Induced Normal Curvature, Spatial Inertia and Retained Orientation

Deriving a connection from spatial bending changes the kinetic structure. For the common nested graph, an induced curvature action gives a positive term involving a spatial derivative of deformation velocity. The squared-linear-curvature comparison has positive energy and bounded phase speeds, but its frozen-coefficient probe equation has an exterior response immediately after a compact excitation. A comparison retaining the orientation as a dynamical variable avoids this particular inverse-inertia equation; it remains a different, conditional completion.

These are local response calculations around a specified geometric background. The frozen model is not an exact global solution of the variable graph equations, and the retained-orientation comparison is not a derivation of the full master field or an integrable embedding.

Calculate the Actual Geometric Symbols

For a time-dependent ordinary spatial graph use its spatial normal frame $N(P)$, with $P=D_xf$. Its temporal connection is $A_0=N^\dagger\partial_tN$. The same projector formula as for spatial curvature gives

$$F_{0i}=S\left[P_t(I+P^\dagger P)^{-1}P_i^\dagger -P_i(I+P^\dagger P)^{-1}P_t^\dagger\right]S.$$

At a background point with $P=0$, let $B_i=\partial_iP$ be the actual second fundamental matrices. Take a real normal deformation polarization $\eta$ with wave covector $k n$, $|n|=1$. Define the normal skew matrices

$$W_i(\eta)=\eta(B_in)^T-(B_in)\eta^T.$$

To leading derivative order, the electric perturbation is proportional to $\omega k W_i$ and the magnetic perturbation is proportional to $k^2(n_jW_i-n_iW_j)$. Hence define positive quadratic forms $K,M,C$ on normal polarizations by

$$\eta^TK\eta=\sum_i\|W_i\|_F^2, \quad \eta^TM\eta=\sum_{i<j}\|n_jW_i-n_iW_j\|_F^2,$$

$$K-M=C,\qquad \eta^TC\eta=\left\|\sum_i n_iW_i\right\|_F^2\ge0.$$

Thus $0\le M\le K$. Keeping a canonical positive deformation term and $\lambda h_d$ times the squared first curvature variation gives the comparison Fourier equation

$$\left[I+\lambda h_d k^2K\right]\omega^2\eta =\left[k^2I+\lambda h_d k^4M\right]\eta.$$

Every squared frequency is nonnegative and at most $k^2$ in the unit-speed normalization, with strict positivity for $k\ne0$. This follows from the quadratic-form inequality, without choosing a polarization eigenvector first.

The common Hodge coefficient is the same declared even-carrier coefficient as before: active at three/four and zero at two/five/six/ten. Its insertion is an action premise, not a consequence of the graph alone. Since the connection is now induced, its disappearance does not produce the independent-connection source constraint.

The Complete Background Hessian Contains More Terms

The displayed comparison does not include every quadratic term of an induced curvature action around a curved graph. If $F=F_{\rm bg}+\delta F+\delta^2F+\cdots$, the magnetic Hessian contains both $\|\delta F\|^2$ and $2\langle F_{\rm bg},\delta^2F\rangle$. The latter changes lower-derivative stiffness and cannot be discarded merely because the leading fourth-derivative symbol is positive.

An independent exact calculation for the fixed-coordinate magnetic energy $\frac12\sum_{i<j}\|F_{ij}\|_F^2$ gives, for the first axial normal polarization,

For example a constant tilt $P=\tau\eta n^T$ at the depth-four origin gives the exact fixed-coordinate magnetic density

$$\frac{19\tau^6+62\tau^4+72\tau^2+32}{(1+\tau^2)^3} =32-24\tau^2+O(\tau^4).$$

Thus the earlier canonical coefficient one is not the full stiffness of that action. Induced spatial metric contractions and volume add further terms; a complete spacetime action must specify its temporal geometry as well. These numbers belong to the stated fixed-coordinate norm and are not promoted to a generally covariant graph action. The background may also need a supporting force. Positivity of the full background dynamics remains unproved.

There is a separate carrier qualification. Arbitrary real normal plane-wave deformations need not preserve the graph's holomorphic class. At depth four, one resulting electric curvature matrix has $\|[W_2,J_N]\|_F^2=4$ for the background normal complex structure. These response variations belong to the real SO6 normal connection, not automatically its U3 subconnection. Preserving the color carrier requires a geometric restriction or a derived response of its complex structure.

Occupied-Sector Controls

In the orthonormal origin frame, choose the first coordinate propagation direction. Both symbols are diagonal on real normal polarizations. The displayed entries include all those polarizations:

The checker also uses four generic directions at each nontrivial depth and three wave-number scales. The numbers follow from the one common embedding, with no independently tuned depth coefficient.

