Retained Normal Rotation Structure and Compatible Transport

A retained orthogonal complex structure supplies a precise unitary normal carrier and a positive local orientation energy. It penalizes the complement of U(q), while leaving SU(q) and its central U1 untouched. It neither selects three/four alone nor resolves the exact-zero dielectric vacuum ambiguity. For the actual induced connection, arbitrary real bending need not admit a parallel continuation of the original J. In particular the reviewed depth-four electric wave fails that continuation already at first order. A projected J-preserving metric connection is explicit, but adopting it changes the ordinary induced normal transport.

This is a bounded geometric comparison, not a solved graph evolution or a particle state. All normal directions below are ordinary spatial directions in the selected ten-dimensional parent. Their endomorphisms and orientation coordinates do not add physical dimensions. Global existence of J on an arbitrary normal bundle is itself a reduction premise; the comparison is local.

1. Precise local action and the two meanings of A

Let N be the real rank-2q normal bundle of the declared even carrier. In an orthonormal normal frame use

$$J^T=-J,\qquad J^2=-I,\qquad A_\mu\in\mathfrak{so}(2q),\qquad K_\mu=D_\mu J=\partial_\mu J+[A_\mu,J].$$

The matrix inner product is $\langle X,Y\rangle_F=\operatorname{tr}(X^TY)$. Use Minkowski signature (+,−,…,−), unit comparison speed and the explicit local comparison Lagrangian

$$\mathcal L=\frac12\partial_\mu u\,\partial^\mu u -\frac12\Omega^2u^2 -\frac{\epsilon(u)}4\langle F_{\mu\nu},F^{\mu\nu}\rangle_F +\frac{f^2}2\langle D_\mu J,D^\mu J\rangle_F, \qquad \epsilon(u)=\kappa h_du^4, \tag{1}$$

where $f,\kappa,\Omega>0$ are common local constants, $F=dA+A\wedge A$, and repeated spacetime indices are summed. Thus the positive static energy addition is $f^2\sum_i\|D_iJ\|_F^2/2$, while the time term is positive kinetic energy. A Euclidean all-plus expression alone would not specify physical time evolution. Equation (1) does. It is an explicitly proposed comparison action, not a reduction already obtained from Psi-infinity.

There are two different variations:

1. Induced A: the ordinary normal connection of a moving spatial graph, $A=N^T dN$, with its actual embedding and ambient metric fixed as the geometric prescription. Changes in A follow changes in the graph; A is not varied independently. A freely retained J adds normal orientation data unless its same-field origin is supplied.
2. Independent A: vary a metric normal connection and J independently in (1). A prescribed ordinary connection metric can realize it spatially, but that realization does not select its dynamics or replace this variation.

The independent comparison permits the full so(2q) connection before retaining J. This enlarges an earlier action already restricted to su(3), and is stated here as a changed action. It does not inherit every old source constraint.

Conclusions obtained by variation in the second interpretation are not graph field equations in the first. This separation also applies to Gauss laws and vacuum admissibility conditions.

2. Orthogonal splitting and exact quadratic eigenvalues

Choose constant $J_0=\operatorname{diag}(j,\ldots,j)$, $j=\left(\begin{smallmatrix}0&-1\\1&0\end{smallmatrix}\right)$. The orthogonal involution $\mathcal C X=-J_0XJ_0$ gives

$$\mathfrak{so}(2q)=\mathfrak h\mathbin{\perp}\mathfrak m, \quad \mathfrak h=\{X:[X,J_0]=0\}=\mathfrak u(q), \quad \mathfrak m=\{X:XJ_0=-J_0X\},$$

$$P_{\mathfrak h}X=\tfrac12(X-J_0XJ_0),\qquad P_{\mathfrak m}X=\tfrac12(X+J_0XJ_0). \tag{2}$$

The involution is orthogonal because J0 is orthogonal. Its ±1 eigenspaces are therefore orthogonal. Commuting with J0 is exactly complex linearity; real skewness then becomes complex skew-Hermitian symmetry. This proves the identification with u(q), not just its dimension. The brackets obey $[\mathfrak h,\mathfrak h]\subset\mathfrak h$, $[\mathfrak h,\mathfrak m]\subset\mathfrak m$, $[\mathfrak m,\mathfrak m]\subset\mathfrak h$, directly by multiplying the involution eigenvalues.

For $L(X)=[X,J_0]$, direct expansion gives

$$L^2X=-2(X+J_0XJ_0)=-4P_{\mathfrak m}X, \qquad L^*L=4P_{\mathfrak m},$$

$$\boxed{\|[X,J_0]\|_F^2=4\|P_{\mathfrak m}X\|_F^2.} \tag{3}$$

The positive commutator quadratic operator has eigenvalue zero with multiplicity q² and eigenvalue four with multiplicity q(q−1). The skew operator L itself has complex eigenvalues ±2i on the complement, each with multiplicity q(q−1)/2. The static penalty is $2f^2\sum_i\|A_i^{\mathfrak m}\|_F^2$ at constant J0; it has no penalty on the full commuting algebra.

