The competing depth-ten branch has two minima of its reference frequency, but those inherited radii have nonzero force in the completed-square action. This calculation varies the finite component eigenvector and the two retained same-field response amplitudes together, then looks for nearby fixed-canonical-norm stationary radii. It tests the assigned harmonic n=23n=23 at depth ten. The action is an inverse construction, not a derivation from the master field.

Action and local continuation

In units R0=c=1R_0=c=1, write Ds=1+R4D_s=1+R^4, Dl=1+C−4R4D_l=1+C^{-4}R^4, and g(R)=γ∑i=s,lwi/Dig(R)=\gamma\sum_{i=s,l}w_i/D_i. The candidate energy per canonical norm is

E=⟨H0(R)⟩+F0(k)+∑i[Di(R)2(qi−JiDi(R))2+βiqik],Ji=rγwi,βi=γwi/r. \mathcal E=\langle H_0(R)\rangle+F_0(k) +\sum_i\left[\frac{D_i(R)}2\left(q_i-\frac{J_i}{D_i(R)}\right)^2+\beta_iq_i k\right], \quad J_i=\sqrt{r\gamma w_i},\quad \beta_i=\sqrt{\gamma w_i/r}.

The qiq_i are retained modes of the same field in this candidate, and r>0r>0 is their source-to-current amplitude ratio. Static elimination gives qi=(Ji−βik)/Diq_i=(J_i-\beta_i k)/D_i and

Feff(k,R)=F0(k)+g(R)k−g(R)2rk2,G=∂kFeff=F0′(k)+g(R)−g(R)rk. F_{\rm eff}(k,R)=F_0(k)+g(R)k-\frac{g(R)}{2r}k^2, \qquad G=\partial_k F_{\rm eff}=F_0'(k)+g(R)-\frac{g(R)}r k.

At each inherited branch current k0k_0, the calculation supplies a local quadratic Taylor germ of F0F_0. Its value and slope reproduce the prior inverse branch. Its curvature is the finite secant estimate from harmonics 22.5 and 23.5, plus g(R0)/rg(R_0)/r; this is an input, not an independently derived current law. At each trial radius the ten-component singular vector supplies kk self-consistently with GG. The frequency includes the complete eliminated-mode scalar offset, rather than only the lowest singular value divided by radius.

The fixed-norm force after this component and retained-mode response is

∂RE=−ϵ/R−GkR+g′(R)k−g′(R)2rk2. \partial_R\mathcal E=-\frac{\epsilon/R-Gk}{R} +g'(R)k-\frac{g'(R)}{2r}k^2.

Solving this equation near each inherited radius gives the following local results. Frequencies and curvatures use the common c/R0c/R_0 units and the stated canonical norm.

rr Branch Shifted R/R0R/R_0 Change from inherited radius Particle frequency Radial curvature
0.01 Inner 1.05638450945 −1.01482% 5.47132994×10−55.47132994\times10^{-5} 3.32019460×10−43.32019460\times10^{-4}
0.01 Outer 14.17892438901 −0.00879661% 2.64995665×10−122.64995665\times10^{-12} 3.80719019×10−133.80719019\times10^{-13}
1 Inner 1.05672321555 −0.983084% 5.47145140×10−55.47145140\times10^{-5} 3.30340813×10−43.30340813\times10^{-4}
1 Outer 14.17892438915 −0.00879661% 2.64995665×10−122.64995665\times10^{-12} 3.80719019×10−133.80719019\times10^{-13}

The force residuals are below 10−5510^{-55} in the high-precision solve. Centered force derivatives at relative radius steps 10−4,10−5,10−610^{-4},10^{-5},10^{-6} agree within about 1.4×10−71.4\times10^{-7} relative. Positive curvature establishes a local minimum only along this self-consistent radius and finite-component path.

Component and spatial controls

At the shifted radii, F0′′−g/rF_0''-g/r remains positive: approximately 3.86×1043.86\times10^4 and 4.16×1094.16\times10^9 for inner and outer branches at r=0.01r=0.01, with similarly positive values at r=1r=1. The nearest other signed finite-component frequency magnitude, divided by the occupied frequency magnitude, exceeds one: about 1.0236 inner and 1.000928 outer. This is a fixed-geometry component-separation diagnostic, not a dynamical spatial-stability theorem.

For a restricted spatial test, suppose the retained same-field amplitudes have a nonnegative gradient stiffness κ\kappa. On a round radius-RR nine-sphere of angular directions in ten ordinary spatial dimensions, a scalar harmonic of degree ℓ\ell has λℓ=ℓ(ℓ+8)/R2\lambda_\ell=\ell(\ell+8)/R^2. The scalar-current Schur curvature in this stipulated extension is

Cℓ=F0′′−γr∑i=s,lwiDi+κλℓ. \mathcal C_\ell=F_0''-\frac{\gamma}{r} \sum_{i=s,l}\frac{w_i}{D_i+\kappa\lambda_\ell}.

For κ≥0\kappa\ge0, Cℓ≥C0=F0′′−g/r>0\mathcal C_\ell\ge\mathcal C_0=F_0''-g/r>0, so this particular scalar-current direction passes for every angular degree under the assumed positive gradient law. Direct evaluations at ℓ=0,1,2,4,8\ell=0,1,2,4,8 and κ=0,1\kappa=0,1 confirm the formula at both shifted radii and both tested rr. The gradient law and round angular metric are model inputs; no ten-dimensional physical spinor or polarization carrier has been supplied for this completed-square action. Its full field, angular, metric, and nonlinear stability therefore remain unknown.

Adding the two shifted current points to the earlier 15 sampled knots preserves the necessary convex Hermite secant inequalities for a radius-independent F0F_0 at each tested rr. The smallest numerical margin is about 1.21×10−71.21\times10^{-7}. These are finite necessary compatibility checks. The inner and outer Taylor germs were specified separately, and no single smooth global F0F_0 has been constructed or shown to produce both states. In particular, the stronger requirement of a shared smooth exact inverse law down to zero current has a different, highly constrained rr, which this continuation did not solve.

The executed script and selected report give the self-consistent roots, force residuals, curvature step checks, finite-component separations, convexity controls, and spatial scalar-current values. A complete test needs one global direct current law and a spatial carrier for this same action, including polarized field and geometry variations. Until then, the outer branch is a supported local possibility in a restricted action, not an established particle or derived physical mass operator.