A projected-contact route to mode and gauge selection

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

The current count hypothesis uses one common fundamental per pattern kind and counts physical portions rather than independently activated resonant levels. The projected and projective formulas below are conditional comparisons. Their spectral labels, gauge notation and normalization do not establish a separate physical arena, occupation rule or mediator-particle interaction.

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. Claim and status

The covariant self-contact construction supplies the exact positive eigenvalue

$$D_{N,d} = S(N+1,d) = \binom{N+d}{d}$$

on the occupied coherent mode line of the cumulative mode bundle. This module tests whether that contact eigenvalue can do a second job: select the next occupied mode by resonance before the sector-dependent mass scales are applied.

The proposed mechanism is

$$\boxed{ \text{projected contact eigenvalue} + \text{ground-relative phase} + \text{retarded cross-sector source} \longrightarrow \text{one resonant mode index}. }$$

It is combined with the exterior-algebra realization of the depth-ten chiral spinor to obtain a candidate gauge-selection block.

The status is mixed.

Exact mathematics developed here

• the driven-resonance theorem once the source frequency is specified;
• the general phase-matching formula for additive, shared-ground, and inclusion--exclusion channels;
• tensor-product contact eigenvalues;
• the canonical decomposition $\mathbb C^4\otimes\mathbb C^4=\operatorname{Sym}^2\mathbb C^4\oplus \Lambda^2\mathbb C^4$ and its $10+3+3$ refinement;
• the projective hyperplane exact sequence producing $35=15+20$;
• the exterior-algebra realization of the chiral $Spin(10)$ spinor;
• the $SU(3)\times SU(2)\times U(1)$ branching and charge table;
• the integer fixed-point calculation selecting depths six and ten from the critical-spinor closure equation.

New physical hypotheses

• the occupied contact eigenvalue contributes the pre-prism source frequency $S(n,d)-1$ after removal of the common ground phase;
• every antisymmetric join contributes one additional phase quantum;
• tensor-contact eigenvalues act as source frequencies for product-form sites;
• the charged-lepton projector selects the determinant-singlet $13$-dimensional contact carrier;
• a rank-$3+2$ polarization split supports the determinant-one gauge subgroup $S(U(3)\times U(2))$ at the fibre/representation level. Its global gauge bundle and relation to the geometric connection remain to be specified; this cannot be the full connection on $T^{1,0}\mathbb{CP}^5$: even its global complex rank-$3+2$ splitting is obstructed (Section 15.1a).

Open calculations

• derive the ground-subtracted source phase directly from the master action;
• calculate every cross-sector projector and prove that no unwanted resonant channel survives;
• establish a topologically admissible global polarization/gauge bundle and derive its rank-$3+2$ reduction; the literal tangent-bundle split, even without the determinant-one condition, is obstructed;
• derive the gauge-connection normalizations and coupling constants from the same projected operator.

The module therefore gives a sharp route rather than declaring the mode and gauge problems closed.

2. Geometric and operator input

Let a particle of real depth $d$ live on the sector manifold $Q_d$. For a positive mode index $n$, write

$$N=n-1.$$

The cumulative tangent-mode bundle is

$$\mathcal E_{N,d} = \operatorname{Sym}^{N} \left( \mathbf1_{\mathbb C} \oplus T^*Q_d\otimes\mathbb C \right).$$

Its complex rank is

$$\operatorname{rank}_{\mathbb C}\mathcal E_{N,d} = \binom{N+d}{d} = S(N+1,d).$$

At an occupied projective mode ray $[z]$, the finite-band projection of a point contact is

$$\Lambda_{N,d}([z]) = S(N+1,d) |z;N\rangle\langle z;N|.$$

Thus

$$\Lambda_{N,d}([z])|z;N\rangle = S(N+1,d)|z;N\rangle.$$

The same integer $d$ is the rank of $T^*Q_d$, the number of independent momenta in the projected principal symbol, and the degree controlling the simplex growth law. Cross-depth interactions act by pullback of the lower-depth field, pushforward of the higher-depth current, and an invariant polarization projector on the shared coordinates.

The mode-selection problem is not the existence of candidate eigenvalues. The free sector operator supplies a candidate band for every positive integer $n$. The problem is to derive why only a finite typed subset is resonantly sourced and remains as an asymptotic particle spectrum.

