Induced Normal Color Geometry on One Nested Spatial Flag

In the four-plus-six spatial reference split, the base has four ordinary spatial dimensions and its perpendicular complement has six. Time is a separate coordinate in the infinite-dimensional physical spacetime. This four-plus-six split concerns ten spatial directions in the matter domain.

A normal connection can be calculated from the bending of actual spatial sectors, rather than introduced as independent matrices. One common nested construction produces a full rank-three unitary normal connection at depth four. Its traceless part is the eight-generator SU3 algebra. Pulling that connection back to depth three retains all eight generators. This is a concrete geometric realization of the proposed color carrier, with the odd-sector pullback and the choice of embedding stated explicitly.

The result does not establish a physical strong force. Higher normal derivatives can generate larger non-Abelian algebras at depth two, and replacing an independently varied connection with an induced one changes the field equations. Those controls identify what a physical sector-selection law still needs to establish.

One Ordinary Spatial Geometry for Every Occupied Depth

Use complex notation for paired ordinary coordinates of Euclidean ten-space; each complex coordinate denotes two real spatial directions. Define the global triangular map

$$\Phi(w)=\left(w_1,w_2,w_3+\frac a2w_1^2, w_4+aw_1w_2,w_5+\frac a2w_2^2\right).$$

The first two coordinates are unchanged, so the inverse subtracts these same polynomials. It is a global diffeomorphism. For each occupied depth $d=2,3,4,5,6,10$, let $Q_d$ be the image of the real coordinate plane whose first $d$ real coordinates can vary. All six sectors are therefore nested in one geometry. No depth is independently optimized, and no direction is compactified. The graph parametrizations retain the original coordinates, so their induced metrics dominate the Euclidean metrics on those coordinates and are complete.

At the origin $D\Phi=I$. The second fundamental forms are the perpendicular components of $D^2\Phi$. The depth-four graph has complex tangent rank two and normal rank three:

$$f(z_1,z_2)=a(z_1^2/2,z_1z_2,z_2^2/2).$$

Its complex second fundamental map at that point is an isomorphism

$$\mathcal B:\operatorname{Sym}^2\mathbb C^2\longrightarrow\mathbb C^3.$$

The count behind this match is $\dim\operatorname{Sym}^2\mathbb C^k=\binom{k+1}{2}$. In a fixed complex-rank-five parent, complete quadratic normal generation would require

$$\binom{k+1}{2}=5-k, \qquad (k-2)(k+5)=0.$$

Thus this particular premise selects $k=2$, real base depth four and six real normal directions. The count alone does not prove a connection algebra or choose the parent. In a variable complex-rank-$M$ parent, the condition instead permits $M=k+\binom{k+1}{2}$ for every positive $k$.

Calculate the Connection Rather Than Assume Its Algebra

For a holomorphic graph $f:\mathbb C^k\to\mathbb C^q$, put $P=Df$, $S=(I_q+PP^\dagger)^{-1/2}$ and

$$N=\begin{pmatrix}-P^\dagger\\I_q\end{pmatrix}S, \qquad A=N^\dagger dN.$$

The columns of $N$ are orthonormal ordinary normal directions. The induced normal connection $A$ is anti-Hermitian. For real coordinate derivatives $P_i=\partial_iP$, its curvature is

$$F_{ij}=S\left[P_i(I_k+P^\dagger P)^{-1}P_j^\dagger -P_j(I_k+P^\dagger P)^{-1}P_i^\dagger\right]S.$$

The checker independently obtains the same expression from $(\partial_iN)^\dagger\Pi_T\partial_jN-(i\leftrightarrow j)$, where $\Pi_T$ is the actual tangent projector. At the origin the equivalent real Ricci formula is

$$(R^N_{ij})_{\alpha\beta} =\sum_l\left(\mathcal B_{\alpha il}\mathcal B_{\beta jl} -\mathcal B_{\alpha jl}\mathcal B_{\beta il}\right).$$

For the depth-four graph these curvature matrices and their commutators span $\mathfrak u(3)$, of real dimension nine. Their traceless projections span $\mathfrak{su}(3)$, of dimension eight. Exact rational elimination confirms the ranks; they do not depend on a numerical singular-value threshold. Since normal parallel transport is unitary, containing this full curvature algebra establishes the full connected U3 normal holonomy for this graph.

