IDOS Energy, Coherence, and Supporting Dynamics — Part 8

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Actual Sobolev fields satisfy $v=0$ almost everywhere on $\{z=0\}$. Strong $H^1$ convergence, these bounds and integration along affine field segments establish $H\in C^1(H^1,\mathbb R)$, with derivative in $H^{-1}$. This is not pointwise smoothness at arbitrary independent vacuum jets, nor a theorem of generic unique evolution. Smooth compact envelopes extend the exact branch weakly: its flux coefficient is $(b_i/2)J\psi$ and produces no boundary delta. Disjoint compact patterns retain independent clocks until contact, unlike a clock depending on their combined global norm.

Occupation Is An Additional Requirement

The standing construction uses $4D-2$ real components. Its nested basis and linked standing coefficients are prepared constraints. At fixed $A$, each linked label has action $A/D$ and assigned branch energy $A\kappa_d$. This is a branch allocation, not a generally additive nonlinear Hamiltonian. Unlinked cosine and sine coefficients would change the mode multiplicity.

A much smaller control exposes what the matrix law does not select. Let $H_d$ be a real Helmert matrix with $H_dH_d^T=1$ and $H_d\mathbf1=0$. For any positive active target $C_d$, set $K_d=\sqrt{d+1}\,C_d^{1/2}H_d$ and use its columns as physical wavevectors $q_l/L_0$. In $d+1$ orthogonal real $J$ planes,

$$e_0=(d+1)^{-1/2}\sum_{l=1}^{d+1}e^{Jq_l\cdot x/L_0}u_l$$

has $a=0$ and the same $C_d$. It therefore has the same isolated clock, energy and translation law with at most 22 real components. It has only $d+1$ linked helices, not $D$ occupied simplex modes. Their wavevectors change with $N$ and need not lie in a prescribed reciprocal lattice. A third spatial moment distinguishes these particular controls, but balanced alternative spectra can remove that distinction. Equality of $C$ also does not imply equality of general perturbations or overlapping-pair responses.

Multiplying any texture by a localized envelope broadens its physical Fourier spectrum. The finite labels of $e_0$ are not automatically an exact finite Fourier band of the complete field. Likewise, rank $d$ of the normalized orientation matrix does not prove that a fully localized envelope involves only $d$ physical directions. These are substantive limits on the IDOS and physical-depth interpretation.

Conservation And Reciprocal Response

Local phase rotation preserves $B$ and sends $a$ to $a+\nabla\theta$. Consequently the full action, including off-rung fields, has

$$\partial_t\rho=\nabla\cdot(\rho b),\qquad |b|\le c.$$

For a suitable conservative solution this bounds the complete action centroid speed by $c$ and confines vacuum-support propagation to a cone of slope $c$. It does not prove perturbation causality on a nonzero background. The complete energy current is $j_{E,i}=-h_{v_i}\cdot\psi_t$; on the exact moving branch it is $h v_i$. Away from that branch the orientation-gradient terms remain in energy transport.

For two orthogonal $J$-invariant component blocks, with $w_1=\rho_1/(\rho_1+\rho_2)$, the exact mixture is

$$a=w_1a_1+(1-w_1)a_2,\qquad B=w_1B_1+(1-w_1)B_2+w_1(1-w_1)(a_1-a_2)(a_1-a_2)^T +\frac{\nabla w_1\nabla w_1^T}{4w_1(1-w_1)}.$$

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