3. The contact-frequency hypothesis

Let

$$P_{[z]}=|z;N\rangle\langle z;N|$$

be the occupied coherent-line projector. Define the ground-subtracted contact operator on that line by

$$\widetilde\Lambda_{N,d}([z]) = \Lambda_{N,d}([z])-P_{[z]}.$$

It obeys

$$\widetilde\Lambda_{N,d}([z])|z;N\rangle = \left[S(N+1,d)-1\right]|z;N\rangle.$$

The subtraction removes the one universal ground occupation already present at $N=0$. The proposed pre-prism source frequency is therefore

$$\boxed{ \Omega_{n,d}=S(n,d)-1. }$$

This is the central new dynamical hypothesis of the module. It must ultimately follow from the time-dependent projected master equation. The subtraction is not inserted into the rest-mass operator: the physical mass remains

$$m(n,d)=\mu_dS(n,d).$$

It applies only to the ground-relative phase used in the mode-production problem.

4. The resonant firing theorem

Let the target pre-prism tower have level operator

$$\widehat N\chi_r=(r-1)\chi_r.$$

Consider a retarded cross-sector source with polarization projector $\mathcal I$:

$$\left(i\partial_\tau-\widehat N\right)\phi = h\,\mathcal I J\,e^{-i\Omega\tau}.$$

Expand

$$\phi(\tau)=\sum_{r\geq1}c_r(\tau)\chi_r$$

and write

$$J_r=\langle\chi_r,\mathcal I J\rangle.$$

Then

$$i\dot c_r-(r-1)c_r = hJ_r e^{-i\Omega\tau}.$$

For $c_r(0)=0$ and $r-1\neq\Omega$,

$$c_r(\tau) = hJ_r e^{-i(r-1)\tau} \frac{1-e^{-i[\Omega-(r-1)]\tau}} {\Omega-(r-1)}.$$

The nonresonant amplitude is bounded. At

$$r-1=\Omega,$$

the limit is

$$c_r(\tau) = -ihJ_r\tau e^{-i\Omega\tau}.$$

The linear growth is the usual secular response of a resonantly driven mode. Nonlinear self-contact must eventually saturate it into a finite-amplitude fixed point.

Using the contact-frequency hypothesis,

$$r-1=S(n,d)-1,$$

so the unique resonant target is

$$\boxed{ r=S(n,d). }$$

This gives the positive firing rule

$$\boxed{ (n,d_{\rm eval}) \longmapsto n_{\rm out}=S(n,d_{\rm eval}). }$$

The representation projector is equally necessary. Exact numerical resonance does not produce a mode when

$$J_r=0.$$

For a shared-depth carrier $F_a$ and matter representations $R_b,R_c$, this condition is controlled by

$$\mathcal V_{abc} = \operatorname{Inv}_{G_{\rm shared}} \left(F_a\otimes R_b^*\otimes R_c\right).$$

The typed firing criterion is therefore

$$\boxed{ \dim\mathcal V_{abc}>0, \qquad r-1=\Omega. }$$

The first equation decides whether the channel exists. The second decides which integer band it occupies.

5. Primary simplex firings

The first four nontrivial simplex values follow immediately:

$$S(3,3)=\binom{5}{3}=10,$$

$$S(3,4)=\binom{6}{4}=15,$$

$$S(4,3)=\binom{6}{3}=20,$$

and

$$S(4,4)=\binom{7}{4}=35.$$

Thus the contact-resonance map produces

$$\boxed{ 3\xrightarrow{d=3}10, \qquad 3\xrightarrow{d=4}15, \qquad 4\xrightarrow{d=3}20, \qquad 4\xrightarrow{d=4}35. }$$

These are the first two neutrino indices, the charm index, and the muon index in the committed tower. The mode values arise from the contact spectrum before any particle mass is used.

6. Multi-source phase matching

Let a nonlinear source contain positive-frequency occupied factors with indices $n_i$, conjugated factors with indices $m_j$, and $q$ degree-one join insertions. Its ground-relative phase rate is

$$\Omega = \sum_i(n_i-1) - \sum_j(m_j-1) + q.$$

Exact resonance with target index $r$ gives

$$\boxed{ r = 1+ \sum_i(n_i-1) - \sum_j(m_j-1) + q. }$$

This one equation contains the tower's additive operations.

6.1 Shared-ground join

For two positive factors and no join insertion,

$$r = 1+(a-1)+(b-1) = a+b-1.$$

Hence

$$\boxed{g(a,b)=a+b-1.}$$

The subtraction is the shared ground occupation.

