$ would force every vector supported on $\mathcal O_0$ to have zero coordinates in $B_3,\ldots,B_6$.

The Hexacode $$\mathcal H_6\subset\mathbb F_4^6$$ is an $[6,3,4]_{\mathbb F_4}$ MDS code. Hence its minimum distance is $4$, and there are no nonzero codewords supported on fewer than four coordinates.

Consequently, if a linear coordinate reduction produced $$\rho(x) = (\rho_1(x),\rho_2(x),0,0,0,0) \in\mathcal H_6,$$ then necessarily $$\rho(x)=0.$$

Thus the union-of-two-tetrads normalization is incompatible with a nontrivial Hexacode image for the present subsystem.

The pattern $$(2,2,2,2,0,0)$$ avoids this collapse by distributing the four distinguished duads over four different tetrads.

The 24 Root Vectors

The $D_4$ root shell contains $24$ roots. Under the explicit embedding $V$, every root remains a minimal Leech vector.

They split into three sets of eight.

Vector Sector

Define $$R_1 = \left\{ \pm4e_{D_k}:1\leq k\leq4 \right\}.$$

Thus $$|R_1|=8.$$

These are the Type-2 vectors of shape $$4^2 0^{22}.$$

First Spinor Sector

Define $$R_2 = \left\{ 2\sum_{k=1}^4s_ke_{D_k} : s_k\in\{+1,-1\}, \quad \prod_{k=1}^4s_k=+1 \right\}.$$

There are $8$ such vectors.

They have shape $$2^8 0^{16}.$$

The central vector $$v_2 = 2(e_{D_1}+e_{D_2}+e_{D_3}+e_{D_4'})$$ belongs to $R_2$.

Second Spinor Sector

Define $$R_3 = \left\{ 2\sum_{k=1}^4s_ke_{D_k} : s_k\in\{+1,-1\}, \quad \prod_{k=1}^4s_k=-1 \right\}.$$

Again $$|R_3|=8.$$

Thus $$R(D_4)=R_1\sqcup R_2\sqcup R_3.$$

Exact Metric Triality

The three sectors have identical internal metric structure.

The total number of unordered pairs is $$\binom{24}{2}=276.$$

The sector decomposition accounts for $$3\binom82=84$$ intra-sector pairs and $$3(8\cdot8)=192$$ inter-sector pairs, giving $$84+192=276.$$

This is the metric realization of the triality symmetry $$\boxed{\operatorname{Out}(D_4)\cong S_3}$$ permuting the three eight-element sectors.

The Type-Sensitive Shell Reduction

A subtlety arises if one attempts to interpret the MOG reduction as a single global linear map on the lattice.

Consider $$r_4=v_1+2v_2+v_3+v_4.$$

Modulo $2$, $$2v_2\equiv0.$$

A naïve global $\mathbb F_2$-linear reduction would therefore imply $$\rho(r_4) = \rho(v_1)+\rho(v_3)+\rho(v_4).$$

But geometrically $r_4$ is supported exclusively on $D_4'$, whereas the three summands are supported on the first three duads. Therefore the ordinary coordinate-support interpretation cannot simultaneously be a global linear map on simple-root coefficients.

The correct object is a shell-level, type-sensitive map $$\boxed{ \bar\rho:R(D_4)\longrightarrow\mathbb F_2^4\times\mathbb F_2. }$$

It records the MOG shell data without claiming a linear extension to the whole lattice.

For $r\in R_1$, define $$\bar\rho(\pm4e_{D_k}) = (\mathbf e_k,0),$$ where $\mathbf e_k$ is the $k$-th standard basis vector of $\mathbb F_2^4$.

For $$r= 2\sum_{k=1}^4s_ke_{D_k}, \qquad s_k\in\{+1,-1\},$$ define $$b_k=\frac{1-s_k}{2}\in\mathbb F_2$$ and $$\bar\rho(r) = ((b_1,b_2,b_3,b_4),1).$$

Thus: $$\begin{align} R_1&\longrightarrow \{e_1,e_2,e_3,e_4\}\times\{0\},\\ R_2&\longrightarrow \{b\in\mathbb F_2^4:\sum b_k=0\}\times\{1\},\\ R_3&\longrightarrow \{b\in\mathbb F_2^4:\sum b_k=1\}\times\{1\}. \end{align}$$

The first map is $2$-to-$1$, because $r$ and $-r$ have the same shell label. The spinor-sector sign patterns are recorded individually.

Root Images in the Ambient MOG Space

Choose an identification $$\mathbb F_2^2\cong\mathbb F_4$$ for each MOG tetrad, together with the standard MOG coordinate convention.

With the duads $D_1,D_2,D_3,D_4'$ aligned with the first four tetrads, the simple-root shell images may be represented by $$\begin{align} H_1&=(1,0,0,0,0,0),\\ H_2&=(1,1,1,1,0,0),\\ H_3&=(0,1,0,0,0,0),\\ H_4&=(0,0,1,0,0,0). \end{align}$$

The fourth Type-2 root satisfies $$r_4=v_1+2v_2+v_3+v_4$$ and is supported on $D_4'$. Its corresponding shell label is $$H_4'=(0,0,0,1,0,0).$$

Thus the four Type-2 directions occupy the four coordinate axes of the active four-column subspace.

