A common background action and its radial stability threshold

The positive-background construction previously canceled the angular force at first order but left the geometric Hessian unspecified. This calculation gives one complete radial inverse action for the maintained finite carrier. It derives the radius force, field--radius mixing and radial curvature from that action, and identifies exactly what remains undetermined for nonuniform geometry. The separate whole-depth conformal-current candidate does specify its metric dependence and has its own calculated two-direction Hessian; the two carriers are not one action and their curvatures cannot be combined.

One action for every harmonic in a sector

Let $H_d(R)=D_d/R+gK_d$ be the Hermitian finite carrier at fixed normal width and fixed hopping strength $g$. The component chain has diagonal entries $(n+j)/R$ and adjacent entries $g h_j$, where the $h_j$ come from the same linked spatial normal-mode calculation for all harmonics at depth $d$. The assigned index $n$ labels the occupied spatial eigenstate; the operators and coefficients are shared.

Choose a reference radius $R_0$, a nonzero retained background value $v$, and a positive background curvature $a$. Write $x=v+\xi$ and define the single sector action through its energy

$$E[\Psi,R,\xi]=\langle\Psi,H_d(R)\Psi\rangle +\frac a2\xi^2 -\frac{R-R_0}{v}(v+\xi) \langle\Psi,C_d\Psi\rangle, \qquad C_d=\left.\partial_RH_d\right|_{R_0} =-\frac{D_d}{R_0^2}. \tag{1}$$

The retained action is $S=\int dt\,[\tfrac i2\Psi^\dagger \overleftrightarrow{\partial_t}\Psi +\tfrac{I_R}{2}\dot R^2+\tfrac Z2\dot\xi^2-E]$, with positive $I_R,Z$. The wave has its canonical first-order time term. The collective kinetic coefficients are inputs whose physical values are not determined here. Equation (1) is an explicit inverse choice. In particular, the background coupling is made from the carrier's reference derivative; its origin in $\Psi_\infty$, the value of $a$, and the selection of $R_0$ have not been derived.

At $R=R_0$, $\xi=0$, the extra term and its wave derivative vanish. Every normalized positive carrier eigenstate therefore retains its standing frequency $\omega_{n,d}>0$ and energy $Q\omega_{n,d}$, where $Q=\langle\Psi,\Psi\rangle$. Neither amplitude nor a supporting energy is substituted for mass.

Full reference variations and radial Hessian

Because $C_d$ is an operator shared by the sector, rather than a number fitted to each particle, the radius and field--radius variations cancel for every wave field at the reference:

$$\partial_RE\big|_*=\langle\Psi,C_d\Psi\rangle -\langle\Psi,C_d\Psi\rangle=0, \qquad D_\Psi\partial_RE\big|_*=0. \tag{2}$$

The background variation also vanishes there. Put $p_{n,d}=-R_0\omega_R/\omega$, with the physical hopping strength held fixed. The finite-carrier inverse identity gives $1<p_{n,d}<d$ and the Hellmann--Feynman relation

$$\langle\Psi,D_d\Psi\rangle=Qp_{n,d}\omega_{n,d}R_0. \tag{3}$$

The three nontrivial second derivatives of (1) are therefore

$$\boxed{G_R=\partial_R^2E\big|_*= \frac{2Qp\omega}{R_0^2},\qquad F_R=\partial_\xi\partial_RE\big|_*= \frac{Qp\omega}{vR_0},\qquad A_\xi=a.} \tag{4}$$

The field--radius mixed block vanishes by (2), including the derivative of the carrier itself. Eliminating the relaxing background yields the actual radial curvature of this action,

$$\boxed{G_{R,\mathrm{eff}}= \frac{2Qp\omega}{R_0^2} -\frac{Q^2p^2\omega^2}{a v^2R_0^2},\qquad a v^2>\frac{Qp\omega}{2}\quad\text{for positive curvature}.} \tag{5}$$

All occupied harmonics at one depth can use one common $a,v,R_0$ only if the inequality holds at the largest $Qp\omega/2$ among them. This is a derived threshold, not a determination of $a$. It is specifically a radial result; it does not test the angular Hessian.

With positive radius and background kinetic coefficients $I_R,Z$, the two retained squared normal frequencies are

$$\Omega_\pm^2=\frac12\left[\frac{G_R}{I_R}+\frac aZ \pm\sqrt{\left(\frac{G_R}{I_R}-\frac aZ\right)^2 +\frac{4F_R^2}{I_RZ}}\right]. \tag{6}$$

Both are positive precisely when (5) holds. The wave perturbations decouple at quadratic order because the mixed block is zero. This gives a conditional finite-carrier linear result, not stability of the full spatial field. The constrained first-order wave Hessian still has a negative direction toward the negative-frequency partner, so the positive-field-block assumption of the earlier Schur inequality is not met for the entire pair. That negative energy direction alone does not imply exponential growth when it is decoupled.

Common-threshold calculation

The extended unified script uses the archived linked-mode hopping integrals and the unchanged particle assignments. It checks the pressure identity independently through the smallest singular vector of the finite carrier, verifies the first radius derivative by a centered difference, and verifies (5) by differentiating the eliminated action. The input hopping values are finite-precision numerical data; extra working digits in the matrix calculation do not certify their last digits. The table reports $\max_n p\omega/2$ at $Q=v=R_0=1$. The two common gaps are dimensionless diagnostics, not fitted physical sector scales; values from different depths cannot be compared as physical stiffnesses without the common kinetic and length units.

These are maxima over the maintained modes within each depth, so no particle gets its own background curvature. A sufficiently stiff background can make the radial block positive for this finite set, while an arbitrarily soft one cannot. The dependence on the chosen gap shows that the present calculation does not select a physical stiffness or a measured length.

Why angular stability remains unselected

The force-canceling rule fixes only the first geometric derivative. For an angular coordinate $q$, write a carrier with the same reference operator and first derivative as

$$H(q)=H_0+qC+\frac{q^2}{2}D+O(q^3),\qquad B(q)=-\frac qv\langle\Psi,C\Psi\rangle. \tag{7}$$

The force and field--geometry mixed block cancel at $q=0$ for every choice of $D$, while the bare angular curvature is $\langle\Psi,D\Psi\rangle$. Changing $D$ changes its sign without changing the frequency or first variation at the reference. This proves that the previously established first-order angular force and transparent background do not determine the angular stability calculation. The full conformally transported spatial carrier, background geometry and their second derivatives must be specified. Choosing $D$ solely to pass the stability test would be a fit.

The same caution applies to (1): adding a term quadratic in $R-R_0$ can change (4) while preserving (2). Equation (1) is the minimal linear-in-radius background completion that was actually tested. Its successful radial cancellation does not establish the master-field origin, photon behavior, a common compatible geometry across depths, nonlinear stability or reciprocal particle coupling.

The unified calculation passed 356 conditions at this radial stage. Its radial-stage report contains every maintained mode, two common gap controls and the exact source fingerprints. The later conformal-current calculation in the same script supplies a separate candidate's two-direction metric Hessian. A derived background stiffness and the complete spatial Hessian remain open.