Energy constraints from scalar response

Summary: The same scalar response fixes the energy-frequency defect, the curvature of energy with canonical norm, and the sensitivity to an interaction change, giving joint inverse constraints on mass ratios and coupling without selecting the physical particle norm.

A common mass scale can survive an energy-frequency difference

The inverse scalar-law construction tests whether complete spatial fields can obey one causal local interaction. Its frequency target needs a further check: the same action determines total energy, and stationary frequency is generally an energy derivative rather than energy per unit norm.

That distinction does not by itself spoil the mass relationships. A common conversion factor can preserve their ratios. The question is whether the factor is common on the actual particle branches after their shapes respond. This note derives tests for that question and connects them to the scalar coupling derivative already present in the full field equation.

Use a fixed physical spatial geometry, measure and canonical time-kinetic normalization. Let the self-adjoint spatial Dirac operator be $\mathscr D$, with $\beta^\dagger=\beta$ and $\beta^2=I$. The candidate action has energy and conserved canonical norm

$$E[\psi]=\langle\psi,\mathscr D\psi\rangle+\int U(J)\,dV, \qquad Q[\psi]=\langle\psi,\psi\rangle, \qquad J=\psi^\dagger\beta\psi, \tag{1}$$

where $U$ is real, $U(0)=0$, and $F=U'$. Its equation is

$$i\partial_\tau\psi=\mathscr D\psi+F(J)\beta\psi. \tag{2}$$

The vacuum subtraction in (1) is fixed for the action. It cannot be changed separately for different particle states. The canonical norm has not been identified with electric charge or particle number.

The general identities below apply to smooth stationary branches for which the indicated derivatives and integrals exist. There are no compactified dimensions. Only the sectors are nested; the two particle fields need not be restrictions of each other.

Compare two branches at the same canonical norm

Write a stationary solution as $e^{-i\omega\tau}\phi$. Its equation is $\nabla_{\bar\phi}E=\omega\phi$. Along a smooth branch parameterized by $Q$, put $h=\partial_Q\phi$. The norm condition and first energy variation give

$$2\operatorname{Re}\langle\phi,h\rangle=1, \qquad \frac{dE}{dQ} =2\operatorname{Re}\langle\omega\phi,h\rangle=\omega. \tag{3}$$

This recovers the local normalization identity. Now compare two independently occupied branches $a,b$ of the same action at one common $Q$. Where $E_a$ and $\omega_a$ are nonzero, differentiation of their energy ratio gives the exact relation

$$\boxed{ \frac{\omega_b}{\omega_a}-\frac{E_b}{E_a} =\frac{E_a}{\omega_a}\frac{d}{dQ}\left(\frac{E_b}{E_a}\right). } \tag{4}$$

Thus equal energy and frequency ratios require the energy ratio to be stationary with respect to their common norm. If both ratios are claimed to equal a target $r$ at $Q_*$, define $\Delta(Q)=E_b(Q)-rE_a(Q)$. Then

$$\Delta(Q_*)=0,\qquad \Delta'(Q_*)=0. \tag{5}$$

Matching both is a requirement of that joint identification, not an extra empirical assumption imposed on every possible mass interpretation.

A sufficient factorization is $E_n(Q)=S_n H(Q)$ with one function $H$. It gives both ratios $S_b/S_a$ wherever they are defined, even though $E_n/(Q\omega_n)=H/(QH')$ need not equal one. This is an algebraic sufficiency condition; the spatial dynamics must still derive it. A frequency match at one norm, by itself, does not imply the factorization or an energy match. The common-norm premise also needs a physical origin.

For the unchanged six-dimensional assignments, the target remains

$$r=\frac{S(35,6)}{S(13,6)} =\frac{\binom{40}{6}}{\binom{18}{6}} =\frac{3515}{17}. \tag{6}$$

No mode or dimensional assignment is varied in this calculation.

The energy defect is an integral of the coupling derivative

Taking the inner product of the stationary equation with $\phi$ yields $\omega Q-E=\int[JF(J)-U(J)]\,dV$. Since $[JF(J)-U(J)]'=JF'(J)$ and the expression vanishes at $J=0$,

$$\boxed{ \omega Q-E =\int dV\int_0^J sF'(s)\,ds =\int J^2\left[\int_0^1 tF'(tJ)\,dt\right]dV. } \tag{9}$$

The second form follows by $s=tJ$ and is valid for negative currents too. The coefficient $F(0)$ cancels: a bare linear contribution to $U$ does not generate this defect.

