The competing depth-ten branch and the self-coupling mass hypothesis

The shared rational reference law has two depth-ten local minima on the exact inverse-current reference frequency curve. Its inner minimum was fitted to the observed tau frequency unit relative to the maintained depth-six unit. The outer minimum is a separate candidate state of the same sector and index, not a new mode assignment. This note tests its occupied branch and asks whether a low selected frequency must imply weak self-coupling.

The frequency and its origin

At fixed depth ten, the common simplex factor cancels between the two radii. With $h=g(R)R/c$, the exact ratio is

$$\frac{\omega_{\mathrm{out}}}{\omega_{\mathrm{in}}} =\frac{R_{\mathrm{in}}}{R_{\mathrm{out}}} \sqrt{\frac{Z_{10}(h_{\mathrm{in}})}{Z_{10}(h_{\mathrm{out}})}}, \qquad Z_{10}(h)=\sum_{j=0}^{9}\frac{h^{2j}}{(j!)^2}. \tag{1}$$

For the fitted positive rational law with $b=1.34$, $C=18.00753138$, and $h_2=500$, the inner branch is at $R_{\mathrm{in}}/R_0=1.06721483$, $h_{\mathrm{in}}=116.63970$; the outer branch is at $R_{\mathrm{out}}/R_0=14.18017176$, $h_{\mathrm{out}}=568.77005$. Equation (1) gives $\omega_{\mathrm{out}}/\omega_{\mathrm{in}} =4.84170206\times10^{-8}$. Calibrating the fitted inner branch to the 1776.93 MeV tau benchmark gives an outer equivalent mass-energy of 86.03366 eV and a cyclic frequency of $2.08028\times10^{16}\,\mathrm{Hz}$. These are conditional values inherited from the fit, not an independent experimental mass measurement.

The large suppression has an exact spectral origin. Writing $Z_{10}(h)=h^{18}T_{10}(h)/(9!)^2$, where $T_{10}(h)\to1$ at large $h$, equation (1) becomes

$$\frac{\omega_{\mathrm{out}}}{\omega_{\mathrm{in}}} =\frac{R_{\mathrm{in}}}{R_{\mathrm{out}}} \left(\frac{h_{\mathrm{in}}}{h_{\mathrm{out}}}\right)^9 \sqrt{\frac{T_{10}(h_{\mathrm{in}})} {T_{10}(h_{\mathrm{out}})}}. \tag{2}$$

Thus a reference-gap increase of a factor about 4.88 is amplified by the ninth power. The outer radius is about 13.29 times the inner radius. The calculation does not establish that either radius is realized by a physical tau structure.

An independent shape grid also varies $b$ and $C$ separately by $\pm0.5\%$ around the reference values, retaining only the depth-two gap anchor and not refitting $C$ to the inner tau/electron unit. All nine trials retain two depth-ten reference-frequency minima; the outer one has positive radial frequency curvature and passes the necessary occupied-slope condition. Its energy after calibrating each inner branch to the tau benchmark spans 72.16–103.00 eV. The inner tau/electron unit ratio simultaneously ranges from 0.8353 to 1.1922 times its benchmark. The grid report therefore supports local branch existence, while the narrow 86.03348–86.03414 eV range under correlated refits is not an unfitted mass prediction. This finite grid does not establish full spatial stability or the physical probability of the state.

A two-channel route to the radius law

The fitted rational shape is exactly a sum of two positive response terms. Put $x=(R/R_0)^4$, $a=b^{-4}$, and $c=C^{-4}$. Then

$$\frac{1+ax}{(1+x)(1+cx)} =\frac{w_s}{1+x}+\frac{w_l}{1+cx}, \qquad w_s=\frac{1-a}{1-c},\quad w_l=\frac{a-c}{1-c}. \tag{2a}$$

Both weights are positive for the fitted $1<b<C$. At $b=1.34$, $w_s=0.68985012$ and $w_l=0.31014988$. The two terms contribute almost equally to the gap at the inner radius, whereas the long-scale term supplies 99.99238% at the outer radius. The two-channel audit checks the identity and the branch fractions for all four fitted shapes in its selected report. This makes a long-scale same-field response a concrete candidate source of the extra branch; it does not derive that response.

