Curvature Generated by Normal Rotation Structure

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.

Result and precise scope

Fix a flat metric reference connection on a real rank-2q ordinary spatial normal bundle, and let J be an orthogonal complex structure. The chosen compatible connection

$$\widehat A=-\frac12JdJ,\qquad \widehat F=-\frac14[dJ,dJ]$$

is the pullback of the canonical connection over the orientation orbit SO(2q)/U(q). Its target curvature and restricted holonomy algebra are u(q) for q=3,4, only the central u(1) for q=2, and zero for q=0. In particular J-only normal transport at depths five/six does not generate the traceless SU2 connection permitted by an arbitrary compatible normal connection. Depth two can still generate full U4 holonomy, even on a two-dimensional base: an explicit analytic local map below proves this through transported curvature jets. Thus this construction gives a geometric distinction between q2 and q3, but does not by itself select color only at three/four.

The reference connection and this functional choice are additional geometric prescriptions. This is a new connection on the spatial normal bundle, generally different from its induced Levi-Civita normal connection in the original embedding. The orientation orbit is a space of possible normal rotations, not a new physical spatial sector. No physical electric charge or observed force is identified here.

1. Canonical connection and curvature, derived directly

Let J0 be multiplication by i on Cq, viewed as R2q. Split real skew endomorphisms as

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

The projections are Ph X=(X−J0 X J0)/2 and Pm X=(X+J0 X J0)/2. Their brackets satisfy [h,h] subset h, [h,m] subset m and [m,m] subset h. Choose a local orthogonal lift g of J=gJ0g−1 and put theta=g−1dg. In that adapted frame,

$$g^{-1}\widehat A g+g^{-1}dg =-\tfrac12J_0[\theta,J_0]+\theta =P_{\mathfrak h}\theta=:a.$$

Indeed −J0[theta,J0]/2=−Pm theta. The canonical connection is therefore exactly the h projection of the moving-frame rotation, rather than an arbitrary h-valued one-form. The Maurer–Cartan identity gives

$$f=da+a\wedge a=-\theta_{\mathfrak m}\wedge\theta_{\mathfrak m}, \qquad f(U,V)=-[\theta_{\mathfrak m}(U),\theta_{\mathfrak m}(V)]. \tag{1}$$

At the reference point, a tangent coordinate X in m changes J by [X,J0]. Direct multiplication gives

$$[[X,J_0],[Y,J_0]]=4[X,Y],$$

so (1) agrees with Fhat=−[dJ,dJ]/4. These formulas use matrix-valued form convention (K wedge K)(U,V)=[K(U),K(V)]. They fix the factors and curvature sign.

Here a is the canonical connection on the principal Uq bundle SO(2q) over SO(2q)/Uq, and also its defining associated real normal-vector connection. For q2 that principal bundle has an ineffective isotropy factor on the base; its full structural group must not be mistaken for its canonical holonomy.

2. Exact bracket algebra and the exceptional q2 case

Every m element is an antilinear map

$$X_Bz=B\bar z,\qquad B^T=-B.$$

Conversely real skewness of an antilinear map implies this complex antisymmetry. Thus m has real dimension q(q−1), with basis X_Eab and X_iEab, where Eab has entries +1 at (a,b), −1 at (b,a). Their bracket is complex linear:

$$[X_B,X_C]z=(B\bar C-C\bar B)z. \tag{2}$$

For the same pair, (2) gives

$$[X_{E_{ab}},X_{iE_{ab}}]=2i(D_a+D_b), \tag{3}$$

where Da is the diagonal coordinate projector. Pairs sharing one index give nonzero real skew off-diagonal matrices; replacing one E by iE gives the corresponding imaginary symmetric off-diagonal matrices. Every off-diagonal u(q) generator is obtained this way when q is at least three. The diagonal pair sums span all q diagonal directions, since for distinct a,b,c,

$$2D_a=(D_a+D_b)+(D_a+D_c)-(D_b+D_c).$$

Therefore the real linear span [m,m] is the entire u(q) for q3 and q4, including its center. This is a curvature-span statement on the full orientation target.

