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even for an initially empty label far above the occupied band. The exact uniform-prefix zero set and the lower bound Q≥4D−2∑Ark2+D−2∑k∉A∥hk∥2Q\ge4D^{-2}\sum_A r_k^2+D^{-2}\sum_{k\notin A}\|h_k\|^2 therefore survive. The full off-branch potential and stiffness values change. There are at most DM+d(M+d−1d)D_M+d\binom{M+d-1}{d} anchor/edge terms, instead of (M+2d2d)\binom{M+2d}{2d} predecessor pairs. The common ambient carrier itself is not reduced: at M=94M=94 it has 52,201,972,702,880 real components. The exact construction remains symbolic rather than a practical simulation or a small fundamental-field proposal. The simplified polynomial and its full amplitude Hessian pass 2,035 checks in principal/sparse-occupation.md and its script.

The externally supplied material frame can also be replaced by retained physical frame components in a conditional extension. Let EjE_j be ten spacelike vectors satisfying Ei⋅Ej=δijE_i\cdot E_j=\delta_{ij} and u⋅Ej=0u\cdot E_j=0. Define the rotational connection Ωμ,ij=Ei⋅∇μEj\Omega_{\mu,ij}=E_i\cdot\nabla_\mu E_j, i<ji<j, and add

LF=κFn2∑i<j[(uμΩμ,ij)2−cF2PμνΩμ,ijΩν,ij],κF>0,0<cF≤1, \mathcal L_F=\frac{\kappa_F n}{2}\sum_{i<j} \left[(u^\mu\Omega_{\mu,ij})^2 -c_F^2P^{\mu\nu}\Omega_{\mu,ij}\Omega_{\nu,ij}\right], \qquad \kappa_F>0,\quad0<c_F\le1,

with multipliers for the frame constraints. These are common parameters and components of the same proposed enlarged field. The connection norm retains rotations and excludes the acceleration component of frame derivatives. Constant rest frames have Ω=0\Omega=0; their complete Lorentz boosts are also constant. Thus the new action and its full first variation vanish on the previously constructed branches, preserving their complete current, stress and energy-momentum relation.

The constrained rest variation is essential. For antisymmetric spatial rotations fijf_{ij}, it gives δEj0=δuj\delta E_j^0=\delta u_j, δEji=fij\delta E_j^i=f_{ij}, and δΩμ,ij=∂μfij\delta\Omega_{\mu,ij}=\partial_\mu f_{ij}. The induced boost derivative cancels in this projected rotational quantity. Consequently the complete principal system appends 45 positive wave blocks of speed cFc_F to the earlier coupled phase/support/carrier system. At cF=1c_F=1 the decoupled block is handled directly rather than substituted into an inverse of its vanishing kinetic-minus-gradient matrix.

The lower-order carrier-frame interaction must remain inside its square:

ϵn8∑i∥Dih−J∑jfij(Tj−μj)e0∥H2, \frac{\epsilon n}{8}\sum_i \left\|\mathcal D_i h-J\sum_jf_{ij}(T_j-\mu_j)e_0\right\|_H^2,

where Dih\mathcal D_i h is the full previous residual variation. Together with the positive frame velocity and gradient energy, this preserves the nonnegative constrained quadratic energy. It does not prove nonlinear stability. Rotating both the spatial texture and its frame gives an exact continuous family; rotating the frame alone generally costs energy. Thus the frame is a physical orientation variable rather than an arbitrary local coordinate choice. All six maintained depths use the same 45 frame channels; unused axes add degrees of freedom rather than a derived observed polarization count. No special depth-five interaction follows.

Physical localization can be represented by Fj=nEjF_j=\sqrt n E_j, for which ∑j∫∣Fj∣2=10S\sum_j\int|F_j|^2=10S in the rest frame. The exact derivative relation

Ωμ,ij=Fi⋅∇μFj−Fj⋅∇μFi2n \Omega_{\mu,ij} =\frac{F_i\cdot\nabla_\mu F_j-F_j\cdot\nabla_\mu F_i}{2n}

follows with the differentiated constraints, including density derivatives. The positive-density coordinate change preserves the full variation; it does not regularize n=0n=0. The constraint u⋅Ej=0u\cdot E_j=0 itself depends on carrier derivatives through uu. Its nonlinear elimination and propagation require further analysis even though the complete linear branch reduction is controlled. Different constant frames can be prescribed in separate core regions with finite transition energy, but overlapping tails cannot simply be pasted into an exact two-particle solution. Pair response, vacuum regularity and general nonlinear causality remain open. The derivation and 306 exact checks are preserved in material-frame-dynamics/report.md.

Selecting Internal Phase Relations With Nearest Edges

Uniform occupation leaves independent phases in the retained real pairs. One common algebraic interaction can reduce this freedom to overall phase and texture translations. In orthonormal spectral coordinates ckc_k, use the specified real bilinear product (a,b)∗(c,d)=(ac−bd,ad+bc)(a,b)*(c,d)=(ac-bd,ad+bc) and define

Vphase,edge=∑k∑j:kj>0pk∥c0∗ck−cej∗ck−ej∥2. V_{\rm phase,edge}=\sum_k\sum_{j:k_j>0} p_k\left\|c_0*c_k-c_{e_j}*c_{k-e_j}\right\|^2.

The product requires common relative identifications of the spectral pair planes. Their quadratic metric, JJ and commuting label operators do not derive that nonlinear tensor. It is an additional common constitutive choice; without it the earlier independent-phase branch remains.

On each uniform occupied prefix the positive gates impose ϕk−ϕ0=ϕej−ϕ0+ϕk−ej−ϕ0\phi_k-\phi_0=\phi_{e_j}-\phi_0+\phi_{k-e_j}-\phi_0 modulo 2π2\pi. Induction gives exactly ϕk=ϕ0+k⋅ϑ\phi_k=\phi_0+k\cdot\vartheta. Empty labels have zero gates, so the boundary does not spuriously require occupation of another shell. The full amplitude first variation vanishes on both occupied and empty labels. The law is invariant under global phase, label translations, axis permutations and common pair conjugation. It also preserves the centered spatial carrier because both products acquire the same phase 2ωt+(k−2q)⋅x2\omega t+(k-2q)\cdot x.

In an affine reference phase, with occupied weight w=1/Dw=1/D, its complete amplitude quadratic form is

Qphase,edge=w2∑k∈A,j:kj>0∥h0+hk−hej−hk−ej∥2+w2∑k∉Abkphase∥hk∥2. Q_{\rm phase,edge}=w^2\sum_{k\in A,j:k_j>0} \|h_0+h_k-h_{e_j}-h_{k-e_j}\|^2 +w^2\sum_{k\notin A}b_k^{\rm phase}\|h_k\|^2.

Here bkphaseb_k^{\rm phase} counts the edges with both eje_j and k−ejk-e_j occupied. This second term comes from opening an empty gate; it reaches

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