The Spinor Field Generates the Middle-Hodge Source

The middle-form source used by the Hodge-contact candidate can be constructed directly from spatial spinor components of the same field. It need not be introduced as an unrelated tensor field. A common Clifford bilinear generates the complete linear middle-form carrier at every comparison depth, and its selfadjoint Hodge filter retains the previously calculated parity rule.

This establishes the source carrier, rather than a physical color symmetry. One spinor squared obeys nonlinear constraints; spatial rotations do not become an independent color action. The construction uses commuting classical field amplitudes. It does not identify the bilinear with an occupied particle state or impose a Grassmann quantum-field interpretation.

1. One Clifford Bilinear at Every Even Depth

Let the ordinary spatial tangent dimension be $d=2m$. Use Hermitian Euclidean Clifford matrices $\gamma_i$ with $\{\gamma_i,\gamma_j\}=2\delta_{ij}I$, full Dirac rank $2^m$, and chiral rank $r=2^{m-1}$. In the standard Pauli tensor basis define

$$C=\gamma_2\gamma_4\cdots\gamma_{2m}.$$

Direct Clifford algebra gives

$$C\gamma_i=(-1)^m\gamma_i^TC,\qquad C^T=(-1)^{m(m+1)/2}C.$$

For a rotation generator $J_{ab}=\gamma_a\gamma_b/2$, these imply $J_{ab}^TC+CJ_{ab}=0$. Hence $C$ defines a complex bilinear Spin-invariant pairing. It is distinct from the positive Hermitian kinetic norm.

For an increasing multi-index $I$ of length $p$, put $\gamma_I=\gamma_{i_1}\cdots\gamma_{i_p}$. Let $\psi$ be the positive-chirality component of the spatial field. Define the degree-$p$ form source by

$$F^{(p)}_I(\psi,\phi)=\psi^TC\gamma_I\phi.$$

The two chiral slots give nonzero matrices only when $p\equiv m\pmod2$. Their transpose parity is

$$(C\gamma_I)^T =(-1)^{p(p-1)/2+mp+m(m+1)/2}C\gamma_I.$$

At middle degree $p=m$, the exponent is $2m^2$, so every matrix is symmetric. The source therefore lies in the symmetric spinor square. Complementary middle blades differ by one common Hodge phase, giving

$$*F^{(m)}=\lambda_m F^{(m)},\qquad \lambda_m^2=(-1)^m.$$

In the executable's chirality and orientation convention, $\lambda_m=1$ for even $m$ and $-i$ for odd $m$. The other chirality supplies the opposite eigenvalue. The parity result does not depend on which chirality is named positive.

Thus the geometric source map is

$$\mathcal B_m:\operatorname{Sym}^2\Sigma^+_{2m} \longrightarrow\Lambda^m_{\lambda_m}(T^*Q_{2m})_{\mathbb C}.$$

It intertwines actual spatial spinor rotations with exterior-form rotations. The checker verifies that relation directly, including rotations involving the last ordinary spatial direction. No normal direction or physical three-plus-three split is inserted.

The standard relation between spinor bilinears and exterior forms is reviewed in Krasnov, Geometry of Spin(10) symmetry breaking, sections II and IV A–C. The normalization, all-depth grade comparison and contact interpretation below are independently calculated here; the paper's unification or Higgs model is not adopted.

2. The Full Linear Carrier and Its Exact Norm

For $p<m$, retain one matrix for each degree-$p$ basis blade. At $p=m$, retain one representative from each complementary pair. Restricted to $\Sigma^+$, these matrices have squared Hilbert–Schmidt norm $r$ and are mutually orthogonal. Their count is $r^2$, so they exhaust the two-spinor matrix space. Symmetric matrices occur at $p=m,m-4,m-8,\ldots$, and alternating matrices at $p=m-2,m-6,\ldots$.

Consequently the familiar symmetric-square decomposition is realized explicitly:

$$\boxed{\operatorname{Sym}^2\Sigma^+_{2m} \cong\Lambda^m_{\lambda_m} \oplus\bigoplus_{j\geq1,\ m-4j\geq0}\Lambda^{m-4j}.}$$

The middle source has full linear rank

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

This is the same rank used by the earlier channel theorem. The all-sector spinor construction now supplies an explicit source map to that carrier, rather than only a matching component count.

