Same-field actions and canonical normalization

A candidate mass mechanism needs a common action whose complete variation sustains the spatial pattern, produces its oscillation frequency and governs its reciprocal response. A count identity, a prescribed spatial eigenfunction and an energy proportional to frequency are different pieces of that problem. Several explicit constructions now connect these pieces under stated assumptions. None yet supplies the complete globally normalized physical mass operator.

The variables below describe one field, $\Psi_\infty$, on the ten-dimensional matter domain plus time, within the infinite-dimensional physical spacetime. They are not mediator particles or additional physical dimensions. Internal field oscillations and coefficient descriptions remain distinct from the electron's literal six-dimensional external trajectory around a nucleus. No probability cloud replaces that motion.

Reviewed Connections as of 5 November 2026

Two separately reviewed inverse-moment candidates extend this action comparison. The centered-only Hamiltonian is

$$H=\Omega\mathcal N+\frac1r\operatorname{tr}\!\left[M^{-1}(\tau K+\alpha V_q)\right],$$

where M is the positive rank-r field moment, K is its gradient moment, V_q its variance moment about the self-determined centre, and tau, alpha are positive supplied coefficients. Its full variation and isotropic motion law differ from the earlier cluster Hamiltonian.

The additional-window action adds a finite smooth centered partition and phase-sensitive graph functional. It has full-space fixed-moment minima and weak translating relative equilibria, with controlled first-simplex weights in the stated limit. The global instantaneous kernel, windows, graph, rank and rest clock are supplied. Neither this law nor its count bounds transfers to the separate focusing density action. Positive quartic contact and focusing density energy have opposite overlap preferences in the same frozen-pair comparison.

The current canonical clock and cluster law

For the current common-fundamental proposal, retain every positive phase action $I_a$ before imposing a total source $P=\sum_a I_a$. On an admitted canonical chart, a proposed changed action is $$\mathcal A=\int\left[p\cdot\dot R+\Theta_{\rm int}(\dot z) -\sum_a I_a\dot\beta_a-H\right]dt, \quad H=\sqrt{c^2|p|^2+B^2}, \quad B=\Omega[P+\hbar(C-1)]+U.$$ The internal one-form excludes the phase terms already written explicitly. The specified finite compact family has $U=T_{\rm rel}+W_0\ge0$, with $W_0$ a shared nonnegative positional well. The complete shape extension must retain its additional polarization, current and kinetic terms. The source leaf $P=\hbar$, changed Hamiltonian and reference state are constitutive inputs. At an admitted $U=0$ reference, $B=C\hbar\Omega$. The older coefficient $m_g$ in relative kinetic energy is not thereby equal to $\hbar\Omega/c^2$.

With $\phi_a=-\beta_a$, independent canonical source coordinates give $$\dot\phi_a=\frac BH\left(\Omega+\partial_{I_a}U\right).$$ On the free rigid centroid branch $d\tau/dt=B/H$. The proper clock is therefore $\Omega$ precisely when the relevant source derivatives vanish. Fixing $P$ before varying would erase this test. Textured polarizations and moving frames retain their connection contributions.

The new covariant rigid reconstruction adds the spin-dependent shift $R=Y+c^2Sp/[B(H+B)]$, where $Y$ is the physical centroid and $S$ is the complete internal angular charge. Its frame one-form yields the canonical form while its field embedding supplies Lorentz contraction and proper phases. This is a supplied reduced action with a rigid field realization; unrestricted local field-action descent remains open.

The combined clock and mass account also proposes a spatial spectral replacement of $C-1$, with explicit joining forces and gap requirements. That newest proposal is provisional. The constructions below retain different kinetic and support premises; they are comparisons rather than omitted pieces of this action.

What an action must determine

The maintained frequency prescription is

$$\omega_{n,d}=\mu_d S(n,d),\qquad S(n,d)=\binom{n+d-1}{d}.$$

Here $d\in\{2,3,4,5,6,10\}$, $\mu_d$ is a dimensional frequency unit and $S$ is a dimensionless cumulative count. Particle mass is this selected frequency in equivalent units, $m=\hbar\omega/c^2$. The physical question is which same-field dynamics produces and sustains the prescribed frequencies and occupied patterns. An arbitrary field amplitude or mode norm is not a replacement observable.

