Complete energy, inertia and causal motion

Mass has the equivalent angular frequency $\omega_m=mc^2/\hbar$. IDWT's dynamical question is what one continuously moving field, $\Psi_\infty$, does to sustain a localized pattern at its selected frequency and produce the complete positive momentum response of that pattern. An assigned clock, a partial carrier norm or a fixed-background wave momentum does not establish that response.

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. The ten-dimensional matter domain is modeled by $\mathbb R^{10}$. Coefficient labels and canonical phase charges add no physical manifold. Internal field oscillations, particle center position and observer registration remain distinct. None of the internal stationary fields below replaces literal deterministic electron trajectories around a nucleus.

Reviewed Connections as of 5 November 2026

The centered-only inverse-moment law has an exact isotropic moving family with complete positive translation coefficient N^2/(2 tau). Here N is the continuous field norm and tau is the positive gradient coefficient; N is not the portion count C. Its carrier rotation is retained. The centered-window law instead gives fixed-momentum minima and exact weak translating relative equilibria. Its real-preparation phase-kick tensor is positive, but is an initial response rather than a proved relaxed dispersion mass. Both actions have a global instantaneous interaction and do not thereby establish physical causal propagation or mass-frequency normalization.

For the separate radial density law, exact balances give E=omega N+gU/10. With delta=m0-omega and R=integral[(g rho^(1/9)-delta)G^2], the finite-gap remainder identity is

$$\frac EN=m_0-\delta\frac{18-d}{20-d}+\frac{2R}{(20-d)N}.$$

Omitting R is a near-threshold approximation. These energies and responses are calculated from their actual laws; they are not identified with the old cluster square root or the observed count-times-fundamental mass.

Common clocks and two different mechanical laws

The current interpretation uses $C$ portions at one fundamental $\Omega$, with aggregate average rate $C\Omega$. Mass in equivalent frequency units is the target $m=C\hbar\Omega/c^2$. This target does not require every candidate's old field energy to equal $C\hbar\Omega$. Its mechanical response must be calculated from its actual action.

For the original compact spinor action, complete canonical reduction gives, on its admitted shape and velocity domain,

$$\inf E(\Pi)=\inf E(0)+\frac{|\Pi|^2}{2C m_g}.$$

This infimum statement requires the same fixed-charge class to be closed under the common translation boost. The connection terms are retained in the one-form; their common-translation contributions cancel. The complete old inertial coefficient is $C m_g$, and its parameter $m_g$ is independent of $\Omega$. Setting $m_gc^2=\hbar\Omega$ would be an extra constitutive relation, not a consequence of unit conversion.

A newer proposed Hamiltonian retains the canonical data and replaces the translation law:

$$H_C=\sqrt{c^2|\Pi|^2+B_C^2},\qquad B_C=\Omega[P+\hbar(C-1)]+U,\qquad B_C>0.$$

The reference action leaf is $P=\hbar$. The remainder $U$ contains the complete relative kinetic, support and polarization energy with the declared reference subtraction, using the new nonnegative pair well. The finite shape theorem gives $U\ge0$ only in its stated restricted class. If that class and all conserved charges admit a zero-remainder reference, the relaxed infimum is

$$\inf H_C(\Pi)=\sqrt{c^2|\Pi|^2+(C\hbar\Omega)^2}, \qquad m_0=\frac{C\hbar\Omega}{c^2}.$$

The coefficient in the new small-momentum expansion is $m_0$. A mixed conserved matrix charge may exclude the aligned reference; then the lower bound survives but equality has not been proved. This qualification matters whenever a variational minimum is offered as a dynamically reachable particle.

The square root, source leaf and count offset are premises of this new construction. Proper-time portion clocks remain common on the aligned reference. Generic polarization or source-dependent $U$ can shift them: for $P=\sum_j I_j$, the reference clock magnitude is $\partial B_C/\partial I_j=\Omega$ only when the additional derivative vanishes. The clock-connectivity note develops this condition and proposes a spectral origin for the offset, with independent review of that newest proposal still pending.

Separate noninteracting clusters require a sum of their Hamiltonians. A single square root applied to total momentum and total rest coefficient loses the energy of relative motion. Smoothly joining clusters introduces derivatives of the joining rule, which contribute forces and cannot be omitted.

An additional proposed covariant rigid-field embedding distinguishes a covariant centroid $Y$ from canonical center $R$:

$$R=Y+\frac{c^2S\Pi}{B_C(H_C+B_C)},$$

where $S$ is its antisymmetric spatial spin tensor. Lorentz contraction and the corresponding density factor preserve total phase action; a frame-dependent connection remains in the canonical one-form. This is a constructed rigid image in the same 10+1 spacetime, not an additional physical manifold. Independent review of that embedding is pending. A covariant rigid image alone does not derive a local field action away from that image, its full field momentum, internal causal bounds or reciprocal interactions. The older exact boost calculations below retain their own actions and domains.

