The curvature chain can be placed on the actual spinor bundle of its proposed surface. This uses the existing Killing-spinor transfer and retains both Dirac branches with their correct multiplicities. The scalar-derived frequency polynomial is unchanged, while the spatial currents differ from the earlier exterior-form comparison.

The result is an inverse linear parent on a specified round field feature. It does not yet derive that feature, its normal extent, radius, gap or nonlinear support from the master field. Mass remains the selected particle frequency in equivalent units. This construction is not an insertion of a non-Hermitian operator into a scalar mass term.

The actual spin connection

Let the surface be S_R^m in ordinary R^(m+2), where m=2h is even. Its full complex spinor rank is r=2^h. A complete orthonormal family of positive Killing spinors gives a pointwise unitary frame U. With the actual spatial Dirac operator D, the previously established identity is

(D2−DR−h(h−1)R2)UF=UΔF,(1) \left(D^2-\frac D R-\frac{h(h-1)}{R^2}\right)UF =U\Delta F, \tag{1}

where Delta acts componentwise on the scalar coefficient vector F. The corresponding Killing connection is flat but is not the ordinary Clifford-compatible spin connection. Equation (1) is a unitary transfer, not an identification of the two connections.

The maintained Transfer of the simplex operator to spatial spinors derives (1) using the actual spin connection. The scalar/Killing construction is also described by Herzog and Shamir, Section 6.2. The present extension applies that identity to the codimension-two surface curvature operator and its causal parent.

Define

BK=D−12R,BK2=UΔU†+k2R2,k=m−12.(2) \boxed{B_K=D-\frac1{2R},\qquad B_K^2=U\Delta U^\dagger+\frac{k^2}{R^2},\quad k=\frac{m-1}{2}.} \tag{2}

This is a local self-adjoint first-order spinor operator. The shift is essential. It makes the polynomial symmetric around the correct center of the two adjacent Dirac levels.

There is a useful complete coordinate realization. At unit radius, let x belong to S^m in R^(m+1), and let Gamma_a be Hermitian ambient Clifford matrices on C^r. In the Killing frame,

BKF=kF−ΓxΓ∇SF.(3) \boxed{B_KF=kF-\Gamma_x\Gamma_{\nabla_S}F.} \tag{3}

The tangent principal symbol is -i Gamma_x Gamma_p for p perpendicular to x. It is Hermitian and squares to |p|^2 I. Direct differentiation of (3) on arbitrary complex spinor values, gradients and Hessians gives B_K^2 F=Delta F+k^2F. A separate stereographic calculation with the full spin connection agrees with (3). Thus this is a spatial spinor equation, not a replacement of spatial coordinates by form indices.

The same curvature product with the full spinor multiplicity

On scalar coefficient harmonics of degree L, set beta=L+k. The two orthogonal spectral projections are

Π±=12(I±RBKβ),RBKΠ±=±βΠ±.(4) \Pi_\pm=\frac12\left(I\pm\frac{R B_K}{\beta}\right), \qquad R B_K\Pi_\pm=\pm\beta\Pi_\pm. \tag{4}

For L=0 only the positive projection is nonzero. At higher degrees, the positive projection is ordinary Dirac level L, while the negative projection is negative Dirac level L-1. Their full dimensions are

dim⁡Π+=r(L+m−1m−1),dim⁡Π−=r(L+m−2m−1),dim⁡Π++dim⁡Π−=r[(L+mm)−(L+m−2m)].(5) \begin{aligned} \dim\Pi_+&=r\binom{L+m-1}{m-1},\\ \dim\Pi_-&=r\binom{L+m-2}{m-1},\\ \dim\Pi_++\dim\Pi_-&= r\left[\binom{L+m}{m}-\binom{L+m-2}{m}\right]. \end{aligned} \tag{5}

These unequal multiplicities must be retained. Treating a scalar and its spinor derivative as two independent copies would overcount. For the depth-six degree-14 feature, for example, the positive and negative B spaces have dimensions 2,720 and 2,240.

