Current mass-operator candidates

The mass problem is to explain how one same-field law produces and sustains the known particle frequencies, their spatial structures and their interactions. Frequency already expresses mass in equivalent units, $mc^2=\hbar\omega$. An assigned frequency, a normalized coefficient vector, a supported core and a complete finite-energy particle are different achievements. Current constructions establish several of these steps conditionally, but no action below yet supplies the whole chain.

The candidates include common-fundamental portion actions, count-feedback actions, finite current hierarchies and material-plus-vacuum constructions. The newer portion route has compact support and a proposed relativistic cluster Hamiltonian; the older routes supply explicit count-frequency equations, polarization-sensitive responses, reference cores, local exits and constructed exteriors. Each has its own kinetic law, geometry and domains. A theorem for one does not establish the same result for another.

Physical spacetime has infinitely many ordinary spatial dimensions plus time. Matter occupies up to ten spatial dimensions; all remaining spatial directions are unbound for every particle. Field components and coefficient spaces do not add physical dimensions. A lower-depth pattern requires a confinement mechanism. No compactification, Kaluza–Klein tower or mediator particle is used. Classical internal field oscillations do not replace an electron's literal six-dimensional trajectory around a nucleus; its proposed external orbit using four spatial directions is a separate motion question. See physical premises, spatial carriers and orbits and binding.

Reviewed Connections as of 5 November 2026

The current comparison now includes several further independently reviewed candidates:

The case's combined-law theorem is a common-action overlap. It still does not supply the entire observed mass-operator chain. Its count bounds do not transfer to the separate radial density law, nor do older cluster boost results transfer into either candidate.

A common-fundamental portion candidate

In the current physical proposal, $C=S(n,d)$ portions share a fundamental angular frequency $\Omega$. Their combined average tick rate, expressed in angular units, is $C\Omega$. Each portion need not have a different resonance, and equal phase spacing is a sufficient special preparation rather than a requirement for additive average counting. Neither a count nor an external tick rate alone determines a canonical Hamiltonian.

One explicit new constitutive choice retains the compact parent's canonical one-form and changes its translation Hamiltonian to

$$H_C=\sqrt{c^2|p|^2+B_C^2},\qquad B_C=\Omega[P+\hbar(C-1)]+U.$$

Here $P$ is total canonical phase action, $p$ is total translation momentum, and $U$ contains relative kinetic, shape, polarization and binding contributions after the stated reference subtraction. They are not interchangeable with the oscillator count. The physical reference leaf $P=\hbar$ is supplied. If $U=0$, then $B_C=C\hbar\Omega$; the resulting mechanical mass agrees with the stipulated count-frequency value. This is an explicit proposal with a checked consequence, not a derivation of its constitutive term or of the action leaf.

The compact-support result and finite shape bound give $U\ge0$ on a restricted admitted shape/velocity class. Attainment additionally requires a reference compatible with all fixed charges; an aligned rank-one reference cannot replace a state with a different conserved matrix charge. In the finite rigid family, a nonnegative pair well can have zero value and gradient at integer-distance configurations. Those include non-simplex arrangements, so the construction does not select a desired $C$, index or depth.

The newer connectivity proposal replaces the fixed component offset by a smoothed graph spectral count. Its connected plateau supplies $C-1$, while disconnected components supply $C-K$. The graph is to be computed from physical field portions; it adds no spatial manifold. Its link law, spectral gap and same-field derivation remain assumptions. Independent review of this newest proposal is pending.

1. The common frequency target and the inverse constraints

With particle index $n\ge1$, let $N=n-1$. The cumulative count is

$$D_{N,d}=S(n,d)=\binom{N+d}{d},\qquad \omega_{n,d}=u_dD_{N,d}.$$

The binomial identity counts available coefficient or spatial channels under the chosen realization. It does not by itself derive their occupation, equal energy, a physical support or a feedback law that makes the count the frequency multiplier. The sector unit $u_d$, action normalization and spatial support must remain explicit. The maintained controls are:

The electron and muon share a depth-six sector unit; neither is that fundamental unit. The non-neutrino numerical prescription uses the electron reference and proposed sector coefficients. Its fitted history remains distinct from a forward derivation. The neutrino unit has a separate solar-splitting calibration. Bottom does not have an established ordinary integer assignment; its mass-count and mixing-count targets cannot both be realized by the unchanged undressed sparse carrier. Tau's depth-ten mass comparison does not resolve its conflict with a universal adjacent-depth transition rule. These qualifications survive every candidate below; see shared frequency scales, outer depth ten and neutrino controls.

