Dimensionless binding and depth-dependent cavity scales

Summary: A dimensionless binding principle selects the core exponent and yields localized depth-five, six and ten profiles with explicit common-size factors, while the gap-to-frequency relation and physical normalization remain inverse conditions.

The spatial scale of a continuously oscillating field

The localized current action separates two equations within one field law. Its current hierarchy supplies the carrier frequency, while its binding variation sustains a localized spatial profile:

$$\omega_N=\omega_{0,d}\binom{N+d}{d},\qquad \mathcal B(\phi)=0. \tag{1}$$

Mass is read from the selected frequency. The physical field magnitude enters the nonlinear spatial fixed point; it is not a second mass variable. At one common canonical norm $Q=1$, every grade at the same depth uses the same profile. Its full inertial coefficient is $\hbar\omega_N/c^2$.

This note derives a useful scaling condition on the binding term, solves its nodeless cores in five, six and ten ordinary spatial dimensions, and computes their geometric size factors. In this specified inverse model,

$$\boxed{L_{\mathrm{rms},d}=\gamma_d\frac{c}{m_c},\qquad (\gamma_5,\gamma_6,\gamma_{10}) \simeq(1.44357039,1.54921602,1.89885917).} \tag{2}$$

The common core coefficient $m_c$ has frequency units. Identifying it with $\omega_{0,d}$ supplies a concrete version of the cavity-size conversion. Their ratio is still an additional dimensionless input: dimensional analysis does not force the two frequencies to agree.

The spatial witness uses flat ordinary $\mathbb R^d$, without an external wall or compactification. It does not determine the global geometry of an IDWT sector. The matrix indices below are field components; they do not add physical coordinates or imply that particles contain other particles.

A dimensionless binding coefficient fixes the exponent

Use $\hbar=c=1$ until units are restored explicitly. The doubled field is $\Psi=(\phi,\zeta)^T$, where each block is an $r_d\times r_d$ matrix. Let $\alpha_a$ be the $d$ Hermitian spatial Clifford matrices and let $\beta$ anticommute with all of them. Define

$$\begin{aligned} A_a&=\sigma_x\otimes\alpha_a,\qquad B_1=\sigma_x\otimes\beta,\qquad B_2=I_2\otimes\beta,\\ J&=\Psi^\dagger B_1\Psi,\qquad K=\Psi^\dagger B_2\Psi,\\ U_{\mathrm{bind}}&=m_c\operatorname{Tr}(J\beta_p) -g\operatorname{Tr}\!\left[J\operatorname{sgn}(K)|K|^a\right]. \end{aligned} \tag{3}$$

The partner involution $\beta_p$ is numerically the same matrix as $\beta$ on the other index. This is the binding term of the existing inverse action, with all its spinor, partner and conjugate variations retained.

The canonical Dirac time term gives the field length dimension $[\Psi]=L^{-d/2}$. Both matrix currents have dimension $L^{-d}$, while an action density has dimension $L^{-(d+1)}$. Consequently

$$[g]=L^{da-1}. \tag{4}$$

If the binding coefficient is required to be dimensionless, this fixes

$$\boxed{a=\frac1d.} \tag{5}$$

This is a declared scaling principle for the inverse law. It does not derive the fractional interaction from the original master kernel. The common gap $m_c$ and the root-current frequency unit $\omega_{0,d}$ remain dimensional coefficients. The law's regularity at zero eigenvalues of $K$ also remains an evolution question.

For even $d$, the representation rank is $r_d=2^{d/2}$. At odd $d$ the same algebra requires $r_d=2^{(d+1)/2}$. A constructive depth-five choice takes five matrices from a six-generator Clifford representation and uses the remaining generator as $\beta$. That last generator is an algebraic insertion, not a sixth spatial derivative. The actual equation has exactly five ordinary spatial directions. A physical channel map between different depths is a further calculation, not supplied by this representation choice.