A sector-coordinate source of real depth $d$ carries the relative level $d-1$. Joining it to mode $n$ therefore gives

$$\boxed{g(d,n)=d+n-1.}$$

The electroweak chain becomes

$$n_W = 1+(72-1)+(5-1) =76,$$

$$n_Z = 1+(76-1)+(6-1) =81,$$

and

$$n_H = 1+(15-1)+(81-1) =95.$$

6.2 Antisymmetric binary join

For two positive factors and one degree-one join insertion,

$$r = 1+(a-1)+(b-1)+1 = a+b.$$

Therefore

$$\boxed{r=a+b.}$$

The charged-lepton and strange/tau relations are

$$n_s = 1+(3-1)+(1-1)+1 =4,$$

$$n_e = 1+(10-1)+(3-1)+1 =13,$$

$$n_\mu = 1+(15-1)+(20-1)+1 =35,$$

and

$$n_\tau = 1+(22-1)+(1-1)+1 =23.$$

The extra unit is interpreted as one phase quantum carried by the antisymmetric pairing node.

6.3 Inclusion--exclusion source

For two positive factors and one conjugated shared factor,

$$r = 1+(a-1)+(b-1)-(c-1) = a+b-c.$$

Thus

$$\boxed{r=a+b-c.}$$

The third neutrino index follows from the cubic source

$$\psi_{\nu_1}\psi_{\nu_2}\overline{\psi_u}$$

as

$$n_{\nu_3} = 1+(10-1)+(15-1)-(3-1) =22.$$

The conjugated factor removes the already-shared up-type seed substructure.

6.4 Three-way additive closure

A fully joined source with three positive factors has two independent binary joins and therefore $q=2$:

$$r = 1+(a-1)+(b-1)+(c-1)+2 = a+b+c.$$

The Higgs closure is

$$n_H = 1+(3-1)+(20-1)+(72-1)+2 =95.$$

The two Higgs routes consequently carry the same source phase:

$$15+81-1 = 3+20+72 = 95.$$

This is a genuine phase co-fixed point if both invariant source projectors are nonzero.

7. Tensor-contact product sites

Let two normalized coherent contacts be

$$\Lambda_A=D_AP_A, \qquad \Lambda_B=D_BP_B.$$

Their tensor product is

$$\Lambda_A\otimes\Lambda_B = D_AD_B(P_A\otimes P_B).$$

It has one nonzero eigenvalue

$$\boxed{D_{AB}=D_AD_B.}$$

Under the same ground-relative resonance hypothesis, a tensor contact selects mode index $D_AD_B$.

7.1 Bottom critical pair

Let

$$W=\mathbb C\Omega\oplus U, \qquad \dim_{\mathbb C}U=3, \qquad \dim_{\mathbb C}W=4.$$

The two-factor contact has eigenvalue

$$\dim(W\otimes W)=4^2=16.$$

It supplies the critical beat anchor

$$\boxed{k_0=16.}$$

One additional ground insertion gives the adjacent partner

$$\boxed{k_0+1=17.}$$

The bottom mass can therefore use the geometric mean of the two adjacent contact eigenvalues,

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

without treating the two levels as unrelated inputs.

The physical derivation still has to show why the projected bottom block is a two-band singular-value problem rather than an ordinary diagonal band.

8. The four-slot contact decomposition and the electron index

The four-slot carrier

$$W=\mathbb C\Omega\oplus U, \qquad \dim U=3,$$

has the canonical tensor decomposition

$$W\otimes W = \operatorname{Sym}^2W \oplus \Lambda^2W.$$

Because

$$\Lambda^2W = (\Omega\wedge U) \oplus \Lambda^2U,$$

we obtain

$$\boxed{ W\otimes W = \operatorname{Sym}^2W \oplus (\Omega\wedge U) \oplus \Lambda^2U. }$$

The dimensions are

$$16=10+3+3.$$

The symmetric piece has

$$\dim\operatorname{Sym}^2W = \binom{4+2-1}{2} =10,$$

which is the first neutrino index. Both remaining pieces have dimension three, but they transform differently under the seed $SU(3)$:

$$\Omega\wedge U\cong U\cong\mathbf3,$$

while

$$\Lambda^2U \cong U^* \cong \overline{\mathbf3}.$$

The determinant contraction

$$U\otimes\Lambda^2U \longrightarrow \Lambda^3U \cong \mathbb C$$

is an $SU(3)$ singlet and is unique up to normalization. This distinguishes $\Lambda^2U$ from the second literal triplet $\Omega\wedge U$.