Earlier inversions of mass, mixing and clock comparisons impose a different set of construction requirements. Nested cumulative bands and an approximately flat occupied spectrum are required by their stipulated isotropic projection and local energy-clock interpretation. These are conditional targets for those models, not requirements already derived for common-frequency portions. For an equipartitioned orthonormal band with kernel $K(x,x)=\sum_j e_j(x)^2$ and measure $w(x)dx$, its energy density is

$$h(x)=\varepsilon K(x,x)w(x).$$

If a local clock law $|\Omega(x)|=h(x)/\sigma_0$ is imposed, one coherent clock requires constant $K(x,x)w(x)$ on participating support, and

$$E/|\Omega|=\sigma_0V.$$

Thus the identification with one action unit requires $\sigma_0V=A_0$. Homogeneous support can supply a constant kernel under its stated geometry; arbitrary bounded ball or Gaussian-band profiles do not. A cumulative homogeneous physical carrier and its dynamics still have to be constructed. A collective clock is a distinct conditional branch. Neither branch permits a nonzero constant participating energy density over infinite volume with finite total energy. The current regular-vacuum construction therefore uses a material-only clock with energetic Cartesian readouts outside participation. See localization and support.

2. Count-feedback actions: an explicit frequency mechanism on supplied geometry

The forward count action begins with a specified ordinary round spatial sphere of radius $R$. Its positive Dirac shell has dimensionless label $h_l=l+d/2$, multiplicity per seed $\binom{l+d-1}{d-1}$, and cumulative band count

$$P_d(h_l)=\sum_{k=0}^{l}\binom{k+d-1}{d-1} =\binom{l+d}{d},\qquad P_d(\chi)=\frac1{d!}\prod_{j=1}^{d}(\chi+j-d/2).$$

The sphere and its operator are additional spatial premises. Their usefulness does not establish a sphere as the physical particle or compactify the containing space.

Using signature $(+,-,\ldots,-)$ for this proper-current comparison, let $\Phi$ be a commuting matrix-spinor field and define scalar current $s=\operatorname{tr}\bar\Phi\Phi$, proper current magnitude $n_c=\sqrt{j^\mu j_\mu}$, and $k=\sqrt{n_c^2-s^2}$. On the chart $n_c,k>0$, introduce an algebraic feedback coordinate $\chi_c$ and write

$$\mathcal L=\mathcal L_{\rm Dirac}-V_d,\qquad V_d=u_dn_cP_d(\chi_c)-\frac{\chi_c}{R}k.$$

The auxiliary coordinate has no propagating degree of freedom. Variation gives $u_dR P_d'(\chi_c)=k/n_c$. On a balanced positive/negative shell pair with real polarization $c$, the complete radius-scaled energy per canonical action $Q$ is

$$RE/Q=h_lc+u_dRP_d(\chi_c)-\chi_cc.$$

The two independent stationary equations are

$$\chi_c=h_l,\qquad c=u_dRP_d'(h_l),$$

so the full field frequency and energy satisfy

$$\omega=E/Q=u_d\binom{l+d}{d}.$$

Spatial kinetic energy is retained before variation; its cancellation on this stationary pair is a consequence of the equations. The count constitutive term was reconstructed by inverse research, so this is a forward consequence of an explicit effective action, rather than a derivation of that term from a fundamental law. A diagnostic radius keeps $0<c<1$; it is not a derived size or measured field amplitude.

The dispersion-count envelope is a separate preferred-time continuation. At each eligible count node let

$$c_l=u_dRP'_d(h_l),\quad m_l=\frac{h_l\sqrt{1-c_l^2}}{c_l},\quad \eta_l=u_dRP_d(h_l)-h_l/c_l,$$ $$F_l(\chi_c)=\sqrt{\chi_c^2+m_l^2}+\eta_l.$$

A compatible smooth maximum of these strictly convex curves has the desired count value and slope and open exact winning arcs. Unique auxiliary elimination produces affine scalar-source neighborhoods. With its specified positive-wave kinetic completion, complete energy is nonnegative and a conditional nonlinear spatial persistence argument applies at all fourteen reference assignments. The finite feedback-label window, nodeless preparation and specified sphere remain essential. The same persistence conclusion does not transfer to the raw proper-current Dirac action, whose complete spatial instabilities have a different scope. Neither version selects only the observed occupied labels or a free localized body in the containing space.