A complete comparison must specify the kinetic normalization, all varied supporting components, the actual spatial domain and boundary conditions. Eliminating a response variable changes both the effective operator and its norm. Squaring an operator after freezing its field-dependent coefficients gives a different action from varying its complete square. Likewise, the rotating generator $E-\omega Q$ is not raw energy $E$.

The known masses may be inputs to inverse construction. The supplied sector units, occupied counts, flat occupations and action scale remain inputs until their dynamical selection is established. The electron-based non-neutrino scale prescription is distinguished from independent empirical calibration; the neutrino unit has its separate splitting calibration. Bottom's ordinary integer assignment remains unresolved, and tau scale/overlap alternatives remain qualified. See shared frequency scales for that normalization problem.

Real counts do not fix a kinetic law

The real spatial count construction uses real homogeneous polynomials of degree $N=n-1$ restricted to a real amplitude sphere $S^d$. This sphere describes shape amplitudes, not physical position. Its parity band is

$$V_N=\bigoplus_{k=N,N-2,\ldots}\mathcal H_k(S^d),\qquad \dim V_N=D_N=\binom{N+d}{d}.$$

With the stated normalized sphere measure its reproducing kernel satisfies $K_N(w,w)=D_N$. Define the positive rank-one point-evaluation filter $\Lambda(w)=\operatorname{ev}_w^T\operatorname{ev}_w$, where $\operatorname{ev}_w f=f(w)$, with $\int\Lambda(w)\,d\sigma(w)=I$, where $d\sigma$ is the uniform measure of total mass one. Its bright eigenvalue is therefore $D_N$. The enlarged shape symmetry and sphere-integral norm are explicit premises; ordinary spatial isotropy alone does not supply them.

For real canonical coordinates $q,p$, the declared Hamiltonian

$$H=\frac{\mu_d}{2}(q^T\Lambda q+p^T\Lambda p)$$

gives bright frequency $\mu_dD_N$. A second-order oscillator with fixed inertia and stiffness proportional to $\Lambda$ instead gives a frequency proportional to $\sqrt{D_N}$. Thus the action, not the dimension count alone, decides the frequency.

An explicit map into ordinary $\mathbb R^d$ is

$$w(x)=\left(\frac{2Lx}{L^2+|x|^2}, \frac{|x|^2-L^2}{L^2+|x|^2}\right),\quad \rho_L(x)=\frac{[2/(1+|x|^2/L^2)]^d}{|S^d|L^d}, \quad (Uf)(x)=\sqrt{\rho_L(x)}f(w(x)).$$

It preserves the real squared field norm. The spatial projector has diagonal $P_N(x,x)=D_N\rho_L(x)$, while its integrated trace is $D_N$. A contact per physical volume retains that spatial density factor. The map supplies smooth trial profiles, not their autonomous support. It also shows why replacing an earlier complex projective measure by real geometry requires rederivation: fourth moments and nonlinear quartic frequencies cannot simply be carried over unchanged.

If $\epsilon$ denotes the energy allocated to each occupied mode, equal participating energy gives $E=\epsilon D_N$; fixed total band energy gives $E=\epsilon$. The frame identity selects neither occupation. Nested bands, flat occupied spectra, uniform participating local-clock energy and energy-density coupling are therefore additional inverse-construction targets, not consequences of counting alone.

A complete positive action with a local current

The homogeneous current construction illustrates how full variation and canonical phase action can be checked. On its specified spatial parent let $K$ be a Hermitian channel acting on retained components, and set

$$\rho=\Psi^\dagger\Psi,\quad j=\Psi^\dagger K\Psi, \quad k=j/\rho,\quad s_k=\sqrt{k^2+\eta^2}.$$

The density is a classical field norm, not a position probability. Complex notation packages real component quadratures. With one common strength $g>0$ and regulator $\eta>0$, the generating functional is

$$E_1=\langle\Psi,H_0\Psi\rangle -g\int\sqrt{j^2+\eta^2\rho^2}\,dV.$$

Its complete real gradient is

$$\mathcal F[\Psi]=H_0\Psi+G(k)K\Psi+C(k)\Psi, \quad G=-\frac{gk}{s_k},\quad C=-\frac{g\eta^2}{s_k}.$$