Define the candidate charged-lepton contact carrier

$$E_e = \operatorname{Sym}^2W \oplus \Lambda^2U.$$

Then

$$\boxed{ \dim E_e=10+3=13. }$$

Equivalently,

$$W\otimes W = E_e \oplus (\Omega\wedge U),$$

so

$$\boxed{16=13+3.}$$

This is the exact carrier identity behind

$$n_s^2=n_e+n_u.$$

It does not say that the electron is composed of a neutrino and an up quark. The spaces above are contact-index carriers. The physical lepton state arises only after the determinant-singlet projector contracts the coloured seed indices.

The unresolved statement is that the projected weak/lepton vertex actually selects $E_e$ rather than another $13$-dimensional subspace. The decomposition makes that selection canonical once determinant-singlet coupling is imposed.

9. Pascal closure as a projective exact sequence

The muon identity

$$35=15+20$$

is the rank statement of a standard hyperplane restriction sequence.

Let

$$\mathbb P^3\hookrightarrow\mathbb P^4$$

be a hyperplane defined by a linear form $\ell$. Then

$$0 \longrightarrow \mathcal O_{\mathbb P^4}(2) \xrightarrow{\times\ell} \mathcal O_{\mathbb P^4}(3) \longrightarrow \mathcal O_{\mathbb P^3}(3) \longrightarrow0.$$

The relevant higher cohomology vanishes, so global sections give

$$0 \longrightarrow H^0(\mathbb P^4,\mathcal O(2)) \longrightarrow H^0(\mathbb P^4,\mathcal O(3)) \longrightarrow H^0(\mathbb P^3,\mathcal O(3)) \longrightarrow0.$$

Their dimensions are

$$\dim H^0(\mathbb P^4,\mathcal O(2)) = \binom{6}{4} =15,$$

$$\dim H^0(\mathbb P^4,\mathcal O(3)) = \binom{7}{4} =35,$$

and

$$\dim H^0(\mathbb P^3,\mathcal O(3)) = \binom{6}{3} =20.$$

Therefore

$$\boxed{35=15+20}$$

is the dimension identity of an exact sequence rather than an isolated Pascal coincidence. In the tower notation,

$$\boxed{n_\mu=n_{\nu_2}+n_c.}$$

The physical projector must identify the three terms of this sequence with the corresponding sector carriers.

10. The third-neutrino pushout

Let

$$A=\operatorname{Sym}^2\mathbb C^4, \qquad \dim A=10,$$

and

$$B=\operatorname{Sym}^2\mathbb C^5, \qquad \dim B=15.$$

Suppose the projected Hopf-chain maps identify a common three-dimensional seed carrier

$$U\cong\mathbb C^3$$

inside both $A$ and $B$. Form the pushout quotient

$$E_{\nu_3} = \frac{A\oplus B}{\Delta U},$$

where $\Delta U$ is the diagonal image of the shared carrier. Then

$$\dim E_{\nu_3} = 10+15-3 =22.$$

Hence

$$\boxed{ n_{\nu_3} = n_{\nu_1}+n_{\nu_2}-n_u =22. }$$

The dimension calculation is exact once the two embeddings of $U$ are given. The open geometric task is to construct those embeddings from the actual cross-sector projectors rather than inserting them abstractly.

11. Separation of the depth-ten coordinate and spinor fibre

The depth-ten particle manifold is proposed to be

$$Q_{10}=\mathbb{CP}^5.$$

A point is represented by

$$q=[Z], \qquad Z\in\mathbb C^6\setminus\{0\}.$$

The coordinate lift $Z$ is not the value of the Dirac spinor. Let

$$L_q=\mathbb CZ.$$

Then

$$T_q^{1,0}\mathbb{CP}^5 \cong \operatorname{Hom}_{\mathbb C}(L_q,L_q^\perp) \cong \mathbb C^5.$$

