A Common First-Derivative Normal Orientation Action

The ray metric and canonical normal connection can be combined in one explicit local comparison action. Unlike curvature induced by a moving spatial graph, its curvature depends on first derivatives of the retained orientation field. The resulting equations have second-order principal derivatives. Every static orientation background has positive principal spatial stiffness and speeds at or below the comparison light speed. A sufficiently rapid homogeneous time-geodesic gives an exact background with a negative spatial principal stiffness. Thus the explicit action has a controlled static causal regime and a concrete dynamical boundary.

This is a constructive scoped result. General-background hyperbolicity, a physical source response, localized particle states and fixed-flux guides for this action remain open. The connection is chosen from normal orientation relative to a fixed flat reference connection; it is generally different from the original embedding's Levi-Civita normal transport.

The Explicit Common Action

On an ordinary real rank-2q spatial normal bundle let J be an orthogonal complex structure and set

$$\widehat A[J]=-\frac12JdJ,\qquad \widehat F[J]=-\frac14[dJ,dJ].$$

With signature (+,−,…,−), define

$$\mathcal L=\frac12\partial_\mu u\partial^\mu u -\frac12\Omega^2u^2 +\frac{f^2}2\langle\partial_\mu J,\partial^\mu J\rangle_F -\frac{\epsilon_d(u)}4\langle\widehat F_{\mu\nu},\widehat F^{\mu\nu}\rangle_F,$$

$$\epsilon_d(u)=h_d(\kappa_2u^2+\kappa_4u^4),\qquad f,\kappa_2,\kappa_4,\Omega>0.$$

The common middle-Hodge prescription gives h=1 at occupied three/four and zero at two/five/six/ten. The odd-sector pullback remains a declared premise. Normal ranks q=4,3,3,2,2,0 correspond to depths 2,3,4,5,6,10 in the selected ten-parent construction. At q0 the normal terms are absent. No field coefficient or orientation coordinate adds a physical spatial dimension.

The derivative of J in the orientation term is relative to the fixed reference connection. It cannot be replaced by D-hat J, which vanishes identically by construction and would delete this kinetic term. In other normal frames the reference connection must transform too. This action is a proposal for projected modes of the same field, not an established reduction from Psi-infinity.

The occupied-rank comparison uses the same local formula on each declared carrier. It does not yet construct one global cross-sector action whose projected orientation fields, reference connections and canonical normalizations satisfy all nesting relations. The existing common nested embeddings supply compatible spatial sectors; additional compatibility of their dynamical J fields and their projections must be derived. An arbitrary local map into an orientation target is not automatically the normal Gauss map of those embeddings.

For q3, the orientation term supplies exactly the complete chiral projective metric: its coefficient is 8f² times the unit-vector Fubini–Study metric. The amplitude and common phase terms of a complex field are separate. The first-order Dirac action is also separate. At q4 the same statement applies only on the pure-ray submanifold.

Energy and the Order of the Equations

In an adapted lift write theta=g-inverse dg and a_mu=theta_mu^m, the component anticommuting with J0. Then

$$\|\partial_\mu J\|_F^2=4\|a_\mu\|_F^2, \qquad \widehat F_{\mu\nu}=-[a_\mu,a_\nu].$$

Each curvature is bilinear in first derivatives. Its squared action has no second derivative hidden inside F. Variation therefore produces second-order equations with derivative-dependent coefficients, rather than the mixed fourth-order spatial response found in the graph-induced comparison.

The exact classical energy density is

$$e=\frac12(u_t^2+|\nabla u|^2+\Omega^2u^2) +2f^2\left(\|a_0\|_F^2+\sum_i\|a_i\|_F^2\right) +\frac\epsilon2\left(\sum_i\|[a_0,a_i]\|_F^2 +\sum_{i<j}\|[a_i,a_j]\|_F^2\right)\ge0.$$

The quartic term is quadratic in velocities, so its Legendre energy has the displayed positive electric and magnetic contributions. With f>0 the orientation time Hessian is strictly positive even when epsilon vanishes. Positive energy and a positive time Hessian alone do not prove hyperbolicity on every spatial background.

