A Three-Channel Hodge Criterion for the Quark Sectors

The maintained down-type quark family occupies three ordinary spatial dimensions, and the up-type family four. Each flavor needs the same color triplet, separate from its spin and flavor. The comparison depths are two, five, six and ten. This investigation asks whether one geometric prescription has a complete three-channel space precisely at the two quark depths.

The result is a conditional dimensional connection. Middle-degree Hodge channels have complex rank three only at depths three and four, over all positive dimensions. An even-dimensional channel restricts isomorphically to its odd-dimensional partner. At four and three, this realizes a chiral four-dimensional structure with a complete three-dimensional shadow. Identifying the channels with color still requires an equivalent-mode interpretation and a common interaction. The rank theorem alone does not make the higher-depth particles color singlets.

A second uniform test reaches the same exceptional pair: the lowest nontrivial adjoint rotation-Casimir channel of a minimal spatial spinor's endomorphisms has rank three only at three and four. These are related geometric tests, not independent experimental observations.

1. One Prescription Across Every Spatial Depth

Let an oriented Euclidean space of even dimension $2m$ contain an oriented hyperplane $V$ of dimension $2m-1$, with unit normal covector $n$. Write the orientation as $\operatorname{vol}_V\wedge n$. A middle form is

$$F=A+B\wedge n,\qquad A\in\Lambda^mV^*,\quad B\in\Lambda^{m-1}V^*.$$

The positive spatial metric gives

$$*_{2m}^2=(-1)^m, \qquad *_{2m}(A+B\wedge n)=(*_V A)\wedge n+(-1)^m *_V B.$$

Choose $\lambda$ with $\lambda^2=(-1)^m$. The map

$$\mathcal E_\lambda(B)=*_V B+\lambda^{-1}B\wedge n$$

satisfies $*_{2m}\mathcal E_\lambda(B)=\lambda\mathcal E_\lambda(B)$. Conversely every eigenform is determined by its $B$ component. Tangential restriction gives $i^*\mathcal E_\lambda(B)=*_V B$, an isomorphism onto $\Lambda^mV^*$. Its Hermitian norm is $2\|B\|^2$; dividing the embedding by $\sqrt2$ is isometric. The two eigenspaces are orthogonal.

For even $m$ the eigenvalues are $\pm1$, the self-dual and anti-self-dual real channels complexified for the field. For odd $m$ they are $\pm i$. This is spatial form duality, not a derivation of fermion helicity.

Both the even channel and its odd-dimensional shadow have rank

$$\boxed{k_m=\binom{2m-1}{m-1}=\frac12\binom{2m}{m}.}$$

The equality counts whether a middle-form basis wedge contains the normal direction. It is Pascal recursion with equal central contributions, not arithmetic on assigned particle indices. The ratio

$$\frac{k_{m+1}}{k_m}=4-\frac{2}{m+1}>1$$

proves strict growth. Since $k_1=1$, $k_2=3$ and $k_3=10$,

$$\boxed{k_m=3\ \Longleftrightarrow\ m=2 \ \Longleftrightarrow\ d=3\text{ or }4.}$$

The larger groups in the last column are the maximal norm-and-volume mixing groups of those component spaces, not predicted physical forces. The prescription is uniform. Choosing an arbitrary three-dimensional subspace of a higher channel is a different rule and removes this criterion.

2. What The Four-Dimensional Shadow Actually Does

At $m=2$, identify a three-dimensional two-form with a vector by its Hodge dual. Each of the two four-dimensional chiral triples is

$$F_\pm(B)=*_3B\pm B\wedge n,\qquad B\in\Lambda^1V^*.$$

Both restrict to $*_3B$. Ordinary rotations of $V$ act identically on the two triples. Rotations mixing $n$ with $V$ act with opposite signs in their three-component representations. Thus a three-dimensional restriction cannot distinguish the two chiral structures through its own rotations, while the additional ordinary spatial direction can distinguish them.

This provides a precise version of a shared three-/four-dimensional geometric origin. It does not require an up-type quark to be present in every hadron. The containing geometry and connection can exist without an up-type constituent. The $\Omega^-$ baryon's $sss$ valence content is an important down-type-only control; see the PDG Omega table.

