CKM mixing from the cross-sector contact operator

Current 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. Particle patterns have different dimensional involvement in that space. The ten-dimensional matter domain is presently modeled by flat $\mathbb R^{10}$. Nested layers describe proposed spatial support, not additional manifolds. Depth counts a pattern's dimensional involvement, not a separate containing space. The familiar three dimensions describe the observer's view, not the ambient dimension. All spatial directions are of the same physical kind. Confinement and persistence of the layers remain dynamical questions.

Mathematical scope: Projective, round-sphere and prescribed global-nesting calculations below retain their stated additional premises. Their curvature, topology and finite-volume normalizations are conditional comparisons; they do not establish the current flat spatial geometry. Field components and mode spaces are distinct from physical position. IDWT has no compactified dimensions or mediator particles; interactions must come from $\Psi_\infty$. See spatial meaning and mathematical scope.

Current Reading and Scope — 1 November 2026

Count ratios and overlap matrices remain conditional mixing comparisons. The current color and charge constructions add explicit representation and source tests, but do not derive a physical CKM matrix or charged-current interaction from this module’s ansatz. Preserve complete normalization, common spatial geometry and the distinction between amplitude overlaps and measured rates.

The new particle-geometry and sector-depths case provides the current derivations, shared assumptions, glossary and reproduction instructions. The twenty-volume research map connects the complete account. Calculations below retain their own stated mathematical premises and do not inherit results from a different action.

1. Result

The present V2 weak-transition operator is

$$P_3D_\infty P_4 \supset g_2\gamma^\mu W_\mu^-T^-P_{(1,1)}\otimes\mathcal C_{34},$$

with the general cross-sector factorization

$$P_dD_\infty P_e = \mathcal I_{de}^{\rm pol}\otimes\mathcal C_{de}^{\rm mode} +\mathcal R_{de}.$$

The existing construction fixes the weak polarization intertwiner and the occupied quark modes, but it does not yet define a nontrivial operator on the three-dimensional generation space.

Under the additional assumption that the leading band-blind weak vertex identifies aligned up- and down-type mass eigenbases, the interaction is generation diagonal. This aligned benchmark gives

$$\boxed{V_{\rm CKM}^{(0)}=I_3.}$$

This is not the observed CKM matrix. A generation-blind vertex in a common interaction basis does not by itself imply this result in the mass basis: in the usual $W^+$ convention, independent left mass diagonalizations give $V=U_{u,L}^\dagger U_{d,L}$. The eigenvalue counts do not fix those eigenvectors or their relative orientation. Thus V2 has not yet predicted either $I_3$ or the observed nontrivial CKM matrix without a further basis/vertex assumption.

The CKM problem isolates a missing piece of the projected master operator:

$$\boxed{ \text{V2 requires a nontrivial generation-space component of } P_3D_\infty P_4. }$$

That component must be derived from the cross-sector coherent contact, the remainder $\mathcal R_{34}$, or both. It cannot be inserted as an arbitrary unitary matrix without destroying the predictive purpose of the construction.

2. The occupied quark bases

The committed up-type occupied modes are

$$(n_u,n_c,n_t)=(3,20,72),$$

at depth

$$d_u=4,$$

while the down-type sector contains

$$n_d=1, \qquad n_s=4,$$

and the bottom state is the critical adjacent-band pair

$$(n_{b,-},n_{b,+})=(16,17).$$

Its mass is

$$m_b =\mu_3\sqrt{S(16,3)S(17,3)}.$$

Thus the physical three-generation spaces may be written schematically as

$$\mathcal F_u =\operatorname{span}\{|u\rangle,|c\rangle,|t\rangle\},$$

$$\mathcal F_d =\operatorname{span}\{|d\rangle,|s\rangle,|b\rangle\}.$$

The first two down states are ordinary occupied coherent bands. The bottom state is already exceptional because it is represented by a two-band singular-value construction. Any CKM derivation must respect that distinction rather than replacing the bottom state by an arbitrary ordinary mode.