Energy, charge and the generator of a stationary field

For an action with fields $y^I$, retain all constitutive and derivative-gate terms when defining

$$\Pi_I^\mu=\frac{\partial\mathcal L}{\partial(\partial_\mu y^I)},\qquad h=\sum_I\Pi_I^0\dot y^I-\mathcal L,\qquad \sigma^i{}_j=\delta^i{}_j\mathcal L-\Pi_I^i\partial_jy^I.$$

The complete canonical energy and phase currents follow from the complete action. A metric variation must also retain its constraints and their response. Removing a retained field from the energy while keeping its force solves a different physical balance.

For a relative equilibrium with two phase symmetries, the actual stationary functional is

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

It is not the raw energy $E$. Positivity of its constrained second variation is different from positivity of the unconstrained raw energy Hessian. Canonical phase action is also different from observed electric charge; the normalization bridge must be supplied or derived explicitly.

The positive count-wave realization illustrates the distinction cleanly. Its action $L_+=\|\dot\Phi\|^2-\|\mathcal F(\Phi)\|^2$ has complete nonnegative energy $E_+=\|\dot\Phi\|^2+\|\mathcal F(\Phi)\|^2$. On a count standing wave, $E_+=\omega\mathcal Q_+$, with $\mathcal Q_+=2\omega\|\phi\|^2$. A common physical value of that action remains unresolved. Its preferred-time constitutive law also needs a complete translation/boost calculation before particle inertia follows.

A complete prepared-state boost result

The local ambient count action supplies a more complete conditional result on flat spacetime with ten spatial directions. It varies support, frame, flow, phase and finite carrier components, and retains all fourteen supplied prefixes across depths two, three, four, five, six and ten. Its exterior coordinates $c_k=e^{-ik\cdot\xi}b_k$ have genuine phase/carrier dynamics; on the prepared fields $c$ is constant and the added residual and first variation vanish.

For $H_0=\mu_d\binom{N+d}{d}$, the complete constrained metric stress and conserved reference quantities are

$$T^{ab}=\chi\sigma H_0V^aV^b,\qquad Q_\theta=eA_0,\qquad E_0=A_0H_0>0,$$ $$E(v)=\gamma E_0,\qquad P(v)=\gamma E_0v,\qquad M_{\rm boost}=E_0/c^2.$$

Here $e=\pm1$ labels the even clock branches, not an identified particle/antiparticle or electric-charge sign. Speed-one units are used in the momentum formula. The full boost transforms every supporting field and interface; Lorentz contraction belongs in the spatial integration. The uniform participating energy density is $\sigma H_0$, with equal prepared allocation $\sigma\mu_d$ per counted mode. The reference exterior has zero energy and stress in this action. That special zero-reference property cannot be imposed on a different action with energetic Cartesian tails.

This result joins supplied count clocks and complete positive inertia. It remains an inverse construction: finite component carrier, occupation potentials, units, common phase-action normalization $A_0$ and reference cutoffs are inputs. Positive inertia does not independently select those inputs or establish the observed particle identities.

The same action's complete reference time form is positive, its static form is nonnegative under its stated common flow bound, and its full bulk characteristic speeds are positive and at most one half in comparison-speed-one units. The positive perturbation energy is the co-rotating second variation of $E-eH_0Q_\theta$. Generic coupled states have the new action's total energy/current.

Its natural exterior perturbation variable is $g=h+R(\zeta)$, including support/frame response. The associated weighted domain cannot be replaced by an ordinary unweighted $H^1$ norm of $h$: coefficients of that change grow at infinity. Finite energy gives matching traces across finite interfaces, but no uniform unweighted coercivity, positive exterior static gap, nonlinear evolution theorem at infinity or general transmission theorem follows. Earlier persistence results from a different action are not transferred.

Regular vacuum changes the complete balance

The later regular-vacuum candidate discussed in the common-action volume instead uses a derivative-gated material law and ordinary Cartesian exterior fields. Its mixed-square version has

$$\mathcal L_3=\chi\mathcal C_++(1-\chi)\mathcal V+p\chi(1-\chi),\qquad \mathcal V=-\frac{k_v}{2}\partial W\cdot\partial W-U,$$ $$U=\frac{k_vm_v^2}{2}|W|^2+\frac{\lambda_v}{4}|W|^4 +\frac{k_{\rm mix}}4(\Phi-\mathrm{Amp})^2, \qquad k_{\rm mix}=4k_vm_v^2.$$

Here $\Phi=|\phi/R|^2$, $\mathrm{Amp}=|v|^2$, and $\chi$ depends on carrier value and the label-gradient determinant. On a legal material component $n=\sqrt{\det B}>0$, with future cofactor orientation. Inactive material terms are smoothly zero extended.