These effective constructions are useful because they show exactly where geometry, constitutive interpolation, canonical norm and kinetic law enter. An operator with the desired eigenvalues is insufficient if its physical propagation, source derivatives or canonical energy belong to a different action. The operator toolkit and energy, inertia and causality keep those distinctions explicit.

3. A common material action retains actual charge and volume together

A more spatially explicit candidate uses ten material labels $\phi^A$ as components of the same field. In signature $(-+\cdots+)$, define

$$B^{AB}=\partial_\mu\phi^A\partial^\mu\phi^B,\quad n_m=\sqrt{\det B},\quad u^\mu=J^\mu/n_m,\quad h=B^{-1},\quad P^{\mu\nu}=g^{\mu\nu}+u^\mu u^\nu,$$

where $J$ is the ten-label cofactor current with future orientation. These expressions are evaluated only on a legal positive Gram chart. They describe material response within the proposed field; the labels are not additional spatial coordinates.

Let $b=v/|v|$ be the finite normalized carrier and $C(b)$ its actual participation/occupation functional. At a prepared uniform prefix $C=D_{N,d}$, while off-prefix occupation penalties and their full derivatives remain part of the action. The response law is

$$F(j)=\nu+\alpha j^2+\Lambda j^4,\quad H=CF,\quad A_c=CF',\quad c=CF'',\quad B_e=C(F+2F'^2/F''),\quad K=c-A_c^2/B_e,$$ $$\lambda=\frac{K}{4[c^2+(726/35)(A_c^2+c^2)]},\quad \tau=4R^2[(18514/315)\lambda(A_c^2+c^2)+K],$$ $$\beta=\frac{\nu\ell^2}{16(1+\operatorname{tr}M_2)},\quad \kappa=\nu/16,\qquad M_2=\sum_k |b_k|^2kk^T.$$

For $s=e^\rho$, the material clock/response terms include

$$sD\theta-sH-B_e(s\log s-s+1) +jD\psi+\frac\lambda2(D\psi-A_c)^2+\frac\tau2(Dj)^2, \qquad D=u^\mu\partial_\mu.$$

Their horizontal current and carrier gradient energies and the shape energy must also be retained. Actual phase charge is not simply $j$: on a rest slice its response charge is $p_\psi=j+\lambda(\dot\psi-CF')$. In the material canonical comparison, writing the charge departure as $y_\psi=p_\psi-j$ and the positive theta-charge density as $r$ gives a rest density of the form

$$C\left[rF+\bar B_e(r\log r-r+1) +F'y_\psi+\frac{y_\psi^2}{2\bar\lambda}\right]$$

plus complete static storage. Here $\bar B_e=B_e/C=F+2F'^2/F''$ and $\bar\lambda=C\lambda$. The particular small common response coefficient, not merely $\lambda>0$, supplies the local convexity needed for the supporting-plane estimate. On the stated fixed-charge canonical class this gives $E\ge A_0D F(j_0)$. Equality also satisfies the full reference field, embedding and free-traction equations. It is not unrestricted nonlinear stability or attraction.

The shape is varied through actual horizontal carrier gradients,

$$q_{AB}=\ell^2\operatorname{Re}\sum_k(E_A b_k)^*E_Bb_k, \quad E_A^\mu=h_{AB}\partial^\mu\phi^B, \quad \chi_s=\frac1{4(1+\operatorname{tr}M_2)},$$ $$S_q=I+\chi_sq,\quad T=(\det S_q)^{-1/10}S_q, \quad V_s=\frac\kappa8\operatorname{tr}[(T^{-1}h-I)^2].$$

Thus $\det T=1$ under all variations, including carrier relaxation. At a prefix $q=M_2$, and an affine reference map $\phi=T^{-1/2}x$ has common physical volume. Both finite bodies and all finite reference widths are ten-dimensional in this construction even when the occupied carrier label is $d<10$. Strong anisotropy at fixed volume is not a derived localized lower-dimensional sheet. The origin prefix has zero moment and isotropic support.