For q>0 this algebra splits orthogonally as

$$\mathfrak u(q)=\mathfrak{su}(q)\oplus\mathbb R J_0, \quad X_{\rm cent}=\frac{\langle J_0,X\rangle_F}{2q}J_0, \quad X_{\rm su}=X-X_{\rm cent}. \tag{4}$$

The real trace condition $\operatorname{tr}(J_0X_{\rm su})=0$ is equivalent to zero complex trace. The central J0 commutes with all of u(q). Neither its removal nor its identification with observed electric charge follows from the penalty. A determinant/traceless restriction is an extra premise if only SU(q) is desired.

3. Orientation kinetics and occupied-depth controls

In one fixed orientation component, orthogonal complex structures form $SO(2q)/U(q)$. Write $J=e^\pi J_0e^{-\pi}$, with $\pi\in\mathfrak m$. The commuting part of an infinitesimal rotation changes no J, so it is omitted. Linearizing around A=0, J=J0 yields

$$D_\mu J=[\partial_\mu\pi+A_\mu^{\mathfrak m},J_0]+O(2), \quad \mathcal L_J^{(2)}=2f^2 \langle\partial_\mu\pi+A_\mu^{\mathfrak m}, \partial^\mu\pi+A^{\mu,\mathfrak m}\rangle_F. \tag{5}$$

Thus there are q(q−1) positive orientation coordinates. With prescribed A=0 they obey the ordinary unit-speed linear wave equation. For independently varied A at a positive constant dielectric, the local gauge J=J0 puts the quadratic term into the complement of A: its squared Proca mass is $4f^2/\epsilon$ in the normalization (1). The commuting connection has no such mass. The orientation coordinates and connection components must not be counted twice as independent physical polarizations in that gauge.

For the actual normal bundle of each even carrier in the selected parent, and the stated pullback to its odd shadow, the comparison is:

At q=0 the normal space, J and its action are empty; there is no central U1. The odd physical sectors have actual normal ranks seven and five. Assigning the table's even-normal J to them uses the declared even-carrier pullback, not an orthogonal complex structure on their whole odd-rank actual normal bundles, which cannot have one.

Equation (5) alone allows SU4 at depth two and SU2 at five/six, with an unpenalized circle in each nonempty case. It consequently does not select only three/four. These are permitted connection algebras, not assertions that every induced graph realizes them: the frozen common quadratic graph has only U1 at two, full U3 at three/four and U2 at five/six. Its explicit higher-jet depth-two control realizes U4 and shows that the smaller quadratic algebra is not enforced by unitary structure or nesting alone. The existing symmetric middle-Hodge coefficient may be retained separately: for even source dimension 2m, $*^2=(-1)^m$ and $*^\dagger=(-1)^m*$, hence $H=(*+*^\dagger)/2$ is zero when m is odd and is the star when m is even. The declared odd-shadow rule gives the displayed h values at the occupied depths. These Hodge identities and their prescription are already derived in the frozen source; they are not derived from the new J penalty. Keeping f>0 also retains orientation dynamics where h=0. Gating f itself by h would be another action choice.

Fixed-six-normal comparison. The earlier prescription keeps the six-plane normal to Q4 as one fixed carrier. At two it still has q=3 and the full eight-generator SU3 commuting with tangent Clifford directions, whereas the actual depth-two even normal bundle here has q=4. At five/six some of that fixed six-plane has become tangent: its tangent-Clifford commutant is only SU2 of dimension three, and the fixed triplet projection erases those new derivative directions. The current q=2 carrier instead uses the four directions still normal to Q6. At ten both prescriptions have no independent normal algebra commuting with the whole tangent symbol. These are distinct carrier prescriptions; (2) does not turn the fixed six-plane into the actual normal bundle at every depth. Neither one makes depth-two force absence a consequence of unitary structure alone.

4. The actual projected metric connection

For any induced metric connection $\nabla^N=d+A$, differentiating the constraints on J gives

$$K_\mu^T=-K_\mu,\qquad K_\mu J+JK_\mu=0.$$

Seek a metric connection $\widehat\nabla=d+\widehat A$ that preserves J. The formula is

$$\boxed{\widehat A=A-\tfrac12J(DJ) =P_{\mathfrak h(J)}A-\tfrac12JdJ.} \tag{6}$$

Indeed $JK$ is skew, so the correction is metric. Since $[JK,J]=2K$, its commutator cancels DJ exactly: $dJ+[\widehat A,J]=K-K=0$. The correction is in m(J). Every other metric connection preserving J differs from (6) by an h(J)-valued one-form. Consequently (6) is the unique such connection whose difference from A has no h part, and is pointwise the nearest preserving metric connection in the Frobenius norm on differences of connections. This proves the projection without appealing to an external connection formula.