The actual constrained orientation equation can also be written directly. Let A=theta^h be the compatible adapted connection and D_A its action on m. A horizontal lift variation eta has delta a_mu=D_A,mu eta because [m,m] lies in h. The equations are

$$D_{A,\mu}P^\mu=0,\qquad P^\mu=4f^2a^\mu+\epsilon[a_\nu,\widehat F^{\mu\nu}],$$

$$\Box u+\Omega^2u+ \frac{\epsilon'(u)}4\langle\widehat F_{\mu\nu},\widehat F^{\mu\nu}\rangle_F=0.$$

These are variations of J and u. They are not the independently varied connection equation D_mu(epsilon F^mu-nu)=j^nu. With matter retained, its orientation variation must be added to this equation rather than imposing that separate Gauss law.

An Exact Local Background and Its Principal Cones

Choose a unit Frobenius vector X in m and take

$$g(x)=\exp(kx_1X),\qquad J(x)=g(x)J_0g(x)^{-1}, \qquad u(t)=u_0\cos(\Omega t).$$

The orientation follows a target geodesic in one spatial direction; all canonical curvature components vanish because only one a_i is nonzero. It solves the source-free orientation equation, while u solves its free restoring equation. This is a smooth local, spatially extended background, not a finite-total-energy particle. At any time epsilon is nonnegative. Freeze its value and the target frame for the principal-symbol calculation.

Let eta be an m-valued tangent disturbance. Put B eta=[X,eta] and C=B-adjoint B, a nonnegative symmetric operator. The complete quadratic principal Lagrangian is

$$L_{\rm principal}^{(2)}= \frac12\langle\eta_t,(4f^2I+\epsilon k^2C)\eta_t\rangle -\frac12\langle\partial_1\eta,4f^2\partial_1\eta\rangle -\frac12\sum_{j>1}\langle\partial_j\eta,(4f^2I+\epsilon k^2C)\partial_j\eta\rangle.$$

There is no magnetic background curvature and hence no omitted term proportional to F-background times delta-squared F in this particular calculation. For a propagation unit vector n and an eigenvalue c of C, the squared characteristic speed is exactly

$$\boxed{v_c^2(n)= \frac{4f^2+\epsilon k^2(1-n_1^2)c} {4f^2+\epsilon k^2c} =1-\frac{\epsilon k^2n_1^2c}{4f^2+\epsilon k^2c}\in(0,1].}$$

The neutral u disturbance has unit principal speed; variations of epsilon do not mix its principal derivatives here because F-background=0. Lower-order target-curvature terms remain and can cause long-wavelength instability of an extended geodesic. The cone result does not assert that every frequency is real or that this background is stable.

For the common unit generator X=X_E12/2, the exact spectrum of C is:

At inactive occupied depths epsilon=0, all retained orientation principal speeds are one. At active three/four the rank-three spectrum applies. At depth ten only the neutral comparison field remains. The distinction is caused by the already declared gate, not by the kinetic term independently selecting those depths.

Electric Curvature Now Costs Orientation Motion

An independently varied connection can have a static electric potential and nonzero electric curvature while its other prescribed geometric variables are static. Under this J-only rule, a time-independent J in the fixed reference frame has F-hat_0i=0. A nonzero electric curvature instead requires both temporal and spatial orientation gradients.

For real skew matrices the sharp inequality is ||[A,B]||_F²<=||A||_F²||B||_F². To see the constant, put A into ordinary real two-plane rotation blocks with rates alpha_r. The nonzero rotation rates of ad_A are alpha_r plus or minus alpha_s for distinct planes. Their squares are at most 2 sum_r alpha_r²=||A||_F². This bounds the norm of ad_A on skew matrices. With E²=sum_i||F-hat_0i||_F² it gives

$$E^2\le\|a_0\|_F^2\sum_i\|a_i\|_F^2, \qquad \boxed{e_J\ge4f^2 E.}$$

The second inequality follows by the arithmetic–geometric mean inequality on the positive orientation energy. It is sharp already in the m algebra: the unit pair X_E12/2, X_iE12/2 has commutator norm one, and equal temporal and spatial magnitudes attain the bound pointwise. Including the electric curvature energy gives e>=4f²E+epsilon E²/2, before the other nonnegative contributions. Thus the old fixed-flux electric energy Q²/(2I) omits necessary orientation energy when imported into this action. Its scalar minimizing profiles and source-separation results cannot be declared solutions here. One must minimize over actual maps J and derive their source equation and allowed curvature together.