For a common reference ball normalization

$$c_0=\frac{A_0}{V_{10}R^{10}},\qquad V_{10}=\pi^5/120,$$

the exact pressureless core has

$$q_\theta=c_0,\quad q_\psi=c_0j_0,\quad h=c_0H, \quad H=Du_d,\quad F(j_0)=u_d.$$

Conditional integration over the full reference volume gives $Q_\theta=A_0$, $Q_\psi=A_0j_0$, $E=A_0H$. Whole-reference boosts then give positive translation inertia equal to that complete reference energy. The retained material core time/static forms and both unit-cone forms are strictly positive under the common shape law at all reference rows. This statement excludes the absent Cartesian amplitude and algebraically locked readout dynamics on the core plateau. A reference-body result does not prove the energy, inertia or evolution of a smooth body plus exterior. See same-field actions and energy, inertia and causality.

4. One regular-vacuum continuation of the material law

The regular common candidate combines the material expression $\mathcal C_+$ with Cartesian components

$$W=(\phi/R,v,j,Z,Y),\quad \Phi=|\phi/R|^2,\quad A=|v|^2,\quad \mathcal N=|W|^2.$$

Here $Z,Y$ are real two-component readouts locked to $e(\theta)=(\cos\theta,\sin\theta)$ and $e(\psi)$ in the material region. The readouts retain regular kinetic energy outside it. The material expression includes the clock/current terms above, carrier kinetics, $-\tau P(dj,dj)/8$, carrier connection storage, $-V_s$, and the lock potential $g_l(|Z-e(\theta)|^2+|Y-e(\psi)|^2)/2$. It also includes horizontal phase stiffness

$$-\frac{c_0n_m}{2}[\gamma_\theta P(d\theta,d\theta) +\gamma_\psi P(d\psi,d\psi)],$$ $$r_{\rm ph}=\frac{sH^2/B_e+\lambda A_c^2}{H},\quad f_{\rm ph}=\frac1{16(1+r_{\rm ph})},\quad \gamma_\theta=f_{\rm ph}s/B_e,\quad \gamma_\psi=f_{\rm ph}\lambda.$$

A flat smooth gate $\chi\in[0,1]$ activates the legal material chart. With $S(t)=0$ for $t\le0$, $S(t)=1$ for $t\ge1$, and the standard flat exponential interpolation between them,

$$\Gamma(v)=S(4(A-1/2))S(4(3/2-A)) S((v_0-\epsilon_0)/\epsilon_0),$$ $$\chi=\Gamma(v)S(4(\det B-1/4)),\qquad \epsilon_0=1/(8\sqrt{64512240}).$$

Material expressions are smoothly zero-extended on inactive collars. The current common action is

$$\boxed{\mathcal L_3=\chi\mathcal C_++(1-\chi)\mathcal V +p_g\chi(1-\chi),}$$ $$\mathcal V=-\frac{k_v}{2}\partial W\cdot\partial W-U_0-U_x, \quad U_0=\frac{k_vm_v^2}{2}\mathcal N+\frac{\lambda_v}{4}\mathcal N^2, \quad U_x=\frac{k_{\rm mix}}4(\Phi-A)^2, \quad k_{\rm mix}=4k_vm_v^2.$$

The square is a common constitutive test input. It changes BOTH opposite-sign label/carrier forces and the derivative-gate load. Its addition leaves the regular zero-vacuum quadratic masses and open-plateau core identities unchanged; it does not prove a positive complete derivative-gated Hamiltonian.

One fixed prescription is $\nu=u_{5,\rm solar}/2$, $\alpha=\nu$, $\Lambda\in\{0,\nu\}$, $R=\ell=1/\nu$, $A_0=\hbar$ supplied, and $p_g\in\{0,c_0\nu\}$. Let $M$ be twice the largest absolute value of either prepared readout rate divided by $\nu$, across all rows and both $\Lambda$ branches. Then

$$k_v=\frac{c_0}{\nu M^4},\quad m_v=\nu M,\quad \lambda_v=\frac{c_0\nu}{M^4},\quad g_l=c_0\nu.$$

Both roots of $F(j_0)=u_d$ are retained:

$$j_0=\pm\sqrt{(u_d-\nu)/\alpha}\quad(\Lambda=0),$$ $$j_0=\pm\sqrt{\frac{2(u_d-\nu)}{\alpha+\sqrt{\alpha^2+4\Lambda(u_d-\nu)}}} \quad(\Lambda>0).$$

No coefficient is retuned by particle, root, radius or boundary trace. The common gaps obey $m_v^2-\omega^2,m_v^2-\Omega^2\ge3m_v^2/4$. The sector scales, finite carrier, occupied reference prefixes and $A_0$ bridge remain supplied.

