Coupling as the source of localization and mass-frequency

The physical proposal is that interactions within one field hold a particle pattern together. The resulting confined motion selects its oscillation frequency and inertial response; mass is that frequency in equivalent units. A self-interaction is one possible sustaining contribution. Coupling to other modes of the same field, or to a supporting field pattern, may also be necessary. The same action must then govern how two particles respond to each other. This proposal does not require a bare interaction coefficient, an observed encounter rate, and a mass-frequency to have the same numerical value.

The canonical scalar-response identities already show the precise relation for a first-order candidate action: on a smooth stationary branch at fixed spatial geometry, $dE/dQ=\omega$, where $Q$ is canonical norm, while the response to another field pattern comes from higher variations of the same energy. Thus frequency and pair coupling are related derivatives of one action, but neither derivative can be inferred from the other without the stationary spatial field and its normalization. The identity is conditional on that first-order action; it is not a universal mass law.

A scale test of localization

For a fixed normalized spatial profile in $d$ ordinary directions, let $\psi_R(x)=Q^{1/2}R^{-d/2}\phi(x/R)$. A first-order positive gradient estimate scales as $A Q/R$. A local density-square contact scales as $B Q^2/R^d$, where $A,B>0$ are profile integrals. If the contact is repulsive, both positive terms decrease as the pattern spreads. If it is attractive, the fixed-profile energy is

$$E_2(R)=\frac{A Q}{R}-\frac{g B Q^2}{R^d},\qquad g>0. \tag{1}$$

For every $d>1$, its only finite stationary radius satisfies $R^{d-1}=d gBQ/A$, and direct differentiation gives

$$E_2''(R_*)=\frac{(1-d)AQ}{R_*^3}<0. \tag{2}$$

It is a maximum, with collapse as $R\to0$ in this ansatz. The result applies to these two scale terms and fixed profiles; it does not rule out other gradients, nonlocal response, curved geometry, or complete same-field support.

A higher positive local density term offers a concrete conditional alternative in the same field. Absorb fixed positive profile and norm factors into $a,B,C>0$, put $y=1/R$, and consider

$$E_3(y)=a y-B y^d+C y^{2d}. \tag{3}$$

The attractive and repulsive terms can arise from one nonlinear energy density, for example $-g\rho^2/2+h\rho^3/3$, with $\rho=\psi^\dagger\psi$; they need not be separate fields. The first finite stationary pair appears when $E_3'=E_3''=0$. Solving those equations gives

$$y_{\rm fold} =\left[\frac{a(d-1)}{2d^2C}\right]^{1/(2d-1)}, \qquad B_{\rm fold} =\frac{2(2d-1)}{d-1}C y_{\rm fold}^{d}. \tag{4}$$

For $B>B_{\rm fold}$, a broad-radius maximum and a smaller-radius minimum occur in this scale function. The minimum first crosses the dispersed reference energy zero at

$$y_{\rm bind} =\left[\frac{a(d-1)}{dC}\right]^{1/(2d-1)}, \qquad B_{\rm bind} =\frac{2d-1}{d-1}C y_{\rm bind}^{d}. \tag{5}$$

The six-depth scale batch checks depths 2, 3, 4, 5, 6 and 10 at $a=C=Q=1$ and $B=1.2B_{\rm bind}$. Its selected report finds one finite scale minimum below the dispersed reference at every depth. The values of the illustrative inverse radius run from 0.930 to 1.019 in the chosen units. This is an existence result only for the scale ansatz; it has not solved a complete spatial field equation or established dynamic stability.

Restoring the norm dependence makes the missing mass step explicit:

$$E_3(Q,R)=\frac{aQ}{R}-\frac{g BQ^2}{R^d} +\frac{hCQ^3}{R^{2d}}, \qquad \omega_{\rm branch}=\left.\frac{\partial E_3}{\partial Q} \right|_{R=R_*(Q)} =\frac a{R_*}-\frac{2gBQ}{R_*^d} +\frac{3hCQ^2}{R_*^{2d}}. \tag{6}$$

The radius derivative drops out by stationarity. In the bound examples of the batch, this first-order frequency shift relative to the chosen dispersed reference is negative. An absolute positive particle frequency, correct inertial response and complete energy require the remaining field and supporting terms. Stable radius selection alone does not prove mass generation.

One interaction can also control a pair encounter

For two pointwise orthogonal field components with densities $\rho_i,\rho_j$, the same nonlinear contact that supplied (3) has the mixed density

$$\Delta U_{ij} =-g\rho_i\rho_j +h\rho_i\rho_j(\rho_i+\rho_j). \tag{7}$$

This follows by subtracting the two isolated densities from $U(\rho_i+\rho_j)$. The coefficients are exactly the same ones that enter each self-energy and its localization equation. Yet the integrated pair response also depends on spatial overlap, polarization, relative phase when the components are not orthogonal, and reciprocal deformation. In particular, little overlap with ordinary matter can give rare encounters even if a pattern's own sustaining coupling is not small. This is a possible route for a neutrino-like weak encounter rate; it has not been calculated for an IDWT neutrino state.

The more specific neutrino hypothesis is that its effective self-coupling itself is small, making its localized oscillation frequency small. That can be tested, but low coupling cannot simply be taken to zero in (3): below the finite threshold (4), this particular scale model has no localized stationary radius. At depth five with $a=C=Q=1$, the threshold is $B_{\rm fold}=1.10616$. A weakly coupled but localized neutrino would therefore need either an effective coefficient still above its profile-dependent localization threshold or a different same-field support mechanism. The toy coefficient is not a physical neutrino coupling, and the observed small neutrino mass is constrained by upper bounds and oscillation differences, not a measured positive absolute value for every state.