The normal spin lift commutes with every tangent Clifford vector. Thus its role as an independent normal rotation at the base does not erase a physical spatial derivative. Neither this fact nor normal holonomy identifies an observed interaction coefficient or occupied particle representation.

Intrinsic Odd Normals and an Even-Carrier Pullback Differ

For odd $d$, compare two connections: the actual normal connection of $Q_d$ inside ten-space, and the normal connection of $Q_{d+1}$ restricted to $Q_d$. The latter is the declared even-shadow carrier. It excludes the one extra tangent direction of $Q_{d+1}$ from the normal bundle, so equality is not automatic.

At the common origin the Lie algebras generated by the actual curvature matrices have the following exact dimensions:

At three the intrinsic curvature acts as SO5 on its five-dimensional first-normal space, with two unused normal directions at this point. It is not the U3 connection. The four-sector normal connection restricted to the same three-sector generates all of U3 and its traceless SU3. At five the analogous distinction is intrinsic SO5 versus a restricted U2 connection. These are geometry calculations, not counts of physical force species.

This prescription also differs from retaining one fixed six-normal SU3 carrier at every lower depth. In particular depth two has four complex normals here; its chosen quadratic graph bends in only one complex normal line and gives an Abelian connection. The different prescription is explicit rather than silently substituted into the earlier fixed-carrier Gauss calculation.

Bending Homogeneity Does Not Supply the Previous Dielectric

At the origin, $\mathcal B$ scales as $a$ and $F$ scales as $a^2$. Therefore $\sum_{i<j}\|F_{ij}\|^2$ scales as $a^4$. At $a=0$ the graph is flat and its induced normal curvature vanishes; there is no independently selectable curvature on that fixed flat graph.

This is quartic curvature energy. It is not a derivation of a separate quartic coefficient multiplying the curvature energy of an independent connection. Treating both expressions as the same term would double count the bending and transfer field equations between different actions.

Nor does geometric origin alone select this flat-ambient graph. The existing connection metric $g=\pi^*g_B+|dy+Ay|^2$ realizes a chosen normal connection in an ordinary curved spatial parent. An embedding and ambient metric must be determined before induced geometry removes the previous independent-connection freedom. The present construction is a specific compatible candidate.

Higher Normal Jets Defeat Universal Quadratic Selection

Depth two admits the complete holomorphic graph

$$f(z)=(z^2/2,z^3/6,z^4/24,z^5/120)\in\mathbb C^4.$$

At each point its curvature has only one independent base two-form. That does not restrict the holonomy algebra to one generator: the curved normal line changes under transport. The checker calculates the actual normal parallel transport along a real coordinate line, transports four curvature matrices back to one point, and obtains U4 dimension sixteen and traceless SU4 dimension fifteen.

An independent exact certificate expands the normal connection and its parallel transport through order six at the origin. The resulting rational covariant curvature jets generate precisely the same algebras. Hence the larger algebra is not numerical rank noise or the result of comparing unrelated normal frames. The quadratic-only control $f=(z^2/2,0,0,0)$ remains Abelian.

The higher-jet control can also coexist with all occupied sectors. Modify the same triangular map by adding $w_1^5/120$ to its second coordinate, $w_1^3/6$ to its fourth and $w_1^4/24$ to its fifth. Recover $w_1$, then $w_2$, then the other coordinates to invert it. Images of the same real flags remain nested, while $Q_2$ has the curve above up to a permutation of its normals. Nesting alone therefore does not exclude this competing depth-two algebra.

This agrees with the scope of the established normal-holonomy mathematics. Console, Di Scala and Olmos prove full U(q) normal holonomy for complete, full, irreducible complex submanifolds of complex Euclidean space in Theorem 2 of their primary paper. Here the explicit curvature certificates establish the particular examples; the theorem is not substituted for checking its hypotheses, and it is not a force-selection theorem.

