IDOS Energy, Coherence, and Supporting Dynamics — Part 45

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$$R_A=H_a^TA_oH_a= \frac4{D^2}\left[S+\frac{q^2}{\ell^2}\mathbf1\mathbf1^T +\frac{N}{(N+d)\ell^2}I\right] +\frac{8q^2N}{D(N+d)\ell^2}(dI-\mathbf1\mathbf1^T).$$

Phase edges annihilate $H_a$ in this contraction, but they can still affect the full inverse. If $r_j$ is the corresponding eigenvalue of $R_A$, inverse Cauchy–Schwarz and the positive anchor graph give

$$c_j\ge\frac{\lambda_j^2}{a_or_j},\qquad c_\parallel\le\frac{D^2}{4a_o} \left(\lambda_\parallel-\frac{dq^2}{D\ell^2}\right),\qquad c_\perp\le\frac{D^2\lambda_\perp}{4a_o}.$$

The scalar map $m_j(c)=\omega/\epsilon-\omega^2/[4(c+1/t_j)]$ is increasing, where $t_j$ is the eigenvalue of $T$. These rational compliance bounds therefore bound the actual inertia. Increasing positive phase-edge stiffness decreases $C$ and $M_A$; at infinite edge stiffness the lower compliance bound is attained by restriction to the affine kernel. This is a finite matrix penalty limit, not a uniform small-wavelength limit.

In the common-unit control $M=94$, $\epsilon=1/(4M^2)$, $\ell=\gamma=\kappa_U=\kappa_F=\omega_{\min}=\mu_d=1$, the exact rational bounds separate both inertia channels for every adjacent same-depth pair among W/Z/Higgs, up/charm/top, the three neutrino labels, and electron/muon. This holds under both common constitutive choices $\kappa=8$ and $\kappa=8/\omega$. The common-unit control does not replace the unequal maintained physical sector units. Down has no affine occupied tangent; bottom remains unresolved.

For positive phase edges, propagation along the occupied all-ones direction has squared speeds $\lambda_\parallel/(4m_\parallel)$ and $\lambda_\perp/[4(d+2)m_\perp]$. The parallel intervals separate the same pairs; their depth-two values increase slightly, whereas depth-four, five and six values decrease. Several transverse intervals overlap and decide no ordering. At zero edges the full occupied-direction spectrum includes nonaffine phases, so these affine restrictions are not generally eigenfrequencies. For an unused direction and either edge choice, the affine quartic coefficient is $c_F^2\kappa_F/(\epsilon m_j)$.

The exact deficit estimate

$$0<1-\frac{\epsilon m_j}{\omega} \le\frac{\gamma\omega r_j}{2\lambda_j^2}$$

also supplies a conditional fixed-action count asymptotic. With $\omega=\mu_dD$ and fixed positive coefficients, provided the same law admits an unbounded prefix sequence retaining drift positivity, the parallel deficit is $O((DN^2)^{-1})$ and the perpendicular deficit is $O(N^{-2})$. Thus $m_j\sim\mu_dD/\epsilon$. Taking the fixed-$N$ wavelength limit first, positive-edge acoustic squared coefficients scale as $N^2/D$, and unused quartic coefficients as $1/D$. Resetting $\epsilon$ when enlarging $M$ changes the action and is a different limit. No infinite-carrier existence theorem follows.

These are internal phase-disturbance inertias at spatial infinity, not complete particle translation inertia or measured pair coupling. They depend on the supplied clock; they do not select it. No special depth-five effect occurs. actual-affine-inertia/report.md and its independent 557-check review preserve the exact derivation, all 26 rational count rows, and separate finite sparse diagnostics up to $D=924$.

Local Evolution For Finite Compact Data Without A Smallness Assumption

The positive-chart SF-DW action has a stronger local result than the small-relative-jet theorem. Its constrained field manifold admits one smooth invertible tangent coframe $T$, common to every spacetime derivative: phase $\langle Je,de\rangle$, horizontal carrier components, $dW$, $\langle E_i,dZ\rangle$, frame rotations $\langle E_i,dE_j\rangle$ and flow components $\langle E_i,dU\rangle$. For material support coordinates, $\langle E_i,dZ\rangle=dz_i+\Omega_{ij}z_j$. This triangular mixing is retained. The coframe need not be integrable.

The complete derivative Hessian is exactly

$$\mathcal L_{q_\mu q_\nu} =T^T\operatorname{diag}_a(G_a^{\mu\nu})T.$$

The phase block is $G_{\rm ph}=(n/b)[(U-h_A)(U-h_A)-b h_{AA}]$.

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