Counting, Frequency Selection and Internal-Motion Inertia

A count becomes a particle mass only when the same dynamics supplies a localized state, its normalization and its complete energy. The construction below makes each step explicit. It is an inverse candidate action on ordinary physical space, not a derivation of the maintained particle spectrum.

Counts From a Common Incidence Rule

Consider complex components of one proposed field on $\mathbb R^d$, with canonical kinetic density $\sum_a(|\partial_t\phi_a|^2-|\nabla\phi_a|^2)$ in units $c=1$. Components are labeled by a Pascal graph. All boundary entries refer to one common component $b$. A cubic incidence couples each child to its two parents. Phase invariance requires

$$q_c=q_p+q_z,\qquad q_{k,0}=q_{0,r}=1, \qquad q_{k,r}=\binom{k+r}{r}.$$

This is a derived graph charge assignment. The graph labels are not spatial coordinates and do not establish physical depth. A separate matched vacuum gap $\mu q_c$ is assumed in the potential

$$V=\mu^2|b|^2+\sum_c\left[(\mu^2q_c^2-\mu^2)|\phi_c|^2+ \left|\mu\phi_c-\frac g\mu\phi_p\phi_z\right|^2\right].$$

Reflected adjacent parents with opposite amplitudes, together with their common child, form an exact invariant three-component configuration. Their counts are $q,q,2q$, where

$$q=\binom{2k+1}{k}=3,10,35,126,\ldots,\qquad k\ge1.$$

Destructive interference makes every exterior classical source vanish. This closure holds at every encompassing graph cutoff. It establishes a useful count family, which differs from the maintained electron, muon and other particle assignments.

Actual Spatial Profiles and Complete Energy

In $\mu=g=1$ units let the parents be $P=\sqrt2F(r)e^{-i\omega t}$ and $Z=-P$, and the child $C=-2H(r)e^{-2i\omega t}$. The profiles obey

$$\Delta F=\delta F-2HF+2F^3,\qquad \Delta H=4\delta H-F^2,\qquad \omega=\sqrt{q^2-\delta},\quad 0<\delta<\frac14.$$

For regular decaying real profiles define

$$I=\int(F^2+4H^2),\quad T=\int(|\nabla F|^2+|\nabla H|^2), \quad P_0=\int HF^2,\quad K=\int F^4.$$

All integrals use the full $d$-dimensional spatial measure. Multiplication of the profile equations and spatial dilation give

$$\delta I=\frac{6-d}{d}T+K,\qquad P_0-K=\frac{2T}{d}.$$

The primitive phase action $J$, graph charge $Q$ and complete rest energy are

$$J=8\omega I,\qquad Q=qJ,\qquad E=\omega J+\frac{8T}{d}.$$

The supporting gradients are part of the particle candidate. Its complete mass-frequency is $E/\hbar$ and rest mass is $E/c^2$. The carrier phase rate $\omega=dE/dJ$ is a collective coordinate rate; assigning it the complete particle energy without checking normalization would omit the supporting contribution.

Internal Motion Enters the Translational Momentum

A boost of speed $u$ changes both envelope and phase. To first order each harmonic becomes $f_l(x-ut)e^{-il\omega(t-ux_1)}$. Direct substitution in the canonical momentum, followed by the virial identity, gives

$$P_1=u\left(8\omega^2 I+\frac{8T}{d}\right)+O(u^2)=uE+O(u^2).$$

Thus the internal oscillation contributes to the same kinetic momentum as translation even when the unboosted center is at rest. For a computed depth-three $q=3$ profile with $\delta=0.109762540262$, the two displayed contributions are approximately $29816.50124$ and $60.92649$, totaling $29877.42773$. Translating only its envelope omits nearly all the inertia. This is a result within the declared Lorentz-covariant action and its stationary stress balance.

Conditional Action Selection

The parent mean and relative phases $(\varphi,\beta)$ have the identification $(\varphi,\beta)\sim(\varphi+\pi,\beta+\pi)$. An untwisted collective wavefunction has integer labels $m,n$ with $m+n$ even. Relative-neutral states therefore require even $m$. Conditional quantization at fixed depth gives

$$8\sqrt{q^2-\delta}\,I_d(\delta)=m\eta, \qquad \eta=\frac{\hbar g^2}{\mu^{5-d}}.$$

One common $m=10000$, $n=0$, $\eta=1$ gives the following depth-three boundary-value solutions:

The charge condition was solved with the profile equations. These are conditional large-action examples, not elementary-particle predictions. Quantization, the physical action unit and the desired spatial assignments remain to be justified.

A regular weak cubic endpoint has $I_d\sim C_d\delta^{2-d/2}$ only for $d<6$. Its action tends to zero at depths two and three, a nonzero limit at four, and infinity at five. The Pohozaev identity excludes this particular endpoint at six and ten. Depth five also has $\eta=\hbar g^2$, independent of $\mu$: changing the sector frequency scale cannot evade an action threshold there. This is a specific dimensional consequence of this candidate, not a neutrino prediction.

Stability and Interaction Belong to the Same Action

Exact closure does not guarantee stability. Higher $q=10,35$ examples have growing incoming-component modes in refined finite-domain calculations. At exactly opposing parent phases the lowest $q=3$ pattern instead has a positive conserved exterior quadratic energy. Small departures in its internal relative phase couple a disturbance to an opposite-sign continuum; outgoing calculations at depths three and five support growth. There is no established finite stable phase-acceptance interval.

For well separated copies of the same grade, the canonical stress and phase current produce leading interaction terms proportional to $\cos\Delta\varphi$ and $\sin\Delta\varphi$, respectively. A quarter-cycle mean-phase difference can suppress the leading force while retaining phase-action transfer. Those initial tail calculations do not establish a full collision. Mean phase between two patterns and relative phase inside one pattern are different variables.

The scale-covariant contact action changes the interaction law to obtain exact complete-energy scaling and a different exterior stability result. Its results must not be assigned to the original action above. The subsequent field evolution tests reciprocal contact under that amended action. Classical profiles still form a continuous family; no physical frequency-selection law has been completed.