At q2 there is only one complex antisymmetric coordinate B=bE12. Equation (2) then yields a scalar multiple of iI2 for every pair. Hence

$$\operatorname{span}[\mathfrak m,\mathfrak m]=\mathbb R J_0, \qquad P_{\mathfrak{su}(2)}[\mathfrak m,\mathfrak m]=0. \tag{4}$$

The traceless isotropy su2 commutes with all of m: this follows either from H E12−E12 conjugate(H)=tr(H)E12 for skew-Hermitian H, or by the three explicit generators checked in the code. Thus it is ineffective on the two-dimensional orientation orbit. The central circle acts on B with weight two and supplies its nonzero canonical curvature. One orientation component SO4/U2 is the two-sphere: the orbit has real dimension two and the effective rotation group is SO3 with circle stabilizer. The associated normal-vector holonomy remains the central circle acting on both complex normal directions; the traceless SU2 factor contributes no curvature. Merely having U2 as structural group would have overcounted it.

For completeness the equality between these curvature spans and target restricted holonomy can be seen without assuming it from a group label. Parallel transport around a small rectangle is I−area times f(U,V) plus higher-order terms, so every curvature generator lies in the holonomy Lie algebra. For q3/q4 they already span h, while transport preserves J and supplies the matching h upper bound. At q2, all curvature takes values in the central line in every adapted frame. On a simply connected target patch the su2 projection is flat and has trivial restricted holonomy, giving the central upper bound as well as the nonzero lower bound. This concerns the connected/restricted group; no assertion about additional discrete global holonomy is needed.

3. Occupied-sector comparison and fixed-six carrier

These normal ranks use the normal bundle of the even carrier in a selected ten-dimensional parent. The odd depths use its declared pullback. Their actual full normal bundles have odd real ranks and cannot carry this orthogonal complex structure on the whole normal space. At q0 there is neither J nor a central circle.

The earlier fixed-six-normal prescription instead retains the six-plane normal to the four-sector. Where that six-plane is genuinely retained as a carrier it has q3 orientation geometry, including at depth two; it does not become the q4 actual depth-two normal bundle in the table. At five/six part of the fixed six-plane is tangent, so its possible Clifford commutant and fixed triplet projection must be assessed under that older prescription. The q2 canonical result here concerns only the four directions still normal to the six-sector. In particular it does not replace an earlier fixed-carrier calculation by relabeling its SU2 commutant. At ten no independent normal algebra commutes with the complete tangent symbol under either prescription.

The table gives the full target possibility. A particular spatial J map can have zero curvature, reduced holonomy or full target holonomy. For a base of real dimension d, the pointwise curvature span has at most binomial(d,2) directions. At d2 it has at most one, so the target's sixteen-dimensional span cannot be interpreted as sixteen curvature directions at one spatial point.

4. An actual two-dimensional analytic pullback with full holonomy

The following map certifies that a low-dimensional base can nevertheless realize the complete target canonical holonomy. For q3 or q4 set

$$X=X_{E_{12}},\qquad Y(s)=\sum_{r=0}^{L-1}s^rY_r,\qquad L=1+4(q-2),$$

where Y0=X_iE12 and, for k=3 through q, append the four matrices

$$X_{E_{1k}},\ X_{iE_{1k}},\ X_{E_{2k}},\ X_{iE_{2k}}$$

in that order. Define an ordinary orthogonal matrix and an orientation field by

$$g(s,t)=e^{sX}e^{tY(s)},\qquad J(s,t)=g(s,t)J_0g(s,t)^{-1}. \tag{5}$$

This is a concrete real analytic map on a small open rectangle in R2. Every generator is real skew, so J is orthogonal and squares to −I everywhere. Along t0,

$$\theta_s=X,\qquad\theta_t=Y(s),\qquad a_s=a_t=0, \qquad f_{st}(s,0)=-[X,Y(s)]. \tag{6}$$

The vanishing of a_s means adapted frames along that line are related by actual canonical parallel transport. Therefore all the matrices in (6) are already expressed at one common transported fiber. Their covariant s jets at s0 are exactly

$$(\nabla_s)^rf_{st}(0,0)=-r![X,Y_r],\quad 0\le r<L. \tag{7}$$

Equivalently evaluate (6) at L distinct arbitrarily small s values; an invertible Vandermonde matrix recovers the same coefficient span. Small loops based at those points, conjugated back by the line transport, put these matrices in one holonomy Lie algebra. Derivatives also lie in that finite-dimensional closed Lie algebra by taking difference quotients. This justifies using the jets as a holonomy certificate.

