The exact carrier pressure and a common cavity size

The same geometric action that gives positive supporting inertia does not, by itself, hold every occupied particle branch at one common radius. This can be tested directly with the finite carrier. No amplitude is identified with mass, and no individual particle radius is fitted.

An exact finite-carrier pressure bound

Keep the physical normal width $L$, coupling $g>0$, and wave speed $c$ fixed while varying the tangential radius $R$. The maintained frequency is

$$\omega(R)=\frac cR\,s_{\min}(T(\gamma)),\qquad \gamma=gR/L, \tag{1}$$

where $T$ is the bidiagonal mass carrier with a fixed assigned index. Its diagonal entries are $n+l$, and its adjacent entries are $\gamma h_l$. The absolute entries of the physical inverse $(cT/R)^{-1}$ are positive constants times $R^{i-j+1}$ for $i\ge j$. Their powers lie between one and $d$. Diagonal unitary signs remove the alternating inverse signs without changing singular values. Both surface signs have the same property.

Let $K(R)$ denote this nonnegative inverse, and let $u,v$ be its positive normalized leading singular vectors. Differentiation gives

$$\frac{d\log\|K\|}{d\log R} =\frac{\sum_{ij}(i-j+1)u_iK_{ij}v_j} {\sum_{ij}u_iK_{ij}v_j}. \tag{2}$$

The right side is a weighted average of the integer powers. Since $\omega=1/\|K\|$, this proves

$$\boxed{1\le p(R):=-\frac{d\log\omega}{d\log R}\le d.} \tag{3}$$

The bound is exact for the finite carrier, not just its large-hopping approximation. In that approximation $\omega\propto R^{-d}$, so $p$ approaches $d$.

Compare the wave with bare geometric tension

In the thin-tube area approximation, let $E_{\rm geom}=A R^m$, with $m=d-2$ and one common $A>0$ at a depth. Normalize the wave to one common canonical phase charge, so its energy is $\omega$ in the chosen units. A stationary radius obeys

$$\boxed{mE_{\rm geom}=p\omega.} \tag{4}$$

At one common radius, the left side is identical for every occupied branch. Equation (3) then limits the ratio of any two frequencies that can satisfy (4) to at most $d$. Several maintained within-depth mass ratios greatly exceed this bound. Bare area tension cannot sustain that common-radius pattern.

For $d>2$, choose $A$ once so the first maintained index has equilibrium radius one, and solve the complete finite-carrier equation for the other indices with the same $g,L,A$. In the large-hopping limit the result is

$$\frac{R_n}{R_{n_0}}\simeq \left[\frac{S(n,d)}{S(n_0,d)}\right]^{1/(2d-2)},\qquad \frac{\omega_n}{\omega_{n_0}}\simeq \left[\frac{S(n,d)}{S(n_0,d)}\right]^{(d-2)/(2d-2)}. \tag{5}$$

These are consequences of the trial supporting action, not alternative mass assignments. For example, the depth-six muon radius becomes about $1.70431$ times the electron radius, while the frequency ratio becomes $8.43704$, instead of the maintained simplex ratio $206.76471$. At depth four, the corresponding top/up frequency ratio becomes about $43.27283$, instead of $81030$. The stationary radial minima are positive, but they fail the tested spectral ratios. Their unequal radii also fail the common-size subclass; unequal radii generated by one law are permitted in the wider search and are not an independent rejection reason.

At depth two, $m=0$: the leading area term has no radial derivative to oppose the nonzero carrier pressure. Curvature terms or a different same-field sustaining contribution must be assessed explicitly there.

At the large-hopping equilibrium, the total energy in this normalized trial obeys $E_{\rm total}/\omega=1+d/(d-2)$. It therefore also requires a complete physical normalization before being identified with the particle's mass-frequency. The scale problem cannot be repaired by silently adding supporting energy after reading mass from the wave.

The earlier common-cavity analysis gives the broader fixed-point context. The present result supplies an exact pressure bound for this specific carrier. It does not exclude a common radius under an occupied same-field action with further derived contributions, or solve the complete curved geometry.

The bounded angular completion has the same leading $R^{-d}$ frequency dependence at fixed physical $g,L$. Its common radial coefficient therefore does not remove this bare-tension obstruction in its controlled large-separation regime. The exact finite-matrix inequality (3), however, is only proved here for the bidiagonal carrier; it is not automatically an all-scale inequality for the completed spatial operator.

Verification

linked_radius_pressure_checks.py passes 830 checks at all six depths and both surface signs. It verifies the exact singular-vector derivative, its positive weighted-average bound, the common-parameter equilibrium roots, their radial stiffness and their large-hopping scaling. The timestamped report preserves every assigned control and the executed source hashes. The 830-check pressure report includes the exact finite-frequency and equilibrium comparisons.