The scalar term $C$ comes from varying the denominator of $k$. Omitting it changes the equation. This degree-one homogeneous gradient defines the distinct positive action

$$L_+=\|\dot\Psi\|^2-\|\mathcal F[\Psi]\|^2, \qquad \ddot\Psi+(D\mathcal F)^*\mathcal F=0.$$

If $\mathcal F[\phi]=\lambda\phi$, real-gradient homogeneity gives the exact standing solution $\Psi=e^{-i|\lambda|t}\phi$. With $N=\|\phi\|^2$,

$$E_+=2\omega^2N,\qquad Q=2\omega N,\qquad E_+=Q\omega.$$

This $Q$ is phase action, not identified electric charge. For the stated Dirac principal symbol the complete spatial energy current satisfies $|\mathcal J^\nu|\le ce$, where $e=|\dot\Psi|^2+|\mathcal F[\Psi]|^2$, $c$ is its propagation-speed coefficient and the superscript labels a unit spatial normal. The source extends continuously at a field zero, but its higher differentiability there is limited; the flux bound is not a global evolution theorem. The supporting geometry, component channel and scales remain prescribed. These results cannot be merged with a later action without rechecking its equations.

The current law also constrains the count dependence. In its leading finite-chain regime, a homogeneous interaction $-a\rho|k|^p$ yields

$$\omega\propto S(n,d)^{p/[1+d(p-1)]}.$$

For $d>1$, linear simplex dependence in this family requires $p=1$. This is a scoped statement about that current family and leading regime, not uniqueness of all possible same-field interactions.

A varied material carrier on a prescribed ball

The material pullback action supplies another explicit common law. On the weighted ball, let

$$L_0=-w^{-1}\operatorname{div}[w(I-xx^T)\nabla], \quad w=\sqrt{1-r^2},\quad \lambda_N=N(N+d+1).$$

A supplied positive polynomial obeys $P_d(\lambda_N)=\binom{N+d}{d}^2$. For a smooth orientation-preserving clamped material map $\phi$, put

$$\rho_\phi=\det(D\phi)w(\phi),\quad A_\phi=(D\phi)^{-1}(I-\phi\phi^T)(D\phi)^{-T}, \quad L_\phi=-\rho_\phi^{-1}\operatorname{div}(\rho_\phi A_\phi\nabla).$$

Changing variables transports both the operator and its weighted norm. The complete reduced action is

$$S=\frac12\int\rho_\phi[|f_t|^2-\mu_d^2f^*P_d(L_\phi)f]\,dx\,dt +\int\left[\frac\kappa2\int|\phi_t|^2dx-E_{\rm el}(\phi-x)\right]dt.$$

Varying the determinant and inverse gradient is essential. At $\phi=x$, balanced real quadrature pairs yield exact joint wave/carrier branches with $\omega_N=\mu_dD_N$ and $E=Q\omega_N$. For $\Psi=A(x)\cos\omega t+B(x)\sin\omega t$, the exact stationary-carrier conditions are

$$\nabla(|A|^2-|B|^2)=0,\qquad \nabla(A\cdot B)=0.$$

A rotating scalar spatial dipole does not generally satisfy them. Component phase rotation and rotation in physical space are distinct.

The full energy obeys $E\ge\mu_d|Q|$; the ground branch minimizes it at fixed phase action on this clamped diffeomorphism domain. Excited states need their own persistence analysis. This model also retains high-order unbounded-speed dispersion, a supplied active ball and an unproved master-field realization. It is a constructive normalization comparison, not a causal freely localized particle.