3. Where CKM lives in the projected operator

Restrict the cross-sector weak block to the physical generation spaces:

$$C_{34} = P_{\mathcal F_d}\mathcal C_{34}P_{\mathcal F_u}.$$

In chosen orthonormal mass bases,

$$(C_{34})_{ij} = \langle d_i|\mathcal C_{34}|u_j\rangle,$$

where

$$(d_1,d_2,d_3)=(d,s,b),$$

$$(u_1,u_2,u_3)=(u,c,t).$$

The charged-current interaction then has the form

$$\mathcal L_W \supset \frac{g_2}{\sqrt2} \overline{d_{Li}}\gamma^\mu (C_{34})_{ij}u_{Lj}W^-_\mu +\text{h.c.}$$

Canonical normalization is determined by the kinetic metrics, not by an arbitrary polar or singular-value decomposition of the vertex. For a transition matrix $K$ before canonical normalization and positive kinetic metrics $Z_u,Z_d$, define

$$K_c=Z_d^{-1/2}KZ_u^{-1/2},\qquad V_-=L_d^\dagger K_cL_u,$$

where $L_u,L_d$ diagonalize the canonically normalized left-handed mass operators. For already orthonormal mass bases, these transformations are identities. The three-generation charged current is unitary only if

$$K_c^\dagger K_c=I \quad\Longleftrightarrow\quad K^\dagger Z_d^{-1}K=Z_u.$$

This is an additional property to derive, not a permission to discard the singular values of $K$. A deficient-rank kernel cannot be repaired by invertible kinetic normalization; its polar unitary completion is nonunique.

The rows here are $(d,s,b)$ and columns $(u,c,t)$, so $V_-$ is the adjoint of the usual up-row/down-column CKM matrix: $V_{\rm CKM}=V_-^\dagger$. Below, identity and zero-mixing conclusions are independent of this convention. See the normalization audit.

4. Leading-order no-go theorem

A Geometric Defect Test for the Overlap Part

For an actual overlap vertex, first embed the two kinetic-normalized generation frames $U_d,U_u$ into one specified field Hilbert space. Then $K_c=U_d^\dagger U_u$ and $P_d=U_dU_d^\dagger$ obey

$$I-K_c^\dagger K_c =U_u^\dagger(I-P_d)U_u\ge0.$$

For equal finite ranks, unitarity requires equality of the two frame images, while their mass bases can still differ. The defect has a physical interpretation as overlap with omitted receiving channels. Replacing the vertex by a polar unitary would erase that information. This is derived with the contact and connection in the spatial band construction. A general weighted interaction vertex needs its own operator; it is not automatically this pure-overlap example.

The Aligned Leading Ansatz

Assume the following particular leading ansatz. Generation preservation is an extra interaction assumption, not a consequence of the diagonal mass law:

1. each occupied quark mass eigenstate is localized on its own coherent mode line;
2. the scalar contact coefficient is band blind;
3. the weak polarization intertwiner $\mathcal I_{34}^{\rm pol}$ acts only on the weak/color polarization factors;
4. the leading mode contact does not mix distinct occupied generation bands;
5. no generation-dependent contribution from $\mathcal R_{34}$ is retained;
6. the kinetic-normalized charged current satisfies the isometry condition in Section 3.

Then on the physical generation space

$$\mathcal C_{34} \propto \sum_{i=1}^3 |d_i\rangle\langle u_i|,$$

up to the kinetic metrics and removable phases. Physical generation-dependent vertex strengths cannot simply be declared state normalizations.

After canonical normalization,

$$C_{34}\longrightarrow I_3,$$

hence

$$\boxed{V_{\rm CKM}^{(0)}=I_3.}$$

The Jarlskog invariant is then

$$\boxed{J^{(0)}=0.}$$

This conditional diagonal, isometric ansatz cannot reproduce inter-generation transitions or CKM CP violation. Without the isometry assumption, it gives a diagonal vertex, not necessarily $I_3$; diagonality still supplies no mixing. Band blindness alone proves neither diagonality nor isometry.

