Normal Charge Geometry and the Matter-Depth Availability Test

The charge-availability test

A geometric charge comparison leaves the maintained matter-sector assignment not ruled out. It supplies a conditional compatibility result: one common exterior construction gives the relative charge pattern, and one stated occupation rule keeps the assigned matter charges available across depths three, four, five, six and ten. Under that rule depth ten retains only the charge −1 singlet. The calculation does not establish a four-of-six-sector preference or a unique complete assignment.

For a four-dimensional sector inside a ten-dimensional one, the ordinary normal space has six real directions. Assume a compatible complex structure and write this same spatial normal bundle as a Hermitian complex rank-three bundle $W$, with determinant line $D=\det W$. Choose the associated carrier

$$E=\Lambda^{\mathrm{even}}W\oplus\overline{\Lambda^{\mathrm{odd}}W} \cong(\mathbf1\oplus\Lambda^2W)\otimes(\mathbf1\oplus D^{-1}).$$

Conjugation on the odd degrees is a geometric hypothesis; it does not follow from nesting alone. The construction introduces no additional physical spatial dimensions. For the central frame rotation $W\mapsto e^{i\phi}W$, the four blocks have integer weights $0,2,-3,-1$ and multiplicities $1,3,1,3$. Normalizing the determinant weight three to one charge unit gives

$$(q_0,q_2,q_3,q_1)=(0,2/3,-1,-1/3).$$

The two triplets have the same SU(3) representation type. The determinant of $E$ is trivial, so the multiplicity-weighted trace vanishes. Both paired transitions are sections of the same line $D$ and have charge difference one. These are consequences of the stated carrier, not separately fitted particle charges. Related exterior-charge patterns have published precedents in Stoica's exterior construction and Furey's number-operator construction; the comparison here concerns their geometric realization in ordinary spatial normal directions.

For a complete exterior carrier of complex rank $n$, the block multiplicities are $\binom nk$ and its total rank is $2^n$. The two-singlet/two-triplet target therefore requires $2^n=8$, hence $n=3$ and six ordinary normal directions. Among the maintained depths, only four inside ten has that gap. Other integer depths also permit two inside eight or three inside nine. This is a normal-gap constraint within the stated carrier family, rather than a derivation of every absolute sector depth.

To test an explicit link to the sector assignments, let

$$S_d=T M_d\cap W_{\mathbb R},\qquad A_d=S_d\cap J S_d,\qquad B_d=A_d^\perp\subset W.$$

These definitions apply at a common point of the nested sectors along the depth-four base; they define subbundles on regions of constant rank. Let $r_d=\dim_{\mathbb C}A_d$. Exterior components have occupation $p$ in $A_d$ and $q$ in $B_d$, multiplicity $\binom{r_d}{p}\binom{3-r_d}{q}$, and charge $(-1)^{p+q}(p+q)/3$. Apply the same additional rule at every depth: retain components with $p=r_d$, so all available complex tangent directions in this normal bundle are occupied.

For the explicit aligned nested flag, $r_d$ at depths $(2,3,4,5,6,10)$ is $(0,0,0,0,1,3)$. The resulting comparison is:

Thus one uniform rule accommodates the assigned matter charges without separately adjusting a coefficient at each depth. Its partial triplet components at depth six do not preserve a complete SU(3) triplet; their availability is not identification with physical particles. Tilting the depth-six real plane relative to $J$ changes its maximal complex rank from one to zero in the tested control, restoring the full charge table there while still allowing −1. Depth ten still retains the charged singlet. A common rotation of both geometry and $J$ preserves the result covariantly.

The same calculations expose the limitation. Depths three and four both have $S_d=0$, so every uniform selector using only this normal intersection and $J$ gives identical spectra there. These inputs cannot uniquely select their different quark charges. Seven occupation predicates, two ways to choose a complex subspace and two compatible flags give 28 comparisons; none selects the complete maintained assignment. Some retain every required charge while also allowing additional ones. This excludes those selectors as complete assignment laws while leaving the displayed compatibility result intact.

The defensible contribution to the sector case is therefore specific: the tested charge geometry does not exclude the maintained matter assignments, its full charge multiplicities fit the four-in-ten normal gap, and one explicit common occupation hypothesis selects the charged singlet at depth ten. It remains a conditional consistency result, not experimental confirmation or proof of preference at the other depths. The physical identification of the determinant connection as electromagnetism remains open. A complete particle simulation is not needed to assess these geometric statements.

The shared normal geometry and normal character calculation give the connected frame and charge identities. The charge checker and recorded results preserve the explicit selector controls.