The five-complex-dimensional tangent polarization is Clifford-quantized by the exterior algebra

$$S_{10} \cong \Lambda^\bullet\mathbb C^5.$$

The two chiral halves are

$$S_{10}^+ \cong \Lambda^{\rm even}\mathbb C^5,$$

and

$$S_{10}^- \cong \Lambda^{\rm odd}\mathbb C^5.$$

Each has complex dimension

$$1+10+5=16.$$

Globally, the spinor bundle can be written in Kähler form as

$$\Sigma Q_{10} \cong \Lambda^{0,\bullet}T^*Q_{10} \otimes K_{Q_{10}}^{1/2}.$$

Since

$$K_{\mathbb{CP}^5}=\mathcal O(-6),$$

one may take

$$K_{Q_{10}}^{1/2}=\mathcal O(-3).$$

The bridge is therefore

$$\boxed{ \mathbb C^6 \longrightarrow \mathbb{CP}^5 \longrightarrow T_q^{1,0}\mathbb{CP}^5\cong\mathbb C^5 \longrightarrow \Lambda^{\rm even}\mathbb C^5\cong\mathbb C^{16}. }$$

There is no requirement for a direct identification $\mathbb C^6\cong\mathbb C^{16}$. The first space supplies homogeneous coordinates; the second exterior algebra supplies the chiral spinor fibre.

12. The rank-$3+2$ polarization reduction

Let

$$V=T_q^{1,0}Q_{10} \cong \mathbb C^5.$$

The required gauge branch begins with a polarization splitting

$$\boxed{V=V_c\oplus V_w,}$$

where

$$\dim_{\mathbb C}V_c=3, \qquad \dim_{\mathbb C}V_w=2.$$

The ranks match the complex dimensions of the depth-six and depth-four Kähler sectors:

$$\dim_{\mathbb C}Q_6=3, \qquad \dim_{\mathbb C}Q_4=2.$$

They are also the two Kähler factors whose cohomology ring supplies the finite deposit grid. This numerical agreement does not produce a global tangent splitting: Section 15.1a proves that the literal complex tangent bundle of $Q_{10}=\mathbb{CP}^5$ admits no rank-$3+2$ splitting at all. A different globally admissible carrier and its dynamical selection law are required.

The unitary stabilizer of the split is $U(3)\times U(2)$. Its intersection with $SU(5)$, after separately imposing determinant one, is

$$S(U(3)\times U(2)) = \left\{ (g_c,g_w): \det g_c\det g_w=1 \right\}.$$

Its global group form is

$$S(U(3)\times U(2)) \cong \frac{SU(3)\times SU(2)\times U(1)}{\mathbb Z_6}.$$

A central generator commuting with $SU(3)\times SU(2)$ has the form

$$Y=y_cI_3\oplus y_wI_2.$$

The determinant condition gives

$$3y_c+2y_w=0.$$

Minimal conventional normalization fixes, up to overall sign,

$$\boxed{ Y=-\frac13I_3\oplus\frac12I_2. }$$

13. One chiral depth-ten fibre contains one family representation

Decompose the positive-chirality spinor as

$$S_{10}^+ = \Lambda^{\rm even}(V_c\oplus V_w).$$

Then

$$S_{10}^+ = \bigoplus_{\substack{0\leq p\leq3\\0\leq q\leq2\\p+q\ {\rm even}}} \Lambda^pV_c\otimes\Lambda^qV_w.$$

On the $(p,q)$ component, the Abelian generator acts with weight

$$\boxed{ Y(p,q)=-\frac p3+\frac q2. }$$

The complete branching is

The dimensions add to

$$1+3+6+1+2+3=16.$$

Equivalently, under $SU(5)\subset Spin(10)$,

$$\boxed{ \mathbf{16} = \mathbf1 \oplus \overline{\mathbf5} \oplus \mathbf{10}. }$$

Several selection rules are immediate.

• Lepton components have colour exterior degree $p=0$ or $p=3$, so they are $SU(3)$ singlets.

• Quark components have $p=1$ or $p=2$, producing $\mathbf3$ or $\overline{\mathbf3}$.

• Weak doublets are exactly the $q=1$ components.

• Weak singlets have $q=0$ or $q=2$.

• The neutrino component has

  $$Q=T_3+Y=\frac12-\frac12=0.$$

• The charged-lepton component has

  $$Q=-\frac12-\frac12=-1.$$

• The quark doublet with $Y=1/6$ has charges

  $$Q_u=\frac12+\frac16=\frac23, \qquad Q_d=-\frac12+\frac16=-\frac13.$$

The representation is chiral because only one of the two $Spin(10)$ Weyl spaces is selected. A separate projected-Clifford calculation must connect this spatial spinor chirality to the observed spacetime handedness.