The two triples are alternatives, or require a common diagonal symmetry. Treating them as two unrelated color groups would add another premise. The embeddings concern bundles and geometry, not one particle solution being the restriction of another particle's solution.

3. Why Three Channels Are Not Yet Color

Three equivalent complex amplitudes with a Hermitian norm and a preserved complex volume have maximal mixing group $SU(3)$. This supplies its fundamental representation and eight traceless Hermitian generators. But real rotation of a geometric triple gives $SO(3)$, with only three generators. Its induced real geometric connection remains in that smaller algebra unless further field-dependent mode geometry supplies the missing directions.

There is also a spin distinction. On a fixed spatial triple the rotations act irreducibly as spin one. Schur's lemma leaves only scalar complex operators commuting with every such rotation, not an independent color algebra. The eight traceless endomorphisms split into spin-one and spin-two spaces of dimensions three and five under those rotations. They cannot simply be relabeled eight internal color directions. This use of Schur's lemma is standard; see Woit, section 2.1.

A constructive relative-frame completion avoids that particular obstruction. Let $R$ be a geometric orthonormal channel frame and $B$ its complex pattern amplitudes. Set $c=R^TB$. A spatial rotation $O$ acts on both:

$$(R,B)\longmapsto(OR,OB),\qquad c\longmapsto c.$$

Color can instead act as

$$c\longmapsto Uc,\qquad B\longmapsto RUR^TB,\qquad U\in SU(3).$$

These actions commute on the joint geometry. Infinitesimally, $\delta R=KR$ and $\delta B=KB$ cancel in $\delta(R^TB)$. The color Casimir is unchanged by $R$. A separate spinor factor then has commuting actions $S_i\otimes I_3$ and $I_2\otimes T_A$. More generally the same construction works at channel rank $k_m$; it does not select rank three on its own.

For a varying frame, the relative kinetic derivative is

$$\partial B-(\partial R)R^TB=R\,\partial c.$$

Its Hermitian norm permits unitary mixing of equivalent relative modes. The frame must come from the existing geometry or same field, and its motion must be included. Freezing it and discarding its variation would restore the spin-one obstruction. Additional allowed invariants can reduce the symmetry: a term $|c^Tc|^2$, for example, is rotation invariant and phase neutral but is not $SU(3)$ invariant. A physical enhancement needs justification from the actual interaction, not just a choice of frame.

4. A Connection From Equivalent Spatial Modes

Let $\Phi_a$ be orthonormal modes of one actual spatial operator, with the same spin, frequency and other noncolor properties. The required color space is their multiplicity space, not three polarizations within one spin-one representation. A varying mode frame has projected connection and curvature

$$\mathcal A=\Phi^\dagger d\Phi,\qquad \mathcal F=d\Phi^\dagger(I-\Phi\Phi^\dagger)\wedge d\Phi.$$

The columns describe modes of the same field. They introduce no additional physical spatial dimensions. A parallel volume or specified determinant connection is needed to isolate $SU(3)$ from the common $U(1)$ phase.

There is no algebraic obstruction to all eight color directions appearing. For a rank-three graph inside a four-mode space, let $Z$ be its one-row complex graph coordinate. At $Z=0$,

$$\mathcal F(v,w)=v^\dagger w-w^\dagger v.$$

Complex mode variations span $\mathfrak u(3)$, and their traceless parts span $\mathfrak{su}(3)$. Real variations span only $\mathfrak{so}(3)$. This is a mechanism for supplying the missing directions, conditional on deriving the mode projector. It is not evidence that such a projector is selected only at the quark depths. The rank theorem and the operator selection have separate jobs.

This connection does not need a six-dimensional physical parameter space. On an ordinary two-dimensional spatial patch take $Z(x,y)=xv_1+yv_2+(x^2/2)v_3$, with $v_1=(1,0,0)$, $v_2=(i,1,0)$ and $v_3=(0,i,1)$. At the origin the connection vanishes. Its curvature is $v_1^\dagger v_2-v_2^\dagger v_1$, and its first covariant derivative along $x$ is $v_3^\dagger v_2-v_2^\dagger v_3$. The traceless parts of these two matrices generate an eight-dimensional Lie algebra. The checker computes that closure explicitly. The same pullback extends to every maintained sector by making it independent of the additional coordinates. Thus full $SU(3)$ mode holonomy is available even at depth two; its availability alone does not select the quark depths.