If $a\le F'(s)\le b$ throughout every current segment between $0$ and $J(x)$, then

$$\frac a2\int J^2\,dV \le \omega Q-E \le \frac b2\int J^2\,dV. \tag{10}$$

In particular, a convex potential gives a nonnegative defect. Strictly positive derivative on the relevant segments gives a positive defect for a nonzero scalar current. Convexity is an explicit hypothesis. For example, $U=-J^2/2+J^4/4$ has zero defect at $J=\sqrt{2/3}$ although $F'(J)=1$. Cancellation of the defect alone does not establish absence of response.

The full complex source variation is

$$\delta\!\left(F(J)\beta\phi\right)[u] =F(J)\beta u +2F'(J)\operatorname{Re}(\phi^\dagger\beta u)\,\beta\phi. \tag{11}$$

Equations (9) and (11) use the same function $F'$. A scalar coupling response and an energy correction cannot be chosen independently. For two fields with a common energy-frequency ratio, their fractional defects $(\omega Q-E)/(\omega Q)$ must agree. Nonzero defects remain compatible with a common mass scale when that agreement follows from the dynamics.

Equation (11) is a variation of the one field. During an encounter its total current includes spatial interference and both polarized patterns. The identities here constrain that interaction; they do not yet calculate a complete particle-pair trajectory or an observed coupling rate.

The full static response determines energy curvature

Use the real inner product $(u,v)_{\mathbb R}=\operatorname{Re}\langle u,v\rangle$. Let $\mathcal L$ be the real derivative of the stationary gradient:

$$\mathcal L u=(\mathscr D-\omega)u +F(J)\beta u +2F'(J)\operatorname{Re}(\phi^\dagger\beta u)\,\beta\phi. \tag{12}$$

It is half the real Hessian of $E-\omega Q$ under this convention. It includes both quadratures and satisfies $\mathcal L(i\phi)=0$. Remove the phase symmetry and any additional symmetry zero modes, and assume the remaining inverse exists on the required sources. Differentiating stationarity with respect to $\omega$ gives

$$\mathcal L\partial_\omega\phi=\phi, \qquad \boxed{\chi:=\frac{dQ}{d\omega} =2(\phi,\mathcal L_\perp^{-1}\phi)_{\mathbb R}.} \tag{13}$$

On a regular branch with nonzero finite $\chi$,

$$\boxed{\frac{d^2E}{dQ^2}=\frac{d\omega}{dQ}=\frac1\chi.} \tag{14}$$

An equivalent calculation avoids inverting the norm-frequency relation: solve $\mathcal Lh=\omega_Q\phi$ together with $2(\phi,h)_{\mathbb R}=1$. This bordered system determines the complete norm tangent and $\omega_Q$ together. It remains a useful local equation when an inverse used in (13) is unavailable.

The static susceptibility is not a complete stability criterion. Time evolution uses $\mathbb J\mathcal L$, where $\mathbb J$ represents multiplication by $-i$ in real coordinates. Modes away from zero frequency can grow without a sign change in (14). The special role and limitations of norm-slope tests for nonlinear Dirac equations are discussed in Comech's analysis.

Allow the entire spatial shape to relax

In real coordinates, let $c$ represent $\phi$ and choose orthonormal columns $V$ perpendicular to both $c$ and the phase direction. They span the allowed norm-preserving shape variations after symmetry removal. Set $H_s=V^{\mathsf T}\mathcal LV$ and $b=V^{\mathsf T}\mathcal Lc$. Where $H_s$ is invertible, write $h=c/(2Q)+Vy$. Projecting its equation onto $V$ gives $y=-H_s^{-1}b/(2Q)$, and hence

$$\boxed{ E''(Q)=\frac{c^{\mathsf T}\mathcal Lc-b^{\mathsf T}H_s^{-1}b}{2Q^2}. } \tag{15}$$

This applies the full spatial Schur reduction to the norm response. Holding the shape fixed would instead give

$$C_{\rm fixed} =\frac{c^{\mathsf T}\mathcal Lc}{2Q^2} =\frac1{Q^2}\int J^2F'(J)\,dV. \tag{16}$$

For a positive shape Hessian the correction in (15) is nonpositive. An excited Dirac branch generally has an indefinite shape Hessian, so there is no such sign conclusion. All allowed complementary modes must enter the calculation; retaining one overlap coefficient is insufficient.

