Common cavity size and collective coupling

Summary: A common cavity radius and count-additive energy constrain the balancing terms of an inverse action, while its canonical probe response requires a separate count scaling that can be tested against actual spatial contacts.

Frequency scale and spatial scale

The proposed mass law supplies a frequency unit directly from the fixed particle assignments:

$$\begin{aligned} D_N&=\binom{N+d}{d},\\ N&=n-1,\\ \mu_d&=\frac{m(n,d)}{D_N},\\ \omega_{0,d}&=\frac{\mu_dc^2}{\hbar}. \end{aligned} \tag{1}$$

Different particles in one depth test whether this is a common unit. The mass-frequency conversion is already available. Identifying the unit with a cavity's spatial scale additionally requires its spatial operator. For example, if that operator gives

$$\omega_{0,d}=\gamma_d\frac{c}{L_d}, \qquad \frac{L_d}{L_e}=\frac{\gamma_d}{\gamma_6} \frac{\mu_6}{\mu_d}, \tag{2}$$

then the inverse mass-scale ratio is the length ratio when the geometric factors agree. Here $L_e$ denotes the electron-sector reference length and $\gamma_d$ is a dimensionless factor to derive from the geometry and wave operator. It cannot be independently retuned for each particle.

A common characteristic well size does not imply equal spatial extent of every occupied mode. Nor does the cavity terminology require an external wall, compactification, or particles occupying one shared location. The following inverse actions test the common-size premise; their origin in the same master field remains to be established.

Relative units at the fixed particle assignments

The existing mass-spectrum benchmarks give the following energy units $\mu_dc^2$ and inverse ratios. For the multi-entry quark and boson families, the table uses the geometric mean of their individual $m_ic^2/D_i$ values. The electron itself fixes the depth-six anchor. These are the current repository's stated benchmark inputs, not a new fit of particle indices or a new mass measurement.

The neutrino rows use the unchanged counts $(2002,11628,65780)$. Their separate units are obtained from

$$\begin{aligned} \mu_{5,\mathrm{solar}}c^2 &=\sqrt{\frac{\Delta m_{21}^2c^4}{D_2^2-D_1^2}},\\ \mu_{5,\mathrm{atm}}c^2 &=\sqrt{\frac{\Delta m_{31}^2c^4}{D_3^2-D_1^2}}. \end{aligned}$$

Their difference is the same finite bare-ratio discrepancy recorded in the mass-spectrum test. The tau row is the scale extracted directly from its benchmark; the proposed shared charged-lepton scale instead sets $\mu_{10}=\mu_6$ and retains the documented tau residual.

The last column becomes a physical length ratio only after multiplication by $\gamma_d/\gamma_6$ in (2). Conversely, one common length across all depths would require those geometric factors to carry the direct mass-unit ratios. This is a quantitative condition for the spatial operator, rather than an already measured hierarchy of particle sizes.

spectral_count_checks.py records the individual units and their family residuals using the exact current benchmark source.

An exact condition on two balancing energies

Write $D$ for the cumulative count and suppose a declared positive energy has two terms, with $A,B,a,b>0$:

$$E(D,L)=A D^rL^{-a}+B D^sL^b. \tag{3}$$

The first term grows as the cavity contracts and the second as it expands. The exponents describe this specified model; they do not prescribe the geometry of an IDWT sector. Its unique positive stationary size follows from $\partial_LE=0$. With $L_1=(aA/bB)^{1/(a+b)}$,

$$\begin{aligned} L_*(D)&=L_1D^{(r-s)/(a+b)},\\ E_*(D)&\propto D^{(br+as)/(a+b)}. \end{aligned} \tag{4}$$

The energy diverges at both ends of the positive length axis, so this stationary point is its minimum. A common size requires $r=s$. With this condition, linear energy requires $r=s=1$. Thus, within (3),

$$\boxed{r=s=1.} \tag{5}$$

Both contributions must then carry the same count factor. A fixed confinement energy added to an increasing number of oscillations generally changes the minimizing size.

Equation (5) is not a theorem about all possible actions. An additional term $D^2(L-L_0)^2$ has zero value and first derivative at $L_0$, but changes the restoring stiffness there. Such a term can preserve common size and linear stationary energy while changing the dynamics. Testing the response therefore adds information beyond fitting stationary masses.

