The remaining lengths and coupling in the frequency unit

The curvature-moment rule removes an independent spatial function. It does not fix the remaining dimensional scales. Restoring their units makes the missing physical input explicit and gives a useful constraint on the charged-lepton comparison.

Mass is frequency in equivalent units throughout. This calculation asks which physical geometry and coupling produce that frequency; it does not rederive the conversion $mc^2=\hbar\omega$.

Restore the lengths already present in the carrier

Let $R_d$ be the tangential length parameter and $L_d$ the normal profile's length unit. These are ordinary spatial lengths, common to the particle modes at one depth. Write the dimensionless normal coordinate as $y=x/L_d$.

The physical coefficient in the curvature-moment rule has inverse-length units:

$$h_{\rm phys}=\frac{z_{\rm phys}B_{\rm phys}}2 =\frac1{L_d}h(y). \tag{1}$$

With a common wave speed $c$ and a dimensionless physical coefficient $g_d$, the frequency matrix therefore has diagonal entries $c(\beta+j_z)/R_d$ and adjacent entries $cg_dh_l/L_d$. Equivalently,

$$\boxed{\gamma_d=g_dR_d/L_d,\qquad \omega_{n,d}=\frac c{R_d}\lambda_{n,d}(\gamma_d).} \tag{2}$$

The normal separation parameter is $R_d/L_d$. This is a restoration of the existing parameters, not an additional adjustable length for each particle. In the large-hopping limit the common unit is

$$\boxed{\omega_{0,d}= \frac{c\,d!}{g_d^{d-1}\prod_lh_l} \frac{L_d^{d-1}}{R_d^d},\qquad \omega_{n,d}=\omega_{0,d}\binom{n+d-1}{d} [1+O(\gamma_d^{-2})].} \tag{3}$$

Thus a measured common frequency unit constrains a combination of the lengths, coupling and computed spatial integrals. It does not determine all of them separately. The geometric factor is fixed by the operator, and cannot be dropped when comparing inferred sizes across depths.

A common radius, width and coefficient fail the six/ten comparison

As an optional stronger trial, set $R_{10}=R_6$, $L_{10}=L_6$ and $g_{10}=g_6$. Then both matrices have the same $\gamma$ and the physical frequency ratio equals their dimensionless singular-value ratio.

There is an upper bound valid at every $\gamma\ge0$, not only in the asymptotic approximation. The maintained anchor indices are $n_e=13$, $n_\tau=23$. Take the central six-by-six block of the depth-ten triangular matrix, using normal labels 2 through 7. Its diagonal magnitudes, ordered increasingly, are $25,26,\ldots,30$, whereas the depth-six magnitudes are $13,14,\ldots,18$.

The computed five adjacent integrals of this central block each exceed every depth-six adjacent integral. Reversing and transposing a negative surface-sign matrix orders its diagonal magnitudes without changing its singular values. The same comparison therefore covers all four choices of the two surface signs.

The absolute entries of a triangular inverse are products of its adjacent coefficients divided by products of its diagonal coefficients. Consequently the central inverse block dominates the depth-six inverse entrywise up to the factor

$$C=\prod_{l=0}^5\frac{25+l}{13+l}. \tag{4}$$

The norm of a nonnegative matrix is monotone under entrywise increase. The norm of the full inverse is at least that of its central submatrix. Therefore

$$\boxed{\frac{\lambda_{\tau,10}}{\lambda_{e,6}}\le C,} \tag{5}$$

which is far below the assigned simplex ratio

$$\frac{S(23,10)}{S(13,6)} =\frac{64512240}{18564}\simeq3475.12605. \tag{6}$$

The positive-sign pair has the sharper bound $\prod_{l=0}^5(23+l)/(13+l)=20.29411765$. These are comparisons between separate particle matrices. They do not identify one particle solution with a part of the other.

This excludes the simultaneous equal-radius, equal-width, equal-coupling trial within the current carrier. Equal radius alone is not excluded; the normal length or coupling may depend on depth. The IDWT premise of one common size for all modes at a given depth is also unaffected.

Conditional radius requirements are computable

For illustration, retain a common $L$ and $g$ across the two depths but allow their radii to differ. Put $r=R_{10}/R_6$. Then $\gamma_{10}=r\gamma_6$, and the positive-sign anchor equation is

$$\boxed{ \frac{\lambda_{23,10}(r\gamma_6)} {r\lambda_{13,6}(\gamma_6)} =\frac{S(23,10)}{S(13,6)}.} \tag{7}$$

The small singular value strictly decreases as positive hopping grows: the nonnegative absolute triangular inverse increases entrywise. Thus the left side of (7) decreases from infinity to zero with $r$, giving a unique conditional radius ratio at each stated $\gamma_6$.

The leading expression is

$$r_{\rm lead}= \left[\frac{10!}{6!} \frac{\prod_lh_l^{(6)}}{\prod_lh_l^{(10)}} \frac1{\gamma_6^4}\right]^{1/10}. \tag{8}$$

Full finite-matrix solutions give:

These values impose the assigned anchor ratio under the extra common $L,g$ assumption. They are not a forward derivation of actual particle sizes. They also do not remove the finite within-depth correction: at $\gamma_6=10000$, the computed muon/electron ratio is about $1.10\times10^{-5}$ below its ideal simplex ratio. Matching one cross-depth anchor cannot silently eliminate that remaining correction.

The physical kernel must still determine the shared lengths and coupling and satisfy the normal-complement control needed for the projected spectrum. This dimensional accounting also leaves the supporting state's energy and coherent motion unresolved. Neither issue is a mass–frequency separation.

Verification

curvature_scale_checks.py checks the entrywise spatial comparison, the analytic inverse bounds at multiple hopping strengths and all surface-sign choices, the exact conditional radius roots, and the dimensional reparametrization. It uses the maintained indices and verified spatial integrals. It does not scan alternative particle assignments or fit a separate coefficient for each mass. The 50-check report records these comparisons and the hashes of the input spatial report.