Angular shape, finite width and responsive support of the simplex carrier

The prescribed-surface operator supplies a useful frequency mechanism, but its supporting field and complete spatial stability must be solved. Three different tests are relevant: changing the angular shape, giving the surface finite normal thickness, and allowing the supporting scales to respond. They cannot be replaced by a test of one restricted coefficient family.

The exterior-form construction uses a physical spherical feature $S_R^m$, $m=d-2$, within an ordinary spatial sector of depth $d$. Its angular degree is $l=n+1$. The separate spin construction uses $S_R^{d-1}$, with spin level $n$. Both have the same finite link matrices where applicable, but different fields and interaction actions. Neither sphere represents compactified spatial directions.

1. What the standing angular test establishes

At fixed radius and gap, the weak-charge standing branch of the stipulated nonlinear Hodge action has

$$E(Q)=Q\Omega-\frac{gQ^2}{2\kappa^2\mathcal A}\overline W+O(Q^3), \qquad \mathcal A=\operatorname{Area}(S_R^m),\quad g>0.$$

For a real degree-$l$ harmonic with sphere-average norm $\langle f^2\rangle=1$, and the stationary mixture of the two signed Hodge branches held fixed,

$$\overline W=C_0\langle f^4\rangle,\qquad C_0=s^4+\frac{l(l+m-1)}3s^3t>0.$$

Here $s$ and $t$ are the scalar endpoint and first-row response of that normalized signed mixture. The factor $1/3$ follows from $\int f^2|\nabla f|^2=l(l+m-1)\int f^4/3$.

For the real highest-weight harmonic $Y\propto\operatorname{Re}(X_1+iX_2)^l$, let $Z$ be a normalized orthogonal degree-$l$ harmonic and write $A=\langle Y^2Z^2\rangle$, $I_4=\langle Y^4\rangle$. The normalized path $(Y+\epsilon Z)/\sqrt{1+\epsilon^2}$ has leading energy curvature proportional to $1-3A/I_4$. For the azimuthal weight $l-2$, transverse scalar, first-Jacobi sector, beta moments give

$$1-\frac{3A}{I_4} =-\frac{l[(b-1)l+1]}{(2l-1)(b+l-1)},\qquad b=\frac{d-3}{2}.$$

This direction is outside the earlier four-complex-mode family and outside rigid rotations. Thus the positive restricted Hessian remains correct, while it does not extend to an unrestricted minimum. The negative leading coefficient excludes a strict fixed-charge minimum near zero charge at depths five, six and ten, conditional on a smooth continuing branch. It does not prove exponentially growing modes or particle breakup. The full same-degree scalar/exact block scan includes both signed components and their complex phases. Other form grades, other degrees, normal fields and finite-amplitude corrections require separate tests. Depth three has only the cosine/sine pair at a given degree; depth two's old $S^0$ feature has no angular tower.

2. A different standing shape removes this particular obstruction

The zonal harmonic, concentrated about a pole, is $Y_l(x)\propto C_l^{(m-1)/2}(x)$. It has the same linear frequency but different contact integrals. At depth five, $m=3$, the normalized zonal field is the Chebyshev polynomial $U_l(x)$, with average norm one. The product identity

$$U_l^2=\sum_{k=0}^l U_{2k},\qquad I_4=l+1$$

counts the orthogonal even channels in its square. For every transverse sector $j=1,\ldots,l$,

$$\frac{A_j}{I_4}=\frac1{2j+1}.$$

One derivation uses $h_j(\theta)=\sin^{j+1}\theta C_{l-j}^{j+1}(\cos\theta)$. It solves the Dirichlet problem with potential $j(j+1)\csc^2\theta$ and eigenvalue $(l+1)^2$. Differentiating the eigenvalue in $j$, with $l-j$ held fixed, gives $\langle\csc^2\theta\rangle_{h_j}=2(l+1)/(2j+1)$. The oscillatory part of $\sin^2((l+1)\theta)$ is orthogonal to the remaining polynomial, yielding the overlap formula. The $j=1$ sector is a rotation; all $j\ge2$ have positive energy curvature, at least $2/5$ in this normalized same-ray measure.

The completed signed and complex same-degree calculations also have positive nonsymmetry energy curvature for the maintained depth-five, six and ten zonal cases. Depth-four zonal controls have negative directions, so changing the shape is not a universal stability repair. Zonal symmetry and exactness of $f^2df=d(f^3)/3$ also pass the first cubic resonance projection; this is not a finite-amplitude existence theorem.

Allowing a lower angular degree remains a separate obstruction. At frozen support, the fixed-charge quadratic energy gives

$$\Delta E_Q=\frac{Q}{2\Omega_l} (\Omega_{l-1}^2-\Omega_l^2)\epsilon^2 +O(\epsilon^4,Q^2\epsilon^2).$$

In the high-gap limit, $\Omega_{l-1}/\Omega_l\to(l-2)/(l+d-2)<1$. The negative term is order $Q$; the quartic is order $Q^2$. A regular positive-gapped support background changes this conclusion only at order $Q^2$, if it leaves the leading spectral ordering unchanged. A singular zero-charge limit, a derived constraint or a genuinely changed leading operator lies outside that statement. Energetic saddles need not be dynamically unstable in a conservative theory.