At constant J, (6) is simply $\widehat A=P_{\mathfrak h}A$, but for varying J dropping the $-JdJ/2$ term is incorrect. Differences of connections are frame-covariant tensors; the separate dJ term alone is frame dependent. For a normal vector v, $\widehat\nabla_\mu v-\nabla^N_\mu v=-J K_\mu v/2$. Thus the two normal transports coincide precisely when DJ=0. The projected connection is a valid metric connection on the same spatial normal bundle; it is not generally its induced Levi-Civita normal connection for the original embedding and ambient metric.

For clarity its curvature also differs. With $(K\wedge K)_{\mu\nu}=[K_\mu,K_\nu]$, direct differentiation gives

$$\widehat F=P_{\mathfrak h(J)}F-\tfrac14K\wedge K, \qquad [\widehat F,J]=0. \tag{7}$$

To verify (7), set $B=-JK/2$. Then $DB=-\tfrac12K\wedge K-\tfrac12J[F,J]$ and $B\wedge B=K\wedge K/4$. Hence $F+DB+B\wedge B$ gives (7). Using curvature of (6) in the action, rather than the original induced F, is a changed action and requires a new response calculation.

5. Exact parallel continuation and the real-wave obstruction

For an endomorphism, the curvature identity follows by commuting covariant derivatives directly:

$$(D_\mu D_\nu-D_\nu D_\mu)J=[F_{\mu\nu},J]. \tag{8}$$

Thus DJ=0 on a patch requires $[F_{\mu\nu},J]=0$ there. Pointwise curvature commutation at one point is only necessary: its parallel transports must preserve the chosen J consistently around every loop. On a simply connected patch that holonomy condition permits a path-independent parallel extension. Along one curve there is always the orthogonal parallel-transport solution $J(s)=U(s)J(0)U(s)^{-1}$, where $U'=-A_sU$. This one-dimensional fact does not establish a solution in all spacetime directions. At one point one can also choose derivatives $\partial_\mu J=-[A_\mu,J]$, but their mixed derivatives remain constrained by (8).

The reviewed depth-four real wave uses η=e0, n=e0 and $B_2n=e_2$. Its electric symbol includes

$$W_2=e_0e_2^T-e_2e_0^T,\quad \|W_2\|_F^2=2,\quad \|[W_2,J_0]\|_F^2=4, \quad \|P_{\mathfrak m}W_2\|_F^2=1. \tag{9}$$

An arbitrary nonzero scalar wave factor multiplying W2 does not change the noncommutation. At a static graph background $F^{(0)}_{0i}=0$. If a smooth small-amplitude family admitted a parallel continuation $J=J_0+s\,\delta J+O(s^2)$, (8) would require

$$0=\delta[F_{02},J] =[\delta F_{02},J_0]+[F^{(0)}_{02},\delta J] =[\delta F_{02},J_0]. \tag{10}$$

Equation (9) contradicts (10) for the nonzero electric wave. A varying J cannot absorb this leakage into an exactly parallel continuation near the original J0 on a spacetime patch. The finite penalty permits DJ≠0 and lets J respond dynamically; its amount and the graph response require solving the coupled equations. Projecting to (6) removes this nonunitary curvature only by changing transport and curvature, not by proving the original real wave stayed in the original unitary class.

For a single isolated W2 there are other orthogonal complex structures that commute with it, by taking its active real two-plane as one complex plane. That algebraic fact does not give a small continuation of the original J0, does not address simultaneous background curvatures, and does not solve the spacetime integrability condition. No stronger nonlinear exclusion is claimed.

6. Field equations: finite penalty versus exact restriction

Constrained variations of J are $\delta J=[\xi,J]$. Integrating the J term by parts gives, for either interpretation with J independent,

$$[J,D_\mu D^\mu J]=0. \tag{11}$$

This is the orientation equation, not DJ=0. It allows nonconstant finite energy orientation waves. For induced A it must be coupled to the graph variation of (1), including the dependence of A on that graph. In particular the corrected induced-curvature background Hessian cannot be inferred positive merely by adding a positive square: its earlier omitted curvature-stiffness terms remain, and the full temporal graph prescription is still required.