An earlier material, shape and vacuum candidate

This separate candidate includes participating material, actual spatial shape response and a regular Cartesian exterior in one density:

$$\boxed{\mathcal L_3=\chi\mathcal C_+ +(1-\chi)\mathcal V+p\chi(1-\chi).}$$

Use metric signature $(-+\cdots+)$. Greek indices $\mu,\nu=0,\ldots,10$ label physical spacetime coordinates; label-component indices $A,B=1,\ldots,10$ are internal field indices, not extra coordinates. The flat gate $0\le\chi\le1$ depends on the carrier amplitude, its real origin component and $\det B$, where $B^{AB}=\partial_\mu\phi^A\partial^\mu\phi^B$. The labels $\phi^A$ are field components over ordinary spacetime. On the legal material chart,

$$n=\sqrt{\det B},\quad h=B^{-1},\quad u^\mu=J^\mu/n,\quad D=u^\mu\partial_\mu, \quad P_{\rm sp}^{\mu\nu}=g^{\mu\nu}+u^\mu u^\nu.$$

No inverse chart is evaluated at an inactive singular label configuration; the material terms are smoothly zero-extended across inactive collars. The velocity $u^\mu$ is distinct from the scalar frequency parameter $\nu$. The cofactor current is $J^\mu=\epsilon^{\mu\nu_1\cdots\nu_{10}}\epsilon_{A_1\cdots A_{10}}\prod_{i=1}^{10}\partial_{\nu_i}\phi^{A_i}/10!$, where the epsilon tensors are Levi-Civita symbols and the orientation is chosen future-directed. Its identically conserved cofactor form is varied together with the labels.

Let $b=v/|v|$ be the normalized finite carrier where legal. Its specified smooth occupation/phase functional $C(b)$ takes the value $\binom{N+d}{d}$ on equal occupied cumulative prefixes, with vanishing first variation there. That property retains all fourteen prepared rows but does not select their actual occupation. Define

$$F=\nu+\nu j^2+\Lambda j^4,\quad H=CF, \quad A_c=CF',\quad c=CF'', \quad B_e=C\left(F+\frac{2F'^2}{F''}\right),$$ $$K=c-\frac{A_c^2}{B_e},\quad \lambda=\frac{K}{4[c^2+(726/35)(A_c^2+c^2)]}, \quad \tau=4R^2[(18514/315)\lambda(A_c^2+c^2)+K].$$

Both $\Lambda=0$ and $\Lambda=\nu$, and both actual signs of $j$, remain in the common test. The real response variable $j$ is not an assigned electric charge.

Writing $s=e^\rho$, the material density contains

$$\begin{aligned} \mathcal L_{\rm mat}={}&sD\theta-sH -B_e(s\log s-s+1)+jD\psi +\frac\lambda2(D\psi-A_c)^2\\ &+\frac\tau2(Dj)^2+\frac\beta2|Db|^2 -\frac\tau8P_{\rm sp}(dj,dj) -\frac\beta8P_{\rm sp}\operatorname{Re}\!\sum(\nabla b)^*\nabla b-V_s. \end{aligned}$$

The shape uses actual gradients. With $E_A^\mu=h_{AB}\partial^\mu\phi^B$,

$$q_{AB}=\ell^2\operatorname{Re}\sum(E_Ab_k)^*E_Bb_k, \quad \chi_s=\frac1{4(1+\operatorname{tr}M_2)}, \quad T=[\det(I+\chi_sq)]^{-1/10}(I+\chi_sq),$$ $$V_s=\frac\kappa8\{ \operatorname{tr}(T^{-1}hT^{-1}h)-2\operatorname{tr}(T^{-1}h)+10\}.$$

Here $M_2=\sum |b_k|^2kk^T$, $\beta=\nu\ell^2/[16(1+\operatorname{tr}M_2)]$, $\kappa=\nu/16$, and the carrier derivative is $\nabla_\mu b_k=\partial_\mu b_k-i k_A\partial_\mu\phi^A b_k/\ell$. The finite index $k$ labels coefficients, not spatial positions. A complex component represents two real quadratures. Cartesian readouts $Z,Y$ are locked to $e(\theta),e(\psi)$ through $V_l=g_l(|Z-e(\theta)|^2+|Y-e(\psi)|^2)/2$, where $e(\vartheta)=(\cos\vartheta,\sin\vartheta)$ is a real quadrature pair. The full participating density is

$$\mathcal C_+=c_0n\mathcal L_{\rm mat}-V_l -\frac{c_0n}{2}[\gamma_\theta P_{\rm sp}(d\theta,d\theta) +\gamma_\psi P_{\rm sp}(d\psi,d\psi)],$$

with $r_{\rm ph}=(sH^2/B_e+\lambda A_c^2)/H$, $f=1/[16(1+r_{\rm ph})]$, $\gamma_\theta=fs/B_e$, $\gamma_\psi=f\lambda$. In particular $\gamma_\theta$ depends on $\rho$ and must be varied.

