Historical excerpt: IDWT_09_Spectral.md

Imported 2026-09-08. Historical evidence only; claims are not endorsed by V2. This is a local extract, not a link to or dependency on another repository. Original full-file SHA-256: 1ad0faacd1b8afb987825ab877e47f99868a558aacf38d70f9a725291128df62.

Original source lines 138–166

T2. The Kernel Uniqueness Theorem

Scope. T2 establishes uniqueness within the following assumptions: (1) the interaction is polynomial in $(\xi_d, \xi_{d'})$; (2) it is degree-4; (3) it is invariant under $U(d)\times U(d')$; (4) the coupling matrix is required to have rank 1; (5) the $\ell$-decomposition is restricted to $\ell=0\oplus\ell=2$ by the polynomial degree. Higher-degree interactions, non-polynomial couplings, or different invariance groups are not addressed. "Unique" in what follows means unique under these five assumptions.

Theorem T2. Among all $U(d)\times U(d')$-invariant degree-4 polynomials on $\Xi_d\times\Xi_{d'}$, the cross-sector interaction $(\xi_d\cdot\xi_{d'})^2$ is the unique one satisfying all three conditions:

1. Non-trivial sector mixing: cannot be written as a product of sector-local terms.
2. $\ell=0\oplus\ell=2$ decomposition: on $S^{d-1}$, $$(\xi\cdot\xi')^2 = \underbrace{\tfrac{1}{d}|\xi|^2|\xi'|^2}_{\ell=0,\;\rm mass\;scale} + \underbrace{\tfrac{d-1}{d}|\xi|^2|\xi'|^2 P_2(\cos\theta)}_{\ell=2,\;\text{self-energy}},$$ with the $\ell=0$ part generating sector mass scales (T9) and the $\ell=2$ part generating the second-order self-energy of scale $\varepsilon=1/(280\sqrt{7})$ (T10a). This $\varepsilon$ enters the $\nu_3$ closure ($\delta_{\nu_3}=\varepsilon\cdot g_{33}=1/35$, T11d); the up-type masses carry the confinement-binding correction of Part 2 §11.9.
3. Rank-1 factorisation: the coupling matrix $g_{dd'}=v_d v_{d'}$ is rank-one, as required by the bi-invariant structure under $U(d)\times U(d')$.

Proof sketch. The $U(d)\times U(d')$-invariant degree-4 ring on $(\xi_d,\xi_{d'})$ has four generators: $|\xi_d|^4$, $|\xi_{d'}|^4$, $|\xi_d|^2|\xi_{d'}|^2$, and $(\xi_d\cdot\xi_{d'})^2$. The first three either fail condition 1 (sector-local) or condition 2 (diagonal in the $\ell$-basis). The fourth satisfies all three. $\square$

$\ell$-decomposition coefficients on $S^{d-1}$ for every sector:

The $d=6$ row covers the electron and muon — both are $d=6$ $\mathbb{CP}^3$ sector excitations of $\Psi_\infty$, genuine 6D objects inhabiting six macroscopic spatial dimensions, whose activity in the three observable coordinates is what a $d=3$ observer measures.

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