5. Full variation, local exits and actual exterior solutions

The gate depends on label derivatives. For every real field $y^I$, put $Q=\mathcal C_+-\mathcal V+p_g(1-2\chi)$. The exact conormal is

$$\Pi_I^\mu=\chi(\mathcal C_+)_{I,\mu} +(1-\chi)\mathcal V_{I,\mu}+Q\chi_{I,\mu}.$$

All coefficients, inverse labels, shape and gate derivatives are varied. In particular $U_x$ adds $U_x\chi_{I,\mu}$ to the conormal. Its derivative-gate time Hessian can change at a determinant transition even though $U_x$ itself has no derivatives. The true algebraic density equation is

$$s(D\theta-H-B_e\rho) -\frac12[\gamma_{\theta,\rho}P(d\theta,d\theta) +\gamma_{\psi,\rho}P(d\psi,d\psi)]=0$$

where $\chi>0$. The shortcut $\rho=(D\theta-H)/B_e$ is valid only when the horizontal squares vanish.

On the exact origin subclass $b=e_0$, use $\phi=g(r)e_r$, $v=a(r)e_0$, real $j,z,y$, $Z=z(r)e(\omega t)$, $Y=y(r)e(\Omega t)$. The physical measure is $(\pi^5/12)r^9dr$, irrespective of the nominal depth-three origin assignment. On a positive material chart $l=g'>0,t=g/r>0$, $n_m=lt^9$; the stationary density shell reduces exactly to $\rho=(\omega-F)/B_e$. The five radial Euler equations retain all gate conormals and the label angular load. They close the omitted carrier and angular first variations, so this is a consistent subclass.

There are smooth local exits from an exact pressureless core at any finite positive radius. At the material edge $a_*^2=3/4$, the negative amplitude slope is forced by the full action; it is not a free shooting parameter. The changed mixed potential changes that slope through its gate load. Fixed right readout traces below one avoid the known obstruction at trace one, but do not prescribe the outgoing derivatives required by an exterior. On the separately constructed short actual origin exit, where $\varepsilon=1-\chi>0$, proper density reduction leaves strict cone bounds for the retained material/empty blocks, while five normalized Cartesian blocks are exactly luminal. The full exit cone statement is therefore nonnegative rather than strict. At the core plateau the amplitude is absent and the readouts obey algebraic locks; they are not five propagating luminal core blocks. Normalization by $\sqrt{\varepsilon k_v}$ has a vanishing weight at the rank-changing edge, so this short-interval result is neither a healthy full determinant-transition theorem nor endpoint-uniform evolution control.

The exterior has a stronger constructive result. Write $f=(w,a,j,z,y)$, $w=g/R$, $\mathcal N=|f|^2$. The exact permanently inactive vacuum equations are

$$w''+9w'/r-9w/r^2=[m_v^2+\nu^2\mathcal N+4m_v^2(w^2-a^2)]w,$$ $$a''+9a'/r=[m_v^2+\nu^2\mathcal N-4m_v^2(w^2-a^2)]a,$$ $$j''+9j'/r=(m_v^2+\nu^2\mathcal N)j,$$ $$z''+9z'/r=(m_v^2-\omega^2+\nu^2\mathcal N)z, \quad y''+9y'/r=(m_v^2-\Omega^2+\nu^2\mathcal N)y.$$

For any $b>0$, a smooth finite-energy exterior exists for every finite boundary trace satisfying

$$\boxed{w_b^2\le3/4,\qquad a_b^2\le1/2,}$$

with arbitrary finite $j_b,z_b,y_b$. It is obtained by unrestricted minimization of the exact vacuum functional

$$k_v\int_b^\infty r^9\left[ \frac{|f'|^2}{2}+\frac{9w^2}{2r^2} +\frac12\sum_i m_i^2f_i^2 +\frac{\nu^2}{4}\mathcal N^2+m_v^2(w^2-a^2)^2\right]dr,$$

where $m_w=m_a=m_j=m_v$, $m_z^2=m_v^2-\omega^2$, $m_y^2=m_v^2-\Omega^2$. Weighted coercivity and local one-dimensional compactness prove attainment without assuming large-field convexity.