Count and oscillator energy give different balances

At the unchanged electron and muon assignments,

$$\begin{aligned} (N_e,N_\mu)&=(12,34),\qquad d=6,\\ (D_e,D_\mu)&=(18564,3838380),\\ \frac{D_\mu}{D_e}&=206.76470588\ldots. \end{aligned} \tag{6}$$

Consider the comparison $E=A D/L+B L^6$, with a count-independent positive volume energy. Minimizing it gives a muon-to-electron radius ratio of $2.14181691$ and energy ratio of $96.53705902$. This action does not give the proposed common size or count-additive energy.

If the cavity contains equally occupied ordinary harmonic levels, the energy sum in oscillator-frequency units is instead the exact level moment

$$\begin{aligned} W_N&=\sum_{k=0}^N(k+d/2) \binom{k+d-1}{d-1}\\ &=d\left(\frac{N}{d+1}+\frac12\right)D_N. \end{aligned} \tag{7}$$

The two six-dimensional values are $246636$ and $123376500$, whose ratio is $500.23719165$. Replacing $D$ by $W_N$ in the contraction term gives radius ratio $2.42994570$ and energy ratio $205.86352708$. The latter is near the target ratio, but this is a different balance with a different radius. It does not establish the exact mass law.

The moving-cavity inertia calculation gives an exact count-additive translation coefficient for a specified quadratic kinetic law. Equation (7) explains why that result cannot simply be relabelled as the total energy of the harmonic cavity. A closed relativistic particle must include its confinement contribution and stress. The per-mode kinetic law and the occupied field content need their own derivation.

What a common kinetic gap would have to do

A large common gap is one possible source of nearly equal per-mode energies. Consider the declared positive spectrum

$$\begin{aligned} E_k&=\Delta\sqrt{1+\lambda(k+d/2)},\\ \lambda&=\frac{\Lambda}{\Delta^2},\qquad \Delta>0,\quad\Lambda\ge0. \end{aligned} \tag{7a}$$

Here $\Lambda$ has units of energy squared and the shell multiplicity is $\binom{k+d-1}{d-1}$. This is a spectral comparison. It is not asserted to be the full spinor spectrum of the proposed physical geometry.

Let $\overline E_N$ be the mean of these energies through degree $N$. Equal occupation gives total energy $qD_N\overline E_N$. For any $\Lambda>0$, adding higher shells strictly raises the mean. Thus the normalized energy is not exactly common at the electron and muon bands. At $d=6$ and small relative spread, the exact finite sums give

$$\frac{\overline E_{34}}{\overline E_{12}}-1 =\frac{66}{7}\lambda+O(\lambda^2). \tag{7b}$$

As a stated accuracy budget, requiring this departure to be at most $2\times10^{-5}$ gives the numerical finite-band condition

$$\begin{aligned} \lambda&\le2.12129418\times10^{-6},\\ \frac{\Delta}{\sqrt\Lambda}&\ge686.59328. \end{aligned} \tag{7c}$$

This is an illustrative tolerance, not an uncertainty-weighted experimental bound. No particle indices or wave amplitudes have been changed. The diagnostic evaluates the complete finite sums and checks independent concavity and Taylor bounds on their means.

The gap route can approximate a count-additive energy, but it requires a substantial separation between the common gap and the oscillator energy scale in this model. Its origin must be supplied by the same field. If the spatial scale were $\hbar c/\sqrt\Lambda$, it would also differ from the inverse gap scale $\hbar c/\Delta$. This is another reason to derive the spatial conversion in (2), rather than silently identifying every mass unit with the fundamental of an ordinary harmonic well.

Pressure of a gapped band

The gap approximation also needs a pressure test. Let $s=k+d/2$ and let $\langle\cdot\rangle_N$ denote the normalized cumulative oscillator average. Consider positive common coefficients in the energy

$$\begin{aligned} \mathcal E_N(L)&=D_N\left(\overline E_N(L)+B L^d\right),\\ \overline E_N(L)&=\left\langle\sqrt{\Delta^2+As/L^2}\right\rangle_N. \end{aligned} \tag{7d}$$

The support term is extensive in the count. Its equilibrium condition is

$$\begin{aligned} BdL^{d-1}&=\frac{A}{L^3}f_N(L),\\ f_N(L)&=\left\langle\frac{s}{\sqrt{\Delta^2+As/L^2}}\right\rangle_N. \end{aligned} \tag{7e}$$

At fixed $L$, the function of $s$ on the right increases strictly. Adding higher shells therefore raises that side, while the left side is common. Two different cumulative bands cannot have exactly the same equilibrium length under this specified supporting law.