The exterior variables and potential are

$$W=(\phi/R,v,j,Z,Y),\quad N=|W|^2, \quad\Phi=|\phi/R|^2,\quad\mathrm{Amp}=|v|^2,$$ $$\mathcal V=-\frac{k_v}{2}\partial W\cdot\partial W-U, \quad U=\frac{k_vm_v^2}{2}N+\frac{\lambda_v}{4}N^2 +\frac{k_{\rm mix}}4(\Phi-\mathrm{Amp})^2, \quad k_{\rm mix}=4k_vm_v^2.$$

The numeral four is a common test input. Its positive square supplies opposite-sign label and amplitude forces. It cannot be kept in energy while omitted from variation.

Here $u_d$ denotes the supplied sector fundamental frequency, so a prefix of count $C$ targets $Cu_d$. The scalar reference $\nu=u_{5,\mathrm{solar}}/2>0$ uses the separately calibrated neutrino scale. The length $R$ is the candidate reference spatial scale, $\ell$ is its label-phase length, and $A_0$ is the supplied common phase-action scale. One finite prescription uses $R=\ell=1/\nu$, $c_0=A_0/(V_{10}R^{10})$, $V_{10}=\pi^5/120$, $k_v=c_0/(\nu M^4)$, $m_v=\nu M$, $\lambda_v=c_0\nu/M^4$, $g_l=c_0\nu$, and $p=0$ or $c_0\nu$. The same $M$ covers twice the largest absolute prepared clock rate divided by $\nu$, over all rows and both polynomial branches. Consequently both exterior readout mass gaps are positive. These are fixed common candidate inputs, not a per-particle fit or a derived physical action unit; $A_0=\hbar$ is supplied.

Vary the gate and preserve both charges

Set $Q_g=\mathcal C_+-\mathcal V+p(1-2\chi)$. The exact first variation is

$$\delta\mathcal L_3=\chi\,\delta\mathcal C_+ +(1-\chi)\delta\mathcal V+Q_g\,\delta\chi,$$ $$\Pi_I^\mu=\chi(\mathcal C_+)_{I,\mu} +(1-\chi)\mathcal V_{I,\mu}+Q_g\chi_{I,\mu}.$$

Thus a potential change also changes derivative-gate conormals and principal loads. A positive vacuum potential does not prove a positive complete Hamiltonian or a healthy determinant transition.

For prepared equal-prefix jets, the gate is one on an open neighborhood, $\det T=1$, $n=1$, $s=1$, $F(j_0)=u_d$, $\omega=Cu_d$, $\Omega=CF'(j_0)$. The complete local identities are

$$\mathfrak h=c_0\omega,\quad q_\theta=c_0,\quad q_\psi=c_0j_0,\quad \sigma^i{}_j=0.$$

A conditional affine ball of the supplied reference volume therefore gives $E=A_0\omega$, $Q_\theta=A_0$, $Q_\psi=A_0j_0$. Every occupied mode has participating energy $c_0u_d$. These are exact common-action retention checks at depths 2,3,4,5,6,10, not a proof that a sharp ball is a smooth isolated particle. The material-only clock qualification matters: an infinite energetic Cartesian tail cannot obey a strictly positive constant full-participation energy-density clock law with finite total energy.

