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75. Three-body relative coordinates: an effective domain, not a new particle depth

75.1 The observed-coordinate kinematics

For three positions in a common observed R3\mathbb R^3, an orthogonal Jacobi transformation separates the center coordinate and two vectors ρ,λ∈R3\rho,\lambda\in\mathbb R^3. Their relative configuration space is R6\mathbb R^6.

With the Euclidean metric and R2=∣ρ∣2+∣λ∣2, R^2=|\rho|^2+|\lambda|^2, a fixed-RR surface is S5S^5. Mass-weighted Jacobi coordinates can be used when the kinetic metric requires them. These statements concern three-body configuration coordinates, not six spatial directions available to a single particle.

75.2 Native depth is not configuration-space dimension

V2 assigns the constituent fields to their native manifolds QdQ_d. Observing three positions in a common three-space does not prove that the native higher-depth coordinates disappear from the dynamics.

Consequently neither the six Jacobi coordinates nor their S5S^5 angular surface can be identified with the electron's depth-six manifold or the neutrino's depth-five sector merely by matching dimensions or manifold names. Such an identification needs an explicit map of operators, metrics, bundles, measures, and constraints.

75.3 Degeneracy is not automatically a frequency

For scalar harmonics on the unit round S5S^5, −ΔS5Yℓ=ℓ(ℓ+4)Yℓ,hℓ=(ℓ+55)−(ℓ+35). -\Delta_{S^5}Y_{\ell}=\ell(\ell+4)Y_{\ell}, \qquad h_\ell=\binom{\ell+5}{5}-\binom{\ell+3}{5}. The Laplacian eigenvalue and its degeneracy are different quantities. Neither is automatically the spectrum of the composite Dirac/contact frequency generator. The previous degeneracy scan therefore did not test a specified V2 baryon operator.

75.4 A conditional valence-space representation

After deriving an appropriate observed-coordinate reduction, a three-quark valence model could use a physical subspace of L2(Rρ,λ6)⊗Λ3Ucolor⊗Hspin/flavor, L^2(\mathbb R^6_{\rho,\lambda}) \otimes\Lambda^3U_{\rm color} \otimes\mathcal H_{\rm spin/flavor}, with the total fermionic permutation constraint enforced. This is a possible effective representation, not a derivation of the complete baryon Hilbert space. Native coordinates and other dynamical channels must either be retained or eliminated with their induced spectral terms.

75.5 Predict the whole baryon pole

The primary target is the complete rest-frequency ratio zB=ωB/ωe. \boxed{z_B=\omega_B/\omega_e.} The earlier approximately 309309--310310 MeV quantity was a per-constituent allocation of a residual after subtracting chosen quark contributions. It is not, without an additional decomposition theorem, a separate measured eigenfrequency that the operator must contain.

75.6 Corrected status

The Jacobi construction provides useful observed three-body kinematics. It does not establish a new particle sector, a baryon-to-neutrino identification, or the elimination of native coordinates. Passive projection alone does not generate a larger frequency. A physical frequency shift requires the interacting generator or a specified change of dynamics; exact elimination represents that interaction rather than replacing it with dimension counting.

76. Exact frequency-domain reduction and the color-singlet pair coefficient

This section records the constructive result of the revised audit. The block identities are exact for the finite matrices checked here. Applying them to the full master operator additionally requires specified domains, a stationary background where needed, and a controlled treatment of its infinite-dimensional channels.

76.1 A frequency equation with no fitted field amplitude

Let H\mathsf H be a dimensionless stationary time-evolution generator, defined using the electron reference and the common phase variable τ=ωet\tau=\omega_e t: i∂τΨ=HΨ. i\partial_\tau\Psi=\mathsf H\Psi. An eigenvalue zz is then a frequency ratio ω/ωe\omega/\omega_e. A covariant spacetime Dirac operator is not automatically this generator; its time direction and stationary reduction must first be specified.

For an orthogonal model-space projector PBP_B and QB=I−PBQ_B=I-P_B, define Heff(z)=PBHPB+PBHQB(z−QBHQB)−1QBHPB. \mathsf H_{\rm eff}(z) =P_B\mathsf H P_B +P_B\mathsf H Q_B (z-Q_B\mathsf H Q_B)^{-1} Q_B\mathsf H P_B. The inverse acts on Ran⁡QB\operatorname{Ran}Q_B. Where it exists, the projected resolvent is PB(z−H)−1PB=[zIPB−Heff(z)]−1. P_B(z-\mathsf H)^{-1}P_B =\left[zI_{P_B}-\mathsf H_{\rm eff}(z)\right]^{-1}. Thus the bound-state calculation seeks ker⁡ ⁣[zBIPB−Heff(zB)]≠{0}. \boxed{\ker\!\left[z_BI_{P_B}-\mathsf H_{\rm eff}(z_B)\right]\ne\{0\}.} A determinant equation is available in a finite representation. At excluded complementary eigenvalues, the original system must be checked or the model space enlarged; denominator clearing alone is insufficient.

