Fixed-norm response as an eigenvalue contribution

A fixed canonical norm is a natural way to prevent the support calculation from turning particle masses into adjustable amplitudes. It changes the question, however. The response no longer has to be transparent in the radial field direction, because that direction is excluded from the constrained tangent space. Its radial derivative instead enters the Lagrange multiplier, which is the standing frequency. This note determines what that means for the current positive response model.

The constrained multiplier

Let a reference mode satisfy $Q[\phi_i]=1$, and write a radial extension as $\Psi=\sqrt{s}\,\phi_i$, so $Q[\Psi]=s$. The physical stationary value $s=Q_i$ need not be the same for every solution. Even at fixed global $Q_i$, the local amplitude can vary with the spatial profile. Suppose the bare energy on this ray is

$$E_{0,i}(s)=s\omega_{0,i}. \tag{1}$$

This equation does not identify amplitude with mass. It states the usual linear dependence of the energy on canonical norm for a mode whose eigenvalue is $\omega_{0,i}$. Let a positive response $z$ have energy

$$E_z=\frac12\langle z,Az\rangle -\gamma\langle z,S(\Psi)\rangle, \qquad A>0, \qquad S(\sqrt{s}\phi_i)=sS_i. \tag{2}$$

Eliminating $z$ gives

$$E_{\mathrm{eff},i}(s) =s\omega_{0,i}-\frac{\gamma^2}{2}s^2K_i, \qquad K_i=\langle S_i,A^{-1}S_i\rangle>0. \tag{3}$$

Stationarity of $E-\omega(Q-Q_i)$ at $s=Q_i$ therefore fixes the standing frequency to

$$\boxed{\omega_i=\omega_{0,i}-\gamma^2Q_iK_i.} \tag{4}$$

The fixed-norm constraint removes the radial amplitude direction from the constrained Hessian. It does not remove the response effect. It converts that effect into a stationary phase-frequency shift. That shift affects the nonlinear eigenproblem, but its identification with physical mass needs an additional complete-energy calculation. In the units of this reduction,

$$E_{\mathrm{eff},i}(Q_i)-Q_i\omega_i =\frac{\gamma^2}{2}Q_i^2K_i>0.$$

Thus even at unit canonical norm the energy differs from this phase multiplier. The mass-equivalent frequency remains $\omega_m=E_{\rm rest}/\hbar=mc^2/\hbar$. Establishing that a particular phase rate equals it requires the complete same-field supporting action, its conserved normalization and its energy–momentum response. See mass and energy response.

Amplitude remains in the dynamics: it can affect the phase frequency through $Q_i$, and changes of the normalized spatial profile can change $K_i$. The same amplitude and profile can enter coupling overlaps. Neither the amplitude alone nor this unconsolidated phase multiplier establishes mass.

Equal-norm control against the simplex spectrum

For the current algebraic controls,

$$(S_i)_\ell=\frac{i+\ell+d}{i+2\ell+d+1}, \qquad K_i=\sum_{\ell=2,4,\ldots} \frac{(S_i)_\ell^2}{\mathcal L_d(\ell)}. \tag{5}$$

The converged values of $K_i$ lie between approximately $0.400$ and $1.225$ over all maintained controls. They vary far more slowly than the simplex count

$$\mathcal S(i,d)=\binom{i+d-1}{d}. \tag{6}$$

For the diagnostic choice $Q_i=1$, if a bare simplex spectrum $\omega_{0,i}=\Omega_{0,d}\mathcal S(i,d)$ is to remain a simplex spectrum after the response, with one possibly renormalized scale $\Omega_d$, equation (4) requires

$$\frac{K_i}{\mathcal S(i,d)} =\frac{\Omega_{0,d}-\Omega_d}{\gamma^2} \tag{7}$$

to be independent of $i$ at fixed depth. The tested ratios fail that condition at every depth containing more than one maintained mode. Their largest-to-smallest spreads are:

The one-mode depth-ten sector cannot test proportionality. At the other depths, a nonzero common coupling changes the simplex ratios rather than renormalizing their common frequency unit. The response alone also cannot generate the count law, because $K_i$ is not proportional to $\mathcal S(i,d)$ under these equal-norm frozen-source assumptions.

If the dynamics selects unequal $Q_i$, the corresponding condition is that $Q_iK_i/\mathcal S(i,d)$ obey one common sector rule. Choosing $Q_i\propto\mathcal S/K_i$ after the fact would merely move the spectral fit into amplitude. Deriving such amplitudes from the same stationary equations would instead be a legitimate nonlinear result. The present calculation does not decide between those possibilities because it holds the normalized source vectors fixed.

What the constraint does and does not solve

The canonical constraint is still useful. It removes the radial direction from perturbations at a specified conserved $Q_i$ and turns the support problem into an honest nonlinear eigenvalue problem whose output is a frequency. It does not require equal peak amplitudes, equal profiles or even equal conserved norms unless the common action supplies that rule.

The present equal-norm source controls are not sufficient as that eigenvalue problem. Preserving the established simplex law requires the complete amplitude-dependent Rayleigh contribution to follow one common spectral rule. A compensating mode-dependent bare term would merely move the fit into the bare operator.

The useful next target is therefore a constrained same-field action in which the spatial operator, source, response and amplitude are derived together. At each depth, the particle harmonics must follow one common field law and shared fundamental parameters. That law may generate different sizes, effective stiffnesses, conserved norms, amplitudes and profiles. Equal size is a testable subclass; independent per-particle retuning does not derive a spectrum. The resulting phase eigenvalue remains a spectral diagnostic until its relation to complete rest energy and inertia is established.

Verification

constrained_response_spectrum_checks.py evaluates the infinite even-harmonic norms at 50 and 80 decimal digits, checks the fixed-norm multiplier identity and compares each response norm with its simplex count. The normal and optimized runs each pass 69 checks. The selected report records the converged shifts and ratio spreads. The source components remain algebraic controls, not measured densities or a derived master-field source.