For example, let two retained same-field amplitudes $q_i$ have positive stiffnesses $D_s=1+x$, $D_l=1+cx$, and local energy $D_iq_i^2/2-J_iq_i+\beta_iq_i k$, where $k$ is the normalized current. Choosing $J_i=\beta_i=\sqrt{\gamma w_i}$ and eliminating these amplitudes gives

$$F_{\rm eff}(k,R)=F_0(k) -\frac{\gamma}{2}\sum_i\frac{w_i}{D_i(R)} +\gamma\sum_i\frac{w_i}{D_i(R)}k -\frac{\gamma}{2}\sum_i\frac{w_i}{D_i(R)}k^2. \tag{2b}$$

The coefficient of the linear current is exactly the fitted gap $g(R)=\gamma[w_s/D_s+w_l/D_l]$, with $\gamma=179.0387654$ for the reference fit. However, the eliminated response subtracts $g(R)$ from current curvature, so the direct same-field contribution must satisfy $F_0''>g(R)$ on a convex occupied branch. The conditional lower bounds are 109.294 and 40.110 in the source units at the inner and outer radii. This is a necessary sign test, not a derivation of $F_0$.

The eliminated response also contributes to the radial energy. At a fixed current $k$, its derivative is

$$\left.\partial_R E_{\rm response}\right|_k =2\gamma(1-k)^2R^3\left[ \frac{w_s}{(1+R^4)^2} +\frac{cw_l}{(1+cR^4)^2}\right]>0, \tag{2c}$$

in units where $R_0=1$. At $k=0$, the audit obtains 56.898 at the inner and 1.571 at the outer radius; the occupied currents change these values only slightly. The frequency minima therefore do not establish stationarity of the complete energy. Other same-field terms and shape response must balance this radial stress while preserving the occupied spectrum. The fourth power, channel scales, source amplitudes, canonical kinetic term and such a balancing term still require derivation from the master field.

The preceding expression is an uncompleted response per canonical norm. Its zero-current offset would shift the particle frequency by $-54.647$ at the inner radius and $-20.055$ at the outer radius in $c/R_0$ units for the equal-source choice. The corresponding occupied frequencies are only $5.4732\times10^{-5}$ and $2.6500\times10^{-12}$. This offset cannot be left out of the mass-operator calculation.

A completed-square same-field action

One homogeneous retained-mode alternative includes the zero-current normalization in a positive square. Per canonical norm, consider

$$\mathcal E=\langle H_0(R)\rangle+F_0(k) +\sum_{i=s,l}\left[ \frac{D_i(R)}2\left(q_i-\frac{J_i}{D_i(R)}\right)^2 +\beta_iq_i k\right], \quad J_i=\sqrt{r\gamma w_i},\quad \beta_i=\sqrt{\gamma w_i/r},\quad r>0. \tag{2d}$$

Here $q_i$ are retained modes of the same field, rather than independent particles. Their kinetic terms and actual spatial profiles have not been specified. Eliminating their static values $q_i=(J_i-\beta_i k)/D_i$ gives

$$F_{\rm eff}(k,R)=F_0(k)+g(R)k -\frac{g(R)}{2r}k^2, \qquad g(R)=\gamma\sum_i\frac{w_i}{D_i(R)}. \tag{2e}$$

The square cancels the uncompleted response's zero-current frequency offset and its associated zero-current radial stress. Its Schur contribution to current curvature is $-g/r$, so the direct term must supply $F_0''=f_{\rm target}''+g/r$ on an occupied inverse branch. Without square completion, reducing $r$ lowers the offset but increases the negative curvature: channel by channel, $2|E_{0,i}||F_{i,kk}|=g_i^2$. For general two-channel coefficients, Cauchy's inequality gives $2|E_0||F_{kk}|\ge g^2$.

