A positive oscillator action and its complete initial shape response

A different canonical action can have the same fixed-geometry frequencies and a different geometric response. A useful explicit comparison is

$$\boxed{L_2=\|\partial_t[B(\alpha)c]\|^2 -c^\dagger H_P(\alpha)^2c.} \tag{1}$$

Here $B^\dagger B=I$ embeds the retained coefficients $c$ in the spatial modes. For a fixed self-adjoint retained carrier $H_P$, its frequency operator is $K_P=H_P^2$, so its positive oscillation frequencies are $|\lambda_j(H_P)|$. The energy is nonnegative. Equation (1) is an alternative inverse construction, not a consequence of the existing first-order master law. Its full physical spatial realization and total inertial response remain open. Mass continues to mean the selected frequency in equivalent units.

Projection and squaring do not generally commute. If $H_P=B^\dagger H_{\rm full}B$ and $Q=I-BB^\dagger$, then

$$\boxed{B^\dagger H_{\rm full}^2B =H_P^2+(QH_{\rm full}B)^\dagger(QH_{\rm full}B).} \tag{1a}$$

Thus (1) explicitly defines the retained effective action. It is not automatically the projection of the full spatial operator's square. A forward reduction must retain the extra term or construct a controlled dressed invariant low subspace before squaring. The calculations below use (1); they do not establish that missing physical reduction.

The later spatial complement calculation derives the leading positive kinetic correction from the maintained full parent. The bounded completion and its complete spatial bound then control the small frequency of a specified improved parent. Those results do not retroactively make the particular shape action (1) a derived occupied master-field law.

The comparison is useful because the small signed partner that obstructs the first-order slow law is not a gapped negative-frequency excitation of this new positive oscillator. It belongs to a degenerate positive-frequency doublet. Its response must be retained under that different structure.

The squared source removes the old partner transition

At fixed geometry, writing $H=H_P$ in this finite identity,

$$K_a=\partial_a(H^2)=\{H,H_a\},\qquad \langle r,K_a n\rangle=(\lambda_r+\lambda_n) \langle r,H_a n\rangle. \tag{2}$$

Thus $K_a$ has zero matrix element between the two signed partners $\lambda_r=-\lambda_n$. Copying the first-order response denominator into (1) would give a wrong answer.

For the chiral carrier,

$$H_P=\begin{pmatrix}0&T^\dagger\\T&0\end{pmatrix},\qquad H_P^2=\begin{pmatrix}T^\dagger T&0\\0&TT^\dagger\end{pmatrix}. \tag{3}$$

If $Tu=\lambda v$, $T^\dagger v=\lambda u$, and $\|u\|=\|v\|=1$, every vector $(a u,b v)$, $|a|^2+|b|^2=1$, has squared eigenvalue $\lambda^2$. The projected $K_a$ on this entire doublet is $2\lambda\partial_a\lambda$ times the identity. Relative phase and population do not remove its first frequency source.

Keep all coefficient modes while the spatial basis moves

Write $\Psi=B(\alpha)c$, where $B$ contains the orthonormal normal modes in each chiral block. Expanding the kinetic term of (1) gives

$$L_2=\|B\dot c+(\partial_aB)c\dot\alpha_a\|^2 -c^\dagger K_P(\alpha)c. \tag{4}$$

All retained coefficient modes remain dynamical in (4). It is not a single-eigenray adiabatic action. Choose orthogonal basis transport at the reference configuration, $B^\dagger\partial_aB=0$, in the same polynomial spatial and temporal-connection convention as the moving normal calculation.

Consider a moment with zero shape velocity, while the wave continues its oscillation: $c=A\phi$, $\dot c=-i\lambda A\phi$, $K_P\phi=\lambda^2\phi$, $\|\phi\|=1$. Define the real positive spatial metric

$$G_{ab}=\operatorname{Re}\langle(\partial_aB)\phi, (\partial_bB)\phi\rangle,\qquad f_a=\partial_a\lambda. \tag{5}$$

The shape Euler equation at this moment is exactly

$$2A^2G_{ab}\ddot\alpha_b+2A^2\lambda f_a=0, \qquad \boxed{G\ddot\alpha=-\lambda f.} \tag{6}$$

The coefficient modes' acceleration is also retained in the variation; its cross term vanishes here by the orthogonal transport condition. No arbitrary amplitude is being identified with mass. Its common factor cancels from (6).