The Static Result Extends Beyond a Single Geodesic

The principal cone bound holds at any point where the reference orientation is static, even if its spatial canonical curvature is nonzero. In an adapted frame let a_i be arbitrary m-valued spatial gradients, set A_i eta=[eta,a_i], and use a unit propagation covector n. Define

$$C=\sum_i A_i^\dagger A_i,\qquad A(n)=\sum_i n_i A_i,\qquad M=4f^2>0.$$

The time and spatial principal matrices are

$$\boxed{K=M I+\epsilon C,\qquad G(n)=M I+\epsilon[C-A(n)^\dagger A(n)].}$$

For each tangent eta, write b_i=A_i eta. Direct expansion gives

$$\sum_{i<j}\|n_i b_j-n_j b_i\|_F^2 =\sum_i\|b_i\|_F^2-\left\|\sum_i n_i b_i\right\|_F^2.$$

The right side is nonnegative by Cauchy–Schwarz. Consequently M I<=G(n)<=K, and all generalized principal squared speeds solving G eta=v² K eta lie in (0,1]. This bound does not require the spatial background to be a stationary solution; it is a pointwise principal statement wherever that background is attained.

Here the full quartic second variation contains a background-curvature term, but its rank-one principal contribution is zero: for delta a_i=n_i eta, the term [delta a_i,delta a_j]=n_i n_j[eta,eta] vanishes. Its unrestricted gradient Hessian need not vanish. This is why the rank-one symbol proof is valid without discarding a background term, unlike the earlier graph-induced calculation. Lower-order terms and global evolution still need their own analysis. The scalar u equation has no mixed second-derivative block with J: epsilon depends on u but not its derivatives.

A Sufficient Causal Regime Includes Moving Backgrounds

The static result extends to a precise open set of moving orientation jets. Put A_mu eta=[eta,a_mu], C=sum_i A_i-adjoint A_i and A(n)=sum_i n_i A_i for a unit spatial covector n. The full frozen principal pencil is

$$P(\tau,n)=K\tau^2+B(n)\tau-G(n),$$

$$K=M I+\epsilon C,\qquad B(n)=-\epsilon[A(n)^\dagger A_0+A_0^\dagger A(n)],$$

$$G(n)=M I+\epsilon[C-A(n)^\dagger A(n)-A_0^\dagger A_0].$$

The sharp skew commutator bound gives A0-adjoint A0<=||a0||_F² I. Therefore the sufficient condition

$$\boxed{\epsilon\|a_0\|_F^2<4f^2=M}$$

makes both K and G strictly positive in every propagation direction. Moreover the original symbol has an exact null-covector identity

$$P(\pm1,n)=\epsilon[A(n)\mp A_0]^\dagger[A(n)\mp A_0]\ge0.$$

These inequalities prove real, semisimple characteristic roots with magnitude at most one, including repeated polarizations. A complete matrix proof avoids assuming simultaneous diagonalization. Define

$$\mathcal A=\begin{pmatrix}0&G\\G&-B\end{pmatrix}, \qquad \mathcal H=\begin{pmatrix}G&0\\0&K\end{pmatrix}>0.$$

For z=(eta,tau eta), the generalized symmetric eigenproblem A-calligraphic z=tau H-calligraphic z is equivalent to P(tau)eta=0. Positive H makes all roots real with a complete eigenbasis. Schur complements show A-calligraphic−H-calligraphic<=0 exactly when P(1)>=0, and A-calligraphic+H-calligraphic>=0 exactly when P(−1)>=0. Hence every eigenvalue is in [−1,1]. This is a directional positive symmetrizer for the frozen principal problem, not a proof of global evolution or a necessary velocity bound.

On the pure-ray orbit ||partial_t J||²=4||a0||²=16||partial_t psi||_FS². The same sufficient condition consequently reads

$$\epsilon\|\partial_t\psi\|_{\rm FS}^2<f^2.$$

At q3 this covers the complete chiral projective polarization, linking the kinetic metric and the causal regime under one normalization. The uncalibrated physical coefficient f remains explicit. Static backgrounds satisfy the condition automatically; moving backgrounds may satisfy it as well. It is a sufficient field-derivative regime of the proposed action, not a restriction on the center-of-motion speed of an identified particle.

