A first-order parent for the curvature frequency operator

The conformal surface Hessian already gives the simplex polynomial as a spatial eigenvalue. Its unrestricted higher-order evolution cannot be used as a fundamental finite-speed equation. A local tensor chain supplies an explicit first-order parent whose low-frequency branch approaches that polynomial, with its frequency dependence and canonical norm retained.

This is an inverse linear construction on a specified round field feature. It does not derive that feature, its common radius, the common gap, or its support from the master field. The tensor components must still be realized as modes of that same field. They are not additional interaction particles or additional spatial dimensions. No compactification or Yukawa coupling is used.

Factor the actual surface operator into first-order links

Let m = d-2 be even, with d at least four. Consider a round S_R^m embedded in ordinary R^d. This is a proposed codimension-two spatial feature, rather than the entire d-dimensional sector. Write Delta for the nonnegative scalar Laplacian. The known conformal bending Hessian is

$$\mathcal J_d=\Delta(\Delta-m/R^2) \prod_{j=1}^{m/2-1}\left[\Delta+\frac{j(m-1-j)}{R^2}\right]. \tag{1}$$

On a degree-ell scalar harmonic its eigenvalue is

$$\mathcal J_dY_\ell =R^{-d}(\ell-1)\ell\cdots(\ell+d-2)Y_\ell =\frac{d!}{R^d}D_{\ell-2,d}Y_\ell. \tag{2}$$

Takeuchi's second-variation theorem and the preceding V2 bending note give this geometric operator.[1] The construction below supplies a different issue: a full first-order evolution that approximates it over a controlled range. It does not claim that the nonlinear bending energy or its physical geometric time coefficient has now been derived.

Use the following natural tensor bundles on the surface: scalars E_0, covectors E_1, and traceless symmetric two-tensors E_2. Define

$$G_0f=\nabla f,\qquad G_1v=\sqrt{\frac m{m-1}} \left[\nabla_{(a}v_{b)}-\frac{g_{ab}}m\nabla\cdot v\right]. \tag{3}$$

For each positive constant b in the product (1), also define

$$A_bT=\left(\nabla T,\frac{\sqrt{2m+b}}R T\right), \qquad A_b^\dagger A_b=\nabla^*\nabla+\frac{2m+b}{R^2}. \tag{4}$$

The range in (4) is the direct sum of covector-valued traceless symmetric two-tensors and another traceless symmetric two-tensor. All components are retained. The second entry is an algebraic curvature coefficient; the first contains only one spatial derivative.

Put lambda_ell = ell(ell+m-1). For a unit scalar harmonic, the normalized gradient and traceless Hessian generate an exactly closed chain. The needed identities are

$$\begin{aligned} \|\nabla Y_\ell\|^2&=\frac{\lambda_\ell}{R^2},\\ \|\operatorname{Hess}_0Y_\ell\|^2 &=\frac{m-1}{mR^4}\lambda_\ell(\lambda_\ell-m),\\ \nabla^*\nabla(\operatorname{Hess}_0Y_\ell) &=\frac{\lambda_\ell-2m}{R^2}\operatorname{Hess}_0Y_\ell. \end{aligned} \tag{5}$$

The first two follow by integration by parts and the Ricci curvature (m-1)g/R^2. The third follows by commuting two covariant derivatives of the Hessian and removing its trace. An independent Cartesian calculation below verifies the complete tensor equation, including its complex spatial polarization.

The first-order link list is

$$G_0,\ G_1,\ A_{b_1},\ A_{b_1}^\dagger,\ldots, A_{b_{m/2-1}},\ A_{b_{m/2-1}}^\dagger, \ G_1^\dagger,\ G_0^\dagger, \qquad b_j=j(m-1-j). \tag{6}$$

There are exactly d links. On the scalar-derived harmonic subspace, their positive matrix coefficients are

$$\frac1R\left( \sqrt\lambda,\sqrt{\lambda-m}, \sqrt{\lambda+b_1},\sqrt{\lambda+b_1},\ldots, \sqrt{\lambda-m},\sqrt\lambda\right). \tag{7}$$

Their product is precisely (2). The intermediate tensor fields are actual sections over the same spatial surface; they are not copies introduced in numbers h_(k,d) to insert the simplex weights.