14. Exterior creation operators and the projected gauge block

Let

$$a_i^\dagger,a_i, \qquad i=1,2,3,$$

be exterior creation and contraction operators on $V_c$, and let

$$b_\alpha^\dagger,b_\alpha, \qquad \alpha=1,2,$$

act on $V_w$. Define the exterior-degree operators

$$N_c=\sum_{i=1}^3a_i^\dagger a_i, \qquad N_w=\sum_{\alpha=1}^2b_\alpha^\dagger b_\alpha.$$

Then

$$\boxed{ Y=-\frac13N_c+\frac12N_w. }$$

The non-Abelian generators act as

$$T_c^a = a_i^\dagger \left(\frac{\lambda^a}{2}\right)^i{}_j a^j,$$

and

$$T_w^r = b_\alpha^\dagger \left(\frac{\sigma^r}{2}\right)^\alpha{}_{\beta} b^\beta.$$

A candidate polarization connection is

$$\nabla_\mu^{\rm pol} = \partial_\mu -ig_3G_\mu^aT_c^a -ig_2W_\mu^rT_w^r -ig_1B_\mu Y.$$

These components are connection pieces of the projected geometry. Their propagation laws and kinetic normalizations must be obtained from the master action rather than assumed from the Standard Model.

The leading cross-sector operator is proposed to factor as

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

Here

• $\mathcal I_{de}^{\rm pol}$ is the exterior-algebra intertwiner selecting the allowed gauge polarization;
• $\mathcal C_{de}^{\rm mode}$ is the pushed-forward coherent contact operator supplying the source phase and mode count;
• $\mathcal R_{de}$ collects extrinsic, derivative, and nonlocal corrections.

On the occupied coherent line,

$$\mathcal C_{de}^{\rm mode} \longrightarrow S(n,e).$$

The weak transition blocks have the schematic form

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

for the up-to-down component of the quark doublet, and

$$P_6D_\infty P_5 \supset g_2\gamma^\mu W_\mu^- T^-P_{(3,1)} \otimes \mathcal C_{65},$$

for the neutrino-to-charged-lepton component. The tau block is

$$P_{10}D_\infty P_5 \supset g_2\gamma^\mu W_\mu^- T^-P_{(3,1)} \otimes \mathcal C_{10,5}.$$

Their adjoints contain the reverse $W^+$ transitions.

Colour and the Abelian generator act diagonally on the depth-resolved components:

$$P_dD_\infty P_d \supset \gamma^\mu \left( g_3G_\mu^aT_c^a + g_1B_\mu Y \right).$$

Colour vanishes identically on the lepton components because $T_c^a$ acts trivially on the $p=0$ and $p=3$ exterior sectors.

15. The no-leptoquark holonomy condition

A generic $U(5)$ connection contains off-diagonal components mixing $V_c$ and $V_w$. In an $SU(5)$ reading these are the unwanted colour--weak generators. The projected connection must preserve the rank-$3+2$ split:

$$\boxed{ P_c\nabla^{\rm pol}P_w=0, \qquad P_w\nabla^{\rm pol}P_c=0. }$$

For a unitary connection on an existing global split bundle, this implies

$$\boxed{ \operatorname{Hol}(\nabla^{\rm pol}) \subseteq U(3)\times U(2). }$$

This removes off-diagonal colour--weak connection components. It does not impose determinant one: that would additionally require the determinant connection to have trivial holonomy. Split preservation and determinant-one reduction are distinct conditions.

15.1 Global tangent-bundle obstruction

If the polarization carrier is literally $E=T^{1,0}\mathbb{CP}^5$, the determinant-one reduction proposed in the earlier version is impossible. The canonical line already identified in Section 11 gives

$$\det E=K_{\mathbb{CP}^5}^{-1}=\mathcal O(6), \qquad c_1(E)=6H\ne0, \qquad H=c_1(\mathcal O(1)).$$

Holonomy contained in $S(U(3)\times U(2))\subset SU(5)$ would induce trivial holonomy on $\det E$, hence a global parallel determinant section. This contradicts $c_1(E)=6H$. Equivalently, traceless curvature would give zero real first Chern class. The Euler sequence gives the same result independently: $c(T^{1,0}\mathbb{CP}^5)=(1+H)^6$. See the Euler-sequence calculation.

The pointwise exterior-algebra branching and charge table remain valid. They do not establish this impossible global reduction. A viable completion must specify a separate determinant-trivial gauge bundle, or rigorously distinguish the gauge component from a full geometric connection retaining its central curvature. Neither route is derived from the projected action here; the auxiliary construction below supplies only admissible bundle data. Simply subtracting a local trace does not by itself define a global $SU(5)$ bundle. The existing spinor twist $K^{1/2}=\mathcal O(-3)$ is not a trivialization of $\det E$.

