Real Quartic Spatial Branches and Their Support

One real canonical quartic law admits stationary oscillating profiles at every maintained band. Exact interval calculations establish their existence and nonlinear orbital stability inside those fixed finite bands. Mapping the profiles into ordinary ten-dimensional space produces a different, explicitly weighted contact law. That law also has certified branches at all fourteen assignments. These are constructive compatibility results; their common-size frequencies do not reproduce the complete mass law.

Vary the Profile and the Clock Together

Use the real parity band $V_N$ and normalized sphere measure defined in Real Spatial Counts and Canonical Contact. For real canonical fields $q,p\in V_N$, let

$$H=\frac\kappa4\int(q^2+p^2)^2d\sigma,\qquad J=\int(q^2+p^2)d\sigma.$$

Hamilton's equations are $q_t=\kappa P_N[(q^2+p^2)p]$ and $p_t=-\kappa P_N[(q^2+p^2)q]$. Both $H$ and $J$ are conserved. The rotating profile $q=\sqrt J u\cos\Omega t$, $p=-\sqrt J u\sin\Omega t$ solves them exactly iff

$$\|u\|_2=1,\qquad P_Nu^3=Fu,\qquad F=\int u^4d\sigma,\qquad \Omega=\kappa JF.$$

The physical amplitude remains uncalibrated through $J$; it is not used as a separate mass fit. The normalized reproducing kernel generally fails this profile equation. Solving it changes the electron–muon comparison from the kernel value about $196.32$ to $197.09888365$, still below the count ratio $3515/17$. A fourth moment evaluated on an arbitrary trial shape is therefore insufficient to identify a stationary frequency.

For an exact low-degree example, at $d=3,N=2$, $u=(8t^2-1)/\sqrt5$ and $F=21/5$. More generally the axial $N=2$ form $u=(t^2+s)/\sqrt{\int(t^2+s)^2d\sigma}$ follows from the largest real root of

$$(d+3)(d+5)(d+7)s^3+9(d+3)(d+7)s^2+9(3d+11)s+15=0.$$

The resulting branches satisfy the full band equation, not merely its axial restriction: axial symmetry makes all other projected components zero.

Complete Finite-Band Stability Check

Write $B=P_Nu^2P_N$. In the rotating frame the linear equations are

$$\delta q_t=\kappa J(B-F)\delta p,\qquad \delta p_t=-\kappa J(3B-F)\delta q.$$

At fixed $J$, the phase direction has $b=F$, and shape rotations have $b=F/3$. Every other eigenvalue is strictly below $F/3$ for the computed branches. Thus the constrained energy is a strict local maximum modulo phase and shape rotations when $\kappa>0$. Conservation and finite dimensionality give nonlinear orbital stability within the fixed band.

All angular sectors are included by separating angular degree $\ell=0,\ldots,N$ and radial polynomials with Gegenbauer parameter $(d-1)/2+\ell$. The largest small radial matrix has 48 rows; its angular multiplicity is retained exactly, so a small calculation tests the full band even when $D$ is large. No million-particle simulation is involved.

The certificates use 768-bit real ball arithmetic, exact polynomial moments, and a contraction ball of radius $10^{-8}$ in the axial coefficients and $F/D$. Positive-matrix trace bounds certify the remaining angular eigenvalues; where a weighted trace is insufficient, interval $LDL^T$ pivots certify the full small block. All fourteen root and Hessian certificates pass. These are finite-band theorems, not full spatial-field stability assertions.

