Physical premise (IDWT): Physical spacetime has infinitely many ordinary spatial dimensions plus time. Matter occupies up to ten spatial dimensions; all remaining spatial directions are unbound for every particle. Depth counts a pattern's bound spatial involvement within that space. “Observed” and “unobserved” describe an observer's access, not different kinds of dimension. Older projective and nested-geometry calculations below retain their additional hypotheses; they do not establish a separate physical manifold or compactified direction. See spatial meaning and mathematical scope.
Summary: A determinant-trivial auxiliary bundle realizes the charge branching globally but is not the tangent spinor bundle or a derived family of modes.
1. An explicit globally admissible bundle
Let L=O(1) on CP5 and choose
The determinant is L^(-6+6)=the trivial line. Hermitian metrics and a chosen parallel determinant trivialization reduce its frame bundle to S(U(3) times U(2)). This is a real global construction, not local trace subtraction on the tangent bundle. The group is the usual (SU(3) times SU(2) times U(1))/Z6 subgroup of SU(5).
The integer block charges (-2,3) are the primitive solution of 3a+2b=0, up to simultaneous sign reversal. Dividing by six gives the existing hypercharge generator Y=diag(-1/3,-1/3,-1/3,1/2,1/2). The choice of degree-one L and of this auxiliary carrier is an additional construction, not a background selection derived from the master action.
For a unitary connection a on L and traceless unitary connections A_c,A_w on the displayed trivial multiplicity bundles, a compatible connection is
The expression is interpreted in local line frames with their transition functions; the global tensor-product connection is what is defined. Its trace is zero and its mixed blocks vanish. With A_c=A_w=0, its actual holonomy is only a U(1) subgroup; the allowed structure group is not a proof that every non-Abelian gauge interaction is dynamically generated.
2. The entire charge table is a global exterior bundle
Use F_g=Lambda^even E_g, of rank 16. Its decomposition is
| Exterior component | SU(3) x SU(2) | Line factor | Y | Conventional field label |
|---|---|---|---|---|
| Lambda^0 | (1,1) | L^0 | 0 | nu^c |
| Lambda^2 colour | (bar3,1) | L^-4 | -2/3 | u^c |
| colour tensor weak | (3,2) | L^1 | 1/6 | Q |
| Lambda^2 weak | (1,1) | L^6 | 1 | e^c |
| Lambda^3 colour tensor weak | (1,2) | L^-3 | -1/2 | leptons |
| Lambda^2 colour tensor Lambda^2 weak | (bar3,1) | L^2 | 1/3 | d^c |
The line exponents are just sums of the five primitive weights. Their multiplicities are 1,3,6,1,2,3. This is the same fibre branching as the existing gauge module, now on a specified global bundle. If read as one family of left-handed 4D fields, the usual hypercharge, mixed, and weak-doublet anomaly checks pass; that is a representation check, not a derivation of 4D field content.
3. It is not a disguised tangent or tangent-spinor bundle
The carrier has
which differs from c(T CP5). Even the rank-16 fibre resemblance does not identify F_g with the geometric half-spin bundle. Their second Chern-character components are
For the first value sum one-half the squares of all 16 line exponents. For the second, the half-spin weights give 2 sum(x_i^2), with x_i the tangent Chern roots; sum(x_i^2)=c1^2-2c2=6H^2. Both rank-16 bundles have c1=0, so a nontrivial line twist cannot equate them while preserving c1. The global matter carrier is genuinely additional data.
4. Fibre multiplicities are not zero-mode multiplicities
If one were additionally to use the ordinary spin Dirac operator on CP5 twisted by a line L^q, its index would be
where the binomial is the polynomial extension for negative arguments and K^(1/2)=O(-3). For the charges q=(-4,-3,0,1,2,6) this gives (-6,-1,0,0,0,56), not one mode per representation. This is a conditional diagnostic of a particular twisted-Dirac interpretation, not a compactification premise imposed on V2. It prevents declaring a complete particle spectrum merely from the 16-dimensional exterior fibre.
5. What this does and does not solve
The global group and charge-table construction is now explicit and avoids the tangent splitting obstruction. It does not derive why this bundle is selected, how it is tied to the single master spinor, which differential operator gives occupied modes, gauge kinetic terms, coupling strengths, or the cross-depth charged-current blocks. Treat it as an admissible candidate against which the missing master-to-gauge map can be tested, not a replacement axiom silently added to V2.
Verification
cp5_gauge_bundle_checks.py computes the Chern classes, exterior weights, anomaly bookkeeping, spinor-character mismatch, and the conditional index values exactly.
Related
- Research status — current proof boundaries and correction routes.