15.2 Remaining source and curvature test

Once an admissible bundle is specified, its reduction must be selected by the projected background, not only because it reproduces the desired gauge group.

The decisive calculation is the curvature decomposition

$$F^{\rm pol} = \begin{pmatrix} F_c & F_{cw}\\ F_{wc} & F_w \end{pmatrix}$$

and the proof that

$$\boxed{F_{cw}=F_{wc}=0}$$

on the physical vacuum and all allowed perturbative sectors. Vanishing mixed curvature alone is not generally equivalent to a chosen parallel splitting; the connection-level conditions above and global holonomy must also be checked.

16. The critical-spinor fixed points

The Gegenbauer criticality condition used in the sector endpoint analysis can be written

$$4k=(d-2)^2.$$

For even depth, define

$$r=\frac d2-1.$$

Then

$$k=r^2.$$

The complex dimension of a chiral spinor in real dimension $d$ is

$$\dim_{\mathbb C}S_d^+ = 2^{d/2-1} = 2^r.$$

Impose the additional critical-spinor closure condition

$$\boxed{k=\dim_{\mathbb C}S_d^+.}$$

It becomes

$$r^2=2^r.$$

The positive integral solutions are

$$r=2, \qquad r=4.$$

Indeed, $r=1$ and $r=3$ fail, while for $r\geq5$ the ratio

$$\frac{2^r}{r^2}$$

is already greater than one and increases monotonically because

$$\frac{2^{r+1}/(r+1)^2}{2^r/r^2} = \frac{2r^2}{(r+1)^2} >1 \qquad(r\geq3).$$

Therefore

$$\boxed{ (d,k)=(6,4), \qquad (d,k)=(10,16). }$$

These values coincide with

$$\dim S_6^+=4=n_s$$

and

$$\dim S_{10}^+=16=k_0=n_s^2.$$

Thus depths six and ten are the only even depths at which Gegenbauer criticality closes on the chiral-spinor dimension. This supplies an exact structural relation between the electron and tau sectors once the critical-spinor equality is accepted as the closure criterion.

17. Unified selection statement

The proposed positive mode-selection theorem can be stated as follows.

Let $J$ be a source monomial produced by the projected master interaction. Let $\mathcal I_J$ be its invariant polarization projector, and let $\Omega_J$ be its ground-relative contact phase. A target pair $(r,d)$ fires only if

$$\boxed{ \mathcal I_J\neq0, \qquad r-1=\Omega_J, \qquad (r,d)\text{ lies on an allowed deposit channel}. }$$

The three conditions perform different jobs:

1. the deposit ring fixes the finite topological cell in which a mode may occur;
2. the polarization projector fixes which interaction can populate that cell;
3. contact-phase resonance fixes the integer mode index inside the cell.

The existing parity and radiative-width mechanisms then test whether the resulting resonance survives as an asymptotic particle rather than a transient or dressing mode.

This separates positive generation from negative exclusion:

$$\boxed{ \text{topology supplies the cells}, \quad \text{representation theory supplies the channels}, \quad \text{contact resonance supplies the indices}, \quad \text{stability analysis removes the complement}. }$$

18. What the construction derives without masses

Conditional on the contact-frequency and join-phase hypotheses, the following indices arise without using particle masses:

$$10, 15, 20, 35$$

from simplex contact eigenvalues;

$$13$$

from the determinant-selected $10+3$ carrier inside the four-slot tensor contact;

$$22$$

from a shared-seed pushout;

$$23$$

from one antisymmetric ground join;

$$16, 17$$

from the two-factor tensor contact and its adjacent insertion;

$$72$$

from the three-Kähler tensor contact; and

$$76, 81, 95$$

from shared-ground and joined-source phase matching.

On the coupling side, the depth-ten chiral spinor supplies

$$S_{10}^+ = \Lambda^{\rm even}\mathbb C^5,$$

and the pointwise rank-$3+2$ split, with a separate determinant-one condition, supplies the candidate subgroup

$$S(U(3)\times U(2)),$$

with the complete quark/lepton representation pattern and hypercharges at the fibre level. A globally admissible realization is still required by Section 15.

No numerical particle masses enter either construction.

19. Falsifiable operator tests

The route fails if any of the following occurs.