The complete first variation contains

$$\delta\mathcal L_3=\chi\delta\mathcal C_++(1-\chi)\delta\mathcal V +\mathcal Q\delta\chi,\qquad \mathcal Q=\mathcal C_+-\mathcal V+p(1-2\chi).$$

Consequently $\mathcal Q\chi_{(I,\mu)}$ contributes to every derivative-gate momentum. A nonnegative added potential does not establish positivity of the full derivative-gated Hamiltonian or health through a determinant transition.

For the consistent stationary radial origin subclass let $g(r)e_r$ be the label field, $a$ the origin amplitude, $j$ the current, and $z,y$ real Cartesian readout profiles at rates $\omega,\Omega$. Define $l=g'$, $t=g/r$, $n=lt^9$ on the positive material chart, $\epsilon=1-\chi$, and

$$F=\nu+\nu j^2+\Lambda j^4,\quad A=F',\quad B_e=F+2F'^2/F'',\quad \rho=(\omega-F)/B_e,\quad s=e^\rho,$$ $$G=\tau j'^2/8,\qquad U_s=\frac\kappa8[(l^{-2}-1)^2+9(t^{-2}-1)^2],\qquad V_l=\frac{g_l}{2}[(z-1)^2+(y-1)^2].$$

The $\rho$ shell is exact on this uniform-clock subclass; horizontal clock gradients modify it on general spacetime fields. $\lambda(j)>0$ and $\tau(j)>0$ are the action's response and current-gradient coefficients, not constant background labels. With $T_{\rm kin}$ the full Cartesian spatial-gradient density, including $9g^2/r^2$, and $W_{\rm clock}=k_v(\omega^2z^2+\Omega^2y^2)/2$, the exact densities are

$$h=\chi\{c_0n[sF+B_e(s\log s-s+1) +\tfrac\lambda2(\Omega^2-A^2)+G+U_s]+V_l\} +\epsilon(T_{\rm kin}+U+W_{\rm clock})-p\chi\epsilon,$$ $$q_\theta=\chi c_0ns+\epsilon k_v\omega z^2,\qquad q_\psi=\chi c_0n[j+\lambda(\Omega-A)]+\epsilon k_v\Omega y^2.$$

Both actual current signs, $\Lambda=0$ or $\nu$, and $p=0$ or $c_0\nu$ remain under the same common coefficients. The material response $\lambda(\Omega^2-A^2)/2$ is signed; replacing it by the nonnegative residual square would give a false raw energy. Both charges, all current/readout gradients, the mixed potential and the gate contribution remain. All integrals use the actual ten-dimensional measure $S_9r^9dr$, $S_9=\pi^5/12$, rather than a depth-three measure inferred from the supplied down clock.

On an exact open material plateau at a supplied prefix, $h=c_0H$, $q_\theta=c_0$, $q_\psi=c_0j_0$, and stress is zero. Conditional integration of the reference volume gives $E=A_0H$, $Q_\theta=A_0$. A finite sharp ball is not a global smooth solution. The regular Cartesian exterior has positive energy; it cannot inherit a prior action's zero-reference tail or be subtracted away to enforce the mass relation.

Stress limits and finite-cut normalization

For an actual smooth radial solution put $H_r=\mathcal L-\sum P_xx'$ and let $I$ be the trace of the complete spatial stress. The exact identities are

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

For this candidate, $\mathcal B$ includes material density and signed response, elastic strain, current-gradient energy, Cartesian gradients, the second readout clock and derivative-gate work. On a pure vacuum tail it reduces to

$$\mathcal B=T_{\rm kin}/5+k_v\Omega^2y^2\ge0.$$

At finite cuts $[b,B]$, with brackets denoting $S_9\int_b^B r^9dr$,

$$E_{[b,B]}-\omega Q_{\theta,[b,B]} =\langle\mathcal B\rangle_{[b,B]} -\frac{S_9}{10}[B^{10}H_r(B)-b^{10}H_r(b)].$$

The constructed inactive exterior has exponential value and derivative decay, finite raw energy and both charges, and $r^{10}H_r\to0$ at infinity. Therefore

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

The two tail quantities are separately positive for a nonzero tail, but they are not equal at a finite matching radius. Their interior compensators differ by the boundary-stress term. Complete source-free conormal and stress matching cancels that term only after an actual inner-to-tail connection is found.