The odd depth in each row uses the existing isometric shadow of its neighboring even carrier. It is not an intrinsic odd-dimensional middle-form Hodge construction. The field's actual extension or reduction supplying that shadow remains a physical premise.

Let $P_m$ be the orthogonal projector onto the middle summand of the symmetric square. Including both complementary middle blades gives

$$\|F^{(m)}(\psi,\psi)\|^2 =2r\|P_m(\psi\otimes\psi)\|^2.$$

The full symmetric Fierz norm is

$$\boxed{\frac{\|F^{(m)}\|^2}{2r} +\frac1r\sum_{j\geq1}\|F^{(m-4j)}\|^2 =\|\psi\|^4.}$$

At four and six there is no lower symmetric summand, so $\|F^{(m)}\|^2=2r\|\psi\|^4$. At ten the vector bilinear must be retained in this identity. Discarding it for a generic spinor changes the source norm; its vanishing is an additional pure-spinor condition. These are arbitrary-normalization identities, not a calibration of physical field amplitude or a mass formula.

3. The Same Field Source Has the Checked Hodge Parity

Applying $H=(*+*^\dagger)/2$ gives

$$HF^{(m)}=\operatorname{Re}\lambda_m F^{(m)}.$$

For every pointwise classical spinor and every phase, the source is retained at even $m$ and cancelled at odd $m$. The instantaneous filtered volume action can therefore be written directly in terms of three same-field pattern sources:

$$E=\gamma\left\| H\mathcal B_m(\psi_1,\psi_1) \wedge_{\rm mode}H\mathcal B_m(\psi_2,\psi_2) \wedge_{\rm mode}H\mathcal B_m(\psi_3,\psi_3) \right\|^2.$$

The slots describe distinguishable spatial patterns or modes of the one field. This energy is twelfth order in their spinor amplitudes. It is positive and phase neutral in each slot. It has a nonzero triplet realization at three/four and the same exact source zero at five/six and nine/ten. Seven/eight remains an active higher-rank control. Whether the master law selects this interaction or suppresses lower allowed terms is still open.

The bilinear transforms as $F\mapsto e^{2i\alpha}F$ when $\psi\mapsto e^{i\alpha}\psi$. This phase weight is not the particle's measured electric charge. The volume norm is phase neutral and does not force a color-inactive pattern to have zero electric charge.

The source may use a chiral field component while the kinetic action retains the full Dirac carrier. Since $P_+\gamma_iP_+=0$, replacing the entire kinetic field by a pointwise chiral compression would kill its first-order spatial symbol. The present source construction does not perform that compression. Actual occupied chiral branches and their dynamics remain to derive.

For one identical Grassmann spinor, a symmetric matrix gives $\psi^TB\psi=0$. The displayed nonzero source uses commuting classical field amplitudes, as permitted in the current field proposal, or would need distinct slots with the correct total exchange symmetry in another interpretation. A bosonic geometric source derived from a classical spinor is not itself a claim about observed particle statistics.

4. At Three/Four, the Triplet Is a Symmetric Spinor Pair

In an orthonormal symmetric-square basis, a two-component spinor $w=(x,y)$ gives

$$c(w)=(x^2,\sqrt2xy,y^2),\qquad c_2^2-2c_1c_3=0.$$

The polynomial constraint is essential: one spinor square does not occupy an arbitrary complex triplet. Its linear span is three dimensional, but its image is a quadratic cone. Generic $SU(3)$ mixing does not preserve that cone. The same field therefore generates the triplet carrier without automatically generating an independent full color action on its local source.

Three such sources can nevertheless have nonzero volume. Exact factorization gives

$$\boxed{\det[c(w_1),c(w_2),c(w_3)] =\sqrt2\prod_{i<j}\det[w_i,w_j].}$$

The volume energy becomes twice the product of the three spinor-pair area norms. It vanishes when any pair is proportional and is nonzero for $w_1=(1,0)$, $w_2=(0,1)$, $w_3=(1,1)$. This supplies a concrete spatial-polarization interpretation of the three-slot contact. It is not a proof that spin-dependent orientation forces reproduce spin-independent observed color.

