IDOS Energy, Coherence, and Supporting Dynamics — Part 26

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partitions $\lambda$ of degree at most $N_{\max}$ and length at most $d$, including the empty partition. If $m_r$ counts parts of size $r$, with $m_0=d-\operatorname{length}(\lambda)$, the orbit multiplicity is

$$M_\lambda=\frac{d!}{\prod_{r\ge0}m_r!},\qquad \sum_\lambda M_\lambda|c_\lambda|^2=1, \qquad C_\lambda=\sqrt{M_\lambda}\,c_\lambda.$$

The $C_\lambda$ are orthonormal real-pair coordinates. The probability-like appearance of a squared norm is only classical coefficient normalization. These variables do not represent electron positions or additional spatial dimensions. Set $p_\lambda=|C_\lambda|^2/M_\lambda$ and $q=d^{-1}\sum_\lambda M_\lambda|\lambda|p_\lambda$.

For lowering a positive part $r$, the exact edge weight is $M_\lambda m_r$. For moving one unit from a coordinate of value $r>0$ to a distinct coordinate of value $s\ge0$, it is $M_\lambda m_r(m_s-\delta_{rs})$. Receiver zeros and directed multiplicities must be retained. With $\lambda^-_r$ and $\lambda^{r\to s}$ denoting the resulting partitions, the complete reduced potentials are

$$V_{\rm anchor}=\sum_\lambda M_\lambda p_\lambda(p_0-p_\lambda)^2, \quad V_{\rm edge}=\sum_{\lambda,r>0}M_\lambda m_rp_\lambda (p_{\lambda^-_r}-p_\lambda)^2,$$ $$V_{\rm phase}=\sum_{\lambda,r>0}M_\lambda m_rp_\lambda \|c_0*c_\lambda-c_1*c_{\lambda^-_r}\|^2,$$ $$V_{\rm mix}=q^2\sum_{\lambda,r>0,s\ge0} M_\lambda m_r(m_s-\delta_{rs}) (p_\lambda-p_{\lambda^{r\to s}})^2.$$

The real pair product is the same constitutive tensor defined above. The spatial variance potential is $W=I[\sum_\lambda M_\lambda(\sum_{r\in\lambda}r^2)p_\lambda-dq^2]/(8\ell^2)$. The common count mass uses $a=d,N=(d+1)q$ and its existing zero completion. No sector-specific interaction adjustment is introduced.

At fixed charge, $\mathcal P=Q(m+\gamma V)+bH_Q+W$ as before. In a horizontal lift of the normalized real-pair coordinates, with $|x|=1$, $x\cdot v=(Jx)\cdot v=0$, the regular equations are

$$\dot x=v,\qquad I\dot v=-I|v|^2x-\nabla\mathcal P +(x\cdot\nabla\mathcal P)x-2QJv, \qquad \dot\theta=m+\gamma V+b\log(Q/n_0).$$

They preserve those constraints and $E=I|v|^2/2+\mathcal P$. Both constant $b$ and $b=m/128$ remain available and coincide at $Q=n_0$. The common phase is reconstructed; horizontal velocities remain independent initial data. This representation uses a redundant mathematical phase lift, not a new physical gauge postulate.

The number of retained pairs is $P_d(N)=\sum_{n=0}^N p_{\le d}(n)$, with generating function $\prod_{j=1}^d(1-t^j)^{-1}$. It gives electron 227, muon 15,121, tau 3,890 and top 59,301 pairs. The largest actual case has about 0.8 million directed edge and redistribution types. This makes sparse evaluations plausible; it does not establish long-time integration accuracy or justify dense Hessians. The complete fourteen-target table and 11,667 exact assertions in permutation-carrier-reduction/report.md verify original-label potentials, first derivatives, connection, inertia and omitted-coordinate tangency at small cutoffs across all six maintained depths.

This subsystem retains the force difference between partitions $(2)$ and $(1,1)$ that broke two-group equality. Generic nonsymmetric perturbations are outside it. It remains a homogeneous frozen-support control, not a localized particle, a spatial encounter, or a solution of supporting-field backreaction. In particular its variance energy can move a uniform prefix; the isolated spatial branch had different derivative data.

Spatial Phase Curvature And Measured Sparse Initial Response

Two distinct calculations now clarify the carrier's available motion. The spatial centered branch has vanishing residuals; the homogeneous frozen carrier generally has a nonzero variance potential. Their forces cannot be identified with each other.

Actual Spatial Carrier

For fixed orthogonal real pairs, write $e_k=\sqrt{p_k}(\cos\phi_k,\sin\phi_k)$, $\sum p_k=1$. The phase connection and curvature in ordinary physical coordinates are

$$a=\sum_kp_k\,d\phi_k,\qquad F=da=\sum_kdp_k\wedge d\phi_k.$$

On a locally fixed subspace of $r$ pairs, $\operatorname{rank}F\le2(r-1)$. The equal-prefix centered label texture has $a_i=F_{ij}=0$. More generally,

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