3. Finite radial width changes the spin carrier

For the distinct spin surface define

$$T_\pm(r)=GU+\frac c r B_\pm,\quad U_{j,j+1}=1, \quad B_+=\operatorname{diag}(n,\ldots,n+d-1), \quad B_-=-\operatorname{diag}(n+d-1,\ldots,n).$$

A minimal radial continuation has positive action

$$S_2=\kappa\int dt\,dr\,d\Omega\, \bigl(|\dot y|^2-c^2|(\partial_r+W)y|^2-|T_\pm(r)y|^2\bigr), \quad y=r^{(d-1)/2}x.$$

The real scalar $W$ is shared by every chain component. Its same-field origin is unspecified. For a common normalized radial profile, let $\mu=\langle1/r\rangle$ and $\sigma^2=\langle1/r^2\rangle-\mu^2$. Exactly,

$$\langle T_\pm^\dagger T_\pm\rangle =T_\pm(\mu)^\dagger T_\pm(\mu)+c^2\sigma^2B_\pm^2.$$

The normal derivative adds $c^2\|(\partial_r+W)f\|^2I$. Replacing inverse radius by its mean misses the positive matrix correction. For $n>0$, no exact solution $f(r)v$ with a fixed nonzero chain vector exists on a radial interval: the coefficient $B^2$ has distinct diagonal entries, forcing a coordinate vector, while $BU+U^\dagger B$ couples every such vector to a neighbor. This excludes that separable ansatz, not a coupled vector profile $v(r)$.

At depth two the projected traces agree, while

$$\det H_- -\det H_+=(2n+1)c^2G^2\sigma^2.$$

For equal weights at radii 1 and 2, $c=G=1$, the exact splits for $n=0,76,81,95$ are $1/16,153/16,163/16,191/16$. These are projection controls, not predicted particle masses. The positive zero-link null vector is fixed; the negative one changes direction with radius. A shared scalar normal derivative, or a shared unitary chain connection preserving their overlap, cannot transport both null lines. Sign-mixing ambient radial operators remain outside the test.

Without $W$, trials escaping to large radius have energy tending to zero; the construction supplies no isolated positive frequency below the continuum. On the bounding circle the spin modes have half-integer rotation weights on the spin cover, whereas vector photon amplitudes have integer weights. A quadratic map can match weights but changes the kinetic metric and is singular as a vector coordinate at a vanishing helicity. Thus depth-two algebra and a two-state degeneracy do not derive photon polarization or propagation.

4. Responsive support can compress the frequency hierarchy

A conditional thin-tube reduction has

$$E=Q\Omega+R^{d-2}T(G),\qquad \Omega=\frac cR\sigma_l(GR/c),\qquad T=T_0+aG^2+bG^{-2}>0.$$

The tension follows from a finite-energy normal texture trial if the unproved projection $G\propto1/\text{width}$ holds. It is not a full coupled field solution. Write $m=d-2$ and $s=d\log\sigma_l/d\log(GR/c)$. Joint scale balance requires

$$mU=Q\Omega(1-s),\qquad \frac{d\log T}{d\log G}=\frac{m(-s)}{1-s}.$$

At high gap the required slopes are $2/3,3/2,12/5,10/3,36/5$ at depths 3,4,5,6,10. The three-term tension has slope strictly between $-2$ and $2$, excluding the high-gap balances at 5,6,10 in this class. Finite-gap equilibria can exist, and must not be excluded by that asymptotic argument.

With one common coefficient choice $c=T_0=a=b=1$, charges 100 and 300 give 26 verified collective equilibria with positive radius–gap Hessians. Their frequencies strongly compress the count ratios. At depth six the muon/electron assigned count ratio is $3515/17$, but the two calculated frequency ratios are about 2.35549 and 2.45191. At depth five the extreme count ratio is $230/7$, versus about 2.02328 and 2.25650. These are failures of these particular collective models, not an exhaustive result for all coefficient choices or responsive field laws.

There is a general local diagnostic. For a mode-independent support $E(P,y)=QP h(y)+U(y)$, $h=h_0e^{-b\cdot y}$, positive total scale Hessian $H$, and fixed charge,

$$\frac{d\log\omega}{d\log P}=1-E_w b^TH^{-1}b<1.$$

If $U_{yy}>0$, this equals $[1+E_w b^TU_{yy}^{-1}b]^{-1}$. It is a continuous-branch response identity; finite discrete ratios and changes between branches need their own calculation. Mode-dependent profiles derived from one law are not excluded. The frozen normal energy also contributes to complete inertia, as treated in the current-support ledger.

Evidence and remaining task

Independent verification recomputes nine exact angular signs, the three depth-five zonal overlap sequences with different quadrature, 34 signed radial projections, and all 26 collective Hessians using direct chain singular values and finite differences of the complete reduced energy. Maximum Hessian discrepancy is below $1.9\times10^{-6}$ in the reported normalized units.

A viable spatial candidate still needs the coupled normal profile, selected frequency, finite-amplitude shape response and complete energy–momentum inertia. The useful constructive alternatives are the zonal surface family and a genuinely vector-valued radial continuation; neither is yet a physical mass operator.