The bound follows by jointly clipping $w,a$. With $x=w^2,y=a^2$, the mass-plus-mixed potential divided by $m_v^2$ is $P(x,y)=(x+y)/2+(x-y)^2$. If both coordinates exceed the bounds, $x=3/4+u,y=1/2+v$, then

$$P(x,y)-P(3/4,1/2)=u+(u-v)^2\ge0.$$

The one-coordinate clipping cases also decrease it. Derivative energy and angular energy do not increase, and the positive $\mathcal N^2$ term decreases. Every unrestricted minimizer therefore remains in the rectangle. It satisfies the full Euler equations with no obstacle reaction. Since $a^2\le1/2$, the material gate and all its derivatives vanish along the exterior. At equality this is flatness of the step, not an open inactive neighborhood. The interior inequalities are strict, giving actual nonreactivation. A separately proved total-norm domain $|f_b|^2<1/2$ is included as a useful simpler case.

The exact summed identity is

$$\frac12(\mathcal N''+9\mathcal N'/r)=|f'|^2+9w^2/r^2 +\sum_i m_i^2f_i^2+\nu^2\mathcal N^2 +4m_v^2(w^2-a^2)^2.$$

Both opposite mixed forces are retained. Weighted finite energy and norm comparison give

$$|f(r)|\le |f_b|(b/r)^{9/2} e^{-(\sqrt3m_v/2)(r-b)}.$$

The equations then give derivative, conormal and stress decay. At a finite later radius the solution enters the common small ball

$$\delta_*^2=\frac{M^2}{16(1+4M^2)}<1/64,$$

where the potential Hessian is at least $9m_v^2/16$. There its remainder equals the unique small exterior with the same trace, including all derivatives and stress. The larger rectangle may be nonconvex and its conormal relation set-valued. Existence does not give arbitrary freedom to choose outward slopes.

This is a constructed exterior response to an inner boundary, not an everywhere vacuum particle. The local inner edge has $a^2=3/4$; the exterior requires $a_b^2\le1/2$. No actual full-action inner path has yet crossed that interval and matched the exterior conormals. Inner existence and exterior existence do not prove their intersection. A lost stationary conormal inverse supplies a compatibility test at that jet, not a universal nonexistence theorem. The fully normalized spatial state remains open.

6. Full energy and both charges are independent tests

For the origin subclass define $G=\tau j'^2/8$, $U_s=\kappa[(l^{-2}-1)^2+9(t^{-2}-1)^2]/8$, $V_l=g_l[(z-1)^2+(y-1)^2]/2$, $U=U_0+U_x$, and

$$T_k=\frac{k_v}{2}[(l^2+9t^2)/R^2+a'^2+j'^2+z'^2+y'^2], \quad W_c=\frac{k_v}{2}(\omega^2z^2+\Omega^2y^2).$$

The complete raw energy and charges are

$$h=\chi\{c_0n_m[sF+B_e(s\log s-s+1) +\lambda(\Omega^2-F'^2)/2+G+U_s]+V_l\} +(1-\chi)(T_k+U+W_c)-p_g\chi(1-\chi),$$ $$q_\theta=\chi c_0n_ms+(1-\chi)k_v\omega z^2, \qquad q_\psi=\chi c_0n_m[j+\lambda(\Omega-F')] +(1-\chi)k_v\Omega y^2.$$

The material response term is signed; it must not be replaced by a positive residual square. Integrating with the full ten-dimensional measure gives

$$E-\omega Q_\theta-\Omega Q_\psi=-\langle\mathcal L_{\rm stat}\rangle.$$

Physical stress is $\sigma^i{}_j=\delta^i{}_j\mathcal L-\Pi_I^i\partial_jy^I$. For radial stress $H_r$ and trace $I$, the exact Euler equations give $(r^{10}H_r)'=r^9I$. On the true stationary shell one may write

$$h+I/10-\omega q_\theta=\mathcal B,$$ $$\mathcal B=\chi c_0n_m[B_e(1-s)+\lambda(\Omega^2-F'^2)/2 +6G/5+U_s-H_s/5] +(1-\chi)[T_k/5+k_v\Omega^2y^2]-QG_\chi/10,$$

where $H_s=\kappa[(l^{-4}-l^{-2})+9(t^{-4}-t^{-2})]/4$, $G_\chi=20\det B\,\chi_{\det B}$, and $Q$ is the complete gate load defined above. The direct mixed potential cancels in this combination but still enters through $Q$. It does not enforce the integrated defect.