Ordinary Ten-Dimensional Volume Changes the Contact

Let $X=(x,y)\in\mathbb R^d\times\mathbb R^{10-d}$, use the stereographic $w(x)$ and $\rho_L(x)$ of the companion note, and set

$$\chi_m(y)=(\pi L^2)^{-m/4}e^{-|y|^2/(2L^2)},\qquad \Phi_u(X)=\sqrt{\rho_L(x)}u(w(x))\chi_{10-d}(y).$$

This is an explicit real norm-preserving map into smooth localized fields. The transverse Gaussian and length $L$ are declared profile choices. The physical Hamiltonian $\kappa\int(Q^2+P^2)^2dX/4$, restricted to these profiles, becomes

$$H=\frac{\kappa C_d}{4L^{10}}\int W_d(t)(q^2+p^2)^2d\sigma, \quad W_d(t)=\frac{(1-t)^d+(1+t)^d}{2},\quad C_d=\Gamma((d+1)/2)2^{(d-12)/2}\pi^{-11/2}.$$

Here $t=w_{d+1}$. The even weight is exact because the quartic density of one parity band is antipodally even. It does not replace the actual asymmetric Jacobian outside band integrals. Stationarity now requires

$$P_N(W_du^3)=F_du,\qquad F_d=\int W_du^4d\sigma,\qquad \Omega=\kappa J C_d F_d/L^{10}.$$

Every assigned band again has a certified root and strict constrained maximum modulo phase. Shape rotation is no longer a zero mode of this weighted fixed-center problem. The dimensionless fourth moments are:

With common $J,\kappa,L$, the muon/electron frequency ratio is $208.42426709$, about $0.803\%$ above the count ratio. The tau/electron ratio is $1.0512349\times10^6$, far above the required $3475.12605$. Thus the same-size comparison fails across charged-lepton depths. The conditional lengths needed to restore their count ratios are $L_\mu/L_e=1.00079975$ and $L_\tau/L_e=1.77040620$; these are inverse requirements, not selected sizes. Quark and boson comparisons additionally require their distinct sector frequency units. None of the formulas gives a special depth-five suppression.

Full Spatial Response of Omitted Modes

For a declared spatial projector $\mathcal P$ onto the chosen transported band and $\mathcal Q=1-\mathcal P$, consider the changed full-field law

$$H=\frac M2(\|\mathcal Q Q\|_2^2+\|\mathcal Q P\|_2^2) +\frac g4\int(Q^2+P^2)^2dX.$$

At unit norm a real stationary profile obeys $(M-\Omega)\psi+g\psi^3=M\mathcal P\psi$. For $g>0,\Omega<M$ the pointwise cubic is strictly increasing. Inverting it and imposing $\mathcal P\psi=\Phi_c$, $\|\psi\|_2=1$ solves the full spatial stationary problem through a small coefficient system. There is no omitted-mode cutoff. Both the asymmetric spatial Jacobian and transverse response enter.

If $A=g\int\Phi_u^4$ is the unperturbed frequency and $r=\mathcal Q\Phi_u^3$, the first correction is $\Omega=A-3g^2\|r\|_2^2/M+O(M^{-2})$. A common gap inferred from this approximation, $M/A_e=556080.64378$, gives a directly solved ratio $206.7976760059$, or $+0.0159457\%$ relative to the count target. The gap was not retuned during the full solve. This is a successful check of an inverse correction, not a parameter-free prediction. Independent quadrature refinements agree to $3\times10^{-12}$ relative. Analytic polynomial moments and 4096-bit interval arithmetic now certify exact full spatial electron and muon roots, including their complete pointwise omitted-mode response. The proof inverts the monotone cubic through $S+qS^3=1$, using

$$[q^k]S^p=(-1)^k\frac{p}{3k+p}\binom{3k+p}{k},\qquad p=2,3.$$

Uniform alternating-series tails and Jacobian bounds give contraction constants below $4.45\times10^{-5}$ in radius-$10^{-8}$ coefficient and scaled-frequency balls. Twelve electron terms and twenty-four muon terms suffice. The self-map bounds sharpen the frequency enclosures to $60.92879248087534\pm4.368\times10^{-11}$ and $12599.932686887933\pm8.170\times10^{-8}$ in the declared common units. These are rigorous model enclosures, not empirical uncertainties.