5. Why the mass hierarchy does not determine CKM

The simplex mass laws determine diagonal mass eigenvalues:

$$M_u =\mu_4\operatorname{diag} \left( S(3,4), S(20,4), S(72,4) \right),$$

while the down sector has the schematic form

$$M_d =\mu_3\operatorname{diag} \left( S(1,3), S(4,3), \sqrt{S(16,3)S(17,3)} \right).$$

These spectra fix eigenvalues but not the relative orientation of the up and down eigenbases.

For any unitary matrices $U_u$ and $U_d$,

$$M_u \mapsto U_uM_uU_u^\dagger,$$

$$M_d \mapsto U_dM_dU_d^\dagger$$

preserves the eigenvalue spectra. The charged-current mixing matrix is

$$V_{\rm CKM}=U_u^\dagger U_d.$$

This is the usual up-row/down-column $W^+$ convention. The down-row/up-column matrix $V_-$ used for the preceding cross-sector kernel is its adjoint, $V_-=U_d^\dagger U_u$; the two must not be given the same matrix label.

Therefore the successful simplex mass hierarchy alone cannot determine CKM. A separate geometric object fixing the relative orientation of the two three-dimensional flavor frames is mathematically necessary.

Any proposal that constructs CKM solely from the six quark masses, with no independently derived cross-sector operator, adds an extra ansatz rather than following from the present V2 mass theorem.

6. The minimal V2 completion

The minimal completion is a generation-space operator

$$G_{34}:\mathcal F_u\to\mathcal F_d$$

appearing in

$$P_3D_\infty P_4 \supset g_2\gamma^\mu W_\mu^-T^-P_{(1,1)} \otimes G_{34}.$$

Its matrix elements must be obtained from the same projected master geometry:

$$(G_{34})_{ij} = \langle d_i| P_3D_\infty P_4 |u_j\rangle_{\rm normalized}.$$

A successful derivation must produce all of the following without fitting CKM data:

• a dominant $1\leftrightarrow1$, $2\leftrightarrow2$, $3\leftrightarrow3$ structure;
• a much larger $1\leftrightarrow2$ transition than $2\leftrightarrow3$;
• a strongly suppressed $1\leftrightarrow3$ transition;
• one irreducible complex phase;
• unitarity after canonical normalization;
• the same universal weak coupling $g_2$ for all generations.

In schematic magnitude form the target hierarchy is

$$|V_{us}|\sim2\times10^{-1}, \qquad |V_{cb}|\sim4\times10^{-2}, \qquad |V_{ub}|\sim4\times10^{-3}.$$

The hierarchy between these three off-diagonal amplitudes is itself a stringent geometric test.

7. Candidate geometric sources already present in V2

The current V2 construction contains several structures capable in principle of generating $G_{34}$, but none has yet been evaluated as a three-generation matrix.

7.1 Cross-sector coherent contact

The general factorization already contains

$$\mathcal C_{34}^{\rm mode}.$$

If its normalized matrix elements between distinct occupied coherent bands are nonzero,

$$\langle d_i|\mathcal C_{34}^{\rm mode}|u_j\rangle\neq0, \qquad i\neq j,$$

then CKM mixing can arise directly from the pushed-forward contact kernel.

The present files specify only its role in source phase and mode counting, not these off-diagonal matrix elements.

7.2 Extrinsic and derivative remainder

The cross-sector operator contains

$$\mathcal R_{34},$$

which collects extrinsic, derivative, and nonlocal corrections. A generation-dependent normal connection or second-fundamental-form term could rotate the quark mass frames:

$$G_{34} = G_{34}^{(0)} +\delta G_{34}^{\rm extrinsic}.$$

If the leading term is diagonal, the observed small CKM angles make this a mathematically natural perturbative possibility. It is not yet a derivation.

