Same-field current support, causal response and complete energy

An interaction can change the time kinetic coefficient of an oscillating field while its supporting components respond reciprocally. A useful local candidate uses a current built polynomially from derivatives of those components. This avoids division by a current that can vanish. It supplies testable equations, but its local characteristic calculations and imposed background wells do not establish a self-supported particle.

1. Canonical frequency and complete energy

For a stationary symmetry orbit with reduced action $L_\omega=\omega^2K[u]-V[u]$, define its actual Noether charge by

$$Q=2\omega K,\qquad E=\omega^2K+V,\qquad E-\omega Q=V-\omega^2K.$$

On a differentiable stationary branch, $dE/dQ=\omega$. This does not generally imply $E=\omega Q$: the derivative of a nonlinear energy function differs from its ratio to charge. Equality is an additional normalization target when a model identifies this particular phase frequency and charge with the complete particle energy. A positive $E-\omega Q$ tests that identification; by itself it is not a no-go theorem for localized massive states.

The physical conversion $\omega_m=E_{\rm rest}/\hbar=mc^2/\hbar$ is unchanged. A classical candidate must establish that its chosen oscillation has this physical identification with the appropriate action and charge normalization. No arbitrary rescaling or omission of supporting energy establishes it. Complete boost inertia also requires the complete stationary stress and energy, not a wave contribution on frozen coefficients.

For a quadratic wave on a static neutral supporting texture, $E_{\rm total}=Q\omega+E_{\rm support}$. If an admissible common amplitude variation decomposes the action into homogeneous terms of degree $p$, its exact amplitude virial gives

$$E-\omega Q=V_0+\sum_p(1-p/2)(V_p-\omega^2K_p).$$

A positive spatial quartic with quadratic time norm therefore contributes $-V_4$ to this difference. The inherited weak standing Hodge interaction has the opposite sign in this ledger. These are different actions and cannot be substituted silently. A constrained unit target may forbid the amplitude variation entirely.

2. A regular local kinetic interaction

Let $Y^i$, $i=1,\ldots,d$, be neutral components of the same field in ordinary physical spacetime, with metric signature $(+,-,\ldots,-)$. Define the current by the dual of their derivative volume form, $j=* (dY^1\wedge\cdots\wedge dY^d)$, with indices raised by that metric. More general polynomial amplitudes multiplying the form are possible, but need not preserve current conservation. The minimal block is

$$\mathcal L= \frac{\kappa_x}{2}|\partial X|^2+ \frac{\kappa_s}{2}\sum_i|\partial Y^i|^2 -\frac\lambda2 j_\mu j^\mu +\frac\alpha2|j^\mu\partial_\mu X|^2-U(X,Y),$$

where all displayed coefficients are positive. The complex wave is a two-real-component field; the phase generator acts on $X$ and leaves $Y$ neutral. Every component is varied. This is a proposed local block of one field, not a new exchanged particle or a derived master action.

On a static support, $j=(J,0)$, the wave time coefficient is $\kappa_x+\alpha J^2$, while its canonical spatial coefficient is $\kappa_x$. The frozen wave slows. This does not settle the coupled principal symbol because perturbations of $Y$ change $j$.

The prior inverse support target asks for two different coefficient gradients. On a round feature $m=d-2$, a pinned current with $J\propto R^{-m}$ gives $\partial_{\log R}\log(\kappa_x+\alpha J^2)=-m$ at $\alpha J^2=\kappa_x$. This is a partial match only. It does not supply the required spatial gradient or gap response, and that reference equality must arise from the normal equations.

A related two-scalar coefficient construction already saturated at speed $c$ cannot have a nonzero derivative of its speed ratio and remain bounded by $c$ on a two-sided open neighborhood. Determinant cofactor kinetics can provide causal margin, but the tested fixed tangent stretches give a radius derivative of their spatial/time ratio below 2, versus the high-gap inverse target $2d$. These scoped coefficient tests do not exclude anisotropic or derivative-mixed completions.

