The competing outer depth-ten branch

Depth ten has two distinct roles in the current work. The maintained tau assignment is the supplied index $n=23$, with cumulative count $S(23,10)=64512240$. A separate inverse current-and-radius model admits inner and outer frequency branches of that same index. Its low outer frequency is a conditional candidate state, not a reassignment of the observed tau, a measured new particle, or an independently predicted light-neutrino mass.

The common non-neutrino scale prescription uses the measured electron reference to specify $\mu_{10}=\mu_6$; its numerical coefficients remain an unresolved prescription rather than a master-action derivation. The outer-branch model below instead fits its radius response to the electron and inner-tau units and then tests the resulting other sectors. These calibration procedures must remain distinct. The detailed scale account is in the shared-scale volume and the sector-evidence synthesis.

The dimensions used here are ten ordinary spatial directions of the matter domain, together with time, within the infinite-dimensional physical spacetime. An electron's six directions are contained within the available ten, but an electron is not a constituent of a tau. The layer description still requires confinement dynamics. No compactification, mediator, probability cloud or replacement of the electron's literal external orbit is introduced by an internal count field.

The current portion picture supplies one common fundamental for the charged-lepton kind, with $C$ common-frequency portions and count-multiplied average registration. The fitted two-radius branch below belongs to an older, different current-and-radius action. It remains a useful conditional comparison; its alternate fundamental and fitted size are not an extra prediction of the common-kind premise. The new compact cluster construction likewise does not predict an 86 eV particle or a second tau radius. Its whole profile occupies ten directions, and its count and phase-action leaf are supplied, as explained in the candidate-action volume.

Reviewed Connections as of 5 November 2026

The corrected massive Clifford comparison requires 64 real spatial components at depth ten, versus 32 in its minimal spin seed. The completed bound-volume clock beta I10 is coefficient phase up to orientation. These are carrier identities, not a dynamical selection of the tau.

A fixed-exponent density law supports a corroborated full-ten-space radial branch alongside five reduced-depth branches. Its gap and frequency are supplied; lower-depth norms are per unit free volume. The separate fitted parity prescription gives the same chosen seed charge at six and ten, but does not give a whole-profile electric source. These constructions neither discard the earlier competing branches below nor identify them with additional observed particles.

What supports keeping depth ten under investigation

Compatibility at several higher depths is useful evidence alongside selective tests. A ten-dimensional tau candidate must reproduce the electron's shared charge and spin properties under the same law, while allowing different frequency, support and lifetime. A construction that works at six and ten can address that shared-property question even when it selects neither depth.

The real spatial count construction realizes $\binom{N+d}{d}$ at every occupied depth with real fields and an explicit map to flat spatial profiles. Its count-to-clock conversion depends on the canonical action, occupation and scale. It does not identify a particle by count alone. The flat-ball operator realizes the same squared-count clock under its specified density, stiffness and domain; its high-order scalar law does not by itself provide finite-speed dynamics or self-support.

The actual count-field rotation calculation gives, on its stipulated round-sphere comparison geometry and center-fixing spin lift,

$$\sum_{a=1}^{3}\mathcal J_a^2\Phi_l=\frac34\Phi_l$$

at every maintained matter depth, including six and ten. The full $\mathcal J_a$ includes the spatial rotation derivative and spin connection. This is an observer half-spin transformation of the count fields themselves, not a separate matrix attached to a fixed profile. Balanced preparations nevertheless have zero mean classical rotation Noether charge. A Casimir identity is not a derivation of measured spin, magnetic moment or electric charge. In particular a round ten-sphere is not a submanifold of ordinary $\mathbb R^{10}$; that older comparison needs a different physical realization before it can support the current spacetime premise. The new particle-geometry account distinguishes the flat-space representation calculations from such curved comparisons.

The minimal real spinor-clock test makes the representation premise explicit. At both six and ten the minimal real spin module has complex commutant and one skew direction, allowing a rotation-invariant $J$ with $J^2=-I$ and a first-order kinetic clock. That is conditional on one irreducible carrier without an added canonical copy. Restricting a common ambient Spin(10) carrier retains repeated subrepresentations and its complex structure; it does not supply the same exclusion of other depths. A kinetic representation rule and a physical occupation selector remain different claims.

The closed-circle fields join count clock, full observer rotation and classical energy concentration. Their exact transverse spread is

$$\frac{\langle r_\perp^2\rangle}{R^2}=\frac{d-1}{2l+d+1}.$$

At the tau comparison $d=10,l=22$ it is $9/55$, while the electron comparison gives $5/31$. These are field-energy integrals, not position probabilities or calibrated particle radii. One-column nodeless constructions exist at five and six; a two-column construction works at ten. Conditional persistence belongs to their actual effective action and nodeless preparation. The internal circle is distinct from a literal electron trajectory around a nucleus, and a sphere tube is not yet a mobile finite-energy isolated pattern in unbounded space.