Bounded Phase Speed Does Not Settle Probe Causality

For a diagonal axial polarization write $a=K_{rr}$, $b=M_{rr}$. A one-coordinate frozen reduction obeys

$$(1-\lambda a\partial_x^2)\partial_t^2\phi -\partial_x^2\phi+\lambda b\partial_x^4\phi=s(t,x).$$

At zero initial perturbation and velocity, its initial acceleration is

$$\partial_t^2\phi=(1-\lambda a\partial_x^2)^{-1}s.$$

For $\lambda a>0$, the inverse kernel is

$$G(x)=\frac{e^{-|x|/\ell}}{2\ell},\qquad \ell=\sqrt{\lambda a}.$$

A nonnegative nonzero smooth source supported in $|x|<1$ therefore gives positive acceleration at $x=2$ immediately. The checker integrates this response independently for every distinct active axial pair. Even when $a=b$ and all homogeneous phase speeds are exactly one, the specified local source couples through this spatial inverse inertia.

An unforced compact-data control is also available when $a>b$. For initial displacement $\phi_0$, zero initial velocity and no source, write $\alpha=\lambda a$ and let $\gamma$ be the local gradient coefficient. Outside the support,

$$\phi_{tt}(0,x) =\frac{\gamma-b/a}{\alpha}\,(G*\phi_0)(x).$$

For the displayed comparison $\gamma=1$, a nonnegative nonzero compact displacement therefore has an immediate exterior acceleration whenever $a>b$. The same formula accommodates a specified background stiffness correction. The homogeneous $a=b,\gamma=1$ branch is the exceptional wave equation; its specified local-force response still has the inverse-inertia tail above.

This is an exact statement for the displayed frozen reduction. The one-coordinate source is not a finite-total-energy source over all transverse space. Nor does freezing coefficients prove an exact global counterexample for the evolving curved graph. The result identifies the acceleration operator and source reduction that a complete geometric Cauchy analysis must address. It prevents claiming causality from the dispersion bound alone.

Retain Orientation Before Imposing Exact Derivative Locking

A simple local comparison replaces the identity $\theta=\partial_x\phi$ with a dynamical orientation variable and a finite restoring term. For an active axial pair choose

$$L=\frac12\dot\phi^2-\frac{c_0^2}{2}\phi_x^2 +\frac{\lambda a}{2}\dot\theta^2 -\frac{\lambda b}{2}\theta_x^2 -\frac r2(\theta-\phi_x)^2.$$

Here $r>0$ and $c_0^2>0$ are common constitutive choices. The energy is the sum of the corresponding five positive squares. Its principal propagation speeds are $\sqrt{c_0^2+r}$ and $\sqrt{b/a}$. Choosing $r=1/3$, $c_0^2=2/3$, $\lambda=1$ gives speeds at most one for every active axial branch, using the already derived $b\le a$.

The full frequency poles obey

$$[(c_0^2+r)k^2-\omega^2] [r+\lambda b k^2-\lambda a\omega^2]-r^2k^2=0.$$

At $k=0$ the orientation gap is $\omega_{\rm orient}^2=r/(\lambda a)$. This is a deformation clock of the comparison action, not the particle mass-frequency or a selected sector fundamental. The two poles have nonnegative squared frequencies at every tested wave number; the positive energy gives the general constant-coefficient stability argument.

The retained comparison also has a direct finite-propagation proof. Put $\zeta=\theta-\phi_x$. Its positive energy density is the sum of the five squares in the action, and its local energy flux is

$$J=-\phi_t(c_0^2\phi_x-r\zeta)-\lambda b\theta_t\theta_x.$$

Cauchy–Schwarz gives $|J|\le c_* e$ with $c_*=\max\{\sqrt{c_0^2+r},\sqrt{b/a}\}$. The local energy balance then confines compact initial data and local sources to that cone. This proof concerns the complete retained comparison; no corresponding complete geometric action is inferred.

Eliminating $\theta$ spectrally retains the denominator $r+\lambda b k^2-\lambda a\omega^2$. The physical poles are the zeros of the full determinant above, not the zeros of this denominator alone. Replacing the retained mode by instantaneous derivative locking discards a response branch and changes the response. Taking an infinitely stiff lock with $c_0$ fixed also makes the first principal speed diverge. It is not a causal approximation uniformly over all frequencies. The finite choice above changes the low-frequency deformation speed to $c_0$; it does not preserve the original frozen law exactly.

An independently evolving orientation can describe ordinary spatial frame deformation, but a general such field need not be the tangent plane of a spatial graph. Its integrability, origin among retained modes of the same field and complete nonlinear dynamics remain to derive. The comparison demonstrates one controlled way to retain the missing time response, rather than adding a claimed physical particle or mediator.

Reproduction

The checker and report retain 270 checks across 12 cases. They use the exact executed nested-geometry source, verify the exterior-square identity independently, compare all occupied sectors and active axial branches, calculate the compact-source and compact-data frozen tails, and check complete two-mode poles and principal-speed bounds. These are geometry and linear-response calculations, with no particle simulation or fitted observed coupling.

The independent review additionally preserves 185 exact background-Hessian and carrier checks with its source manifest; it supplies the stiffness correction and finite-propagation argument above.

A Retained Normal Complex Structure Makes the Carrier Question Explicit

The normal-structure response derives the exact unitary/complement splitting, positive orientation kinetic term and compatible connection. The original real electric wave obstructs a parallel continuation near the original structure because its background electric curvature vanishes. A finite orientation penalty does not generally impose exact holonomy reduction. It also leaves commuting SU3 vacuum histories untouched, so it does not repair the independently varied zero-dielectric vacuum.