Radiation balance selects a spectral shape, but leaves its action amplitude free
A charged harmonic probe driven by a common cubic electric-noise spectrum has mean energy approaching K times its frequency in a narrow-resonance limit. Radiation damping cancels the probe’s charge and mass in that limit. The coefficient K remains the chosen field amplitude: damping also admits the quiet field, and the effective model has no finite-energy cutoff-free limit at fixed damping. This is Q02’s conditional mechanism test, not a derivation of quantum structure. Written review is recorded separately in the Q02 review.
1. Physical premises and observable
Consider a one-dimensional dipole coordinate x with mass m>0, nonzero charge q, and natural angular frequency w>0, in SI units. Use the stable effective law
\[m\ddot x+m\gamma\dot x+mw^2x=qE(t),\qquad \gamma=\frac{q^2w^2}{6\pi\epsilon_0mc^3}>0.\]
The damping rate follows cycle-averaged dipole radiation: the Larmor power q² times mean acceleration squared divided by 6π epsilon_0 c³ equals m gamma times mean velocity squared on a harmonic orbit. It is an on-resonance, weak-damping approximation, with gamma/w much smaller than one; it is not a point-charge radiation-reaction equation valid at arbitrarily high frequency. For each finite cutoff below, the stable equation defines the tested model.
Supply a centered stationary Gaussian field, independent of probe initial data, with two-sided angular-frequency spectral density defined by
\[\langle E(t)E(0)\rangle=\int_{\mathbb R}\frac{d\omega}{2\pi} S_E(\omega)e^{-i\omega t},\qquad S_E(\omega)=\frac{K}{3\pi\epsilon_0c^3}|\omega|^3 \mathbf1_{|\omega|\le\Lambda},\quad \Lambda>w,\quad K\ge0.\]
K has action units. The cubic shape is a candidate radiation preparation; stationarity and the oscillator equation do not select it. The common field uses the same K and cutoff for all probes. A finite laboratory cutoff specifies a frame; no exact Lorentz invariance is asserted for this regulated model. The external harmonic potential, Gaussian reservoir and dipole approximation are supplied physical premises. Gaussian tails do not impose a hard speed bound.
Measure the ensemble mechanical action
\[J(t)=\frac{m}{2w}\langle\dot x(t)^2+w^2x(t)^2\rangle.\]
This is mean oscillator energy divided by angular frequency, not the earlier long-window displacement observable. For a deterministic undamped orbit it is (1/2π) times the closed phase-space integral of p dx. No phase or commutator postulate enters its definition.
2. Exact stationary response of the effective model
At fixed positive gamma and finite Lambda, homogeneous solutions decay. The stationary forced solution has finite position and velocity variance; finite-second-moment initial transients vanish. Thus J(t) tends to J_* as physical time tends to infinity. In the underdamped regime the homogeneous amplitude decays as exp(-gamma t/2); the reservoir determines the limiting covariance, rather than that covariance being fixed by the damping alone. Fourier response gives the exact model identity
\[J_* =\frac{q^2}{2mw}\int_{-\Lambda}^{\Lambda}\frac{d\omega}{2\pi} \frac{(\omega^2+w^2)S_E(\omega)} {(w^2-\omega^2)^2+\gamma^2\omega^2}. \tag{1}\]
For the supplied spectrum this becomes
\[\frac{J_*}{K}=\frac{\gamma}{\pi w^3} \int_0^\Lambda \frac{\omega^3(\omega^2+w^2)} {(w^2-\omega^2)^2+\gamma^2\omega^2}\,d\omega \quad(K>0).\tag{2}\]
The prefactor uses q²/(3π epsilon_0 c³)=2m gamma/w². All quantities in (2) are dimensionless after integration: gamma/w³ has units of time squared.