For q3 the five jets have linear rank five and Lie closure rank nine. They give both real-skew and imaginary-symmetric off-diagonal generators for links 13 and 23, together with i(D1+D2). The links generate all su3; the last generator has nonzero trace, completing u3. For q4 the nine jets give links from 1 and 2 to 3 and 4. This connected graph similarly generates all su4, and i(D1+D2) supplies the center. Exact elimination in the frozen checker gives rank sixteen. The upper bounds are u3 and u4 because the connection preserves J, proving equality for both pullbacks.

At each individual point of this two-dimensional surface there is still just one curvature two-form. Full holonomy comes from spatially changing curvature transported to one fiber and its commutators. The q4 map supplies a depth-two local dimension control; the q3 map can be extended constantly in extra coordinates to a three- or four-dimensional spatial base. These are explicit J maps on trivial ordinary normal bundles, not proofs that an arbitrary selected nested graph induces precisely this J-only connection. Their purpose is to separate a realizable pullback holonomy statement from a target-dimension count.

5. The same J, pure-spinor ray and central Spin lift

At q3 the chiral Spin6 module has complex dimension four and every nonzero chiral spinor is pure, as established in the frozen normal-charge note. In the selected orientation component, its projective ray determines J and conversely. This does not identify that parameter space with an actual six-dimensional electron spatial submanifold.

On a local Spin lift the normal spin module is

$$S=\Lambda^*W\otimes(\det W)^{-1/2}, \qquad \rho(H)=\Lambda^*(H)-\tfrac12\operatorname{tr}_{\mathbb C}(H)I. \tag{8}$$

The half-determinant notation is local or on the lifted Uq bundle; an arbitrary Uq bundle need not admit that half-line independently. For central H=i alpha Iq, (8) gives the infinitesimal eigenvalue

$$i\alpha(r-q/2)$$

on grade r. In particular the pure vacuum ray has weight −q/2, equal to −3/2 at q3, while grade two has weight +1/2. Omitting this shift would replace the Spin lift by the exterior representation and misstate the central transport.

The canonical compatible connection preserves the J-selected vacuum ray under its Spin lift. Its line curvature is

$$f_{\rm ray}=-\tfrac12\operatorname{tr}_{\mathbb C}f. \tag{9}$$

The traceless su3 part acts trivially on the vacuum ray, while its central trace gives that ray's phase curvature. One can also see (9) by projecting the fixed flat spin connection onto the moving pure-spinor line: m has zero vacuum expectation and the h part acts by −tr/2. This explains the shared orientation and central phase geometry without inferring electric charge.

There is also a global target-bundle check of this same trace relation. On the orientation CP3, let L be the pure vacuum line O(−1), put Q=C4/L, and write H=c1(O(1)). The chosen Spin convention has $S^+=L\oplus Q$. Its ordinary complex normal bundle is $W=\Lambda^2Q$: the opposite complex half is $L\otimes Q$, since $\Lambda^2\mathbb C^4$ splits into those two pieces. Indeed $\det W=(\det Q)^2=L^{-2}$, so the vacuum factor $(\det W)^{-1/2}$ is precisely L. Moreover $\Lambda^2W\otimes L=Q$, recovering the remaining three chiral components in (8).

