Phase-Complete Coefficient Rotation and Polarization Positivity

This construction retains the complete phase of one normal-spinor-valued coefficient field. It is a positive second-order wave model on field values, not a Lorentz Dirac kinetic law. Its fixed norm, selected normal generator and restoring potential are constitutive hypotheses. It tests localization and polarization without claiming observed fermionic spin, statistics or the simplex frequency law.

Field and Common Action

Let the even normal rank be 2q and use the full positive-chiral normal Spin carrier S_plus=Lambda_even(Cq), with its half-determinant Spin shift. Its complex dimension is 2^(q−1) for q positive. Write the coefficient field as Psi=R0 U with U dagger U=1 and unknown constant R0. Fixing the norm is an added target constraint, not an amplitude measurement. The constant R0 enters the kinetic coefficients and is not fitted independently for individual particles.

The ordinary normal generator used in the orientation calculation acts as i(N1+N2−1)/2 on this carrier. Combine it with the common phase generator i/2 to define iG=i(N1+N2)/2. This leaves the vacuum coefficient stationary and gives Hermitian G eigenvalues 0, 1/2 and 1. The phase combination is an explicit additional choice; it is not the unchanged Spin generator or a derived electric charge.

Define z=U dagger G squared U, so 0<=z<=1, and

$$\gamma_{\mu\nu}=\eta_{\mu\nu}-16\lambda R_0^2\operatorname{Re}(\partial_\mu U^\dagger\partial_\nu U), \quad \mathcal L=T(1-\sqrt{\det\gamma/\det\eta})-\frac\kappa2(m^2z-bz^2), \quad \beta=16\lambda R_0^2,\quad \kappa=T\beta.$$

Take T,lambda,R0,m squared positive and 0<b<m squared. The potential is smooth and nonnegative everywhere on the full coefficient sphere. The complete determinant cone and positive energy proof applies to its full induced Riemannian metric. Target coordinates still describe field values, not additional physical spatial dimensions.

The full coefficient sphere includes the phase omitted by a projective ray. At q3 its seven real directions are exactly the six orientation directions plus one phase. At q4 its fifteen real directions also include the two nonpure ray directions; purity is not imposed in this comparison. At q2 it is a three-dimensional coefficient target. These dimensions are not physical spatial depths.

Radial Branch and Changed Equations

Choose U=(cos F(r),sin F(r) exp(i omega t),0,...), where the first coefficient has G weight zero and the second has weight one. This real three-coordinate sphere is a totally geodesic subtarget. The potential's transverse first derivatives vanish there. Its induced metric is dF squared+sin squared F d alpha squared. Consequently A=1+beta F-prime squared, B=1−beta omega squared sin squared F and the complete radial equations, energy and stress have the same functional form as the orientation calculation, now with beta=16 lambda R0 squared.

This is a change of field and potential, not an inheritance of the old J-only solution. The projected root orientation angle is 2F, and its metric alone would have angular coefficient sin squared F cos squared F. The retained phase contributes the additional sin to the fourth F. The chosen full-coefficient potential uses the full Killing norm z=sin squared F; the old projective Killing variance is sin squared(2F). They agree in their small-field mass after their canonical factors are retained, but differ nonlinearly.

The full U tends to the constant vacuum at spatial infinity. Both its phase kinetic energy and spatial gradient energy are finite for an exponentially decaying F. Its first coefficient is static, so its separate phase current vanishes; the rotating second coefficient carries the conserved G charge. This supplies a complete coefficient-field ansatz with no missing projective phase energy. The physical amplitude remains constrained, and a finite-stiffness amplitude completion would require another equation.

Every Occupied Carrier

The exact even-grade weights of G give:

Depth ten has only the trivial coefficient phase and no active normal pair; this does not solve its charged-lepton branch. The coefficient-sphere vacuum is the unit sphere in the weight-zero subspace. It is a circle at q2/q3 and a three-sphere at q4. Thus retaining the full phase removes the earlier claim that q3 has no gapless field direction: it has a vacuum phase. This is a genuine change in the complete field content, while the orientation-only Hessian remains correct within its earlier scope.

Transverse Quadratic Form

Let an unused complex coefficient have weight a in [0,1]. In a frame rotating with G, write

$$U=e^{i\omega Gt}\left(\sqrt{1-|\zeta|^2}(\cos F,\sin F),\zeta,0,\ldots\right).$$

The complete coefficient metric, rather than the projective metric, has second variation

$$\delta g_{tt}=|\partial_t\zeta+i a\omega\zeta|^2-\omega^2\sin^2F|\zeta|^2, \quad \delta g_{rr}=|\partial_r\zeta|^2-F'^2|\zeta|^2, \quad \delta z=(a^2-\sin^2F)|\zeta|^2.$$

Let C0=sqrt(A/B), Cr=sqrt(B/A), Ct=sqrt(A B), y=sin squared F and K=m squared−2b y−C0 omega squared. After dividing by the common positive factor kappa/2, the rotating-frame quadratic energy has positive time term C0|zeta_t| squared and spatial form

$$\mathfrak q_a[\zeta]=\int\left(C_r|\partial_r\zeta|^2+C_t|\nabla_{\rm angular}\zeta|^2+W_a|\zeta|^2\right)d^dx, \qquad W_a=(a^2-y)K-C_rF'^2.$$

The time equation also contains the gyroscopic term from i a omega. Positivity of the complete quadratic energy is a sufficient energetic control of that linear system. This claim requires checking the full determinant expansion and the complete equations, not only the potential.

