A Geometric Contact Envelope with Exact Nonlinear Spatial Persistence

The forward construction can retain the successful open quadratic charts without choosing a smooth cutoff function or patch width. Use the convex envelope of the geometric count tangents themselves. Every maintained assignment lies in an open region where one explicit quadratic term wins. In the preferred-time realization, the complete field equation is exactly linear throughout that region, giving the earlier nonlinear persistence argument a simpler constitutive action.

The count interaction remains a specified effective law. The theorem establishes its mathematical compatibility with the spatial fields and persistence; it does not independently derive that law from Psi_infinity.

1. Count Tangents and a Convexity Criterion

Retain the ordinary sphere geometry, canonical matrix-spinor kinetic term and definitions from the forward action:

$$h_l=l+d/2,\quad S_l=\binom{l+d}{d},\quad P_d(h)=\frac1{d!}\prod_{j=1}^d(h+j-d/2),\quad \epsilon_d=\mu_dR_d.$$

On a continuous chart set c=epsilon P_d'(h), x=sqrt(1-c squared) and f(x)=epsilon P_d(h)-h c. Its derivatives are

$$f'(x)=\frac{hx}{c},\qquad f''(x)=\frac h{c^3}-\frac{x^2}{c^2\epsilon P_d''(h)}.$$

Thus f is strictly convex precisely when

$$h\epsilon P_d''(h)>c(1-c^2). \tag{1}$$

For every d>=3 this holds throughout h>=d/2 and 0<c<1. At h=d/2, P_d'=H_d and P_d''=H_d squared-H_d^(2). For d>=3, the latter is at least its d=3 value, two. Therefore

$$\left.hP_d''-P_d'\right|_{h=d/2}\ge d-H_d>0.$$

Also (hP_d''-P_d')'=hP_d'''>0 on that interval, since all product factors are positive. This proves the full convexity claim without a finite harmonic scan.

Depth two has a genuine different boundary. There P_2(h)=h(h+1)/2, c=epsilon(h+1/2), and

$$f''(x)=\frac1\epsilon-\frac1{2c^3}>0 \quad\Longleftrightarrow\quad c^3>\epsilon/2. \tag{2}$$

Use this contiguous convex chart at depth two. All three maintained massive depth-two assignments lie in it under the shared comparison scale below. This is a declared convexity condition, not an observed particle-selection mechanism or a photon construction.

2. One Explicit Maximum of Quadratic Terms

Let L_d contain every nonnegative integer l in the chosen convex chart with 0<c_l=epsilon P_d'(h_l)<1. This is finite at fixed epsilon>0. Define

$$x_l=\sqrt{1-c_l^2},\qquad m_l=\frac{h_lx_l}{R_dc_l},\qquad \eta_l=\mu_dS_l-\frac{h_l}{R_dc_l}.$$

The potential is

$$\boxed{V_d(\Phi)=\max_{l\in L_d} \left[m_l s+\eta_l\rho\right],\qquad s=\operatorname{tr}\Phi^\dagger\beta\Phi,\quad \rho=\operatorname{tr}\Phi^\dagger\Phi.} \tag{3}$$

Every coefficient follows from the same count and sector scale recipe; none is fitted separately to a particle amplitude. The scalar m_l is a source coefficient, distinct from the observed mass-frequency.

For rho>0, the maximum in R V/rho is the maximum of the tangent lines

$$\ell_l(x)=f(x_l)+f'(x_l)(x-x_l),\qquad x=s/\rho.$$

Strict convexity gives ell_k(x_l)<f(x_l)=ell_l(x_l) for k!=l. Thus l is the unique winner at its intended scalar ratio. There is a strict positive gap to every other line and an open winner interval. This proves that the coefficient charts can coexist in one local action; they have not been independently prescribed as incompatible pieces.

Potential (3) is defined at Phi=0 by zero. At tied nonzero charts it is continuous but generally not differentiable. The persistence theorem below constructs solutions staying inside a unique smooth chart and therefore does not need to assume a global evolution through ties.

3. Exact Spatial Clocks and Nonlinear Persistence

Use the same balanced positive shell E_l and its opposite-sign partner beta E_l. The stationary field is

$$\Phi_l=e^{-i\omega_lt}(a_lI+b_l\beta)E_l,\qquad a_l^2=(1+c_l)/2,\quad b_l^2=(1-c_l)/2.$$

Its pointwise ratio is s/rho=x_l, so chart l wins everywhere. The complete source is exactly (m_l beta+eta_l I)Phi, and

$$\boxed{\omega_l=\mu_dS_l,\qquad E_l=\omega_lQ.}$$

The spatial Dirac energy and the full interaction energy are retained. The canonical normalization Q stays free. These are internal classical field solutions and do not replace deterministic six-dimensional electron trajectories.