The complete first-order evolution

Multiply each link in (6) by one speed c, and call the resulting operator L_j, mapping E_(j-1) to E_j. The notation reuses E_j for the successive bundles in the list; E_d is again the scalar bundle. Introduce canonical fields x_j in E_j for j=0,...,d-1, and y_j in E_j for j=1,...,d. Each field has the usual positive time norm. The intermediate bundles have both an x and a y copy, coupled by one common positive frequency gap G. Define

$$(Tx)_j=L_jx_{j-1}+Gx_j\quad(1\le j<d), \qquad (Tx)_d=L_dx_{d-1}, \tag{8}$$

$$i\partial_t\begin{pmatrix}x\\y\end{pmatrix} =\underbrace{\begin{pmatrix}0&T^\dagger\\T&0\end{pmatrix}}_{H} \begin{pmatrix}x\\y\end{pmatrix}, \qquad Q=\int\left(\sum|x_j|^2+\sum|y_j|^2\right)dA. \tag{9}$$

This is one local first-order equation. Every intermediate equation, both spectral signs and the entire complex tensor field are present. The signs do not supply an antiparticle interpretation. Since H is self-adjoint on its natural closed-surface domain, the full linear evolution conserves Q. This linear fact does not establish nonlinear particle stability or a positive lower bound for the signed Hamiltonian.

Its propagation speed is controlled directly. The principal symbols of G_0 and A_b have norm |p|. For G_1, on a unit tangent covector p,

$$\|\sigma(G_1)(p)v\|^2 =\frac{m}{2(m-1)}|v|^2 +\frac{m-2}{2(m-1)}|p\cdot v|^2\le |v|^2. \tag{10}$$

After grouping variables, the principal matrix of (9) is a direct sum of the Hermitian pairs of these link symbols. The G couplings have no spatial derivatives. Thus its complete symbol norm is c|p|. The usual local norm-flux estimate gives propagation within the surface distance c|t| while retaining all field components.[2] The large gap does not alter that principal speed.

This supplies a causal linear parent on the assumed surface. It does not identify these independent tensor components with the particular spinor and derivative carrier of the master action. Nor does it supply propagation or confinement in the two ordinary normal directions.

The exact branch and its finite-gap correction

On a degree-ell scalar harmonic, (8) is a d-by-d bidiagonal matrix with diagonal t_j equal to c times (7), and superdiagonal G. The positive low frequency is its smallest singular value. Put

$$a_\ell=\frac{\prod_{j=1}^dt_j}{G^{d-1}},\qquad C_\ell=a_\ell T_\ell^{-1}. \tag{11}$$

The matrix C has the explicit bounded entries

$$(C_\ell)_{ij}= (-1)^{j-i} \prod_{a<i}\frac{t_a}{G} \prod_{a>j}\frac{t_a}{G}\quad(i\le j), \tag{12}$$

and zero otherwise. Its upper-right entry has magnitude one. Hence

$$\boxed{\omega_\ell=\frac{a_\ell}{\|C_\ell\|},\qquad a_\ell=\frac{d!c^d}{R^dG^{d-1}}D_{\ell-2,d}.} \tag{13}$$

This is an exact finite-dimensional reduction of the stated spatial chain. It includes the actual normalized low eigenfield. Computing the largest singular value of C avoids losing the very small frequency when subtracting quantities of order G. Independent high-precision decomposition of the original T checks this method.

Define eta = GR/c. For fixed ell and eta large,

$$\frac{\omega_\ell}{a_\ell} =1-\frac{\lambda_\ell}{\eta^2}+O(\eta^{-4}),\qquad \omega_{0,d}=\frac{d!c}{R}\eta^{-(d-1)}. \tag{14}$$

Thus the geometric polynomial is the leading frequency with one common unit per depth. It is not exact at finite gap. Conjugating C by alternating signs makes all its nonzero entries positive. Every entry except the unit corner increases with ell. The positive matrix C^dagger C then has a strictly increasing largest eigenvalue, proving that omega_ell/a_ell decreases strictly with ell. A single finite gap therefore lowers the higher-to-lower frequency ratio below its simplex value in this particular construction.

At eta=10,000, using one radius and gap within depth six,

Their frequency ratio is 206.7622950, compared with the simplex ratio 206.7647059. The relative change is -0.00001166003. These are numerical consequences of the chosen eta, not an experimentally selected gap or a correction fitted to measured masses. The depth-ten degree-24 control has omega/a = 0.9999925600006 at that same dimensionless eta.

The high-degree spectrum is different: when ell greatly exceeds eta, the diagonal first-order links dominate the fixed gap, and the full frequencies grow linearly with ell. The polynomial approximation then fails. This is how the parent avoids promoting the order-d operator to an unrestricted fundamental dispersion law.

Retain the response frequency and canonical metric

Let P select x_0 and y_d, and let Q select all intermediate fields. For a complex frequency z off the intermediate spectrum, their exact elimination gives

$$\Sigma(z)=H_{PQ}(z-H_{QQ})^{-1}H_{QP},\qquad [zI-\Sigma(z)]u=0. \tag{15}$$

Taking z in the upper half-plane gives the retarded resolvent. The endpoint block of the complete resolvent equals [zI-Sigma(z)]^(-1). Dropping its frequency dependence would change both the response and the physical projection norm.