On the stationary real origin subclass, $\phi=g(r)e_r$, $v=a(r)e_0$, $Z=z(r)e(\omega t)$, $Y=y(r)e(\Omega t)$, the true rho shell is $\rho=(\omega-F)/B_e$. Away from this subclass its horizontal phase terms remain. Put $l=g'$, $t=g/r$, $n=lt^9$ on the future positive chart, and

$$G=\tau j'^2/8,\quad U_s=\kappa[(l^{-2}-1)^2+9(t^{-2}-1)^2]/8.$$

Write the raw canonical energy density as $\mathfrak h=\sum_I\Pi_I^0\partial_0y^I-\mathcal L_3$, reserving $h_{AB}=(B^{-1})_{AB}$ for the inverse label matrix. The full raw energy retains the signed response contribution $\lambda(\Omega^2-F'^2)/2$, supporting gradients/strain, lock energy, positive exterior energy and the signed gate term. Its charges are

$$q_\theta=\chi c_0ns+(1-\chi)k_v\omega z^2, \quad q_\psi=\chi c_0n[j+\lambda(\Omega-F')] +(1-\chi)k_v\Omega y^2.$$

Neither response charge is discarded. The exact stationary Legendre identity is

$$E-\omega Q_\theta-\Omega Q_\psi=-\int\mathcal L_3\,d^{10}x.$$

Let $\mathcal L_{\rm stat}$ be the full density evaluated on this stationary subclass, $P_x=\partial\mathcal L_{\rm stat}/\partial x'$ for $x=(g,a,j,z,y)$, and $L_t=\partial\mathcal L_{\rm stat}/\partial t$, with the stretch $t=g/r$ treated independently in that partial. Define the radial and tangential physical stresses $H_r=\mathcal L_{\rm stat}-\sum_xP_xx'$, $K_t=\mathcal L_{\rm stat}-tL_t/9$, and their trace $I=H_r+9K_t$. All five Euler equations imply

$$(r^{10}H_r)'=r^9I,\qquad \mathfrak h+I/10-\omega q_\theta=\mathcal B.$$

The normalization density is explicitly

$$\begin{aligned} \mathcal B={}&\chi c_0n\left[B_e(1-s)+\frac\lambda2(\Omega^2-F'^2) +\frac65G+U_s-\frac15H_s\right]\\ &+\epsilon\left[\frac15T_{\rm kin}+k_v\Omega^2y^2\right] -\frac1{10}Q_gG_\chi, \end{aligned}$$

where $\epsilon=1-\chi$, $G_\chi=l\chi_l+t\chi_t$, and $Q_g$ is the gate load above, evaluated on the actual stationary fields. Here

$$H_s=\frac\kappa4[(l^{-4}-l^{-2})+9(t^{-4}-t^{-2})],\qquad T_{\rm kin}=\frac{k_v}{2}\left[\frac{l^2+9t^2}{R^2}+a'^2+j'^2+z'^2+y'^2\right].$$

These quantities include the actual radial and angular label gradients, current and both readouts; none is a prescribed core value.

On a finite tail cut beginning at $b$, with the verified zero stress limit at infinity,

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

The finite boundary stress cannot be deleted. Only a complete smooth match with zero stress limits at both ends yields the global virial and $E-\omega Q_\theta=\int\mathcal B$. A normalized state must then satisfy both $Q_\theta=A_0$ and $\int\mathcal B=0$. Local core identities do not enforce either condition on the complete state.

What is constructed and what is still missing

The unchanged common action has actual smooth local five-equation exits from the origin core, with a forced amplitude slope and fixed readout departures below one. A separate variational construction supplies smooth permanently inactive exteriors. Its larger accepted trace domain includes

$$(g_b/R)^2\le3/4,\qquad a_b^2\le1/2,$$

with arbitrary finite other three traces. The minimizers solve the full action there, decay exponentially and meet the unique small tail. Their actual conormal response can be set-valued in the larger domain; it is not arbitrary slope freedom. Both positive tail energy and its positive normalization-density integral are retained separately.

Exact label traction and first-turn energy-component bounds constrain a fully solved connection. They do not construct it. The finite-amplitude inner-to-exterior passage, full source-free matching, complete normalization, determinant-transition evolution and positive complete particle inertia remain open. The radial origin problem has ten-dimensional support; it does not derive physical depth-three confinement or the other anisotropic interfaces. No accepted global stationary particle follows from the current local/exterior results.

The common-action requirement is productive precisely because all these tests constrain the same equations. A successful continuation must retain the actual clocks, all varied supporting fields, both charges, complete stress and the same coefficients across the maintained depths. The candidate-action volume develops the spatial construction, energy and inertia explains the complete response requirements, and same-field coupling separates reciprocal interaction from prescribed-target overlaps.