1. The time-dependent projected contact term does not generate the ground-relative source phase $S(n,d)-1$.
2. The invariant projectors admit additional resonant targets outside the committed mode set.
3. Tensor contacts do not enter the source operator multiplicatively.
4. The charged-lepton projector does not select the determinant-singlet $13$-carrier.
5. The specified polarization/gauge bundle does not admit the required global reduction. The literal determinant-one reduction of $T^{1,0}\mathbb{CP}^5$ already fails this test (Section 15).
6. Off-diagonal colour--weak curvature components survive in the physical background.
7. The projected weak generators fail to connect the designated depth pairs with universal normalization.

These are binary structural tests rather than parameters that can be retuned against the mass data.

20. Decisive master-action calculation

The mode part of the master calculation must produce, after projection and linearization about the condensate,

$$\boxed{ \left(i\partial_\tau-\widehat N\right)\phi_d = \sum_J h_J\mathcal I_J \exp(-i\Omega_J\tau) + \text{nonresonant terms}, }$$

with

$$\Omega_J = \sum_i(n_i-1) - \sum_j(m_j-1) + q_J$$

and with every single-parent contact source obeying

$$\Omega_{n,d}=S(n,d)-1.$$

The gauge part must produce

$$\boxed{ P_{10}D_\infty P_{10} = \slashed D_{\mathbb{CP}^5} + \gamma^\mu \left( G_\mu^aT_c^a + W_\mu^rT_w^r + B_\mu Y \right) + \Phi_{10}, }$$

with

$$\operatorname{Hol}(\nabla^{\rm gauge}) \subseteq S(U(3)\times U(2)),$$

on a separately specified admissible gauge bundle, not as the full holonomy of the tangent bundle, and cross-depth blocks whose exterior-algebra intertwiners reproduce the weak doublet transitions while forbidden colour and Abelian channels vanish.

A successful derivation would join mode selection and gauge selection in one projected operator:

$$\boxed{ \text{symmetric-power contact} \otimes \text{exterior-algebra polarization}. }$$

The symmetric factor determines the resonant integer and mass multiplier. The exterior factor determines charge, colour, weak chirality, and whether the vertex is present.

21. Result

The completed proposal is

$$\boxed{ \begin{aligned} &\text{finite-band locality} \Longrightarrow \Lambda_{N,d}=S(N+1,d)P_{[z]},\\ &\text{ground subtraction} \Longrightarrow \Omega_{n,d}=S(n,d)-1,\\ &\text{retarded resonance} \Longrightarrow n_{\rm out}=S(n,d),\\ &\text{source phase algebra} \Longrightarrow \text{additive and inclusion--exclusion indices},\\ &\text{tensor contact} \Longrightarrow 16\text{ and }72,\\ &\text{depth-ten exterior algebra} \Longrightarrow \mathbf{16}\text{ with one-family gauge representations},\\ &\text{rank-}3+2\text{ split + determinant-one fibre condition} \Longrightarrow S(U(3)\times U(2)). \end{aligned} }$$

The exact algebra substantially narrows the remaining physics. The open task includes deriving the source-frequency law and specifying a topologically admissible global gauge construction before deriving its projected dynamics:

$$\boxed{ \Omega_{n,d}=S(n,d)-1 }$$

for the occupied contact source, and

$$\boxed{ \operatorname{Hol}(\nabla^{\rm gauge}) \subseteq S(U(3)\times U(2)) }$$

for a specified determinant-trivial gauge bundle. This is not a permissible condition on the full $T^{1,0}\mathbb{CP}^5$ connection. Section 15 records the obstruction and the unresolved distinction between gauge and geometry.

Sources and companion modules

Primary companion modules:

• A covariant self-contact mass operator
• The mass spectrum selects particle depth
• Particle depth as spatial geometry
• Coupling as higher-dimensional polarization
• Bound within, free without
• IDWT_02_Mass.md
• IDWT_03_Forces.md
• IDWT_07_Mode_Selection.md
• IDWT_09_Spectral.md

Primary mathematical cross-checks:

• Ivan Todorov, Clifford Algebras and Spinors — Clifford factorization, spinor dimensions, and low-dimensional spin-group identities.
• Idrish Huet and Julieta Medina, A projective Dirac operator on $\mathbb{CP}^n$ and extended SUSY — Dirac and $Spin^c$ structures on complex projective spaces.
• Nathan Berkovits and Sergey A. Cherkis, Higher-Dimensional Twistor Transforms using Pure Spinors — projective pure-spinor geometry and the spaces $SO(2n)/U(n)$.