Exterior-form realization of the curvature chain

The first-order curvature parent also has a realization on one common geometric bundle. It uses exterior forms on the actual proposed surface, with their exterior derivative and metric adjoint. This removes the need for different tensor bundles at successive links. It does not derive a master-spinor field feature or the particle geometry.

For even containing depth d, its leading scalar-derived operator is the same conformal curvature Hessian as before. The odd-depth continuation is a specified inverse polynomial construction. No corresponding odd-surface bending energy or neutrino mechanism is inferred from that continuation.

A full first-order geometric operator

On S_R^m, with m=d-2, let d_S be the exterior derivative and delta_S its L2 adjoint. Retain all complex exterior forms, rather than only the scalar component. The Hodge–Dirac operator and form grading are

$$D_H=d_S+\delta_S,\qquad \Gamma\big|_{\Lambda^p}=(-1)^p,\qquad \{D_H,\Gamma\}=0. \tag{1}$$

Put k=(m-1)/2 and define the local first-order operator

$$B=D_H+\frac{k}{R}\Gamma, \qquad B^2=\Delta_H+\frac{k^2}{R^2}. \tag{2}$$

The form indices are tensor components over those same ordinary spatial directions. They are not extra spatial dimensions. Exterior forms are a Clifford module; that fact alone does not identify a physical fermion's spinor transformation or its master-field origin.

If Y_ell is a normalized scalar harmonic and lambda=ell(ell+m-1), the complete subspace consisting of Y_ell and R d_S Y_ell/sqrt(lambda) is invariant. In that basis,

$$B_\ell=\frac1R \begin{pmatrix}k&\sqrt\lambda\\\sqrt\lambda&-k\end{pmatrix}, \qquad \operatorname{spec}(B_\ell)= \left\{\frac{\ell+k}{R},-\frac{\ell+k}{R}\right\}. \tag{3}$$

The identities delta_S d_S Y=lambda Y/R^2 and d_S^2 Y=0 establish closure without omitting a generated higher form. Every other form family remains in the full operator (2); equation (3) is a closed subspace used for this particular calculation.

Geometric shifts and the cumulative polynomial

Use d copies of the same form bundle at the links of the canonical chain in A first-order parent for the curvature frequency operator. Set

$$L_j=c\left(B+\frac{a_j}{R}\right),\qquad a_j=j-\frac{d+1}{2},\qquad j=1,\ldots,d. \tag{4}$$

The shifts are symmetric about zero. On the positive eigenvalue in (3), the link coefficients are exactly

$$t_j=\frac cR(\ell-2+j),\qquad \prod_{j=1}^dt_j=\frac{d!c^d}{R^d}D_{\ell-2,d}. \tag{5}$$

At even d, the symmetric polynomial in B contains only B^2. Substituting (2) in its scalar block recovers the curvature operator (1) of the preceding note. At odd d, it is an odd polynomial. For example, d=5 has m=3, k=1 and

$$P_5(B)=B(B^2-R^{-2})(B^2-4R^{-2}). \tag{6}$$

On the positive scalar-derived pair, its eigenvalue is R^(-5)(ell-1)ell(ell+1)(ell+2)(ell+3). This algebra is exact. Its odd-depth physical energy and mode-selection law remain unspecified.

The full first-order principal symbol of B is i(epsilon(p)-iota(p)), where epsilon is exterior multiplication and iota is contraction. Their algebra gives

$$\sigma(B)(p)^\dagger=\sigma(B)(p),\qquad \sigma(B)(p)^2=|p|^2I. \tag{7}$$

All complete form components therefore obey the same principal speed bound. The shifted links (4) and the algebraic common gap do not change it. The parent retains both signed Hodge branches and both signs of its full Hamiltonian. No identification with antiparticles is made.

Keep the parent rather than only the polynomial

The exact low positive frequency is again a_ell/||C_ell||, with a_ell=product(t_j)/G^(d-1) and the full scaled inverse in equation (12) of the preceding note. Reversing the signed Hodge eigenvalue reverses the absolute link list. Its smallest singular value is unchanged, so the two form branches remain present at the same low positive frequency. This is a degeneracy of the specified linear construction, not a dynamically selected clock or particle multiplicity.

At fixed ell and eta=GR/c large, the finite-gap correction is

$$\frac{\omega_\ell}{a_\ell} =1-\frac{(\ell-1)^2+(\ell+m)^2}{2\eta^2} +O(\eta^{-4}). \tag{8}$$

The difference from the tensor-link realization is physical information about the parent, even though their leading even-depth operator agrees. Their full intermediate fields and finite-frequency responses differ. Agreement of a limiting spatial polynomial does not select its unique dynamical realization.

For the depth-five cutoffs associated with the existing counts, the surface degrees are 11, 16 and 23. At eta=10,000 the complete chain gives

These are unchanged-index mathematical controls. They do not determine the very small depth-five frequency unit. With common G and common R, the leading unit ratio would instead be omega_(0,5)/omega_(0,6)=eta/6. A suppressed depth-five unit therefore needs additional derived geometry or dynamics within this class.

Nor does this continuation automatically suppress the endpoint response: its coefficients in the table approach one, as in the other depths. The actual spatial coupling map, its full polarization and both objects' response are still required. No electric, colour or weak vertex has been assigned to these form components.

The full first-pair polarization is explicit

Writing beta=ell+k, the normalized positive eigenvector of (3) has

$$|u_0|^2=\frac{\beta+k}{2\beta},\qquad |u_1|^2=\frac{\beta-k}{2\beta},\qquad \langle\Gamma\rangle=\frac{k}{\beta}. \tag{9}$$

At the first depth-five control those weights are 13/24 and 11/24. At the electron surface degree they are 17/31 and 14/31. The parity matrix element is a form-grading expectation, not electric charge. It must not be used as a neutral projector or as a derived suppression factor without specifying and deriving the corresponding interaction.

The next useful coupling calculation is on these actual spatial forms, keeping their phase gradients and the full current generated by a declared local action. Counting forms or choosing a component alone would not test the geometric compatibility mechanism.

Verification

hodge_curvature_chain_checks.py passes 190 conditions in the 20260913T232552 report. It checks the full exterior algebra, Hermitian Clifford symbols and form grading through m=8, exact shifted-polynomial identities, the complete scalar/exact-form chain equation, both signed form branches, canonical norm and finite-gap asymptotics. It retains the full first-pair eigenfield arrays and exact source hashes.

The surface, its ordinary normal extent, the common radius and gap, the nonlinear sustaining action and complete translation inertia remain open. This note supplies a local geometric operator construction and its controlled response, not a finished physical mass operator.

Related research

• Spatial mass operator: results and remaining dynamics — connects this calculation to its premises, later corrections and the complete source question.