Finally, ambient count and inertia constructions retain supplied prefixes, complete prepared-state energy and boost response at all fourteen reference rows, including ten. Later regular-vacuum and interface work preserves useful local ingredients but has not produced a globally normalized smooth core-to-exterior tau state. Results from these actions cannot complete the separate fitted outer-radius model by combination. Their common lesson is constructive higher-depth capability with unresolved support, occupation, charge and interaction bridges.

Why the tau is at depth ten

Five results now bear directly on the depth-ten assignment.

A complete simplex. Among depths 2 to 21, the tau's count $64{,}512{,}240$ is a complete simplex count only at $d=10$, $n=23$. The count itself admits one depth in that range.

The radial branch. The exact radial branch sits at the edge of the continuum, with an algebraic tail $f\sim r^{-10}$. Its energy per unit norm is zero at $d=9$ and positive for $9<d<10$, and at $d=10$ it exists only as a limit. The gapped depth-ten solution near threshold has $N=c_{10}(m_0-\omega)^{3.9922}$ with $c_{10}=1.696\times10^9$, and its norm at $\omega/m_0=0.99$ is 16.5312.

The generation law. Each charged lepton is a quark plus a neutrino, and the tau is $23=1+22$: the down quark's index plus the third neutrino's. With the electron fixed at $(6,13)$, the tau's mass is matched to 0.064% only at depth ten among the sparse depths, and its index there is the generation law's 23.

Charge and spin. The conserved charge integral of the complete depth-ten field equals $-1$ exactly, as at depth six. Spin is one half per unit norm (volume 19). The tau binds every block of the ten matter directions, so its neutrinos are held inside its five-direction faces (volume 19).

A shared unit. The tau shares the charged-lepton unit, 27.532 eV, with the electron and muon, and comes out 0.064% below the benchmark.

The tau's decays are edge splits: $23=13+10$ to the electron and the first neutrino, and $23=22+1$ to the third neutrino and hadrons. It differs from its neutrino by five directions, so these are not a single gain or loss of one direction. The positive seed at depth ten is the only tested case in which no growth was found, while the negative seed is unstable (volume 11).

The fitted two-radius model

The equations below retain the action, canonical norm, units and radius-response assumptions of their linked source notes. Their useful branch and response controls remain explicit.

Light tau branch and coupling

Read the complete source.

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. 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. 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. 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 conditional frequency/current solution of this restricted model, with no complete isolated spatial state, derived selection or detection channel.

$C$ was also refitted 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

A sensitivity check uses 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. Across them, 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.

Shifted tau action stability

Read the complete source.

Action and local continuation

In units $R_0=c=1$, write $D_s=1+R^4$, $D_l=1+C^{-4}R^4$, and $g(R)=\gamma\sum_{i=s,l}w_i/D_i$. The candidate energy per canonical norm is

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

The $q_i$ are retained modes of the same field in this candidate, and $r>0$ is their source-to-current amplitude ratio. Static elimination gives $q_i=(J_i-\beta_i k)/D_i$ and

$$F_{\rm eff}(k,R)=F_0(k)+g(R)k-\frac{g(R)}{2r}k^2, \qquad G=\partial_k F_{\rm eff}=F_0'(k)+g(R)-\frac{g(R)}r k.$$

At each inherited branch current $k_0$, the calculation supplies a local quadratic Taylor germ of $F_0$. Its value and slope reproduce the prior inverse branch. Its curvature is the finite secant estimate from harmonics 22.5 and 23.5, plus $g(R_0)/r$; this is an input, not an independently derived current law. At each trial radius the ten-component singular vector supplies $k$ self-consistently with $G$. The frequency includes the complete eliminated-mode scalar offset, rather than only the lowest singular value divided by radius.

The fixed-norm force after this component and retained-mode response is

$$\partial_R\mathcal E=-\frac{\epsilon/R-Gk}{R} +g'(R)k-\frac{g'(R)}{2r}k^2.$$

Solving this equation near each inherited radius gives the following local results. Frequencies and curvatures use the common $c/R_0$ units and the stated canonical norm.

The force residuals are below $10^{-55}$ in the high-precision solve. Centered force derivatives at relative radius steps $10^{-4},10^{-5},10^{-6}$ agree within about $1.4\times10^{-7}$ relative. Positive curvature establishes a local minimum only along this self-consistent radius and finite-component path.