Only a complete smooth state with the appropriate origin/infinity stress limits has global virial $\langle I\rangle=0$. It must then separately satisfy

$$\boxed{Q_\theta=A_0,\qquad \langle\mathcal B\rangle=0,} \quad\Longrightarrow\quad E=A_0\omega.$$

At a finite exterior cut the correct identity is

$$E_{\rm tail}-\omega Q_{\theta,\rm tail} =\langle\mathcal B\rangle_{\rm tail} +\frac{\pi^5}{120}b^{10}H_r(b).$$

Both tail quantities are separately positive for a nonzero exterior; they need not be equal at the cut. Their interior compensators differ by boundary stress. Raw massive-tail energy cannot be subtracted to impose the mass-frequency relation. Likewise a reference core's positive boost inertia does not establish the full smooth state's inertia before its sustaining stresses and exterior are solved.

Future orientation matters. On every participating component $n_m=\sqrt{\det B}>0$; a signed formula $lt^9$ cannot be carried through orientation reversal as negative material density to cancel charge. A normalized future-oriented completion of an exact core radius $r_0$ requires $r_0<R$, because its core alone contributes $A_0(r_0/R)^{10}$ and a nonzero exit adds positive theta charge. This is a necessary bound, not a radius-selection law.

7. Finite-chain current reconstruction remains a distinct comparison

A finite component chain supplies an exact diagnostic of how a causal carrier differs from a desired count law. For a real lower bidiagonal matrix

$$(B_n)_{jj}=n+j,\qquad (B_n)_{j,j-1}=b_j>0, \qquad P_{n,d}=\det B_n=\prod_{j=0}^{d-1}(n+j)=d!S(n,d),$$

the smallest singular value $\epsilon_n$ obeys

$$\frac{\epsilon_n}{P_{n,d}}=\frac1{\|\operatorname{adj}B_n\|_2}.$$

At fixed positive hopping list and radius, the adjugate norm increases strictly with $n>0$; hence $\epsilon_n/S(n,d)$ decreases strictly. A finite fixed chain therefore does not give an exact count clock. This restricted theorem does not exclude a nonlinear current-dependent edge, a derived mode-dependent radius or another spatial action.

A useful repaired depth-two example can be written explicitly. For $P=n(n+1)$ and edge $G>0$,

$$\epsilon(P,G)=\tfrac12[\sqrt{G^2+4P+1}-\sqrt{G^2+1}].$$

For a normalized local current $k$, complete variation of $\rho f(k)$ gives $G=f'(k)$, scalar offset $C_f=f-Gk$, and frequency $\lambda_n=\epsilon+C_f$. With reference $g>0$, $\mu=(1+g^2)^{-1/2}$, set

$$G(P)^2=g^2-4P,\qquad k(G)=\tfrac G2[\mu-(1+G^2)^{-1/2}],$$ $$f(k(G))=\frac{\mu(g^2+G^2)}4-\frac1{2\mu} +\frac1{2\sqrt{1+G^2}}.$$

On its finite invertible branch this one function gives $f'(k)=G$ and exactly $\lambda_n=2\mu S(n,2)$, without a separate particle coefficient. It has $f''>0$. Eliminating a stable retained response that couples affinely to $k$ would instead produce a concave effective function, so that simple elimination is not its physical origin. Retained dynamics, nonlinear coupling or a direct convex term are different possibilities.

The all-depth inverse problem likewise requires a finite invertible branch. If $\lambda_n=\mu_d(g)P_d(n)$, its necessary envelope equation is

$$\partial_n\epsilon_n(G_d(n))=\mu_d(g)P_d'(n), \quad \mu_d(g)=\frac1{(d-1)!} \left[\sum_{i=0}^{d-1}\frac{g^{2i}}{(i!)^2}\right]^{-1/2}.$$

Where the actual current $k_d(n)=\partial_G\epsilon_n$ is one-to-one, the inverse function

$$f_d(k_d(n))=\mu_dP_d(n)-\epsilon_n(G_d(n))+G_d(n)k_d(n)$$

has derivative $G_d$ and returns the target frequency. It is depth dependent and does not derive the cross-depth scale. Finite branch and component/spatial scans provide conditional controls, not complete field or geometry stability.