The same-action batch tests whether one radius-independent $F_0(k)$ could at least match both sampled inverse-current bands. It maps target values and derivatives to $F_0$ at harmonics 5, 20, 22.5, 23, 23.5, 26 and 30 on each radius, plus $F_0(0)=F_0'(0)=0$. All adjacent secant slopes lie between their endpoint derivatives for tested $r=0.001,0.01,1,100$ and the zero-current matching ratio below. The selected report records the samples. These are necessary convex-interpolation checks, not a constructed smooth $F_0$, a full spatial state, or a test at the other depths.

The same proposed action has a radial force test. At fixed canonical norm and on the stationary component field, its derivative is

$$\partial_R\mathcal E =-\frac{\epsilon/R-Gk}{R} +g'(R)k-\frac{g'(R)}{2r}k^2, \qquad G=g(R)-\frac{g(R)}r k+F_0'(k), \tag{2f}$$

where $\epsilon$ is the dimensionless finite-chain singular frequency and $G$ is the physical current edge. At the inherited inner and outer radii, this derivative is positive for all five tested positive $r$. At $r=1$ it is $3.248\times10^{-6}$ and $4.749\times10^{-16}$ in the corresponding $c/R_0$ energy-per-norm units. This completed-square action therefore does not make either inherited radius an exact stationary state. At $r=1$ the outer residual is small relative to its frequency, leaving a nearby shifted equilibrium possible; the complete field, radius and metric Hessian has not been solved.

There is a sharper condition if the exact inverse current law must continue smoothly from each branch to $k=0$. One radius-independent direct term must then have the same $F_0''(0)$ at both radii:

$$f_{{\rm inv},\mathrm{in}}''(0)+\frac{g_{\rm in}}r =f_{{\rm inv},\mathrm{out}}''(0)+\frac{g_{\rm out}}r. \tag{2g}$$

The exact zero-current inverse formula gives physical curvatures $2.31215\times10^{12}$ and $2.69108\times10^{17}$ in $c/R_0$ units. Equation (2g) fixes $r=2.570865\times10^{-16}$, rather than leaving the source/current amplitude ratio free. Its sampled convexity inequalities still pass. But its fixed-norm radial forces at the inherited radii become $4.548\times10^6$ and $2.162\times10^{-9}$, respectively. Thus the simple completed-square action cannot simultaneously keep the inherited radii stationary and realize one smooth zero-current direct law under that full inverse continuation. A shifted solution, additional same-field stress, or a law that applies only on occupied finite bands remains possible. The magnitude is a diagnostic in this action's chosen normalization, not a measured force. The action is a candidate construction with specific force and spatial tests remaining, not a derived physical mass operator.

A local continuation of this action finds nearby stationary radii with positive restricted radial curvature on both depth-ten branches. Its scalar-current angular controls pass under an assumed nonnegative retained-mode gradient stiffness; the complete polarized field and geometric spatial variations remain open.

Absolute spectrum across the maintained sectors

The rational law was fitted to the depth-six electron and inner depth-ten tau frequency units; the other depths were only checked for local roots and necessary inverse slopes. To test the candidate as one common mass operator, fix the remaining physical scale $c/R_0$ with the observed electron mass-energy. With no further depth-dependent conversion, every assigned mode then has the definite prediction

$$m_{n,d}^{\rm trial}c^2 =m_ec^2\frac{u_d}{u_6} \frac{S(n,d)}{S(13,6)},\qquad S(n,d)=\binom{n+d-1}{d}. \tag{3}$$

The selected whole-spectrum calculation records all 16 maintained mode values under this one calibration. Representative results are:

The W discrepancy is over two orders of magnitude at the selected $h_2=500$ fit, and the neutrino discrepancy is much larger, so the common rational law with this reference normalization fails the full assigned mass spectrum. The neutrino bound cited is a measurement of an effective electron-neutrino mass, whereas the maintained depth-five assignments are model labels. The KATRIN result is an upper bound on an effective electron-neutrino mass, not a measured nonzero value or a lower bound on any neutrino mass. The MeV trial values are incompatible with treating these assigned modes as the observed light neutrinos without an additional, derived mixing or suppression mechanism. A dimension-dependent spatial conversion could change (3), but it would need to follow from the common field and geometric operator. Assigning a new fitted conversion to each depth would not repair this candidate as a predictive law for all known assignments. These comparisons do not exclude a separate, possibly unseen exotic state. They only limit the claim that this one fitted law also explains the named W and neutrino states. The outer depth-ten branch remains a mathematical solution with no derived selection or detection channel.

The same script refits $C$ at depth-two reference gaps 220, 300, 500, and 1000 with $b=1.34$. Across those four trials, the lightest assigned depth-five neutrino ranges from 2.286 to 10.389 MeV, and the assigned W from 0.791 to 337.6 TeV. This finite scan shows the failure is not confined to $h_2=500$. It is not a proof that no other common spatial law or normalization can work. The $h_2=220$ trial also has the independently observed depth-two $n=95$ finite spatial growth, so lowering that gap is not a verified stability remedy.

Occupied branch and local controls

The reproducible batch checks four neighboring shared-law shapes $b=1.332,1.34,1.35,1.36$. For each shape the one outer scale $C$ is refitted to the same inner tau/electron unit ratio, so this variation is a local fit sensitivity test, not an independent prediction. In the selected report, the conditional outer mass-energy ranges from 86.03348 to 86.03414 eV. All four outer points are strict local minima of the reference frequency curve and pass the assigned necessary occupied-slope inequality.

At $b=1.34$, exact finite-chain inverse roots exist at sampled harmonics 1, 5, 20, 23, 26, and 30 on the outer branch. Their edge and negative normalized current decrease in order, as required by the sampled convex inverse-current construction. At the assigned index 23, the outer normalized current is $-5.9474\times10^{-13}$. The inner branch current is $-4.5331\times10^{-6}$. The sampled controls do not prove invertibility across every intermediate point.

The inner depth-ten frequency minimum and its following maximum occur at $1.0672R_0$ and $1.2557R_0$. That maximum is only $1.02684$ times the inner minimum. The second minimum is at $14.1802R_0$, with a much lower frequency. This maximum is not a calculated particle decay barrier. The complete radius energy, supporting field and collective radius inertia have not been supplied. Changing $b$ over the four cases changes the frequency maximum over the inner minimum from 1.00222 to 1.14515, even though the outer frequency remains close under refitting.

The necessary occupied-slope limit for the assigned depth-ten index is $h>35.21386344$. Along the outward side of the $b=1.34$ outer branch, the shared law reaches this limit at $R/R_0=54.72550$. Thus the inverse occupied branch is not established out to arbitrarily large radius. The reference-frequency curve beyond this limit cannot be interpreted as the same physical particle branch without a different current continuation. Neither the local outer minimum nor this finite domain proves global binding.

The complete signed finite-component matrix at fixed geometry gives the nearest other component magnitude divided by the occupied frequency as 1.02401 for the inner branch and 1.000928 for the outer branch. Both exceed one. The local inverse-current curvature is positive on the sampled branch, so its rank-one real-component correction also has the favorable sign. This is a finite component control only: the outer branch has not passed a depth-ten spatial or full field–metric stability calculation.

Does low mass imply weak self-coupling?

