IDOS Energy, Coherence, and Supporting Dynamics — Part 3

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Support can preserve IDOS scaling through a multiplicative contribution. For volume-like $p=d$ and inverse-length $q=1$, support is fraction $1/(d+1)$ of the total. These are constitutive assumptions, not a derived spatial support action. If the reduced preparation $I_jv_j=b_0$ is instead held fixed across cutoffs, equilibrium gives $E_{\rm total}\propto D^{p/(p+q)}$. Common physical energy per mode must therefore be selected jointly with the supporting geometry. A genuinely moving spatial basis also contributes a canonical dilation connection, which a spatial completion must retain.

The occupation and support mechanisms also admit one explicit combined reduced Hamiltonian. Set $b_j=v_jI_j$, $B=\sum b_j$, $G=\kappa F\sum(b_j-B/D)^2/(2b_{\rm ref})$, with resonances defined using $v_j$. Then

$$H=\frac{P_L^2}{2M_L}+aL^p+L^{-q}(B+G)$$

conserves $B$, $G$ and total modal action. Uniform $b_j=b_0$ stays uniform, so the physical wave energies $b_0L^{-q}$ remain equal during breathing. For $K=B+G>0$, conservation gives $(K/H)^{1/q}\le L\le(H/a)^{1/p}$, while the positive mismatch bounds modal energy deviations. This proves conditional persistence in the combined finite-dimensional action. It still requires the spatial dilation connection, common preparation and particle-clock bridges described above; those do not follow from the reduced model's conservation laws.

What Has Become More Concrete

The IDOS interpretation has a conservative persistence construction, an exact spatial trace profile and an explicit support-energy condition. These results do not yet form one derived local particle action. They make the remaining target precise: select the common complete energy per counted state, the occupied cutoff and the actual particle clock under one same-field law. Neither counting alone nor converting total energy to equivalent frequency replaces that last dynamical equation. The same tests cover depths two, three, four, five, six and ten without a special depth-five coefficient.

Reproducible Checks

The ball-density checker passes 140 conditions; the equal-energy checker passes 143; the coherence checker passes 480; and the combined support checker passes 109. The support-count comparison adds 42 exact checks across all fourteen assignments. The displayed derivations establish the general claims; finite checks verify their implementation and boundary controls.

Nested Overlaps and Actual Mixing Dynamics

For homogeneous nested real bands with dimensions $D_1<D_2<D_3$, normalized kernel states at one common anchor have overlap $\langle k_i,k_j\rangle=\sqrt{D_i/D_j}$ for $i<j$. This supplies a deterministic field realization of a count-ratio overlap. It does not require random coefficients. At different anchors the numerator is the smaller band's off-diagonal kernel, so geometry remains present.

Put $a=\sqrt{D_1/D_2}$ and $b=\sqrt{D_2/D_3}$. Their Gram matrix is $G=\left(\begin{smallmatrix}1&a&ab\\a&1&b\\ab&b&1\end{smallmatrix}\right)$, with determinant $(1-a^2)(1-b^2)$. It is not a unitary mixing matrix. Kinetics in this nonorthogonal basis retains $G$: linear frequencies solve $\det(K-\omega G)=0$. Distinct eigenfrequencies of one self-adjoint linear operator cannot have these nonorthogonal eigenvectors. Nonlinear state-dependent patterns remain a separate possible realization.

The interaction operator matters as well. An integrated identity vertex gives the overlap above, while a point contact at the common anchor gives $\sqrt{D_iD_j}$. Neither expression alone derives a transition amplitude. Opposite-parity sphere bands have zero identity contact, but on normalized $S^d$ the states $u_0=1$, $u_1=\sqrt{d+1}x_1$ satisfy $\langle u_0,x_1u_1\rangle=1/\sqrt{d+1}$. Thus a parity-changing spatial coefficient can connect them; an exclusion requires the actual contact's parity law. Such a coefficient must itself follow from the same field.

An explicit three-angle unitary completion can use the fitted count fractions $x=1/20$, $y=3/1771$, $z=1/81030$ as squared angle sines. This is an added interpretation of the fits. It gives $|V_{ts}|^2=(1-x)y+x(1-y)z+2\sqrt{x(1-x)y(1-y)z}\cos\delta$, not an exact equality with $|V_{cb}|^2=y(1-z)$. Full unitarity therefore does not imply the two-angle approximation as an exact identity. Actual detuned Hamiltonian exchange additionally depends on frequency difference and encounter time. The nested-contact calculation passes 88 checks. No empirical mixing mechanism is claimed.

A Coherent Carrier and Its Actual Count Clock

The combined inverse target is stronger than a mode-count identity: the occupied bands must nest, their actual coherent field must have the density required by the clock, and the same action must supply the carrier rate and reciprocal response. The following explicit constructions meet parts of this target together. Their finite cell, polarization selection and occupations remain specified inputs, not a derivation of particle confinement.

On a periodic comparison cell in ordinary flat space, define $D_j=-J\partial_j$, where $J^2=-1$ acts on real field-component pairs. These commuting self-adjoint operators have integer joint Fourier labels. For $(z)_a=z(z-1)\cdots(z-a+1)$, put

$$A_a=\prod_j(D_j)_{a_j},\qquad Q_N=\sum_{|a|=N+1}\binom{N+1}{a}A_a^*A_a.$$

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