Rapid Orientation Motion Can Destroy Hyperbolicity

The same source-free action has an exact local time-dependent comparison background

$$g(t)=e^{vtX},\quad J(t)=g(t)J_0g(t)^{-1},\quad u(t)=u_0\cos(\Omega t).$$

It is a target geodesic and has F-hat=0, so the neutral and orientation equations hold. It is homogeneous and spatially extended, not a finite-energy particle. For a tangent curvature eigenvalue c of C_X=ad_X-adjoint ad_X, the frozen principal matrices and speed are

$$K=M I,\qquad G=M I-\epsilon v^2 C_X, \qquad v_c^2=1-\frac{\epsilon v^2c}{4f^2}.$$

For the unit X used above, c_max=1 at q2,3,4. The maximum-curvature polarization therefore loses positive principal spatial stiffness at epsilon v²=4f² and has negative squared characteristic speed above it. Arbitrarily short waves then have imaginary leading frequencies; this is a local principal failure, stronger than a long-wavelength instability. Positive complete energy does not prevent it, because a moving background changes the quadratic spatial response.

This failure also occurs on exact smooth finite-total-energy backgrounds. Choose smooth compact cutoffs chi and zeta, both one on an ordinary spatial ball. Let phi solve the free wave equation with phi(0,x)=0 and phi_t(0,x)=v chi(x), and let u solve the free restoring equation with u(0,x)=u0 zeta(x), u_t(0,x)=0. Define

$$J(t,x)=e^{\phi(t,x)X}J_0e^{-\phi(t,x)X}.$$

Then a_mu=(partial_mu phi)X and every curvature commutator vanishes identically. The constrained orientation equation reduces to X Box phi=0, while u retains its free equation. These are globally smooth free-field solutions embedded in the common action, with finite conserved positive energy. In the interior of the initial ball the principal response is exactly the time-geodesic response above, and epsilon(u0)v² can exceed 4f². Thus the moving failure is not caused by infinite total energy or off-shell data. Existence of this special solution does not provide a wellposed Cauchy problem for its transverse orientation disturbances.

At active occupied three/four this threshold can be crossed when u and v are large enough. At inactive two/five/six epsilon=0 under the existing gate, so this particular failure is absent. The threshold is not a predicted particle speed or measured coupling. A complete constitutive action must control the motion, restrict its valid regime or modify its large-derivative terms. The static result is useful progress, while the same calculation excludes a global causal-completion claim for the unrestricted action.

Contribution and Reproduction

This combines the canonical connection, the full rank-three projective metric and a common neutral dielectric in one explicit action. It removes independent connection histories at fixed J and supplies a conventional second-order principal comparison with exact bounded speeds throughout the static-orientation regime. It also exhibits an exact moving background where that regime fails. The changed Gauss/source variation, electric orientation cost and large-motion threshold are concrete conditions for a more unified geometric color model.

The checker and report verify 110 exact conditions: the full orientation tangent Gram matrix, nonnegative curvature-response spectra at ranks two/three/four, three coefficient choices, three propagation angles, the cone identity and every occupied-sector assignment. No particle simulation, global hyperbolicity theorem, observed charge normalization or confinement claim is included.

The static and moving principal checker and report preserve 384 exact conditions. They test noncommuting spatial gradients, full tangent Gram identities at three propagation angles, the vanished rank-one background term, positive factorized static stiffness, the moving threshold on both sides and at equality, and full mixed moving pencils with exact positive LDL factors and null-covector identities at three dielectric coefficients. The earlier 276-condition execution remains preserved in source history. An independent action-level review confirms the extensions and their factors.

That independent review preserves a further 38 exact conditions, the constrained Euler equations, direct electric momentum variations, the spatial Jacobi sign, sharp energy saturation and the finite-energy exact background construction. Its complete derivation and exact frozen source evidence are retained with the private research record.

A separately hashed independent addendum proves the sufficient moving-background bound, including repeated characteristic roots, using the same positive symmetric linearization. The original independent submission and executable hashes remain unchanged.

A Distinct Causal Constitutive Alternative

The timelike orientation determinant retains this target metric and canonical connection while replacing the derivative constitutive law. Its complete mixed-gradient principal symbol has one cone inside the ordinary light cone throughout its timelike domain. It also supplies exact codimension-two maps carrying determinant curvature and complete transverse inertia. The changed fourth-order terms do not equal this article's curvature-squared term, so its Hodge gate and independently varied flux-guide results do not transfer. A static dilation proof and a divergent width-mode norm delimit what the new action can localize. The rapid-motion counterexample above remains a valid limitation of the original action.