7.3 Holonomy of a generation bundle

If the three occupied up modes and three occupied down modes define rank-three bundles over a shared observer or contact base, a relative transport operator could be

$$U_{34}(\gamma) = \mathcal P\exp \left( -\int_\gamma A_{34} \right).$$

Then

$$V_{\rm CKM} = U_d^\dagger U_u$$

would be a genuine relative holonomy. The CP phase would be an invariant phase of this transport, not an inserted parameter.

However, the present V2 documents define normal holonomy for gauge reduction, not a rank-three generation connection $A_{34}$. This is therefore a concrete missing construction.

7.4 Bottom two-band structure

The bottom state already involves an off-diagonal singular-value problem between the adjacent bands $16$ and $17$. This proves that V2 is not restricted in principle to purely diagonal band operators.

The CKM calculation should therefore examine whether the same projected master kernel that produces the bottom singular value also generates weak transition matrix elements between the occupied up- and down-sector bands.

This is the most economical place to search for nontrivial generation mixing because it uses an off-diagonal mechanism already required elsewhere in V2.

8. A finite matrix problem

The CKM calculation can be reduced to a finite operator computation.

Let the up-sector normalized occupied states be

$$|U\rangle = \left( |3,4\rangle, |20,4\rangle, |72,4\rangle \right),$$

and let the down-sector states be

$$|D\rangle = \left( |1,3\rangle, |4,3\rangle, |b,3\rangle \right),$$

where $|b,3\rangle$ is the normalized singular vector of the $16$--$17$ bottom block.

Define

$$K_{ij} = \langle D_i| P_3D_\infty P_4 |U_j\rangle.$$

Remove the derived universal polarization and weak-coupling factors and use the independently derived kinetic metrics and mass-basis rotations:

$$\boxed{ V_{\rm CKM}^{\rm V2} =\left(L_d^\dagger Z_d^{-1/2}KZ_u^{-1/2}L_u\right)^\dagger. }$$

This is a standard unitary CKM candidate only after the isometry test in Section 3 passes. Taking an arbitrary polar factor is not this calculation.

No quark mass values are required to choose its mixing angles once the occupied mode projectors and master kernel are fixed.

This $3\times3$ calculation is the decisive CKM test.

9. Falsification criteria

The CKM sector falsifies or constrains V2 in several clean ways.

Failure A: exact band diagonality

If the kinetic-normalized current in the physical mass bases is diagonal,

$$K_{ij}=0 \qquad(i\neq j),$$

it gives no generation mixing. If it also satisfies the isometry condition, its removable diagonal phases give $V_{\rm CKM}=I_3$, which is experimentally false. A diagonal nonisometric vertex is not a unitary CKM matrix at all.

Failure B: real transition matrix only

If every element of the normalized mass-basis charged-current matrix can be made simultaneously real by rephasing the quark states, then

$$J=0,$$

and V2 fails to reproduce observed quark CP violation.

Failure C: generic order-one mixing

If the normalized off-diagonal elements are generically of the same order as the diagonal ones, the observed hierarchical CKM structure is not reproduced.

Failure D: fitted generation rotation

If an arbitrary $U(3)$ rotation must be introduced after the projected operator is calculated, then CKM has not been derived. The model has merely reintroduced the Standard Model mixing parameters.

Success criterion

A successful V2 derivation must calculate a unique or discretely selected $K$ from the master projection and recover both the approximate CKM hierarchy and a nonzero rephasing-invariant CP phase.

10. Status

The present CKM calculation therefore has a definite outcome:

$$\boxed{ \text{stated generation-diagonal, isometric leading ansatz} \Longrightarrow V_{\rm CKM}=I_3, }$$

which is not physical.

This does not falsify the full projected-master framework because the existing V2 documents explicitly leave the cross-sector projector and remainder $\mathcal R_{34}$ uncalculated. It does falsify the stronger claim that the already-written leading band-blind weak contact is sufficient to explain quark mixing.