3. Negative signed binding can coexist with a causal local response

Freeze a diagonal support-gradient patch $\partial_iY^j=s_i\delta_i^j$, a wave velocity $v$, and zero wave spatial gradient. Put $S=\prod_i s_i$ and $c_i=\prod_{h\ne i}s_h$. After removing determinant second variations that are constant-coefficient divergences, the exact quadratic principal Lagrangian is

$$\begin{aligned} L_2={}&\frac{\kappa_x}{2}(x_t^2-|\nabla x|^2) +\frac{\kappa_s}{2}\sum_i(b_{i,t}^2-|\nabla b_i|^2)\\ &+\frac\lambda2\left[\sum_i c_i^2b_{i,t}^2- \left(\sum_i c_i\partial_i b_i\right)^2\right] +\frac\alpha2\left[Sx_t+v\sum_i c_i\partial_i b_i\right]^2. \end{aligned}$$

The Hamiltonian cancels the mixed time-space term. Its support spatial Fourier matrix is

$$\kappa_s|k|^2I+(\lambda-\alpha v^2)c(k)c(k)^T, \qquad c(k)_i=c_i k_i.$$

For nonzero cofactors it is positive for every direction whenever

$$\alpha v^2<\lambda+\frac{\kappa_s}{\max_i c_i^2}.$$

The time matrix is positive. For a timelike real covector every canonical principal contribution is positive, the minor term is nonnegative by Cauchy–Schwarz, and the last term is a square. Thus there is no real superluminal characteristic. The positive quadratic Hamiltonian supplies real frequencies for this constant-coefficient block.

Consequently the interval

$$\lambda<\alpha v^2<\lambda+\frac{\kappa_s}{\max_i c_i^2}$$

permits a negative signed current-pair contribution $(\lambda-\alpha v^2)J^2/2$ to $E-\omega Q$, while retaining positive perturbation energy and causal local propagation. It is not negative total energy or negative inertia.

One exact common witness at every maintained depth has $s_i=\kappa_x=\kappa_s=\lambda=\alpha=1$, $v^2=3/2$. The signed contribution is $-1/4$; the coupled squared front speeds are $1$ and $1/8$, with transverse support modes at $1/2$. The second real wave quadrature has its own positive slowed block. The $1/8$ here is a characteristic-speed ratio in this chosen patch, not the earlier Hodge directional contact coefficient or a neutrino rate. No special depth-five term was inserted, and this example has no depth-five-specific effect.

4. Localization and concentration are different tests

On a prescribed static current, the massive scalar wave obeys

$$(-\Delta+M^2)u=\omega^2[1+(\alpha/\kappa_x)J^2]u.$$

A trial satisfying $\int|\nabla u|^2<(\alpha/\kappa_x)M^2\int J^2|u|^2$ has frequency below the exterior continuum and supplies a sufficient bound-state test for a smooth localized coefficient. This is not a solution of the reciprocal support equations. A compact kinetic bump alone has no positive-frequency square-integrable massless scalar state in an otherwise free exterior.

An explicit smooth trial $Y_i=sx_i e^{-r^2/(2L^2)}$ has $J=s^d e^{-dr^2/(2L^2)}(1-r^2/L^2)$. For a Gaussian wave of width $\sigma$, $t=\sigma^2/L^2$, its weighted current is

$$\frac{\int J^2u^2}{\int u^2} =s^{2d}\frac{1+dt+d(d+2)t^2/4}{(1+dt)^{d/2+2}}.$$

This realizes a conditional well with regular fields. Adding only canonical neutral support energy nevertheless allows a fixed-charge concentration sequence with energy tending to zero. A positive current minor blocks that first sequence, but for $d>2$ another exists even with any fixed finite polynomial wave potential. Take common compact shapes, support amplitude $b=L^{1/2+\delta}$, wave amplitude $A=L^{-2d\delta}$, and

$$0<\delta<\min\{(d-2)/(4d),1/(4q),1/4\},$$

where $2q$ is the largest wave-potential power. The positive energy terms scale as $L^{2d\delta}$, $L^{d-2-4d\delta}$, $L^{d-4qd\delta}$, and $L^{d-1+2\delta}$, all tending to zero. This excludes a positive-energy global minimum in that unrestricted variational class. It neither excludes local stable states nor proves that the sequence stays inside the full causal field domain. A bounded target removes this particular large-amplitude path; global coercivity and the actual stationary equations still need proof.