An extended neutrino pattern is another distinct possibility. If one spatial carrier obeys $\omega_d=\alpha_d c/R_d$, where $\alpha_d$ is its derived dimensionless eigenvalue, then

$$\frac{R_\nu}{R_e} =\frac{\alpha_\nu}{\alpha_e} \frac{\omega_e}{\omega_\nu}. \tag{8}$$

The large observed electron-to-neutrino frequency contrast could then reflect a large support ratio, a small spectral factor, or both. This relation is conditional on each state having a meaningful characteristic radius and a common propagation speed; a diffuse or differently localized neutrino need not obey that simple carrier law. An extended physical support is also distinct from the width of a neutrino wave packet during propagation.

Equation (1) gives a useful limit on one simple explanation of such extension. At depth five, the magnitude of a local density-square self-contact relative to the gradient estimate is $gBQ/(A R^4)$. Making $g$ weaker and $R$ larger suppresses that ratio in both ways. Thus this particular local contact cannot by itself hold a very broad, weakly coupled state. A supporting interaction with a different scale law, a suppressed gradient term, or a different spatial solution would have to be derived from the same field. This is a restriction on the fixed-profile contact model, not on the proposal that neutrinos may be much larger.

An isolated composite can likewise have rest energy that differs from the sum of its components' bare rest energies because its complete interacting field contains kinetic, stress and interaction energy. This is consistent with the measured proton-mass problem and its lattice QCD decomposition, but no IDWT proton or neutron structure has been solved here. A positive interaction term alone does not determine whether binding raises or lowers the composite's total energy relative to physically isolated constituents.

The proposed strong inter-quark coupling has a clear conditional test in (7). Where the two densities satisfy $h(\rho_i+\rho_j)>g$, that local mixed term is positive: the interaction contribution to a composite can exceed the sum of the isolated self terms. Where the inequality reverses it is negative. Thus the same nonlinear interaction can be attractive in one density range, helping localization, and add positive cross energy at another. The total proton or neutron mass would still require its complete spatial kinetic energy, supporting stress and field solution. The couplings and density ranges for actual quark patterns have not been derived in IDWT.

The direct test of the proposal is therefore one canonically normalized same-field action with a finite, stable particle profile; the action must give its selected frequency, full positive inertial response, and reciprocal pair response without an independently fitted interaction rule. The two-term scale obstruction and three-term example identify what such a derivation must address, rather than providing the mass operator themselves.

A single saturated interaction: self frequency and pair response

To test one law for both roles, take the same local interaction density $U(\rho)=-g\rho^2/[2(1+\rho/\rho_s)]$, with $g,\rho_s>0$. Its variation gives the self-action $U'(\rho)\psi$ in a canonical first-order field equation. For a weak second density $\eta$ in an orthogonal component, the mixed energy begins as $U'(\rho)\eta$; its differential pair response is $U''(\rho)$. Thus the self and pair terms are derivatives of one interaction function, without a separately chosen pair coefficient. The local second derivative is $$U''(\rho)=-g(1+\rho/\rho_s)^{-3}.$$

For a uniform-density scale control, absorb its volume factor and $\rho_s$ into units, set $y=1/R$, and stipulate a positive first-order carrier cost $Qy$. The complete scale energy is $$E(Q,y)=Qy-\frac g2\frac{Q^2y^d}{1+Qy^d}. \tag{9}$$ This density ansatz is not a smooth solved spatial field. At $Q=1$, the first compact branch becomes bound relative to dispersed energy zero when $$y_{\rm bind}^{d}=d-1,\qquad g_{\rm bind}=\frac{2d}{(d-1)^{(d-1)/d}}. \tag{10}$$ For $g>g_{\rm bind}$, its compact stationary radius is a local minimum. At that radius the first-order standing-frequency shift is $$\omega=E_Q=y\frac{d-2-y^d}{d}<-\frac yd<0. \tag{11}$$ The strict inequality follows from binding, since the compact branch then has $y^d>d-1$. This one-law interaction really does bind a scale profile, but it does not supply a positive rest frequency on this canonical branch without additional dynamics.

The overlap curvature at fixed radius and its value after allowing the radius to respond are $$K_{\rm fixed}=E_{QQ}=-\frac{g y^d}{(1+Qy^d)^3},\qquad K_{\rm relaxed}=E_{QQ}-\frac{E_{Qy}^2}{E_{yy}} \leq K_{\rm fixed}<0. \tag{12}$$ The second expression is the stationary-radius Schur complement. It shows that the same attractive self-interaction also generates an attractive infinitesimal overlapping-density response, strengthened here by radius relaxation. It is not a measured two-particle force; separation, polarization, relative phase and full field deformation have not been calculated.

The single-law script evaluates (9)–(12) at depths 2, 3, 4, 5, 6 and 10 with $Q=1$ and $g=1.2g_{\rm bind}$. Its report contains all six rows and finite-difference checks of the relaxed curvature. Representative values are:

This exact self/pair link tests the proposed mechanism but cannot be substituted for the IDWT mass operator. The carrier cost and the saturation scale are input, the bound frequency has the wrong sign for a positive rest frequency under this first-order interpretation, and the stipulated profile has no demonstrated full spatial stability or canonical inertial response. A successful shared action must derive its carrier and interaction together and pass those tests across the maintained sectors.