The central Type-1 root is $$H_2=(1,1,1,1,0,0).$$

The ambient span is therefore $$V_\rho = \operatorname{span}_{\mathbb F_2}\{H_1,H_2,H_3,H_4'\} = \mathbb F_2^4\times\{0\}^2.$$

It has dimension $4$ and contains $16$ elements.

Its weight enumerator is $$\boxed{ W_{V_\rho}(z) = 1+4z+6z^2+4z^3+z^4 = (1+z)^4. }$$

The 12 Nonzero Root Images

Under the binary shell labeling, opposite roots have the same image. Hence the $24$ roots produce $12$ distinct nonzero labels.

Their weights are:

Thus $$8+6+8+2=24.$$

Inside the $15$ nonzero vectors of $$\mathbb F_2^4\times\{0\}^2,$$ there remain exactly three labels not represented by a $D_4$ root: $$\begin{align} u_1&=(1,1,0,0,0,0),\\ u_2&=(1,0,1,0,0,0),\\ u_3&=(0,1,1,0,0,0). \end{align}$$

They satisfy $$u_1+u_2=u_3.$$

Consequently $$V_3=\{0,u_1,u_2,u_3\} \cong\mathbb F_2^2.$$

With respect to the binary dot product, $$\langle u_i,u_i\rangle=0,$$ while $$\langle u_i,u_j\rangle=1 \qquad(i\neq j).$$

This gives the intrinsic three-point symplectic core associated with the triality complement.

Triality and the $S_3$ Action

The three sectors $$R_1,\qquad R_2,\qquad R_3$$ are the vector and two half-spinor sectors of $D_4$.

The outer automorphism group $$\operatorname{Out}(D_4)\cong S_3$$ permutes these sectors.

At the combinatorial level, a transposition of the two spinor sectors is represented by changing the sign of one duad. Such a transformation preserves the vector sector while interchanging even and odd sign parity.

A three-cycle can be represented, after choosing the standard four-dimensional realization, by the corresponding Hadamard-type transformation $$H_4 = \frac12 \begin{pmatrix} 1&1&1&1\\ 1&-1&1&-1\\ 1&1&-1&-1\\ 1&-1&-1&1 \end{pmatrix}.$$

The role of this matrix here is representation-theoretic: it expresses the familiar triality interchange between the vector and spinor descriptions. The present construction does not require an explicit element of $Co_0$ realizing every outer automorphism.

Interaction with the Hexacode

The Hexacode is the $[6,3,4]_{\mathbb F_4}$ MDS code $$\mathcal H_6\subset\mathbb F_4^6.$$

Its weight distribution is $$\boxed{ W_{\mathcal H_6}(z)=1+45z^4+18z^6. }$$

In particular, there are no nonzero codewords of weights $1,2,3$.

The Spinor Sectors

Every vector in $R_2\cup R_3$ has entries $\pm2$ on the same octad $\mathcal O_0$. Modulo $2$, the signs disappear: $$+2\equiv-2\equiv0\pmod2$$ at the level of the doubled lattice representative, while after the standard Construction-A normalization the underlying binary support is the characteristic vector of $\mathcal O_0$.

Because $$|\mathcal O_0\cap B_i| = \begin{cases} 2,&1\leq i\leq4,\\ 0,&i=5,6, \end{cases}$$ its MOG image is a weight-four Hexacode word.

With the chosen coordinate convention, $$\rho(v_2) = (1,1,1,1,0,0) \in\mathcal H_6.$$

The scalar multiples give the line $$L_{\mathcal H_6} = \mathbb F_4(1,1,1,1,0,0) \subset\mathcal H_6.$$

The Type-2 Sector

The vectors in $R_1$ have entries in $4\mathbb Z$. Consequently their ordinary binary Construction-A image is zero.

This is why the type-sensitive shell reduction and the ordinary Golay projection must not be conflated.

The shell reduction retains the four transverse duad directions, while the binary Golay reduction sends them into the parity kernel.

Thus: $$\boxed{ \text{shell information}\neq\text{Golay information}. }$$

Hexacode Complementarity

Let $$W = \operatorname{span}_{\mathbb F_4}\bigl(\rho(D_4)\bigr) = \mathbb F_4^4\times\{0\}^2.$$

Then $$\dim_{\mathbb F_4}W=4.$$

Since $$\dim_{\mathbb F_4}\mathcal H_6=3$$ and $$W\cap\mathcal H_6 = \mathbb F_4(1,1,1,1,0,0),$$ we have $$\dim_{\mathbb F_4}(W\cap\mathcal H_6)=1.$$

Therefore $$\dim_{\mathbb F_4}(W+\mathcal H_6) = 4+3-1 = 6.$$

Since $$\dim_{\mathbb F_4}\mathbb F_4^6=6,$$ it follows that $$\boxed{ W+\mathcal H_6=\mathbb F_4^6. }$$

Hence the embedded $D_4$ coordinate subsystem and the Hexacode are complementary in the ambient MOG space, with precisely one common one-dimensional sector: $$\boxed{ W\cap\mathcal H_6 = \mathbb F_4\cdot\rho(v_2). }$$

This is the intrinsic algebraic bridge between the $D_4$ subsystem and the Hexacode.