This sign can be traced to the free spatial spectrum in the quartic control. Let $\chi_n$ be a simple normalized eigenfield within the chosen invariant symmetry class, with eigenvalue $\lambda_n$. For $E_g=\langle\psi,\mathscr D\psi\rangle+g\int J^2d\nu/2$, define

$$N_n=J_{\chi_n}\beta\chi_n,\qquad A_n=\int J_{\chi_n}^2d\nu,\qquad B_n=(P_\perp N_n,(\mathscr D-\lambda_n)_\perp^{-1}P_\perp N_n)_{\mathbb R}. \tag{17}$$

Here $P_\perp$ removes $\chi_n$; the complementary spectral inverse is assumed to exist. The first profile correction is $\xi_n=-(\mathscr D-\lambda_n)_\perp^{-1}P_\perp N_n$. Expanding the normalized energy, its kinetic contribution at second order is $B_n$, while the interaction contributes $2(N_n,\xi_n)_{\mathbb R}=-2B_n$. Consequently

$$e_n(g)=\lambda_n+\frac g2A_n-g^2B_n+O(g^3), \qquad w_n(g)=\lambda_n+gA_n-3g^2B_n+O(g^3). \tag{18}$$

For one fixed quartic law, varying the norm at $Q=1$ gives

$$E''=gA_n-6g^2B_n+O(g^3),\qquad C_{\rm fixed}=gA_n-4g^2B_n+O(g^3). \tag{19}$$

The leading shape correction is therefore $-2g^2B_n$. The denominators in (17) have both signs. This is a property of the response spectrum; neither sign is assigned an antiparticle interpretation.

A usable derivative for reconstructing the common law

Consider a small common action change $U_\epsilon=U+\epsilon W$, with $W(0)=0$, at fixed physical norm and geometry. Suppose the stationary branch varies smoothly with $\epsilon$. Its fixed-norm first energy change has no implicit shape term:

$$\boxed{\partial_\epsilon E\big|_0=\int W(J)\,dV.} \tag{20}$$

Put $k=\partial_\epsilon\phi|_0$. Differentiating stationarity gives $\mathcal Lk+W'(J)\beta\phi=\omega_\epsilon\phi$, with $(\phi,k)_{\mathbb R}=0$. Pair this equation with the complete norm tangent $h$ from (14). Self-adjointness and $\mathcal Lh=\omega_Q\phi$ remove the term involving $k$, giving

$$\boxed{\partial_\epsilon\omega\big|_0 =2(h,W'(J)\beta\phi)_{\mathbb R}.} \tag{21}$$

Equations (20) and (21) give distinct, computable sensitivities of the energy and frequency targets to the same change of interaction. For example, the variation of $E_b/E_a$ is $[\int W(J_b)dV-(E_b/E_a)\int W(J_a)dV]/E_a$. The corresponding frequency-ratio variation uses (21) on both branches. Their shapes, physical density units and norm tangents determine the weights; there is no independent adjustable law for each state.

Any proposed change also changes the coupling derivative by $\epsilon W''(J)$ in (11). These formulas provide a local inverse method for an already specified candidate action. They do not prove that a law meeting all the nonlinear constraints exists, or that a large correction can be extrapolated from a first derivative.

Verification

mass_energy_response_checks.py checks the ratio identity exactly and the local sum rule for convex, saturating and nonconvex laws at currents of both signs. Full complex source derivatives and common basis changes are tested at spinor ranks two, eight and thirty-two. Exact algebra also checks the common-bare-mass free-limit bound and a double-zero control that preserves the distinction between an isolated match and an identity on an open interval.

The spatial calculation solves the same quartic law at three common coefficients and nearby physical norms. It compares energy differences, frequency differences, the bordered norm response, the phase-removed full complex susceptibility and the spatial Schur correction. An independent free spectral sum verifies (17). Both real response quadratures remain present in the dynamical matrix.

The inverse sensitivities (20) and (21) are checked by changing the potential to $U+\epsilon W$ with either $W=gJ^2/2$ or $W=gJ^3/(3J_c)$, using the same declared density scale $J_c=10^6$ on both branches. New fixed-norm spatial solves at $\epsilon=\pm0.05$ verify the predicted energy and frequency changes. These are sensitivity controls. Their independently differentiated spatial residuals are below $4.4\times10^{-11}$ at the largest cutoffs.

Paired cutoffs $44,66,90$ and $110,150,190$ are enlarged separately. Actual curved derivatives on independent refined quadrature grids check the stationary field and its norm tangent. Coarse residuals are recorded, not treated as converged because their compressed equations are solved. The largest-cutoff norm-response residuals are below $2.8\times10^{-12}$. These controls supply numerical convergence evidence within the invariant radial class, with the general identities conditional on their stated functional assumptions.