For independently varied A, cyclicity of trace gives $\langle K,[\delta A,J]\rangle_F=\langle[J,K],\delta A\rangle_F$. The source-free connection equation for (1) is therefore

$$D_\mu(\epsilon F^{\mu\nu})+f^2[J,D^\nu J]=0. \tag{12}$$

A finite f at positive dielectric does not require DJ=0: the curvature term balances its current. At exact ε=0 on an open source-free vacuum, however, (12) does force DJ=0. Since K anticommutes with J, $[J,K]=2JK$, which is zero only when K=0. This exact algebraic consequence of independently varying A in the degenerate vacuum must be distinguished from treating a finite penalty as an exact holonomy restriction for induced A, or at ε>0. With other matter currents the corresponding equations change again.

7. The commuting vacuum histories survive

Let q=3 and let j_r be j on normal plane r and zero on the other planes. Set

$$T=\tfrac12(j_1-j_2),\quad \|T\|_F^2=1,\quad [T,J_0]=0,\quad \operatorname{tr}(J_0T)=0.$$

This T is already in su(3); removing the central circle does not help. In temporal gauge choose the same common Cartan history as the frozen source,

$$A_2=b(t)x_1T,\quad A_\mu=0\ (\mu\ne2),\quad J=J_0,$$

$$F_{12}=b(t)T,\qquad F_{02}=\dot b(t)x_1T,\qquad DJ=0. \tag{13}$$

Take u=0 while a smooth flat-ended b rises from zero to any settled B before the later probe. All histories have identical initial raw data. Both the dielectric curvature term and its u derivative vanish there, and the new J term, its orientation equation and its connection current are identically zero. They solve (11)–(12), have zero vacuum energy and satisfy the Bianchi identity because they arise from A. Thus (1) retains the source-free local Cauchy ambiguity in the entire commuting branch.

After b settles, the identical neutral probe p(t)u yields the same equation as before,

$$\ddot u+\Omega^2u+2\kappa h_d B^2u^3=p(t), \quad \mathcal E=\tfrac12\dot u^2+\tfrac12\Omega^2u^2 +\tfrac12\kappa h_dB^2u^4. \tag{14}$$

The J contribution stays zero. The connection equation still holds because the settled magnetic curvature is constant in space, ε depends only on time, and all matrices commute. At active three/four the subsequent responses therefore remain distinguishable. These are local homogeneous histories, not finite-total-energy localized particles. In the induced interpretation an arbitrary b(t) is not a permitted independent variation; whether an actual graph and J can realize such a history is a different geometric problem. The flat fixed graph has zero induced F, as the source already establishes.

8. Positive energy and causal scope

For the fully specified independent local comparison (1), the classical energy density, on the gauge constraint surface, is

$$\mathcal E=\tfrac12u_t^2+\tfrac12|\nabla u|^2+\tfrac12\Omega^2u^2 +\tfrac\epsilon2\left(\sum_i\|F_{0i}\|_F^2+ \sum_{i<j}\|F_{ij}\|_F^2\right) +\tfrac{f^2}2\left(\|D_tJ\|_F^2+\sum_i\|D_iJ\|_F^2\right)\ge0. \tag{15}$$

On a smooth patch with ε bounded strictly above zero, the principal equations are the scalar wave, the wave-map orientation equation and the gauge-fixed ordinary connection wave equation. Their common characteristic speed in (1) is one. Gauge constraints still have to be imposed. This scoped local principal statement does not imply global regularity, does not repair the degenerate ε=0 evolution, and is not a causality proof for the induced graph action. Equation (13) directly prevents a uniqueness claim in that vacuum. If A is prescribed, the J sector alone has positive wave-map energy and a unit-speed principal operator; work by a time-dependent prescribed A must be included before claiming its standalone energy is conserved.

The actual geometric extra premise is now precise: either restrict the evolving induced geometry so its normal holonomy preserves a parallel J, derive the finite J response and accept nonunitary induced curvature, or explicitly adopt the different compatible connection (6) and rederive its action and coupling. Separate Hodge/occupation selection and a rule fixing commuting vacuum connection histories are still required for the original color and dielectric claims.

Reproduction

The exact checker and report verify occupied normal ranks, commutator spectra, compatible connection and curvature, the actual depth-four electric obstruction and the unchanged commuting vacuum history. The original independent execution includes 1,548 checks with exact frozen-input integrity checks. The maintained executable retains the mathematical checks and records its own executed hash; it needs the installed system Python with SymPy. These are geometric and local-action comparisons, not a coupled graph evolution or particle simulation.

A Connection Determined by Orientation Has Different Dynamics

The canonical normal-structure calculation specializes the compatible connection to a fixed flat reference. Its actual rank-two holonomy has no traceless SU2 curvature, whereas rank three and rank four admit full U3 and U4 through explicit low-dimensional pullbacks. The combined orientation action retains the exact ray metric, positive electric orientation cost and static principal cone bound. It also exhibits rapid-motion loss of hyperbolicity, and its constrained orientation variation does not supply the earlier independent-connection Gauss law.