For a real eigenfrequency outside the intermediate spectrum, the full field reconstructed from endpoint values u has norm

$$\boxed{Q=u^\dagger Z(\omega)u,\qquad Z(\omega)=I-\Sigma'(\omega) =I+H_{PQ}(\omega-H_{QQ})^{-2}H_{QP}.} \tag{16}$$

At zero frequency this metric is diagonal. Its entries are

$$Z_L=1+\sum_{j=1}^{d-1}\prod_{a=1}^j(t_a/G)^2, \qquad Z_R=1+\sum_{j=1}^{d-1}\prod_{a=d-j+1}^d(t_a/G)^2. \tag{17}$$

The leading canonically normalized frequency is therefore a_ell/sqrt(Z_L Z_R), with the remaining frequency dependence still specified by (15). Equation (17) is the norm of the actual intermediate field, rather than a discretionary normalization of an amplitude.

An endpoint potential V(x) has, between two normalized low branches, the exact projected matrix element

$$\langle\Phi_\ell,PVP\Phi_{\ell'}\rangle =K_{\ell\ell'}\langle Y_\ell,VY_{\ell'}\rangle, \quad K_{\ell\ell'}=x_{0,\ell}^*x_{0,\ell'} +y_{d,\ell}^*y_{d,\ell'}. \tag{18}$$

The same-degree coefficient is the endpoint weight in the table. It is also the complete eigenfrequency derivative under a uniform endpoint potential, independently checked by varying the full matrix. At eta 10,000 the mixed depth-six coefficient is 0.9999917899833. No frequency division has been used to make these coefficients similar.

The spatial matrix element in (18) still matters. It retains the angular patterns, orientation and phase, and it may vary strongly or vanish. A common endpoint coefficient alone is not a common physical vertex or a neutrino suppression mechanism. A reciprocal interaction must arise from the same nonlinear field action, with both objects allowed to respond.

Scale and physical scope

The radius-to-frequency conversion in this construction is explicit:

$$\frac{\omega_{0,d}R}{c}=d!\eta^{-(d-1)}. \tag{19}$$

At depth six and eta=10,000 it is 7.2e-18. Thus a small correction to the simplex ratio comes with a strong gap-to-radius hierarchy. It cannot be combined silently with the different inverse assumption R proportional to c/omega_0 from a previously solved core.

Equal radius and equal G across depths would give omega_(0,10)/omega_(0,6) = 5040/eta^4. Their shared charged-lepton unit would then require eta=5040^(1/4), outside the large-gap regime needed for the assigned degrees. Different radii by depth remain allowed, but their relation must be derived from the sustaining geometry; it is not fixed by the tensor chain alone.

The construction provides a natural spatial polynomial, a complete causal linear parent, controlled frequency corrections and an actual canonical response coefficient. The physical origin of the surface, its radius and gap, its nonlinear action and mode selection, its master-spinor realization, and the complete translation inertia are still open. Energy equals omega Q for a normalized eigenmode of this quadratic equation on its fixed surface; that does not calculate the inertia of a self-sustaining field feature moving through ordinary R^d. Mass remains the selected frequency in equivalent units throughout.

Verification and sources

causal_curvature_chain_checks.py passes 352 conditions in the 20260913T230830 report. It independently computes Cartesian covariant gradients, traceless Hessians, their divergence and their full rough Laplacian on complex harmonic fields. It checks common-rotation covariance, all link principal singular values, the geometric product, complete eigenfields and independent 100-digit singular-value results. The old report includes a smallest-singular-value diagnostic below floating-point resolution; that unused diagnostic is omitted from the current implementation. Its frequency uses the well-resolved largest singular value and passes the independent precision check.

curvature_chain_response_checks.py passes 124 conditions in the 20260913T231429 report. It checks the exact internal field, full Schur equation, canonical norm, zero-frequency metric, probe variation, retarded resolvent and unchanged-assignment frequency comparison. The reports preserve the exact source versions and complete numerical field arrays. The proofs and scope above accompany those checks.

[1] Yuya Takeuchi, On the second variation of the Graham-Witten energy, 2018, revised 2020. The geometric Hessian is its contribution; the first-order chain and its response are derived here.

[2] Alan McIntosh and Andrew J. Morris, Finite Propagation Speed for First Order Systems and Huygens' Principle for Hyperbolic Equations, 2012. This supplies the general first-order propagation framework; equation (10) verifies the principal bound for the displayed tensor system.

Related research

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