The problem has now been reduced to one explicit missing object:

$$\boxed{ K_{ij} = \langle D_i|P_3D_\infty P_4|U_j\rangle. }$$

Computing this finite matrix from the master kernel is the next CKM calculation. The comparison must use the kinetic-normalized mass-basis current and verify its unitarity, mixing magnitudes, and rephasing-invariant CP violation without fitting CKM data. Reality of an unnormalized matrix alone is not a CP test if the other required operators are complex.

11. The asymmetric $4\to3$ geometric-loss test

The preceding discussion treated the cross-sector block abstractly. The simplest literal implementation of the dimensional asymmetry can be tested directly.

A depth-four cumulative mode carrier is

$$W_4=\mathbb C\oplus(T^*Q_4\otimes\mathbb C), \qquad \dim_{\mathbb C}W_4=5,$$

while after one real geometric direction is discarded the depth-three carrier is

$$W_3=\mathbb C\oplus(T^*Q_3\otimes\mathbb C), \qquad \dim_{\mathbb C}W_3=4.$$

Choose the lost direction as a one-dimensional summand $L$ so that locally

$$W_4\cong W_3\oplus L.$$

The occupied mode bundle at degree $N$ then branches exactly as

$$\operatorname{Sym}^{N}(W_4) \cong \bigoplus_{r=0}^{N} \operatorname{Sym}^{N-r}(W_3)\otimes L^{\otimes r}.$$

This is the mathematical statement that the four-dimensional object can lose occupation carried by the discarded direction, while the three-dimensional object does not gain a direction.

If the lost direction is traced out rather than coherently retained, a normalized coherent state with lost-direction weight $p$ induces the binomial mixture

$$\rho_{N,4}\longmapsto \sum_{r=0}^{N} \binom Nr(1-p)^{N-r}p^r\,\rho_{N-r,3}.$$

Hence a pure one-direction-loss map obeys the exact support rule

$$\boxed{N_{\rm out}\leq N_{\rm in}.}$$

Because the particle label is $N=n-1$, this is equivalently

$$\boxed{n_{\rm down}\leq n_{\rm up}.}$$

For the occupied quark modes,

$$(n_u,n_c,n_t)=(3,20,72),$$

and

$$(n_d,n_s,n_{b,-},n_{b,+})=(1,4,16,17).$$

The resulting support table is

The $u\to s$ row is decisive. The observed Cabibbo transition $u\leftrightarrow s$ is nonzero, but the literal one-direction-loss map gives

$$\boxed{K_{su}=0}$$

because it would require

$$N_u=2\longrightarrow N_s=3,$$

which is an increase of cumulative mode degree after the fourth direction has been removed.

Thus the simplest possible identification

$$\boxed{ \text{CKM mixing}=\text{partial trace of one geometric direction} }$$

is ruled out by the committed V2 quark mode indices.

This is a support obstruction for pure loss between the stated occupied-degree states, independent of $p$ and diagonal rescalings or phase conventions. Arbitrary basis mixing or a polar completion may change displayed entries, but is not the same degree-resolved loss channel. A partial trace is a quantum channel, not by itself a coherent CKM amplitude.

The result also clarifies what additional geometry would be required. A successful $4\to3$ weak map must do more than marginalize one coordinate. It must contain an operator that can change cumulative mode degree after or during projection,

$$P_{N_d,3}\,D_{34}\,P_{N_u,4}\neq0$$

for cases with

$$N_d>N_u,$$

in particular

$$P_{3,3}\,D_{34}\,P_{2,4}\neq0$$

for $u\to s$.

Such a term can arise only from a nontrivial derivative, extrinsic, resonant, or nonlocal part of the cross-sector master operator. It is not supplied by dimensional loss alone.

Therefore the geometric asymmetry remains physically relevant, but the CKM matrix cannot be identified with bare $4\to3$ dimensional reduction. The experimentally required Cabibbo element proves that the weak cross-sector operator must combine dimensional reduction with genuine mode-changing dynamics.