The full spatial equation and its auxiliary functional

The upper-only branch satisfies $\phi^\dagger\phi=\phi\phi^\dagger=\rho I$. Its complete lower-component equation is

$$\mathcal B(\phi)=D\phi+m_c\beta\phi\beta_p -g\beta\phi f(\phi^\dagger\beta\phi)=0, \qquad D=-i\alpha\cdot\nabla, \quad f(K)=\operatorname{sgn}(K)|K|^a. \tag{6}$$

It is useful to obtain (6) from a spatial functional. Define

$$\begin{aligned} \mathcal W[\phi]&=T+m_c S-V,\\ T&=\int d^dx\,\operatorname{Tr}(\phi^\dagger D\phi),\\ S&=\int d^dx\,\operatorname{Tr}(\phi^\dagger\beta\phi\beta_p),\\ V&=\frac{g}{1+a}\int d^dx\, \operatorname{Tr}|\phi^\dagger\beta\phi|^{1+a}. \end{aligned} \tag{7}$$

For fields with the stated decay and differentiable matrix currents, $\delta\mathcal W/\delta\phi^\dagger=\mathcal B(\phi)$. The derivative of the last trace is the full matrix derivative. This argument makes no restriction to variations of its eigenvalues alone.

The functional $\mathcal W$ is a tool for solving the spatial fixed point. It is not extra particle energy. In the full doubled action, the spatial kinetic term and both terms in (3) connect upper and lower blocks. Their values vanish on $(\phi,0)^T$, although their variations sustain (6). The full stationary energy remains $E_0=\hbar\omega_N Q$; adding $\mathcal W$ to it would be an error.

A virial identity shows why the common gap matters

Vary the entire spatial field by a constant real factor. Since $T$ and $S$ are quadratic while $V$ has degree $2(1+a)$, stationarity gives

$$T+m_c S=(1+a)V. \tag{8}$$

Independently use the norm-preserving spatial dilation $\phi_\lambda(x)=\lambda^{d/2}\phi(\lambda x)$. Under this variation, $T\mapsto\lambda T$, $S\mapsto S$ and $V\mapsto\lambda^{da}V$. Thus

$$T=daV,\qquad \boxed{m_c S=[1-(d-1)a]V.} \tag{9}$$

At the dimensionless exponent (5), these become

$$\boxed{T=V,\qquad m_c S=\frac Vd.} \tag{10}$$

For $g>0$ and a nontrivial profile in this class, $V>0$. A zero core gap cannot satisfy (10). This is an analytic restriction on finite-integral solutions of (6), not a nonexistence claim about all possible field actions or all particle geometries. It explains why the earlier zero-gap critical core instead used $a=1/(d-1)$.

At (5), $\mathcal W$ is flat along this one norm-preserving dilation of a solution because $T=V$. The dilated fields at fixed nonzero $m_c$ do not all solve (6). Flatness of this restricted functional is therefore neither a scale family of solutions nor a positive-energy stability proof.

Solve the profile without large central values

Set $m_c=1$ temporarily and choose the radial matrix ansatz

$$\phi(x)=f(r)I+i h(r)\alpha_{\hat x},\qquad \rho=f^2+h^2,\qquad r=|x|. \tag{11}$$

Clifford multiplication gives $\beta\phi f(\phi^\dagger\beta\phi)=\rho^a\phi$ and $\beta\phi\beta_p=fI-ih\alpha_{\hat x}$. Therefore (6) retains both radial components:

$$f'=-(1+g\rho^{1/d})h,\qquad h'=(g\rho^{1/d}-1)f-\frac{d-1}{r}h. \tag{12}$$

To calculate the shape, set its central value to one and solve for an unknown coefficient $p$ in place of $g$:

$$\begin{cases} f'=-(1+p\rho^{1/d})h,\\ h'=(p\rho^{1/d}-1)f-(d-1)h/r,\\ f(0)=1,\qquad h(0)=0. \end{cases} \tag{13}$$

Regularity gives $h(r)=(p-1)r/d+O(r^3)$ and $f(r)=1-(p^2-1)r^2/(2d)+O(r^4)$. The decaying linear tail has $f\propto r^{-\nu}K_\nu(r)$ and $h/f=K_{\nu+1}(r)/K_\nu(r)$, where $\nu=d/2-1$. The solver retains higher origin terms and uses this tail ratio as its outer boundary condition.

Shooting brackets a nodeless branch before boundary-value collocation refines the full profile. These calculations establish particular numerical solutions, not uniqueness or a classification of excited cores.

Let $q_d$ be the canonical norm of this central-normalized shape, including all matrix partners and the ordinary spatial angular measure:

$$q_d=r_d\,\frac{2\pi^{d/2}}{\Gamma(d/2)} \int_0^\infty r^{d-1}(f^2+h^2)\,dr. \tag{14}$$

Dividing the entire profile by $\sqrt{q_d}$ produces $Q=1$. The required single dimensionless coefficient for that depth is then

$$\boxed{g_d=p_d q_d^{1/d}.} \tag{15}$$

No coefficient or field magnitude is retuned for a different mass grade at the same depth. The computed values are:

The displayed precision summarizes numerical collocation and quadrature; it is not an experimental uncertainty. The six-dimensional norm and radius also agree with the earlier independently normalized core solver.