3. The cancellation and its order of limits
Take gamma/w to zero with w, K and finite Lambda>w fixed, after reaching the stationary state. This can be implemented by reducing q²/m while retaining the same imposed field. In a neighborhood of omega=w, put u=omega-w. The numerator in (2) approaches 2w⁵ and its denominator is w²(4u²+gamma²) to leading order. Since
\[\int_{-\infty}^{\infty}\frac{du}{4u^2+\gamma^2} =\frac{\pi}{2\gamma},\]
the resonant contribution to the integral is πw³/gamma. Outside any fixed neighborhood of w the integral stays bounded as gamma tends to zero; its prefactor tends to zero. Taking that neighborhood smaller after the damping limit proves
\[\lim_{\gamma\downarrow0}J_*=K.\tag{3}\]
More generally, for a fixed even field spectrum continuous at w and integrable on a finite cutoff, the same argument gives
\[J_*\longrightarrow\frac{3\pi\epsilon_0c^3S_E(w)}{w^3}.\]
Consequently requiring the same limiting J_* over a band of oscillator frequencies requires S_E(w) proportional to w³ on that band. This explains what the radiation clock contributes beyond mass-only composition: it selects a spectral shape conditional on equal actions, and cancels q and m. It does not dynamically enforce equal actions or set the common amplitude. The convergence is pointwise in probe parameters, with resonance strictly inside the cutoff; no uniformity over all frequencies or arbitrarily small masses is claimed. Dipole and nonrelativistic validity also require appropriate probe amplitudes, for example Kw/(mc²) much smaller than one in the resonance limit.
Taking q to zero at finite observation time instead removes forcing and damping, retaining the initial oscillator energy. Relaxation time diverges as gamma tends to zero. The stationary-first limit is essential.
4. Quiet preparation and ultraviolet countertests
For any fixed cutoff and probe parameters, replacing E by aE replaces K and J_* by a²K and a²J_*. The deterministic damping and natural frequency do not change. At K=0 the damped oscillator approaches rest; arbitrarily small positive K gives arbitrarily small positive stationary action. A nonzero reservoir spectrum supplies positivity, but no lower bound independent of preparation. Even a symmetry condition on field covariances that preserves multiplication by a positive constant cannot fix that constant.
Cutoff removal gives a separate obstruction in this effective model. At large omega the integrand in (2) is asymptotic to omega. At fixed gamma>0,
\[\frac{J_*}{K}\sim\frac{\gamma\Lambda^2}{2\pi w^3} \quad(\Lambda\longrightarrow\infty).\tag{4}\]
Thus stationary energy diverges for K>0. Position variance separately has a logarithmic ultraviolet divergence; velocity variance supplies the quadratic energy divergence. Taking weak damping first at every finite cutoff gives (3), whereas removing the cutoff first diverges. A joint limit needs control of the ultraviolet tail as well as the narrow resonance; (3) supplies none. Equation (4) diagnoses failure of extending the viscous approximation and the cubic preparation together. It is not a no-go theorem for an extended-charge, causal field-plus-particle model with frequency-dependent response.
5. Strategic decision
Radiative response is a useful physical explanation for a conditional mass/charge cancellation and an energy-frequency relation. This linear, externally maintained reservoir fails to select a positive universal action: its amplitude is freely rescalable, its quiet solution is admitted, and its cutoff-free extension fails. Gaussian classical states persist; the calculation supplies no quantum measurement or phase structure and no physical energy gap.
Park linear radiation-balance variants. A return requires an explicit dynamical reservoir/backreaction law that breaks amplitude rescaling and excludes the quiet state, with a controlled ultraviolet response. The next mechanism test should ask whether autonomous nonlinear energy transfer can supply that missing selection, using its zero-field solution and stability as the first countertest. No prescribed nonzero noise amplitude may stand in for that test.
Source recovery used B23, Zwanzig’s equations (16)–(17), (23), (25)–(30): forcing, friction and reservoir preparation must be accounted for separately. These supply the reduction perspective, not the radiation spectrum. The composition note §4 supplies the preparation-label comparison. Radiation-specific source coverage and written proof status are in the review; this note is a completed exploratory test and has no new accepted claim ID.