At three/four an oriented real channel also supplies a trivial complex determinant after complexification, because its transition matrices lie in $SO(3)$. Extending the frame group to $SU(3)$ is then possible globally on that channel bundle. Its induced connection still lies in $SO(3)$; actual mode transport must supply the enlargement and preserve the volume.

5. Color Neutrality Is A Representation Condition

With $T_A=\lambda_A/2$ in the fundamental triplet,

$$\operatorname{tr}(T_AT_B)=\frac12\delta_{AB},\qquad \sum_A T_A^2=\frac43 I_3.$$

Therefore every triplet state has positive squared total generator norm:

$$\sum_A\|T_A\psi\|^2=\frac43\|\psi\|^2.$$

No orientation or phase makes a nonzero isolated triplet a color singlet. Even $\rho=I_3/3$ has zero mean generators but Casimir $4/3$. A genuine singlet satisfies every $T_A\psi=0$. Correlated products can achieve that:

$$\delta_{i\bar j}\in3\otimes\bar3,\qquad \epsilon_{ijk}\in3\otimes3\otimes3.$$

The checker verifies all eight total generators annihilate these tensors. Both quark families must have the same color representation type. Assigning one $3$ and the other $\bar3$ would prevent the ordinary $uud$ and $udd$ three-quark states from having a color singlet. The mixed product $3\otimes3\otimes\bar3$ has Casimirs $4/3$, $10/3$ and $16/3$, with no zero.

The standard empirical color targets are $C_F=4/3$, $C_A=3$ and $T_R=1/2$. Jet observables constrain those factors; the PDG QCD review, sections 9.1 and 9.3.2.1 describes the algebra and experimental comparisons. These constrain a geometric model without prescribing an IDWT mediator-particle mechanism.

The same algebra gives pair weights $-2/3$ and $1/3$ for the antisymmetric and symmetric $qq$ channels, and $-4/3$ and $1/6$ for the singlet and octet $q\bar q$ channels. They can be used in a derived common spatial overlap interaction. Their signs alone do not establish a physical force without that interaction and its energy kernel.

6. Uniform Controls And The Remaining Selection Problem

The commutant comparisons demand commutation with all retained spatial rotations or Clifford generators, respectively. They are stronger than commutation only with the observer's rotations. None of these availability counts selects physical states. The real-spinor classification is the standard periodic Clifford classification; see Baez, The Octonions, section 2.4 and table 4.

For the positive Spin(10) carrier, restriction multiplicities are $(8,8)$, $8$, $(4,4)$, $4$, $(2,2)$ and $1$ at the displayed depths. An independent Clifford-blade count reproduces their squared-sum commutant dimensions. This admits color at five as well as three/four. Full tangent Clifford compatibility must use the complete parent Dirac carrier, because individual ambient Clifford generators exchange its chiral halves.

The rotation two-form criterion is also an all-dimensional result: $\Lambda^2\mathbb R^d$ is the adjoint of $\mathfrak{so}(d)$, simple at $d=3$ and all $d\ge5$; four splits into two three-dimensional ideals and two has one abelian direction. Keeping a four-dimensional chiral triple inside a larger ordinary tangent space does not make it invariant under all larger-sector rotations. The exact squared leakage summed over mixed rotation generators is $6(d-4)$. A selected four-plane remains an available subpattern; it does not represent the complete higher-dimensional rotation channel.

Fibre rank is not spectral multiplicity. On a unit round sphere, the first exact two-form family has eigenvalue $2(d-1)$ and multiplicity $\binom{d+1}{2}$. The first coexact family, for $d\ge3$, has eigenvalue $3(d-2)$ and multiplicity $\binom{d+1}{3}$. The lower value, or their sum when equal, gives the table. At four, each chiral half of the first 20-dimensional space has dimension ten, not three. This control follows the standard form spectrum in Ikeda and Taniguchi (1978), theorem 4.2. Round spheres supply one compatible nested metric family; they are controls, not adopted IDWT sector topologies or compactification.