A Common Positivity Identity

For a smooth localized root branch with 0<F<pi/2 and F decreasing to zero, put h0=cos F and h1=sin F. The full radial equation gives L0 h0=0 and L1 h1=0, where La=−div(Cr radial derivative)−Ct angular Laplacian+Wa. Both functions are strictly positive at finite radius. The corresponding ground-state transformations, initially for compactly supported smooth perturbations, are

$$\mathfrak q_0[\zeta]=\int C_r h_0^2|\partial_r(\zeta/h_0)|^2+C_t|\nabla_{\rm angular}\zeta|^2\,d^dx\ge0,$$

and the same formula with subscripts one. Boundary terms must vanish on the stated domain. The h1 zero mode is localized; h0 approaches one and its global symmetry variation is not normalizable on infinite space.

Since Wa=(1−a squared)W0+a squared W1, the complete spatial forms obey

$$\mathfrak q_a=(1-a^2)\mathfrak q_0+a^2\mathfrak q_1\ge0\qquad(0\le a\le1).$$

This controls every unused coefficient polarization, including the q4 nonpure directions, without needing a separate eigenvalue scan for each one. It does not by itself control the root radial amplitude, whose fixed-charge index criterion must still be checked. The two active phase coordinates have nonnegative spatial gradient forms; their kinetic and gyroscopic mixing must remain in the full constrained-energy calculation.

For clarity, the stationary first-coefficient phase has radial coefficient kappa Cr cos squared F and angular coefficient kappa Ct cos squared F. The rotating second-coefficient phase has radial coefficient kappa sin squared F/sqrt(A B) and angular coefficient kappa C0 sin squared F. All are nonnegative. The second phase mixes gyroscopically with the root amplitude, which is why the conserved-charge reduction and radial index are still necessary. At q4 the additional weight-one coefficient gives a normalizable symmetry zero mode sin F; its extra weight-zero coefficient gives a massless vacuum direction. Neither is a negative quadratic-energy direction or a selected observed polarization.

The exact checker independently verifies 36 conditions: the actual Spin shift, all six occupied carriers, complete coefficient metric jets and potential variation, the full determinant quadratic expansion, complete transverse energy and both positive comparison solutions. The proof concerns the indicated quadratic-form domain; it does not prove nonlinear global stability.

Changed Radial Branches and One Fixed-Charge Failure

The complete-field profiles are recomputed with beta=1, T=4, kappa=4, m squared=1 and b=1/4. These retain the earlier effective lambda R0 squared=1/16 and T, while changing the field and nonlinear potential as stated. Every coefficient is common across ordinary depths; no individual particle amplitude or mass is fitted. The same bounded radial solver uses radius 80/tolerance 1e-8 and radius 110/tolerance 2e-10, independent branch sensitivity and 1100/2200 interval radial spectral meshes.

All ten refined boundary problems converge. Center/energy/charge changes are below 8.5e-10 relative and normalized virial residuals below 1.9e-12. The minimum timelike lapse is approximately 0.0591, so every profile stays strictly in the causal domain. The full energy and charge obey E'=omega Q' in the sensitivity calculation. The reported values describe this finite execution rather than certified numerical bounds.

Each computed unconstrained radial Hessian has one negative direction and positive second eigenvalue. Its translation mode approaches zero under mesh refinement. The fixed-charge rank-one update therefore removes that negative direction in the nine cases with Q'<0, under the tested index hypothesis. It retains a negative constrained-energy direction in the depth-three omega=0.97 case, where Q'=782.360866 and Q'/Q_omega|fixed shape=1.20950362. In contrast, the same depth at omega=0.95 has Q'=−6940.99280 and ratio −6.05508592. The earlier J-only action passed both of these sampled frequencies. This difference is a result of the changed complete action, not a numerical reinterpretation of the old orientation solution.

Five of these ten cases also exceed the free charged-wave threshold E/Q=m=1. Favorable local quadratic tests therefore cannot establish global binding of those states. The other five pass this particular threshold, which remains insufficient to exclude every competing finite-amplitude state or fission process. No lifetime or observed decay channel is calculated.

The profiles have finite complete coefficient energy, conserved charge and supporting stress. Their spherically integrated stresses satisfy the von Laue identity and give the complete ordinary-sector boost response E(v)=gamma E(0), P(v)=gamma v E(0). Motion through unused containing directions remains an additional problem. The unit norm remains an imposed constraint; its finite-stiffness amplitude replacement must be solved rather than assumed.

Reproduction and Evidential Scope

The exact checker and exact report contain the 36 conditions above. The profile driver and profile report contain ten recomputed branches and the empty depth-ten control. The driver imports the audited radial solver, retaining both executed source hashes. Its energy, charge and Hessian normalizations are rederived for kappa=4; the inherited unit-kappa fixed-shape susceptibility is restored explicitly. Running each script creates a dated report alongside its executed source.

This positive wave comparison demonstrates a way to include the full phase and every available coefficient polarization without an additional independent source field. It is not a Lorentz Dirac model, a derivation of fermionic spin or statistics, physical electric charge, strong-force selection, the depth-ten lepton, or a discrete mass-frequency spectrum. Its fixed norm and normal/phase generator remain explicit physical premises. It supplies a constructive full-coefficient test and a useful analytic stability tool shared across sectors; it does not add a new preference for the observed dimensional assignments.