The same frozen gap has also been solved numerically at all fourteen assignments, without retuning norm, length or coupling. The tau branch continues to $\Omega_\tau=6729860.46923<M$, with omitted norm about $0.3008482$. Its electron-relative frequency remains $31.7843$ times the target count ratio. Thus failure of its large-gap approximation did not imply loss of the actual branch, but the full branch does not rescue this common-size mass comparison. All depth-five branches remain regular under the same law, without a special suppression.

For electron and muon, all angular sectors of the full continuum slow linear cluster pass numerical constrained-energy tests, together with a positive high-cluster estimate. The smallest numerical margins are about $1.78574$ and $62.2142$. Quadrature and contour refinements retain these signs. This is numerical continuum linear-stability evidence; the rigorous root theorem does not turn it into an interval or nonlinear stability theorem. The prescribed spatial projector still supplies the band.

Multiplying the exact stationary equation by $\psi$ gives

$$\Omega=M\|\mathcal Q\psi\|_2^2+g\int\psi^4, \qquad H=\frac\Omega4+\frac M4\|\mathcal Q\psi\|_2^2.$$

Hence this completion changes the earlier common energy/frequency ratio $H/\Omega=J/4$. The imposed spatial projector still needs a physical origin and a moving, self-supported realization.

Support Must Be Varied in the Same Action

The exact physical gradient factor of a normalized parity profile is

$$G_d=\int\left[(1+t^2)|\nabla_{S^d}u|^2 +\frac{d(d-2)}4u^2+\frac{d(d+2)}4t^2u^2\right]d\sigma +\frac{10-d}{2}.$$

Thus $\int|\nabla\Phi_u|^2dX=G_d/L^2$. Positive gradient plus positive quartic energy decreases strictly as $L$ grows at fixed $J$. It cannot select a stationary finite size, even after varying the profile.

A constructive changed local law with attractive quartic and positive sextic terms gives

$$E=\frac{aJG}{2L^2}-\frac{bJ^2C_dF_d}{4L^{10}} +\frac{sJ^3T_d}{6L^{20}},\quad a,b,s>0,$$

where $T_d=L^{20}\int\Phi_u^6dX$. Put $A=aG$, $B=5bJC_dF_d/2$, $C=10sJ^2T_d/3$, $x=L^2$. Length stationarity is $Ax^9-Bx^5+C=0$. A strict dilation minimum exists precisely when

$$C<\frac49B\left(\frac{5B}{9A}\right)^{5/4};$$

it is the smaller of the two positive roots. This is a trial-profile dilation result; the full changed-action shape equation still needs to be solved. Its phase rate also changes to $\Omega=aG/L^2-bJC_dF_d/L^{10}+sJ^2T_d/L^{20}$. Finite support cannot be added while silently retaining the quartic-only clock or stability theorem.

Reproduction and Scientific Status

The real-energy-count research collection of 14 October 2026 stores the exact scripts, source hashes and dated reports. The principal certificate is real-quartic-certify.py --label all --hessian, with --flat selecting the physical-volume action. The stationary solvers use installed NumPy and SciPy; the certificate uses python-flint Arb. The full spatial gap comparison uses continuum-gap-solve.py and an independent quadrature cross-check. Separate reviews verify the complete angular decomposition, weighted contraction bounds, normalization and gradient formula.

The positive result is the existence of explicit real nonlinear branches at all fourteen maintained bands, with rigorous finite-band persistence and a direct physical-space map. The open physical tasks are support, band and norm selection, complete translational response, common sector scales, and the observed frequency family under one action. These internal field calculations do not replace deterministic electron center trajectories.

The portable stationary solver, flat-volume solver and interval certificate are stored together with their frozen coefficient inputs. Run from the exported repository root. The certificate reads those named inputs; new solver outputs are dated separately. Uniform-measure certificate and physical-volume certificate record the original successful runs and exact source provenance.