For containing depth d=m+2, use the d first-order links

Lj=c(BK+j−(d+1)/2R),j=1,...,d.(6) L_j=c\left(B_K+\frac{j-(d+1)/2}{R}\right),\quad j=1,...,d. \tag{6}

At even d their product is an even polynomial in B_K. Substitution of (2) gives the transferred conformal bending Hessian exactly:

∏j=1dLj U=cdUJd,R,∏jLjΠ±=d!cdRdDL−2,dΠ±.(7) \prod_{j=1}^dL_j\,U =c^d U\mathcal J_{d,R},\qquad \prod_jL_j\Pi_\pm= \frac{d!c^d}{R^d}D_{L-2,d}\Pi_\pm. \tag{7}

Insert these links into the complete canonical chain with one common gap G. Each link is self-adjoint, and the full Hamiltonian has the Hermitian off-diagonal pair T and T^dagger. Its propagation speed is c. Both signed B spaces and both signs of the parent remain present. The positive low frequency is the same stable singular-value expression as in the Hodge comparison, including its finite-gap correction:

ωL=d!cRη−(d−1)DL−2,d[1−(L−1)2+(L+m)22η2+O(η−4)],η=GR/c.(8) \omega_L=\frac{d!c}{R}\eta^{-(d-1)} D_{L-2,d}\left[1- \frac{(L-1)^2+(L+m)^2}{2\eta^2}+O(\eta^{-4})\right], \quad \eta=GR/c. \tag{8}

The full exterior-form parent had additional coexact form families. They belong to that different field representation. Equations (3)-(7) instead retain the complete actual surface spinor bundle and its correct adjacent Dirac multiplets.

The eigenvalue in (7) equals the simplex count polynomial. The number of available surface harmonics is still the multiplicity in (5), with cumulative growth L^m. It is not automatically the proposed d-dimensional IDOS with growth L^d. Neither an occupation rule nor a microscopic count interpretation follows just from equality of these polynomials.

The spatial spinor polarization is different

For a scalar harmonic Y and a constant coefficient spinor z, the actual eigenfield is Pi_+(Yz) or Pi_-(Yz), normalized with its full surface measure. Write u=a+i b for an oriented orthonormal plane. For Y proportional to (u dot x)^L,

BK(Yz)=(L+k)Yz−L(u⋅x)L−1ΓxΓuz.(9) B_K(Yz)=(L+k)Yz -L(u\cdot x)^{L-1}\Gamma_x\Gamma_u z. \tag{9}

If Gamma_u z=0, the complete field is already on the positive branch. Generic complex spinors have both projections. For unit Y and z, define s=z^dagger i Gamma_a Gamma_b z. Their norms are

∥Π±(Yz)∥2=12[1±k−LsL+k].(10) \|\Pi_\pm(Yz)\|^2 =\frac12\left[1\pm\frac{k-Ls}{L+k}\right]. \tag{10}

For the real standing harmonic sqrt(2) Re Y, the orbital expectation vanishes and (10) becomes [1 +/- k/(L+k)]/2. This normalization retains the whole complex spinor; it does not identify its magnitude with mass.

The local tangent Clifford current is

Ja=χ†[−iΓxΓPxea]χ,Px=I−xxT.(11) J_a=\chi^\dagger[-i\Gamma_x\Gamma_{P_xe_a}]\chi, \qquad P_x=I-xx^T. \tag{11}

A real scalar harmonic does not generally make (11) zero. Its projected spinor has a spatial polarization texture from (9). Direct full-component integration gives nonzero current squares for both signed projections of the real assigned-degree controls. Thus the exact zero-self-current standing solutions of the exterior-form trial cannot simply be carried over to actual spinors. A nonlinear fixed-point calculation is required for the declared quartic interaction on these fields.

For clarity, (11) is the form of the spinor Clifford bilinear on the surface. Its coefficient in the full parent's conserved kinetic flux is still the chain coefficient derived in The kinetic current and the chosen endpoint interaction. Choosing to use it equally at the endpoints defines a different interaction operator and needs a master-action origin.

Verification and remaining physical requirements

curvature-spinor-parent-20260914T010237_864118Z.json passes 401 checks. Arbitrary complete spatial spinor jets verify (2)-(3); a separate stereographic spin-connection evaluation checks their geometric meaning. Full common spatial and spin rotations, principal speed, both branch multiplicities and the exact shifted product are checked.

For the assigned degrees, independent surface quadrature verifies the complete projected norms and nonzero currents. A positive cubature for the remaining ambient angular directions retains their second and fourth moments. The spinor source has polynomial dependence in those directions, so this integrates the displayed norm and current square without replacing their polarization by an isotropic density.

This improves the field representation of the inverse parent. It does not yet identify all chain components as projected modes of the one master spinor, derive the two ordinary normal profiles, select a common radius or gap, solve nonlinear support and mode selection, or obtain the complete translational inertia. Those remain physical requirements for a mass-operator derivation.