For scale, the bare depth-six ratio is $3838380/18564=3515/17\approx206.76470588$, against the benchmark $206.76828299$. Correcting a finite chain toward the bare count rule is therefore not the same as improving that empirical residual. The finite-chain source retains the precise bounds, sampled branches and canonical-normalization tests. These comparisons constrain a proposed mechanism; they do not supply the current common material action by substitution.

8. A simpler density-current hierarchy preserves core response

The density-current simplification is another useful explicit action, with a canonical first-order spatial principal operator. For a matrix field $\Psi$, total density $\rho=\operatorname{Tr}\Psi^\dagger\Psi$, scalar matrix current $C$, partner rank $s_p$ and $q=1/(s_p+1)$, its mass factor replaces

$$F_q=\sqrt{qs_p\operatorname{Tr}C^2+(1-q)(\operatorname{Tr}C)^2}$$

by

$$\widetilde F=\frac{\rho^2+F_q^2}{2\rho},\qquad \widetilde F-F_q=\frac{(\rho-F_q)^2}{2\rho}\ge0.$$

Potential and source are assigned zero at zero field. Scalar-current zero in a nonzero field is therefore regular. On a conformal-unitary coherent core, $F_q=\rho$ and $\delta F_q=\delta\rho$ for every complete field variation. The difference starts at fourth order, so the complete core Hessian and third mass-potential variation agree. The binding term's total-density replacement preserves its full source and Hessian on the same core. These statements concern complete field/partner variations, not only the occupied coefficient ray. They do not make the nonlinear interactions identical away from it.

For a finite graded hierarchy let $R_k$ be the retained scalar-insertion current, $\rho_k=\sum_{N\ge k}\|\Psi_N\|_F^2$, and $h_{k,d}=\binom{k+d-1}{d-1}$. Its mass action is

$$\widetilde U_{\rm mass}=\frac{u_d}{2}\sum_k h_{k,d} \left[\rho_k+ \frac{qs_p\operatorname{Tr}R_k^2+(1-q)(\operatorname{Tr}R_k)^2}{\rho_k}\right].$$

Empty active channels contribute zero. The partner rank is not replaced by the generally larger retained-current matrix size. Coherent isolated cores have

$$\omega_{N,d}=u_d\sum_{k=0}^Nh_{k,d} =u_d\binom{N+d}{d}.$$

The weights are an inverse constitutive prescription, not a derived spatial operator. At an initially empty channel above the occupied grade, a Lipschitz source need not have a linear derivative at zero; there is no uniform infinite-grade differentiability theorem.

The same law supplies a polarization-sensitive pair energy. For grades $N\ge M$, local densities $\rho,\sigma$ and coefficient-ray overlap $\tau$, subtracting isolated energies gives

$$V_{\rm mass}=-u_dq\frac{\rho\sigma}{\rho+\sigma} \sum_{k=1}^M h_{k,d}(1-\tau^k).$$

Here overlap is a field-coefficient contraction, not a position probability. The finite-overlap interaction differs from the earlier square-root law, although the weak-probe limit and complete core Hessian agree. Setting $q=0$ would remove the polarization-sensitive contribution while retaining prescribed grade masses; a mass-only test would miss that loss. Neither expression is an observed gauge vertex or an encounter rate. Pair response still requires both actual spatial patterns, their own isolated frequencies, relative phases, orientations and reciprocal deformations under the same action. See same-field coupling and particle-pair encounters.

9. What these candidates establish

There are explicit actions in which the cumulative count becomes the selected frequency, and there are common material reference states with retained energy, both charges, pressure and positive reference inertia. The regular-vacuum candidate further has exact local exits and finite-amplitude exteriors with proved decay and nonreactivation. These are constructive conditional results across the maintained preparations.

The remaining physical bridge is still substantial: an actual smooth inner-to-exterior spatial state, full charge and raw-energy normalization, healthy transition dynamics and complete positive inertia; physical lower-depth confinement; occupation/index selection; and common sector/action scales derived from the same field law. Persistence must be assessed for the intended particle: a relatively stable electron and a finite-lived tau need not meet an identical stability condition. No unstable comparison alone derives an observed decay channel or lifetime. Later raw-energy and global-construction proposals whose scientific review is incomplete do not supply established solutions to these gaps.

The count, constitutive action, spatial support, canonical normalization and empirical calibration must therefore remain visible in every claimed mass mechanism. The observational and proof targets state how a complete candidate would be tested.