For an arbitrary fixed positive coefficient $g$, the same nodeless shape instead has

$$Q=\left(\frac{g_d}{g}\right)^d. \tag{16}$$

Thus the common $Q=1$ condition constrains a prospective master-derived coefficient. The calculation does not quantize $Q$ or derive the displayed $g_d$ values from one infinite-dimensional master law. These sustaining coefficients are not identified with observed electromagnetic or weak couplings. Their numerical closeness at depths five and six is not a discrete identity.

Restore the gap and retain the geometric conversion

Let $\phi_1$ be a unit-norm solution at unit gap in natural units. Equation (6) with (5) has the exact scaling law

$$\phi_{m_c}(x)=m_c^{d/2}\phi_1(m_c x),\qquad Q[\phi_{m_c}]=1. \tag{17}$$

The coefficient $g_d$ stays fixed. Every derivative and source in the spatial equation acquires the same factor $m_c^{d/2+1}$. This changes a common depth scale, not the masses of different grades by amplitude tuning.

To restore units with dimensionless $g_d$, write the spatial equation as $cD\phi+m_c\beta\phi\beta_p-cg_d\beta\phi f(K)=0$. The scaling argument uses $\kappa=m_c/c$ in place of $m_c$ in (17). Define

$$\gamma_d^2= \frac{\int_0^\infty r^{d+1}(f^2+h^2)\,dr} {\int_0^\infty r^{d-1}(f^2+h^2)\,dr}. \tag{18}$$

This proves the size conversion (2). If the additional inverse relation $m_c=\omega_{0,d}$ is imposed, the assigned electron and muon share $L_{\mathrm{rms},6}=1.54921602\,c/\omega_{0,6}$. Under the proposed shared charged-lepton frequency unit, the depth-ten to depth-six ratio is

$$\frac{L_{\mathrm{rms},10}}{L_{\mathrm{rms},6}} =\frac{\gamma_{10}}{\gamma_6} =1.22569038\ldots. \tag{19}$$

The existing tau mass residual remains unchanged. Across arbitrary depths, the general ratio also retains $m_{c,e}/m_{c,d}$, or $\omega_{0,e}/\omega_{0,d}$ when the same identification is made. These are conditional field radii. Their projection into a measured interaction form factor has not been derived.

The unchanged fixed counts are $(2002,11628,65780)$ at depth five, $(18564,3838380)$ at depth six, and $64512240$ at depth ten. Each can be put on the corresponding common spatial profile in (1). This does not identify the coefficient grades with actual cavity harmonics, select the occupied grades, or explain rare neutrino interactions.

The later pair and smooth-binding calculation shows that two such regular isolated currents can cancel. A smooth comparison removes that specific binding singularity while preserving the isolated simplex frequencies, but changes the core profile and size factor. Thus the numerical factors in this note belong to the stated fractional law; they are not universal consequences of the frequency counts.

Verification and physical scope

scaled_gap_core_checks.py checks 89 conditions. It increases the outer endpoints from 24 to 32 at depths five and six, and from 32 to 40 at depth ten, while tightening the collocation tolerance from $2\times10^{-10}$ to $2\times10^{-11}$. It verifies both virials using an independently integrated spatial kinetic derivative, all ordinary Clifford directions, fixed assignments, and the unit norm and radius scaling at gaps $0.4$, $1$ and $2.3$.

Independent Cartesian differences test the complete sustaining equation. Their residual is normalized by the core scale times the field norm, so a large particle-frequency coefficient cannot conceal a discarded lower-component spatial residual. Separate spacetime differences verify the complete boosted equation and every stress component at each depth. The raw data preserve the profiles, derivatives and full stress matrices. The solver is scaled_gap_core.py.

These results extend the localized fixed point and its common-size law. They do not extend the earlier finite depth-six stability tests to all angular modes or to depths five and ten. The original action's carrier requirements, the polarized propagation limitation of its proposed rest-frame completion, and nonsmooth-current evolution also remain. The outcome is a concrete inverse sustaining law and size conversion, with its remaining physical inputs stated explicitly.