A Parallel Quadratic Identification Would Lose Full Color

Requiring $\mathcal B:\operatorname{Sym}^2T\to N$ to be a parallel isomorphism would make normal parallel transport the symmetric-square image of tangent parallel transport. A complex-rank-two tangent bundle has unitary holonomy of dimension at most four. Its symmetric-square image cannot supply full U3 or SU3.

The actual graph avoids that restriction because its second fundamental form is not parallel. Its necessary integrability condition would be

$$R^N_{ij}\mathcal B(X,Y) -\mathcal B(R^T_{ij}X,Y)-\mathcal B(X,R^T_{ij}Y)=0.$$

At $a=1$, the sum of the squared real components of this obstruction, summed over $i<j$ and all ordered tangent arguments in the orthonormal origin frame, is exactly $1536$. Thus constant polynomial coefficients in graph coordinates do not mean covariantly constant bending.

There is a related pointwise warning. The isotropically normalized quadratic graph $(\sqrt2 z_1^2/2,z_1z_2,\sqrt2 z_2^2/2)$ has only a four-dimensional curvature-generated algebra at the origin, but nine at the tested neighboring point $z_1=2/5,z_2=0$. A special point can hide directions present in the connection holonomy.

Physical Scope and Reproduction

The common construction supplies an actual spatial U3 carrier and a conditional four-dimensional spatial base/six-normal quadratic dimension criterion. It keeps every occupied depth in one nested geometry and makes the additional three-sector pullback visible. To turn it into a color mechanism, a common action must select the geometric class, determine its dynamics and show how the surviving connection couples to the same field.

In particular, restricting variations to induced connections removes independent $A_0$ variation. The earlier source constraint $\mathcal D_iD_i=\rho$ and its exact-zero consequences must be derived anew; they cannot be carried over from the independently varied action. Likewise neither a U2 curvature algebra nor its absence in a selected graph establishes the presence or absence of another observed force.

The checker and report preserve 689 checks across 13 cases, plus 174 checks in the exact Clifford dependency. They include rational curvature algebra certificates, actual transported higher-jet holonomy, normal-frame curvature checks, all occupied sectors, one common nesting map and its inverse, spin compatibility and bending homogeneity. No particle evolution or fitted force strength enters these tests.

The Induced Connection Changes the Inertia Operator

The induced-response calculation uses the actual graph second fundamental forms to construct a squared-first-curvature comparison with positive mixed space-time inertia and magnetic symbols bounded by the electric ones. The complete background Hessian also contains nonzero curvature-stiffness terms, so this comparison does not establish full graph positivity. Arbitrary real bending also need not preserve the U3 carrier. Its frozen probe equation has an immediate exterior tail. A separate finite-lock orientation comparison retains a second response branch with positive energy and a direct finite-propagation proof; its graph integrability and master-field origin remain open.

Parallel Curved Geometry and Retained Normal Structure

The parallel quadratic comparison realizes the same four-dimensional spatial base/six-normal rank match in an ordinary curved spatial parent, with a complete compatible occupied flag. Its normal holonomy is Sym²U2 and its traceless group is SO3, carrying an additional symmetric pair invariant. Thus pointwise quadratic rank and smooth nesting do not select full SU3. The retained normal-structure calculation gives an explicit compatible metric connection and shows why the reviewed real electric wave cannot continue the original normal complex structure in exact parallel fashion. Adopting that projected connection changes the induced transport and its variation.

Canonical Orientation Transport Is an Explicit Alternative

The canonical orientation connection is determined by a normal rotation structure relative to a fixed flat reference. Explicit two-dimensional maps realize full U4 and U3 holonomy, confirming that low base dimension cannot by itself exclude the larger depth-two algebra. This connection is generally different from the induced graph transport above. Its common action has a distinct source variation and electric orientation energy, with a proved static principal cone bound and an explicit moving-background hyperbolicity boundary. The geometric color synthesis compares those routes under their actual assumptions.