A Scale-Covariant Action for Count Scaling and Shared Contact

This local action is an inverse construction. It combines exact scaling, localized profiles and reciprocal contact under one rule. Its count weights and source projection remain physical assumptions. The original Pascal action provides the notation and spatial profile identities; the changes below define a different model.

One Scaling Rule

For each nonboundary child let $B_c=\phi_p\phi_z$, $q_c=q_p+q_z$, and

$$M_c=\frac{\mu q_c}{2},\qquad g_c=\gamma M_c^{(5-d)/2}.$$

Here $\mu>0$ is a common sector scale and $\gamma>0$ is one dimensionless coupling. Reflection $\mathcal R$ exchanges graph labels $(i,j)$ and $(j,i)$; $\Pi_+=(1+\mathcal R)/2$ acts on the vector of quadratic sources. Retain every canonical field kinetic term and use

$$V_{\mathrm{proj}}=\mu^2|b|^2+ \sum_c\frac34\mu^2q_c^2|\phi_c|^2+ \left\|M\phi-\frac gM\Pi_+B(\phi)\right\|^2.$$

The vacuum squared gaps remain $\mu^2q_c^2$. Reflection symmetry alone does not require this projection: selecting the even source channel is an additional interaction assumption. It acts on component labels, not on an asserted physical reflection axis.

For an opposing compact triad with counts $q,q,2q$, put $M=\mu q$, $\alpha=(d-1)/2$ and $\omega_0=\sqrt{1-\delta}$. The fields are

$$P=\frac{\sqrt2 M^\alpha}{\gamma}F(Mr)e^{-iM\omega_0t},\quad Z=-P,\quad C=-\frac{2M^\alpha}{\gamma}H(Mr)e^{-2iM\omega_0t}.$$

Canonical integration gives

$$\boxed{A_q=\frac{8\omega_0 I}{\gamma^2},\qquad E_q=\frac{\mu q}{\gamma^2}\left(8\omega_0^2 I+\frac{8T}{d}\right).}$$

The phase action is independent of count, complete energy is proportional to count, and radius scales as $1/(\mu q)$ on a common base profile. A separately justified common action condition could therefore select one profile across counts. It does not yet select that condition, the profile or the absolute physical scale.

The powers follow directly from canonical scaling: $1+2\alpha-d=0$ for action, cubic coefficient power $2-\alpha=(5-d)/2$, and quartic power $2-2\alpha=3-d$.

At depth five the cubic coefficient is count independent; depth three has a count-independent quartic coefficient. These are classical dimensional identities, not quantum protection of the ratios.

A Combinatorial Exterior Stability Bound

The source projection removes the anomalous incoming pair responsible for the tested higher-triad instability in the unprojected action. Remaining exterior quadratic terms are normal complex bilinears and positive squares. A rotating quadratic energy alone need not be positive, but each connected normal block also has a conserved phase current $N$.

For compact index $k\ge1$, its exterior chain has counts $b,q+b,3q+b$, where

$$q=\binom{2k+1}{k},\qquad b=\binom{2k+1}{k-1}, \qquad \frac bq=\frac{k}{k+2},\qquad \frac{3q+b}{b}=4+\frac6k\le10.$$

Choose the mathematical energy shift $\nu=-3\mu b/2$. With $\Omega=\mu\sqrt{1-\delta}$ and $0<\delta<1/4$, completing kinetic squares in $E_2-\nu N$ leaves

$$\frac34\mu^2q_j^2-(\Omega q_j+\nu)^2 \ge\left[\frac34-\left(\frac{17}{20}\right)^2\right]\mu^2q_j^2 =\frac{11}{400}\mu^2q_j^2>0.$$

The boundary retains an additional positive vacuum term. This supplies a conserved positive exterior quadratic energy on the stated finite-energy domain. The rational margin is a proven bound, not a measured coupling. It is linear isolated-state control, not nonlinear orbital stability.

The occupied fixed-charge criterion is separate. At radius 60, refining 600 to 1200 radial cells gives lowest constrained eigenvalues $0.0954466\to0.0954627$ at depth three and $0.0559417\to0.0559439$ at depth five on the tested base profiles. These finite-volume results are not continuum spectral enclosures.

One Density Term Supplies Self-Contact and Pair Contact

The projected action alone makes different opposing grades an exact direct sum. To obtain nonzero interaction, define each odd reflected component $v_j=(\phi_{ij}-\phi_{ji})/\sqrt2$, its scale $M_j=\mu q_j$, and add

$$\Delta V_\rho=\lambda\gamma^2 \left[\sum_j M_j^{(3-d)/2}|v_j|^2\right]^2,\qquad\lambda>0.$$

Also retain the common relative-parent lock

$$\Delta V_{\mathrm{lock}}=\kappa\gamma^2\sum_{\{P,Z\}} (\mu q_P)^{3-d}|P+Z|^2(|P|^2+|Z|^2),\qquad\kappa>\lambda,$$

with each reflected pair counted once. Both terms belong to the same amended potential. Examples use $\lambda=1/4$, $\kappa=1$.

The density term changes the isolated profiles. With $a=1+4\lambda$ their equations and exact map from an old base solution are

$$\Delta F=\delta F-2HF+2aF^3,\qquad\Delta H=4\delta H-F^2,$$ $$F(x)=a^{-1}F_0(x/\sqrt a),\quad H(x)=a^{-1}H_0(x/\sqrt a), \quad\delta=\delta_0/a.$$

Thus the self-support is recalculated consistently. Complete count scaling survives. The occupied constrained Hessian gains a positive rank-one term under the corresponding unitary dilation. The condition $\kappa>\lambda$ controls the relative-parent block for the regular nodeless profiles. The added exterior terms are positive normal density terms, so the isolated exterior bound survives. This does not prove stability of a finite encounter.

For two different grades the exact cross energy is

$$U_{kl}(R)=2\lambda\gamma^2(M_kM_l)^{(3-d)/2} \int |v_k(x)|^2|v_l(x-R)|^2\,d^dx.$$

Translation invariance gives reciprocal forces. Radially decreasing initial densities repel; this term is blind to their global relative phase. For fixed lighter count and fixed separation, the concentrated heavy limit gives $U\propto q_h^{(1-d)/2}$. A larger complete mass can therefore accompany smaller integrated contact at that separation. It is not an observed neutrino interaction rate.

Physical Scope

The supported count subsequence $3,10,35,126,\ldots$ does not reproduce the full maintained assignments. The inherited depth-six sample is unbound. The photon zero-index case, physical polarization, cross-depth contact, quantum corrections and the spatial origin of the graph projection remain open. The complete rest-energy ratios hold classically on a common selected base branch; frequency selection itself remains conditional. Reciprocal field evolution tests the amended action beyond its initial forces.