In the proposed normalized-current density $\rho f_d(k)$, the same function controls both the frequency shift and a local current response. On the exact inverse branch, $f_d'(k)=G$ and $f_d''(k)=dG/dk$. A symmetric secant between harmonics 22.5 and 23.5 at depth ten gives $f_{10}''\approx4.1340\times10^4$ on the inner branch and $5.8948\times10^{10}$ on the outer branch, in the stated dimensionless $c/R$ normalization. The latter local curvature is about $1.426\times10^6$ times larger, although its oscillation frequency is about $4.842\times10^{-8}$ as large. The outer current magnitude is only $1.312\times10^{-7}$ of the inner one. Hence neither current magnitude nor current curvature alone is a universal mass proxy in this candidate. These derivatives are not a physical two-particle vertex or encounter rate.

There is nevertheless an exact conditional relation between the current action and a pair interaction. Suppose two patterns overlap at one position, their field components are pointwise orthogonal, and their cross current vanishes, so that their densities and currents add: $\rho=\rho_i+\rho_j$ and $j=\rho_i k_i+\rho_j k_j$. If both see the same local function $f$, their contact contribution relative to isolated patterns is

$$\Delta U_{ij}=(\rho_i+\rho_j) f\!\left(\frac{\rho_i k_i+\rho_j k_j}{\rho_i+\rho_j}\right) -\rho_i f(k_i)-\rho_j f(k_j). \tag{4}$$

For convex $f$, Jensen's inequality gives $\Delta U_{ij}\le0$, with equality when the currents agree or the law is affine between them. This is a negative mixed contribution to the stated generating density under the orthogonality assumptions. Its relation to complete pair energy and force requires the canonical action and both particles' spatial response. Actual patterns need not be orthogonal at each point, and their relative phase, geometry and reciprocal deformation can change the result. The inner and outer tau radii also use different local reference gaps, so (4) cannot be applied across those branches without deriving one spatial action that defines $f$ through the intervening field configuration.

A testable version of the self-coupling hypothesis asks whether one canonically normalized same-field interaction supplies both the selected self-frequency and the response between two occupied spatial patterns. For example, a scalar contact $U[\Psi]=(\lambda/2)\int(\Psi^\dagger\Psi)^2$ has a self term $(\lambda/2)\int\rho_i^2$. For mutually orthogonal field components at overlapping positions it also has a pair term $\lambda\int\rho_i\rho_j$. The common $\lambda$ does not make the self frequency equal to that pair coefficient: they involve different density integrals, canonical norms, and the stationary field equation. Relative polarization and phase can change the pair term for nonorthogonal patterns. This is an algebraic illustration, not a derivation of IDWT's particle profiles.

The neutrino and nucleon examples motivate different checks. Rare neutrino encounters in ordinary matter do not, by themselves, measure the neutrino pattern's self-coupling. Electron and muon weak vertices are closely constrained despite very different lepton masses, so a universal rule equating mass to an external vertex strength is not supported by those observations; the one-field theory would need to derive both quantities from state-dependent spatial patterns. See the Particle Data Group electroweak review. For a nucleon, energy stored in interacting quark and gluon dynamics accounts for most of its rest energy in a lattice QCD mass decomposition. That supports testing a composite field's complete energy against its constituents, but it does not identify an IDWT self-coupling coefficient or imply that every bound composite is heavier than the sum of the isolated constituents' total energies.

The exact inverse current interaction is quadratic-homogeneous in field amplitude, while a pure scalar quartic contact is fourth-order. The fixed linear projection obstruction in the preceding note still applies. A nonzero same-field background, nonlinear reduction, fixed norm, or additional master invariant must supply the needed effective interaction, its positive current curvature, and its spatial support. The rational gap shape can be written as a positive source divided by a positive stiffness, but choosing those radius functions to reproduce the fit is an inverse construction, not their derivation from the master field.

Observable channels

The 86.03 eV value is the equivalent rest energy of the outer candidate. It does not by itself predict a light-emission line at 86.03 eV. A transition from the fitted inner branch to the outer one would release approximately 1776.929914 MeV before recoil and any other products. Production or emission predictions require the same-field coupling vertex, conservation laws, line width, and a channel connecting the states. The current model gives none of those quantities. A spectral feature near 86 eV cannot be assigned to this branch by its energy alone.