This is standard Feshbach-Schur reduction, not a newly derived V2 interaction. See Dusson, Sigal, and Stamm, The Feshbach-Schur map and perturbation theory, Theorem 1.2 and equations (1.15)--(1.16). The independently checked two-channel example in Section 74 fixes the sign convention used here.

76.2 Derive the color factor from the specified generators

The V2 gauge-block proposal uses fundamental generators Ta=λa/2T^a=\lambda^a/2. With tr⁡(TaTb)=δab/2\operatorname{tr}(T^aT^b)=\delta^{ab}/2, define Cij=∑a=18TiaTja=12(Xij−13I), C_{ij}=\sum_{a=1}^8T_i^aT_j^a =\frac12\left(\mathsf X_{ij}-\frac13I\right), where Xij\mathsf X_{ij} exchanges the two color slots. The generator normalization and completeness identity are standard; see Haber, Useful relations among the generators in the defining and adjoint representations of SU(N).

On the determinant singlet, Xij=−I\mathsf X_{ij}=-I. Therefore PϵCijPϵ=−23Pϵ,Pϵ∑i<jCijPϵ=−2Pϵ. \boxed{P_\epsilon C_{ij}P_\epsilon=-\frac23P_\epsilon,} \qquad \boxed{P_\epsilon\sum_{i<j}C_{ij}P_\epsilon=-2P_\epsilon.} The sum contains three unordered pairs. These are operator eigenvalues from representation theory, not field amplitudes and not an inserted hadron-mass fit.

76.3 What this fixes in an actual pair interaction

If the derived dimensionless interaction has the form Vpair(z)=∑i<jCij⊗Kij(z), \mathsf V_{\rm pair}(z)=\sum_{i<j}C_{ij}\otimes K_{ij}(z), with Kij(z)K_{ij}(z) acting on the remaining factors, its singlet restriction is Vpair,B(z)=−23∑i<jKij(z). \boxed{\mathsf V_{{\rm pair},B}(z)=-\frac23\sum_{i<j}K_{ij}(z).} It reduces to −2K(z)-2K(z) only when the three remaining pair operators actually coincide. It does not supply a universal frequency ratio of two or 2/32/3. The sign is retained; whether this is an attractive contribution depends on the remaining operator convention.

This form is a conditional template for matching to the V2 master interaction. The audit has derived its color coefficient, not the existence, radial dependence, spin dependence, or normalization of KijK_{ij}.

76.4 What has and has not been solved

The static/spectral Schur distinction, the projection-versus-frequency error, and the singlet pair coefficient have been resolved exactly. A normalized eigenvector may be used to represent a mode, but no measured field amplitude is used as an input.

Electron anchoring fixes the unit. It does not, by itself, determine every dimensionless interaction coefficient. The next calculation must derive the remaining blocks and their spectral dependence from the V2 dynamics, retaining induced terms rather than guessing an integer near the desired mass. No numerical baryon-frequency prediction is claimed by this revision.

77. Canonical Closure and Exact Normal-Kernel Controls

The canonical action calculation supplies an explicit response freedom: a single analytic change of the reconstructed count constitutive function leaves every maintained single-shell mass clock and stationary field unchanged, while the restricted internal shape frequency obeys Ωt2=Ω02/(1+t)\Omega_t^2=\Omega_0^2/(1+t). It does not identify that disturbance with the common hadronic clock. A separate two-mode candidate derives phase locking and its energy accounting with the missing spatial matrix elements explicit.

The normal-response calculation supplies an exact finite-width Hartree solution with curvature/depth 2/3\sqrt{2/3} and frozen bound spacing/depth 1/21/2. A local-power action attains curvature ratio two but has only one frozen bound level. The sech-power classification extends the positive finite-width kernel construction to all exponents p>1p>1 and proves Rcurv≤2R_{\rm curv}\le\sqrt2 in this exact finite-measure convolution family. This is a scoped family result, not an exclusion of other profiles or nonlinear laws.

These explicit actions sharpen the remaining spectral problem. They do not derive the common clock, the reciprocal coefficient or a complete baryon pole, and their differing canonical energy identities cannot be transferred between actions without a common reduction.

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