For any perturbation staying inside that winner chart, the co-rotating equation is exactly

$$i\partial_t\delta\Phi=\mathcal A_l\delta\Phi,\qquad \mathcal A_l=D_R+\frac{h_lx_l}{R_dc_l}\beta -\frac{h_l}{R_dc_l}I. \tag{4}$$

This is self-adjoint on the complete ordinary sphere Dirac domain. Its full spectrum is

$$\frac1{R_d}\left[\pm\sqrt{(m+d/2)^2+(h_lx_l/c_l)^2} -h_l/c_l\right],\qquad m=0,1,2,\ldots.$$

The stationary zonal field has a strictly positive density minimum: its scalar Gegenbauer polynomial and derivative cannot vanish together, and the massive mixing matrix is invertible for c_l>0. Choose an integer Sobolev index s0>d/2. The graph norm built from powers of A_l is equivalent to H^(s0) and is conserved by (4). Sobolev embedding keeps a sufficiently small disturbance uniformly small for all time. The density minimum then keeps its scalar ratio uniformly close to x_l, inside the positive winner gap. This closes the argument: the exact linear solution never leaves the chart, and hence solves the exact nonlinear envelope action for all time. It gives conditional nonlinear Lyapunov persistence on this carrier.

This uses the same analytical estimates as the earlier affine-patch proof. The new contribution is the globally compatible geometric envelope and its automatic open charts, which remove arbitrary cutoff shape and width. It does not establish localized support in an unbounded sector.

4. The Common Comparison Across All Occupied Depths

Use epsilon_d=3/[10 max_l P_d'(h_l)], with the maximum over maintained assignments in that sector. This is a shared diagnostic geometry scale, not a calibrated radius or fitted physical amplitude.

All fourteen maintained massive assignments are included. The scalar-ratio margin to a chart boundary is positive for each. It is about 1.1e-9 for the up comparison, 1.8e-6 for the electron, 5.9e-3 for the muon and 1.3e-2 for the tau. These margins quantify this diagnostic's neighborhood; they are not measured scattering tolerances or lifetime predictions.

At depth two the W convexity condition and Higgs chart endpoint give the nonempty sufficient scale interval

$$\frac{2}{153\sqrt{153}}<\epsilon_2<\frac2{191}.$$

The displayed comparison scale belongs to that interval. It is a conditional radius-window constraint under the assumed common sector geometry, rather than a measured length prediction.

5. Covariant Density Choices Retain Their Current Dynamics

Replacing rho by the proper current norm n produces a different, Lorentz-scalar envelope. It has the same balanced stationary fields but also the proper-current source derivatives. The preceding unitary and nonlinear persistence results do not transfer to it.

Alternatively, replace rho by q=u dot j, with u supplied by a retained timelike matrix block B of Psi_infinity. On a unique winner chart the potential is exactly m_l s+eta_l q. Both full equations and the complete stationary ledger are therefore those of the responding-clock construction, including the B source and E=omega_l Q_Phi+Omega Q_B. Its existing ground linear orbital theorem retains its stated load and preparation. There is no new nonlinear covariant or assigned excited-shell persistence theorem.

The original matrix carrier remains explicit. In particular, a single minimal commuting spinor at depths two and four obeys j_mu j^mu=s squared, so the raw inverse action's k invariant vanishes. Matrix columns there are a substantive field-component premise. Their observed spin, charge and reduction from the single master field remain separate questions.

Verification and Evidential Scope

A positive wave completion now retains the exact count fields and has nonnegative complete energy. One common compact convolution preserves every eligible count node, while an exact all-spectrum nonresonance test permits a separate nonlinear persistence theorem for nodeless fields at all maintained assignments. This changes the kinetic law; the first-order contact attraction is not transferred.

The actual two-branch beam contact has an exact positive layer-overlap decomposition and maximizes attraction at alignment in the stated orthogonal-column preparation. Its aligned electron/muon control has negative complete energy. The local persistence theorem remains valid, while this strong-overlap result constrains the action as a global physical encounter model.

The closed-circle construction now gives exact count fields with calculable tube concentration. A small zonal direct sum makes them globally nodeless at every occupied depth; generic coherent real additions also prove one-column nodeless existence at five and six. The positive density floor permits the preferred-time nonlinear theorem above. This remains a construction on the assumed sphere, rather than a containing-space mobile particle or a transferred covariant stability theorem.

An exact polynomial continuation now proves closure of a smaller eight-dimensional family at depths two, three, four, six and ten. Its depth-ten member is axial and remains closed despite the larger family leakage. At the stated ground preparation, every real scalar curvature retains growth at six and in this explicit depth-ten family. Depth five needs a larger coupled space. These results constrain the proper-current continuation, not the preferred-time nonlinear theorem or assigned excited particles.

A direct proper-current spatial comparison now keeps the affine chart current Hessian on both carriers. The stated full-matrix ground controls grow at every occupied depth in an exact reducing family; the minimum-carrier finite test retains growth at two, four, five and six, while its depth-ten space fails closure. These are ground controls, not the maintained excited assignments or a change to the preferred-time nonlinear theorem.

The reduced-carrier construction retains the same strictly positive spatial density and the same columnwise self-adjoint operator. Consequently this preferred-time nonlinear persistence argument also applies at its minimum column ranks. The responding-background Hessian theorem has different dependencies and is not transferred by that observation.

convex-count-envelope.py and its report pass 257 evaluations over every maintained assignment. At 100-digit precision, the actual maximum over all admissible tangents has the intended unique winner and a nonempty open chart. Finite scalar-ratio displacements and independent matrix-induced invariant variations remain inside it. Independent Hermitian pair diagonalization matches the full spatial spectral formula. The nonlinear persistence conclusion also uses the domain and Sobolev proof above.

This is a concrete simpler effective realization of exact count clocks and conditional nonlinear spatial persistence. Deriving its geometric contact law, physical scales and localized carrier from the master field remains open. The same construction permits other convex target spectra, so it does not independently select the assigned sector depths.