Four-dimensional curvature self-duality gives another version of the three/four relation. With one spatial direction labeled $4$, $F_{ij}=\pm\epsilon_{ijk}F_{k4}$. If the field is independent of that direction and $\Phi=A_4$, then $F_{k4}=D_k\Phi$, giving the three-dimensional Bogomolny equation $F_3=\pm *_3D\Phi$. Its form degrees match only in three; ordinary two-form self-duality matches only in four. This reduction works for arbitrary gauge algebra and does not select $SU(3)$. Reducing further to two retains two additional connection components, and embedding a four-dimensional solution in higher dimensions remains possible. Neither operation proves absence of color at those depths. Parallel quaternionic triples likewise recur at three/four, seven/eight and higher pairs.

The important unsuccessful stronger hypothesis is that every known particle occupies the full fundamental middle-Hodge channel. That would give nontrivial $SU(10)$-type response at five/six and $SU(126)$-type response at ten, rather than deriving the required color neutrality. Scalar or contracted singlet branches are available, but their occupation needs a common operator rule. Three selected channels can also be chosen inside larger spaces; that changed premise no longer singles out three/four. A maximal allowed group is not proof of a generated interaction or of fundamental-state occupation.

7. A Second Uniform Operator Test

Let $\Sigma_d$ be a minimal complex spatial rotation spinor, using one chiral half when $d$ is even. Its rank is $r_d=2^{\lfloor(d-1)/2\rfloor}$. The scalar endomorphism always exists; its traceless endomorphisms have rank $r_d^2-1$, equal to three only at depths three and four. To avoid selecting an entire large space by hand, use the canonical positive adjoint rotation Casimir

$$\mathcal C_d(A)=\sum_{a<b}[S_{ab},[S_{ab},A]],\qquad S_{ab}=-\frac{i}{2}\gamma_a\gamma_b.$$

For a Clifford blade of degree $k$, a generator contributes only when exactly one of its two indices belongs to the blade. Each such double commutator returns the blade with coefficient one. Consequently

$$\mathcal C_d(A_k)=k(d-k)A_k.$$

At odd depth the even Clifford grades $0,2,\ldots,d-1$ form the complete endomorphism space. At even depth, complementary even grades are identified on a chiral spinor; grades below the middle have their usual binomial ranks, and an even middle grade has half its usual rank. Their total is $r_d^2$.

For odd $d\ge3$, the lowest positive channel has eigenvalue $d-1$ and rank $d$, from grade $d-1$. For even $d\ge6$, it has eigenvalue $2(d-2)$ and rank $\binom d2$, from grade two. Four is the exceptional half-middle case with rank three. Two has no nontrivial channel. These formulas prove the three/four uniqueness over all positive dimensions, including depth one's scalar-only boundary case; the checker verifies depths two through sixteen and the blade-generator counts.

This gives a concrete operator-based dimensional test without presupposing middle degree in every sector. Its carrier differs from the middle-Hodge prescription at higher depths, so the two must be compared separately. At three/four they recover the same vector or chiral two-form geometry. They do not count three flavors or add spatial coordinates.

The physical hypothesis would be that a particle's relevant response uses its lowest nontrivial polarization channel. The operator and ranks are fixed by geometry; occupation of that channel is not yet derived. Its components transform under spatial rotation and still need the relative-mode construction to become independent color. Scalar endomorphisms at higher depths are allowed, but a pure polarized spinor density is not proportional to the identity when its rank exceeds one. Scalar availability therefore does not establish a color-neutral lepton state.