For example, set $\Delta=1$, $A=10^{-6}$ and choose one common $B$ so that the electron band has $L_e=1$. Independent minimization gives $L_\mu=1.11676579$. The optimized energy ratio differs from the count ratio by only $8.32\times10^{-6}$, while the radii differ by 11.7 percent. Thus an accurate count-like energy does not ensure the common size. A supporting field with a different derived response may satisfy both conditions; this volume-energy comparison does not.

An extensive collective action with a common radius

A useful inverse family has a collective radius coordinate $R$ and action

$$\mathcal L_D=\frac D2\dot R^2-D f(R). \tag{8}$$

The displayed units absorb common kinetic and energy scales. If $f'(R_0)=0$ and $f''(R_0)>0$, every count has the same equilibrium radius, stationary energy $D f(R_0)$ and complete nonlinear equation

$$\ddot R+f'(R)=0. \tag{9}$$

For $f(R)=R^{-1}+R^6/6$, the equilibrium is $R_0=1$ and the small oscillation frequency squared is seven. Direct integrations starting at $R=1.03$, $\dot R=0$ for counts $1$ and $D_\mu/D_e$ agree in the actual, unscaled radius trajectory to $3.01\times10^{-14}$. The complete conserved energies differ by the count factor.

This supplies a consistent collective pattern to seek in the master-field reduction. It does not supply the reduction itself. In particular, $D$ in both terms of (8) is a declared hypothesis about the field content, not a fit of an eigenfunction amplitude.

The physical coupling must be normalized independently

Let a probe coordinate $z$ couple near equilibrium through $C_Dz\,\delta R$, where $\delta R=R-R_0$. Write the quadratic reciprocal action as

$$\begin{aligned} \mathcal L^{(2)}={}&\frac D2\dot{\delta R}^{\,2}+\frac12\dot z^2\\ &-\frac D2\omega_R^2\delta R^2-\frac12\omega_z^2z^2\\ &-C_Dz\,\delta R. \end{aligned} \tag{10}$$

Both responses follow from the same energy. In the canonical coordinate $x=\sqrt D\,\delta R$, the physical off-diagonal vertex is

$$g_D=\frac{C_D}{\sqrt D}. \tag{11}$$

This is kinetic normalization, not an arbitrary division of a measured response by mass. Eliminating the radius at frequency $\Omega$ produces a probe stiffness correction with magnitude

$$\left|\frac{C_D^2}{D(\omega_R^2-\Omega^2)}\right|. \tag{12}$$

For $C_D\propto D^p$, the three distinct scalings are:

Consequently $C_D\propto\sqrt D$ gives a common canonical coupling and probe response. The choice $C_D\propto D$ gives the same radius motion under a fixed drive, but a probe response proportional to $D$. Equal motion alone is not equal coupling.

The diagnostic verifies this distinction using both eigenvalues of the full reciprocal two-coordinate problem, independently computed with the original kinetic metric and in canonical coordinates.

A spatial contact supplies an instructive scaling test

There is a direct comparison with the previously derived spatial oscillator contact. For normalized Hermite modes of common length $L$, its central trace $K_{N,d}(0)=\sum_{|\alpha|\le N}|h_\alpha^L(0)|^2$ is

$$\begin{aligned} K_{N,d}(0)&=\pi^{-d/2}L^{-d}C_{N,d},\\ C_{N,d}&=\binom{\lfloor N/2\rfloor+d/2}{\lfloor N/2\rfloor}. \end{aligned} \tag{13}$$

As $N$ grows at fixed $d$, this behaves as $N^{d/2}$, whereas $D_N$ behaves as $N^d$. Hence $K_{N,d}/\sqrt{D_N}$ tends to a constant.