Allowing two distinct spinor slots gives

$$c(u,v)=\left(u_1v_1,\frac{u_1v_2+u_2v_1}{\sqrt2},u_2v_2\right).$$

Every binary quadratic factors over $\mathbb C$, so this map fills the entire triplet carrier. Its norm is

$$\|c(u,v)\|^2=\frac12\left(\|u\|^2\|v\|^2+|u^\dagger v|^2\right).$$

This gives a constructive alternative to a single pointwise square: correlated or projected pairs of same-field spatial modes can supply arbitrary triplet amplitudes. The field action must derive those modes and their degeneracy. An arbitrary $SU(3)$ action on the resulting triplet is not automatically a linear action on the two underlying spinors.

Physical rotations act through the symmetric spin-one representation, with only scalar complex commutant on a fixed triplet. Independent color still requires the earlier relative-frame or equivalent-mode construction and its complete kinetic dynamics. The new bilinear gives that proposal a specific same-field source and explicit algebraic constraints to satisfy.

5. The Isotropic Third Slot Can Come from Spinor Polarization

A uniform unit spinor $w\in\mathbb C^2$ already gives exactly the third-slot covariance used in the triplet pair contraction. Write $w=(\sqrt t,e^{i\phi}\sqrt{1-t})$, with $t$ uniform on $[0,1]$ and $\phi$ uniform on $[0,2\pi]$. The common phase is immaterial. Its triplet source has diagonal second moments

$$\int_0^1[t^2,\ 2t(1-t),\ (1-t)^2]dt =\left[\frac13,\frac13,\frac13\right].$$

The off-diagonal moments vanish by phase integration. Thus

$$\boxed{\langle c(w)c(w)^\dagger\rangle=I_3/3.}$$

The ensemble is confined to the spinor-square cone; it need not fill all triplet states to have this covariance. Contracting its volume slot therefore gives the same pair operator $(I-\mathsf S)/3$. With unit underlying spinors $u,v$, its actual source value is

$$\frac13\left(1-|u^\dagger v|^4\right),$$

because $c(u)^\dagger c(v)=(u^\dagger v)^2$. The independent pair-area calculation yields $\langle|\det(u,w)|^2|\det(v,w)|^2\rangle=(1+|u^\dagger v|^2)/6$ and the same result.

This supplies a specific geometric realization of the earlier isotropic covariance premise, with the same mathematical pair-Casimir operator. Uniform polarization and the physical amplitude scale remain stated preparation assumptions. It does not turn the constrained source ensemble into an independently derived $SU(3)$ multiplet or establish observed spin-independent strong response.

Verification

The single checker and its report preserve 5,578 passing checks across 13 cases with exact executed provenance. They construct Clifford matrices through depth twelve, verify complete orthogonal bilinear grades and their symmetry, exact middle-form chirality, actual rotation covariance, full Fierz norms, filtered-source parity, triplet constraints, pair-area volume factorization, distinct-pair surjectivity and norm, and the uniform-spinor third-slot contraction. The source-only calculation retains all full-field kinetic Clifford directions. No particle simulation or independent amplitude fit is used.

Independent Color Can Reside in the Ordinary Normal Spinor Factor

An explicit full ambient Clifford comparison supplies a different three-component carrier: the degree-two spinor factor of the ordinary six-normal space. It retains a complete four-dimensional Dirac spinor and gives an independent color action at depths three/four, with standard pair-Casimir contractions and a real metric normal connection. This normal triplet is not the symmetric-square spin-one source above. The same fixed projection loses added kinetic directions at five/six and ten, while a depth-two triplet remains kinematically possible. Joining the instantaneous Hodge filter to this independent color carrier requires a complete common source and action; matching component counts alone does not perform that join.

A Lower-Order Hermitian Color-Current Alternative

The Hermitian adjoint-current construction uses a bilinear with one conjugate spinor and separate normal color matrices. Its Hodge-filtered norm is fourth order in field amplitude, compared with the twelfth-order three-source holomorphic volume above. Both are proposed interactions with different source symmetries and dynamics. The new current has exact spin-dependent Fierz and pair formulas, preserves independent color at three/four, and retains the full kinetic field. A real sawtooth gives its density source, whereas a positive-frequency complex comb gives a different density spectrum. Their response tables must be calculated separately.