The minimal carrier is an additional physical choice. The maintained paired spatial Dirac construction has ranks two, four, four, eight, eight and thirty-two at the comparison depths. Its unprojected endomorphism space does not pass this rank-three selector. At even depth the full Clifford grades $0,\ldots,d$ occur; the lowest positive channel comes from grades one and $d-1$, with rank $2d$ for even $d\ge4$; at two those grades coincide and give rank two. At odd depth two equivalent minimal rotation spinors multiply every endomorphism multiplicity by four, so the lowest rank is $4d$. This gives 2, 12, 8, 20, 12 and 20 in the table. The checker verifies the dimension sums and lowest channels through sixteen. Thus the second criterion needs a justified minimal rotation branch or a revised field carrier. It is not a property automatically inherited by the full maintained Dirac field. The first middle-Hodge criterion does not use that spinor projection, but still needs its own physical mode selection.

Why Would The Higher Channels Have No Strong Forces?

That absence is not explained by either rank theorem. The counts five, fifteen and forty-five are geometric response components, not established colors. The same caution applies to the three-component channels. The lowest channels have the following commutator behavior:

At five, the lowest grade-four blades are dual to vectors. Their commutators produce grade-two blades, outside the five-dimensional channel; together these generate fifteen directions. At six and ten the lowest channels already consist of the closed rotation bivectors. These algebras admit non-Abelian interactions if supplied with appropriate dynamics. The script checks the blade commutator closure through sixteen. At three/four the native algebra is $\mathfrak{su}(2)$, not the required eight-generator color algebra. Relative complex mode mixing is a separate enhancement in every sector where it is invoked.

A geometric frame connection does not automatically constitute an independent strong force. Nor does absence of an $SU(3)$ full channel exclude a different strong interaction. A successful common physical rule must establish a dynamical response at three/four and either suppress the corresponding response or select genuinely invariant occupied states elsewhere. An interaction coefficient set to zero by sector, or assigning higher particles to scalar channels without deriving their selection, would merely insert that answer. The present calculations constrain this missing rule; they do not supply it.

One narrower interaction hypothesis gives another discriminating test. Let $V=\mathbb C^k$ be a complete equivalent-mode channel with full $SU(k)$ mixing. An invariant trilinear amplitude in three fundamental slots can exist only when $k$ divides three: the center element $z=e^{2\pi i/k}I$ acts on the three slots as $z^3$. For nontrivial channels, this selects $k=3$, where the invariant is

$$D(a,b,c)=\epsilon_{ijk}a_i b_j c_k.$$

Thus a common three-pattern orientation vertex would be available at three/four and absent on the larger full middle-Hodge channels. Rank one has a scalar product but no non-Abelian orientation. The three slots must be distinguishable modes or pattern data from the same field; $D(c,c,c)=0$ for commuting amplitudes. A phase-neutral energy such as $|D(a,b,c)|^2$ is sixth order. It also needs compatibility with electric charge, the actual spatial overlap, and the single-field action.

This is a promising conditional selection rule, not yet an explanation of force absence. Ordinary quartic terms survive at every $k\ge2$:

$$E_4(a,b)=|a^\dagger b|^2.$$

For unit $a=e_1$ and $b=(e_1+e_2)/\sqrt2$, its projected complex gradient with respect to $a^*$ is $b(b^\dagger a)-a|b^\dagger a|^2=e_2/2$. Its squared norm is $1/4$, independent of $k\ge2$. This is a nonzero relative orientation response, not a force calculation until spatial dependence is specified. Consequently trilinear determinant selection does not suppress generic higher-sector overlap interactions. A physical rule making that vertex the relevant strong channel must explain why these other allowed terms do not create analogous strong responses.

8. What Would Make This A Particle-Property Connection

A useful common law would derive an equivalent-mode bundle from the sector operator, preserve its full observed spin representation, and make the color action scalar on each occupied mass-frequency branch. For a triplet, color invariance gives

$$M_{n,d}=\mu_d S(n,d) I_3$$

by Schur's lemma. Three colors then share one frequency; degeneracy does not multiply the eigenfrequency by three. If the underlying mass feedback instead uses a total trace counting every color component, that normalization must be rederived rather than silently treating a per-color count as a total.

The concrete progress is an exact full-channel rank criterion selecting three/four and its shared chiral restriction map, together with constructive the lowest nontrivial spinor-endomorphism criterion, and constructive ways of separating relative color from observed spin and generating an eight-direction mode connection. The remaining physical tasks are to derive the equivalent modes and interaction, preserve the full nested spatial geometry, and select singlet branches for the nonquark particles under the same law. No spontaneous dimension-dependent cancellation of a triplet has been established. These linked mathematical consequences are conditional support for investigation, not independent experimental proof of IDWT.