This scaling has an explicit conditional use. Take the positive occupied density $\rho_N=q\sum|h_\alpha^L|^2$ with common occupation $q$, a central density probe, and the collective kinetic term of (8). Its interaction $gz\rho_N(0)$ gives a radius vertex

$$\begin{aligned} C_N&=gq\,\partial_LK_{N,d}(0)\\ &=-\frac{d gq}{L}K_{N,d}(0). \end{aligned} \tag{14}$$

At common $L$, the normalized vertex therefore has the leading count behavior required by (11). The actual electron and muon central coefficients, after removing $\pi^{-3}L^{-6}$, are $84$ and $1140$:

$$\begin{aligned} \frac{g_\mu}{g_e}&=\frac{1140/84}{\sqrt{D_\mu/D_e}}\\ &=0.94381606\ldots. \end{aligned} \tag{15}$$

This is a 5.6 percent difference in this specified density-and-radius model, rather than an exact universal vertex. It does not identify a Dirac scalar or electromagnetic current with positive density.

Nested-sector marginalization changes the result

A probe occupying only $a$ shared ordinary coordinates of a containing $b$-dimensional sector samples a different density. Integrate the normal coordinates of the complete occupied density before applying the local probe. The bilinear marginal rule gives at the shared origin the value $k_N=K^{\rm marg}_{N,b\to a}(0)$:

$$\begin{aligned} k_N&=\pi^{-a/2}L^{-a}c_N,\\ c_N&=\sum_{r=0}^{\lfloor N/2\rfloor}w_r \binom{N-2r+b-a}{b-a},\\ w_r&=\frac{(a/2)_r}{r!}. \end{aligned} \tag{16}$$

The generating function for the dimensionless coefficients is

$$(1-t^2)^{-a/2}(1-t)^{-(b-a+1)}. \tag{17}$$

The first factor counts the even shared-coordinate values at the origin. The second counts every normalized normal mode and performs the cumulative sum. This is a density trace; integrating a linear wave and then squaring would be a different operation.

Equation (17) has leading growth $N^{b-a/2}$. After normalization by $\sqrt{D_b(N)}$, the vertex scales as $N^{(b-a)/2}$. Only $a=b$ has the preceding constant leading behavior. At the actual six-dimensional degrees, the conditional muon-to-electron canonical vertex ratios are:

These compare branches at the same containing depth, common width and common probe rule. The dimensional units of the probe coefficient change with $a$; the rows are not absolute couplings between different sectors.

The complete band trace is isotropic. It is a useful exact control, but does not test the polarized spatial patterns of an actual particle pair. Their spinor vertices, both complex polarizations, relative orientation and reciprocal spatial evolution remain necessary.

What the inverse test supplies

The localized current fixed point supplies a further concrete same-field inverse example. Its depth-six nodeless profile has root mean squared radius $1.549216\ldots c/m_c$, with the same profile at every occupied mass grade. If its common core coefficient is identified with $\omega_{0,6}$, this gives an explicit geometric factor in (2). That coefficient identification and a common unit canonical norm remain stated inverse premises; neither is derived by extracting a frequency unit from the observed masses. The full matrix action also gives the corresponding canonical inertia, while physical mode-carrier selection and complete stability remain open.

The dimensionless binding calculation extends this spatial fixed point to depths five and ten and supplies the factors $\gamma_5=1.44357039$, $\gamma_6=1.54921602$ and $\gamma_{10}=1.89885917$. With the explicit shared-gap and shared-frequency identifications, the depth-ten to depth-six size ratio is $1.22569038$. These are radii of the reconstructed field profiles; their measured interaction form factors and cross-depth vertices remain to be derived.

The common-size premise constrains confinement as well as oscillation. Equations (3)–(9) supply explicit balances with measurable collective consequences. Equations (10)–(17) show how the requested grouping of coupling responses imposes a further condition on the same reduction. The point-contact comparison is close to that condition within one full spatial sector, but its ordinary nested marginal does not retain it.

The spatial and moving-cavity response resolves these probes by momentum and energy transfer, retaining complex polarizations and consistently moving final modes.

The open derivation is a same-field action that supplies the count-additive kinetic and supporting energies, selects the occupied modes, and produces the physical normalized vertex. These calculations constrain that action; they do not establish a completed mass operator.

The surface-bending frequency calculation tests a concrete geometric restoring energy with the exact simplex eigenvalues. Its conformal scale remains undetermined, and ordinary tension changes the frequency ratios. The associated nonlinear mode response derives a scale vertex from that same energy and distinguishes its absolute canonical strength from a common fractional frequency shift.

cavity_balance_checks.py verifies the independent positive-energy minima, exact energy moments, full nonlinear breathing trajectories, reciprocal canonical spectra and the spatial marginal formula against direct normalized Hermite quadrature.