For a fixed circular wave in (1), the positive canonical phase charge is $\mathcal Q=2\lambda A^2$, and $E=2\lambda^2A^2=\mathcal Q\lambda$. This charge is the action's phase normalization, not an assertion about electric charge. Rescaling the wave changes both quantities while leaving its frequency unchanged.

The instantaneous small branch therefore has

$$\boxed{\frac{\ddot\lambda}{\lambda} =-f^TG^{-1}f\equiv-\Omega_{\rm source}^2.} \tag{7}$$

Equation (7) is a local initial curvature. It does not extrapolate a trajectory beyond the neighborhood where the basis and operator were computed. It tests whether a slowly maintained radial shape is plausible under (4) alone.

Every positive-frequency doublet polarization is covered

At the real reference point, the real and imaginary metric blocks decouple. For a general doublet polarization the real metric is

$$G(q)=qG_u+(1-q)G_v,\qquad q=|a|^2\in[0,1]. \tag{8}$$

The relative phase drops out because the two chiral blocks are orthogonal. The source vector $f$ is the same for all $q$. Cauchy's inequality in the positive metric gives

$$f^TG(q)^{-1}f\ge\frac{f_d^2}{G(q)_{dd}}, \qquad \boxed{\frac{\lambda}{\Omega_{\rm source}(q)} \le\frac{\lambda\sqrt{\max((G_u)_{dd},(G_v)_{dd})}}{|f_d|}.} \tag{9}$$

This is a bound over the entire complex doublet, while retaining every centered deformation. It does not rely on a finite scan of polarizations.

At $\gamma=10000$, equal doublet populations give the following local source times relative to the wave time $1/\lambda$:

Each entry is $\lambda/\Omega_{\rm source}$. The analytic bound (9) is also much smaller than one for all maintained controls, including the original carrier. Choosing another doublet polarization does not restore an initially slow shape under (4).

The large-hopping limit has $\Omega_{\rm source}\sim|\gamma\kappa_d|/\sqrt{G_{dd}^{\infty}}$. Together with $\lambda\propto\gamma^{-(d-1)}$, this gives $\lambda/\Omega_{\rm source}\propto\gamma^{-d}$. The density-gradient carrier at depth ten enters this asymptotic regime later because its first adjacent integral is small. The full finite calculation above is retained; the asymptotic check uses higher hopping rather than claiming the leading expression is already accurate there.

Do not eliminate a slow radial response prematurely

For comparison, restrict a wave to a normalized trial ray $\Psi=a e^{-i\theta}\phi(\alpha)$, with potential energy $a^2\lambda^2$ in the explicitly retained action. Eliminating only its cyclic phase at fixed $\mathcal Q$ gives the exact Routh expression

$$L_{\mathcal Q}=\dot a^2+a^2g_{ab}\dot\alpha_a\dot\alpha_b +\mathcal Q A_a\dot\alpha_a-a^2\lambda^2 -\frac{\mathcal Q^2}{4a^2}, \tag{10}$$

where $g$ is that ray's positive projective metric and $A_a=i\langle\phi,\partial_a\phi\rangle$. Its instantaneous radial equilibrium is $a^2=\mathcal Q/(2\lambda)$, with radial oscillation frequency $2\lambda$. Substituting this equilibrium into $\dot a^2$ would produce the positive term $\mathcal Q\dot\lambda^2/(8\lambda^3)$. That substitution is a further slow approximation. It cannot justify slow geometry when the actual initial source acts on the much shorter times shown above. Equation (4) avoids that premature elimination.

The alternative action thus changes the signed response structure in a real way, but does not by itself provide the missing sustained particle geometry. A further contribution must be obtained from the same field and tested with its energy, canonical normalization and motion retained.

Verification

squared_wave_response.py constructs the complete positive normal quantum tensor. squared_wave_checks.py checks its positivity, its independent curvature contraction, the full initial acceleration, the source scaling, spatial refinements and the exact Routh algebra. Its 554-check report passes for all maintained indices and both surface signs, with three carrier/multiplier controls. The polarization bound (9) is additionally checked by squared_wave_polarization_checks.py. Its 936-check report verifies both pure chiral populations, mixed populations and the common analytic endpoint bound at every maintained control. Raw spatial tensors and executed source hashes are preserved separately.