What the Construction Establishes

The construction establishes the following chain of explicit objects: $$\boxed{ \sqrt2D_4 \hookrightarrow \Lambda_{24} \supset S_{24} \longrightarrow \text{MOG sextet} \longrightarrow \mathbb F_4^6. }$$

More precisely:

1. The explicit matrix $V$ satisfies $$V^TV=2G_{D_4}.$$

2. All four columns of $V$ are minimal Leech vectors.

3. The $24$ embedded $D_4$ roots split into three eight-element sectors $$R_1\sqcup R_2\sqcup R_3$$ with identical metric laws.

4. The three sectors realize the vector and two spinor representations associated with $D_4$ triality.

5. The MOG sextet $$(B_1,\ldots,B_6)$$ places the four distinguished duads in four distinct tetrads.

6. The shell reduction is explicitly type-sensitive and is not claimed to extend linearly to the entire lattice.

7. The central Type-1 sector determines the nonzero Hexacode line $$\mathbb F_4(1,1,1,1,0,0).$$

8. The Type-2 sector supplies the four transverse ambient coordinate directions.

9. The resulting ambient $D_4$ span and Hexacode satisfy $$W+\mathcal H_6=\mathbb F_4^6, \qquad W\cap\mathcal H_6=\mathbb F_4\rho(v_2).$$

Thus the bridge is intrinsic: it uses only the lattice geometry, Construction-A/Golay structure, MOG coordinates, finite-field representation theory, and the $D_4$ triality decomposition.

Epistemic Status

For clarity, the claims of this document have the following status.

Conclusion

The essential mathematical bridge is now completely explicit.

Starting from four concrete minimal vectors in the Leech lattice, we obtain a scaled $D_4$ root system whose $24$ roots split naturally into the three triality sectors $$8_v,\qquad8_s,\qquad8_c.$$

A compatible MOG sextet distributes the four distinguished duads across four tetrads, preventing the otherwise unavoidable Hexacode rank collapse. The resulting shell reduction retains the distinction between the vector and spinor sectors, while the ordinary Construction-A/Golay reduction identifies the two spinor sectors through their common octad support.

The final ambient-space relation $$\boxed{ \operatorname{span}_{\mathbb F_4}(\rho(D_4))+\mathcal H_6=\mathbb F_4^6, \qquad \operatorname{span}_{\mathbb F_4}(\rho(D_4))\cap\mathcal H_6 = \mathbb F_4(1,1,1,1,0,0) }$$ is therefore an intrinsic finite-field statement attached to the chosen MOG realization.

No external numerical modulus is required to obtain this structure.

Audited Numerical Discretization Attempt

The original project also considered a separate numerical construction involving a large prime $P$, decimal truncations of analytic constants, and an elliptic curve $$E_P:y^2=x^3+Ax+B\pmod P.$$

That construction is logically independent of the lattice–MOG–Hexacode bridge developed above.

The relevant numerical claims were audited separately. In particular, the proposed identities $$B=2A, \qquad j(E)=j(-163), \qquad a_P=3$$ do not hold for the published parameters.

Likewise, the proposed CM relation $$4P=9+163y^2$$ is incompatible with the published prime.

The important methodological conclusion is therefore not that the intrinsic lattice/coding construction fails, but rather that the numerical elliptic-curve layer does not follow from it.

Accordingly, this appendix is deliberately separated from the main construction.

The CM Congruence Obstruction

If $$4P=9+163y^2,$$ then necessarily $$4P-9\equiv0\pmod{163}.$$

For the published value of $P$, this congruence fails. Thus the proposed discriminant-$-163$ lock is not an identity satisfied by the published parameters.

The Coefficient Lock

The claimed relation $$B=2A$$ also fails for the published coefficients. Hence it cannot be used as an intrinsic algebraic consequence of the $D_4$, Leech, or Hexacode construction.

The Frobenius Trace

The proposed Frobenius trace $$a_P=3$$ and the consequent point-count identity $$\#E(\mathbb F_P)=P-2$$ are likewise separate arithmetic assertions.

Their verification requires an independent finite-field computation and does not follow from the MOG or Hexacode data.

Interpretation

The numerical layer should therefore be regarded as an independent discretization experiment rather than as a theorem arising from the intrinsic bridge.

The main mathematical result of this document is the coordinate-explicit chain $$\boxed{ \sqrt2D_4 \hookrightarrow \Lambda_{24} \longrightarrow \text{Golay/MOG data} \longrightarrow \mathbb F_4^6, }$$ together with the exact triality and Hexacode complementarity statements.