5. Exact static and helical energy ledgers

For ordinary positive spatial gradients $T$, derivative-free potential $U$, ordinary phase norm $K_0$, current norm $K_J=(\alpha/2)\int J^2h(X)$, and static minor $S=(\lambda/2)\int J^2$, a common spatial dilation of a smooth stationary state gives

$$E-\omega Q=\frac{2T}{d}+2(S-\omega^2K_J).$$

The stronger sufficient local bound $\lambda\ge\alpha\omega^2h$ makes this positive. Section 3 shows that this stronger bound is not necessary for local causal positive-energy perturbations. It must not be used to close the entire current-support route.

There is, however, a stronger bound for this particular static action. Put $\delta=\alpha\omega^2|X|^2-\lambda$ and let $s_i$ be the singular values of the support gradient. Positivity of its spatial Fourier block requires $\delta\max_i c_i^2<\kappa_s$ wherever $\delta>0$. Since $J^2/\max_i c_i^2=\min_i s_i^2$,

$$\delta J^2<\kappa_s\min_i s_i^2, \qquad S-\omega^2K_J\ge-\frac{T_Y}{d}.$$

Here $T_Y=(\kappa_s/2)\int\sum_i s_i^2$. Rank-deficient points have zero negative-pair density. Combining the bound with the stationary virial yields

$$E-\omega Q\ge\frac{2T_X}{d}>0$$

for a nontrivial localized charged wave with canonical wave gradient energy $T_X>0$. Strict principal margin strengthens the inequality on a nonzero-current region. Thus even the enlarged local negative-pair window cannot achieve the additional equality target in this simple static-support action with a positive Hamiltonian. Its supporting gradient energy costs at least what the permitted negative contribution can offset. This preserves the local witness while ruling out an overextension of it to a complete stationary equality. It is not a general exclusion of states with $E>\omega Q$, moving currents, mixed Hodge terms, or a different derivative action.

For a synchronized physically rotating pattern, $Y(t,x)=Y_0(e^{-\omega t A}x)$, $X(t,x)=e^{i\omega t}X_0(e^{-\omega t A}x)$, where $A$ is antisymmetric, the current is $j=(J,J\omega Ax)$. The generator combines phase and physical rotation; call its charge $Q_{\rm hel}$, distinct from phase charge alone. Let $K_{\rm rot}=(\lambda/2)\int J^2|Ax|^2$. Its dilation degree is $2-d$, yielding

$$E-\omega Q_{\rm hel} =\frac{2T}{d}+2(S-\omega^2K_J) -\frac{2(d-1)}d\omega^2K_{\rm rot}.$$

This is a candidate balance, not a rotating solution. In the timelike current rest frame the old sufficient bound becomes $\alpha\omega^2|X|^2\le\lambda(1-|\omega Ax|^2)$. It implies $S-\omega^2K_J\ge\omega^2K_{\rm rot}$, keeping the ledger positive. The negative-pair window is therefore essential to any proposed escape within this architecture. Spatial wave gradients, rank defects, radiative tails and light-cylinder behavior are untested.

For the Hodge square there are additional mixed derivative-gap terms. With a gap scaling as $L^{-s}$, its normalized wave contribution is $2V_2+(1+s)V_1+2sV_0=-\kappa\,d\Omega^2/d\log L$. This is strictly positive on the lowest nonzero finite-chain branch for $0\le s\le1$: every entry of the sign-adjusted inverse increases as $L^{1+(1-s)(j-i)}$. For larger exponents this proof stops. Using this in an ambient virial requires an actual ambient embedding with those dilation powers, not just the surface matrix.

Remaining physical bridge

The constructive result is a reciprocal local current interaction with an explicitly checked causal window, including negative signed binding. It still needs one smooth finite-energy stationary texture and wave, the actual simplex spectrum, the complete principal symbol on that inhomogeneous solution, its relaxed stability, physical normalization, boost inertia and a particle-pair response. The local block does not yet derive a physical mass or coupling operator.

Independent verification checks the characteristic factorization at depths 2,3,4,5,6,10 and the exact negative-pair witness and cofactor bound; the remaining identities above were reviewed by direct variation and homogeneous scaling, without a new PDE run.