IDOS Energy, Coherence, and Supporting Dynamics — Part 22

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stress $n\omega UU$ and complete boost relation survive. Their phase speed is $c_D$, independent of density. The supporting drift matrix is unchanged; the static Schur lower bound remains positive with the same common coefficient choice. The vector support restoring square is smaller but still positive. This preserves the local constrained quadratic-energy argument, not equality of every quadratic coefficient.

The interaction law changes explicitly. Holding that support-coefficient version fixed, write $F=U\cdot a-h-\gamma V$ and let $Q$ be the signed non-clock derivative Lagrangian per unit density. Eliminating density in the original and changed laws gives

$$\mathcal L_{\rm old}=bn_0(e^{(F+Q)/b}-1),\qquad \mathcal L_{\rm new}=bn_0(e^{F/b}-1)+n_0Q.$$

For $F=tF_1,Q=t^2Q_2$, their difference starts at $n_0t^3F_1Q_2/b$, followed by $n_0t^4[F_1^2Q_2/(2b^2)+Q_2^2/(2b)]$. Higher field Taylor coefficients add their corresponding higher terms. Thus matching isolated energies and quadratic derivative terms does not preserve nonlinear contact responses. $Q$ is a signed Lagrangian, not complete conserved energy. The required reciprocal force and inverted mixing response must be derived again from this action.

All six maintained depths use this same ambient law. Spatially uniform participating energy, observed branch selection, economical fundamental components, vacuum and actual pair solutions remain unresolved. The complete derivation and 662 checks are in saturated-flow-audit; density-weight-repair-review independently checks the constrained principal maps, branch energy and contact qualification. Principal's density-weight-contact calculation gives the exact interaction comparison.

Actual Two-Band Density Preparations

The causal positive chart permits a precise local preparation test. For same-depth prefixes $N_a<N_b$, let $x=f_b/f_a$, $C=\sqrt{D_a/D_b}$ and $R=1+x^2+2Cx\cos\Delta>0$. The normalized raw sum has two constant spectral amplitudes after removing its common spatial phase:

$$c_A=\frac{D_a^{-1/2}+xe^{i\Delta}D_b^{-1/2}}{\sqrt R},\qquad c_B=\frac{xe^{i\Delta}}{\sqrt{D_bR}}.$$

The exact current and mean label are

$$\chi-\omega_a=\frac{(\omega_b-\omega_a)(x^2+Cx\cos\Delta)}R, \quad A=\frac{Cx}R(v\cos\Delta+g\sin\Delta), \quad q=q_a+(q_b-q_a)\frac{x^2}R.$$

Here $v=(q_a-q_b)/\ell$, $g=\nabla\log(f_b/f_a)$; stationary envelopes are assumed on the initial slice. Moving envelopes add their time-gradient term. These are actual normalized field derivatives, not an averaged frequency or imposed synchronization.

The occupation penalty is computable without building the huge ambient carrier. With $p_A=|c_A|^2$, $p_B=|c_B|^2$ and $E=d\binom{N_a+d-1}{d-1}$, the sparse occupation sum is

$$V_{\rm anchor}+V_{\rm edge} =(D_b-D_a+E)p_B(p_A-p_B)^2,\qquad V_{\rm mix}=0.$$

The edge count includes only crossings of shell $N_a+1$, not every lower shell. For $N_a\ge1$, the optional phase penalty is $Ep_Bp_A|c_B-c_A|^2$. For $N_a=0$, including the down/strange comparison, it is instead $d[\binom{N_b+d-1}{d}-1]p_B^2|c_B-c_A|^2$.

Specify support $w=w_a+w_b$, $n_0=n_{\rm peak}(w_a+w_b)^2$ and finite constrained support-vector, frame and flow jets. SF-DW reconstructs

$$\log\frac{n_{\rm aux}}{n_0} =\frac{\chi-m(z)-[h(A,z)-m(z)]-\gamma V}{b}.$$

It gives finite positive density at every finite regular local jet, but possibly enormous compression or depletion. The raw-sum norm is $n_{\rm raw}=n_{\rm peak}w_a^2R$; retaining that magnitude as well would require the additional compatibility equation $[\chi-h-\gamma V]/b=\log[R/(1+x)^2]$. It is not automatic. Reconstructing density changes the seed magnitude; retaining a chosen density instead changes the required phase time jet. Neither choice can be described as retaining both arbitrary inputs unchanged.

Near the smaller band, $z-z_a=O(x^2)$ and the flat target interpolant gives $m-\omega_a=O(x^{12})$. The current shifts at first order, while alignment and occupation penalties start at second order. Thus the constant-coefficient example has

$$\log(n_{\rm aux}/n_0) =128C\frac{\omega_b-\omega_a}{\omega_{\min}} x\cos\Delta+O(x^2).$$

The exact finite preparation condition is $|\chi-h-\gamma V|\le\epsilon b$, implying a density ratio between $e^{-\epsilon}$ and $e^\epsilon$. The leading small-$x$ expression does not replace this finite bound. At quadrature its linear term vanishes; the quadratic gradient and occupation terms remain.

Interpolation And Compressibility Are Additional Constitutive Choices

The earlier fourteen-target mass interpolant has normalized weights proportional to $c_A/\|z-z_A\|^6$, where

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