From $c(Q)=1/(1-H)=1+H+H^2+H^3$, let its formal roots be $x_1,x_2,x_3$. The roots of W are $H-x_i$, giving

$$c_1(W)=2H,\qquad c_2(W)=2H^2,\qquad c_3(W)=0, \qquad \operatorname{tr}f_W=-2f_L. \tag{9a}$$

The last equality also follows immediately from the determinant connection, and agrees with (9). This W is not the orientation-space tangent bundle: $T\mathbb{CP}^3=\operatorname{Hom}(L,Q)$ has first Chern class $4H$. Its underlying real rank-six normal bundle is trivial on this target construction; the checks c1 squared−2c2=0 and c3=0 agree with that fact. Conjugating the orientation convention conjugates the relevant bundles and signs. These target characteristic classes concern the chosen normal-orientation map and cannot be read as physical electric charge or as a characteristic number of a lower-dimensional sector without its actual pullback.

The maintained determinant-paired exterior carrier instead has central weights r and r−q, or normalized r/q−b. It is obtained locally from the Spin carrier by opposite half-determinant twists. Those branches and their particle-state selections are additional representation data. A J field determines neither which branch is occupied nor the physical charge normalization.

6. Constant-J vacuum freedom and changed variations

Choosing the functional Ahat[J] with a fixed flat reference connection excludes the old independently prescribed commuting connection histories. At constant J in its flat reference frame, Ahat0 and Fhat0. There is no freely specifiable su3 magnetic history with the same fixed J under this prescription. This removes that independent connection freedom by changing the allowed fields. It does not prove that J itself has unique vacuum dynamics. Every compatible metric connection still has the form Ahat+H with H in h(J); allowing that extra H restores the freedom. With a nonflat retained reference connection, even constant reference-parallel J can retain its commuting reference curvature.

The variation also changes. In an adapted frame g=e^pi, pi in m near J0, direct series multiplication gives

$$a=P_h(e^{-\pi}de^\pi) =-\tfrac12[\pi,d\pi]+O(\pi^3). \tag{10}$$

Counting each pi and d pi as first order, the displayed h remainder actually starts at fourth order; the weaker cubic remainder is sufficient here. Thus delta a0 for every first-order orientation variation about pi0. Curvature begins at second order in orientation perturbations, and curvature squared begins at fourth order. A curvature-only J action therefore has no ordinary quadratic orientation kinetic term at that constant background; a separately retained positive reference-orientation term |dJ| squared supplies one if adopted.

There is a simple conditional source certificate. Let an associated Uq coefficient field phi have the minimally coupled comparison action |(d+a)phi| squared minus m squared |phi| squared, together with any canonical curvature-squared term and an optional positive reference-orientation term |dJ| squared. At pi0 take phi=v exp(−i omega t+i k dot x), omega squared=|k| squared+m squared. This solves its free field equation and has a nonzero Uq Noether current when omega and v are nonzero. For q at least two, choosing v=(1,0,...,0) also gives a nonzero traceless current component along i diag(1,−1,0,...,0), with magnitude 2|omega| times the squared amplitude in the conventional kinetic normalization. Its orientation first variation vanishes because delta a0; the curvature and sigma first variations vanish there as well. Hence pi0 is stationary despite that current, including when a separate Hodge coefficient removes the curvature term. This is an explicit comparison action and local source statement, not an observed charged-particle solution.

The old independent-A Gauss law would have imposed its own current constraint when its curvature coefficient vanished. That constraint does not follow here, because a0 and its variations are constrained by J; there is no independent Uq temporal connection variation. Induced graph connections similarly require graph variations rather than importing independent connection equations. Any complete J-only action must derive its own orientation equation, same-field source response and remaining constraints. No causal completion or physical force law is claimed by the algebra above.

7. Verification and Reproduction

The standalone exact checker and report retain 330 mathematical conditions, exact bracket and holonomy ranks, explicit common-frame curvature jets, central/traceless decomposition and determinant-shift weights. Run the system Python with SymPy installed and pass a new output path with --report. The original independent execution additionally verified four frozen input hashes, for 334 conditions. Its exact executable, inputs and dated report remain preserved as private research evidence.

The principal reviewed the derivation, bundle factors and changed variation independently. The ray-metric calculation further identifies the complete local projective metric with the normal-orientation metric, with exact factor sixteen. No particle simulation, physical charge normalization or complete force dynamics is inferred from these geometric checks.