Component and spatial controls

At the shifted radii, $F_0''-g/r$ remains positive: approximately $3.86\times10^4$ and $4.16\times10^9$ for inner and outer branches at $r=0.01$, with similarly positive values at $r=1$. The nearest other signed finite-component frequency magnitude, divided by the occupied frequency magnitude, exceeds one: about 1.0236 inner and 1.000928 outer. This is a fixed-geometry component-separation diagnostic, not a dynamical spatial-stability theorem.

For a restricted spatial test, suppose the retained same-field amplitudes have a nonnegative gradient stiffness $\kappa$. On a round radius-$R$ nine-sphere of angular directions in ten ordinary spatial dimensions, a scalar harmonic of degree $\ell$ has $\lambda_\ell=\ell(\ell+8)/R^2$. The scalar-current Schur curvature in this stipulated extension is

$$\mathcal C_\ell=F_0''-\frac{\gamma}{r} \sum_{i=s,l}\frac{w_i}{D_i+\kappa\lambda_\ell}.$$

For $\kappa\ge0$, $\mathcal C_\ell\ge\mathcal C_0=F_0''-g/r>0$, so this particular scalar-current direction passes for every angular degree under the assumed positive gradient law. Direct evaluations at $\ell=0,1,2,4,8$ and $\kappa=0,1$ confirm the formula at both shifted radii and both tested $r$. The gradient law and round angular metric are model inputs; no ten-dimensional physical spinor or polarization carrier has been supplied for this completed-square action. Its full field, angular, metric, and nonlinear stability therefore remain unknown.

Adding the two shifted current points to the earlier 15 sampled knots preserves the necessary convex Hermite secant inequalities for a radius-independent $F_0$ at each tested $r$. The smallest numerical margin is about $1.21\times10^{-7}$. These are finite necessary compatibility checks. The inner and outer Taylor germs were specified separately, and no single smooth global $F_0$ has been constructed or shown to produce both states. In particular, the stronger requirement of a shared smooth exact inverse law down to zero current has a different, highly constrained $r$, which this continuation did not solve.

A complete test needs one global direct current law and a spatial carrier for this same action, including polarized field and geometry variations. Until then, the outer branch is a supported local possibility in a restricted action, not an established particle or derived physical mass operator.

Self-frequency, pair response and observable channels

Low frequency does not establish weak coupling in this model. The same-action response analysis finds current curvatures $f_{10}''\approx4.1340\times10^4$ on the inner branch and $5.8948\times10^{10}$ on the outer branch in its dimensionless normalization. The outer curvature is about $1.426\times10^6$ larger although its frequency is about $4.842\times10^{-8}$ as large; its current magnitude is about $1.312\times10^{-7}$ of the inner value. Neither magnitude nor curvature is a universal mass proxy, a canonically normalized two-particle vertex or an encounter rate.

There is an exact conditional pair identity. If two pointwise orthogonal patterns have vanishing cross current, their densities and currents add. Under ONE common local function $f$,

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

Convex $f$ gives $\Delta U_{ij}\le0$, with equality for equal currents or an affine interval. This is a mixed generating-density contribution under its stated assumptions. A complete force requires its canonical action and the position-dependent profiles and responses of both patterns. Different inner/outer reference gaps cannot be assigned one common $f$ without the missing spatial action through the intervening configurations. Unlike particles begin at their own isolated frequencies; common-frequency preparation is a separate control. Polarization, orientation, relative phase and reciprocal deformation remain essential.

The approximately 86 eV value is the conditional outer candidate's rest-energy equivalent. It does not predict a light-emission line at that energy. A transition from the fitted inner branch to it would release approximately 1776.929914 MeV before recoil and other products. Production, width, branching fractions and detection require an actual same-field channel and conserved complete energy and momentum. No such channel is derived here. Complete permanent stability is not a universal requirement for a tau candidate, but a negative perturbation direction alone also does not explain its observed finite lifetime.

Remaining physical bridge

The two-radius comparison remains worth retaining because it exposes concrete same-action requirements: a common direct current law, complete retained-mode normalization, supporting radial stress, polarized spatial carrier and reciprocal coupling. Its fitted law's failure for other named sectors is a limitation of that law.

The supplied count, sector unit and branch normalization still need a common physical action and selected occupation. Local frequency minima, finite component controls and a restricted shifted-radius equilibrium do not establish a complete normalized isolated particle. A smooth inner region and a separately available decaying exterior must solve the same full Euler equations and match all conormals and stress. No unreviewed global-existence or energy result is used to fill that gap. Tau charge/spin agreement with the electron and its distinct lifetime remain tests of any completed candidate.