9. The Channel Now Supplies a Pair Operator and Charge Tests

The determinant-contact continuation contracts the third slot of $|\epsilon(a,b,c)|^2$ against isotropic same-field covariance. It gives

$$\sigma^2(I-\mathsf S) =\sigma^2\left(\frac23I-2\sum_AT_A\otimes T_A\right)$$

on the two remaining triplet slots. This explicitly connects the geometric volume tensor to the standard pair-Casimir factors. The covariance, interaction sign and spatial kernel remain physical inputs.

Using the same complete channel rank as the color multiplicity of a specified chiral weak family gives $q_u=(k+1)/(2k)$ and $q_d=(1-k)/(2k)$ under neutrino neutrality and the charged-lepton normalization. The global weak-doublet anomaly further requires $k$ odd for an odd number of these generations without added cancellation. The middle-Hodge rank is odd exactly when $m$ is a power of two. Among the maintained non-Abelian channels this accepts three/four; seven/eight also passes as a higher control.

The same continuation compares a scalar volume amplitude with the full three-form norm. The latter exists for every $k\ge3$ and gives pair coefficient $(k-2)\sigma^2$. Thus the interaction-level selection problem remains explicit. The anomaly test constrains the displayed chiral family, rather than proving absence of all higher-sector forces.

Reproducibility

The single exact checker uses Python and SymPy. Run it with python3 color-depth-algebra-checks.py. It creates a new UTC-dated result directory containing its executed source and JSON report. The reviewed run passes 36,489 algebraic checks, including maintained depths and neighboring controls through sixteen. These are finite exact relation checks, not particle simulations or independent empirical confirmations. The JSON source hash identifies the executed bytes.

10. A Corrected Paired-Mode Neutral Branch

The paired spatial-mode calculation gives an explicit cancellation operator on a carrier and its dual: the total quadratic generator is $k(I-P_{\mathrm{trace}})$. The trace branch is exactly neutral, while an identical pair at rank three has no singlet. Oscillator realification permits even-shell scalar states at the maintained charged-lepton levels at six and ten, but the corresponding all-depth controls do not select their physical occupation. The same pairing on the complete Hodge channel also supplies a trace scalar at the quark ranks. Electric charge can survive in a separate common geometric line character; a fully dual central phase would cancel. These constructions narrow the required occupation rule and must retain the full-band contact normalization. They do not yet derive higher-sector force absence.

The paired continuation also tests a complete positive fast-band action: opposite matched mixed curvatures cancel a velocity cross-channel while induced inertias add. A product ray and its conjugate has zero Berry curvature but Casimir $k-1$, so this geometric cancellation cannot be identified with color neutrality without the invariant occupation rule. See the exact paired response.

11. A Common Contact Filter and Its Physical Premises

The symmetric-Hodge feedback calculation tests a specific common interaction kernel. The selfadjoint part of the raw real middle-form Hodge map is active at three/four and zero at two, five/six and nine/ten. Applied before the complete mode-volume norm, it preserves the triplet pair Casimirs while cancelling the larger-channel operator. Seven/eight remains active. The complementary skew map instead permits a velocity interaction. Kernel selection, time-reversal data, independent spin/mode interpretation and source formation before lower-sector projection remain explicit physical premises; neither generic reciprocity nor sector nesting forces them.

12. The Channel Has a Spinor-Bilinear Source

An explicit Clifford-bilinear source now maps the spatial spinor symmetric square onto the middle-Hodge channel at every depth. Three/four gives the full symmetric square of a two-component spinor, while ten retains a separate vector summand in addition to its 126-component middle form. The construction derives the carrier and its canonical norm from the same field. A pointwise spinor square obeys a quadratic triplet constraint; distinct spinor pairs can fill the carrier, but full independent color symmetry is not thereby derived. Uniform spinor polarization already realizes the isotropic third-slot covariance needed by the existing pair operator.