For a complete regular state with proved zero stress limits at both ends, the global virial is $\langle I\rangle=0$, and normalization requires independently $Q_\theta=A_0$ and $E=A_0\omega$, equivalently $\langle\mathcal B\rangle=0$ with that charge condition. A local inner solution and an exterior existence theorem do not prove these global conditions. Positive components needed to transfer shell traction are not automatically bounded by the full signed energy. No complete raw-energy theorem or globally normalized particle is inferred from those component estimates.

Causal propagation belongs to a complete action

The finite-speed count chain illustrates an alternative to the high-order flat-ball scalar stiffness. Retaining its form and intermediate chain components gives a complete local parent with a stated speed bound. Its common finite-window example approximates all fourteen supplied clocks within one part per million, but contains additional heavy and form branches. With $\kappa_d=\mu_dM^{d-1}/d!$, a speed ceiling $c$ requires $R\le c\,d!/(b\mu_dM^{d-1})$. Accuracy and branch separation therefore have explicit scale costs.

The moving finite-speed action adds the actual supporting motion under its stated field and constraint hypotheses. No speed, positivity or boost result transfers merely because another action shares its stationary frequency list. In the regular-vacuum candidate, local constrained health near a particular smooth inner exit does not prove determinant-transition health, endpoint-uniform transmission or long-time persistence.

These limitations are checks on a proposed law at actual tested clocks. An unhealthy hypothetical off-reference branch is not automatically a rejection of the maintained particle states. Conversely, a growing mode on an actual state needs its channel, timescale and energy-momentum balance before it explains physical decay. Electron persistence and finite tau-like lifetime are different targets; universal permanent stability is not imposed.

Further exact conditional inertia and source-energy tools

The following inverse spinor and quartic identities belong to their stated actions. They retain useful complete-energy tests and derivations; they do not supply the missing global connection or positivity theorem for the derivative-gated candidate above.

Localized current inertia

Read the complete source.

Frequency, a sustaining field and inertia in one action

Mass already has the equivalent frequency $\omega_m=mc^2/\hbar$. The dynamical question is whether a continuously oscillating, localized field can sustain itself and have the corresponding inertia under translation. The construction here answers that question for one explicit inverse action. It retains the simplex current hierarchy and adds local terms whose complete variations sustain a spatial core.

For its displayed branch, the exact results are

$$\boxed{\omega_N=\omega_{0,d}\binom{N+d}{d}},\qquad E_0=\hbar\omega_N Q,\qquad M_{ab}=\frac{\hbar\omega_N Q}{c^2}\delta_{ab}. \tag{1}$$

Here $Q$ is the dimensionless canonical field norm. One common $Q=1$ condition makes the inertial coefficient the mass-frequency value. No field magnitude is fitted separately for the electron and muon. Their spatial fixed point is the same, while their coefficient grades differ.

This is reconstructed dynamics. The original scalar quartic master term has not been shown to generate this action, its carrier or its weights. The oscillator coefficient grade has also not yet been identified with the actual spatial harmonics of a derived particle cavity. The existence and inertia calculation therefore closes a conditional field problem; it does not complete the physical IDWT mass-operator derivation.

A localized spatial zero field gives an exact frequency branch

The detailed intermediate core, stress and symplectic equations in this subsection use natural units $\hbar=c=1$; the frequency/energy/inertia summary restores them explicitly. In even spatial dimension $d$, take $r=2^{d/2}$ and Hermitian matrices $\alpha_a,\beta$ satisfying $\{\alpha_a,\alpha_b\}=2\delta_{ab}I$, $\{\alpha_a,\beta\}=0$, $\beta^2=I$. The doubled-spinor principal and insertion matrices are

$$A_a=\sigma_x\otimes\alpha_a,\qquad B_0=\sigma_z\otimes I_r,\qquad B_1=\sigma_x\otimes\beta,\qquad B_2=I_2\otimes\beta,$$

where $\sigma_x,\sigma_z$ are Pauli matrices on the doubled block. These field-component matrices add no spatial directions. Each $B_j$ anticommutes with every $A_a$. The spatial kinetic operator is $-iA_a\partial_a$, and the canonical time term is $\tfrac i2\operatorname{Tr}(\Psi^\dagger\dot\Psi-\dot\Psi^\dagger\Psi)$.

The partner currents are $C=\Psi^\dagger B_0\Psi$, $J=\Psi^\dagger B_1\Psi$ and $K=\Psi^\dagger B_2\Psi$. The fixed partner involution $\beta_p$ is the same numerical matrix as $\beta$ acting on the partner index. The supplied grade-independent core coefficient is $m_c$, the binding strength is $g$, and the spectral matrix function is $f(K)=\operatorname{sgn}(K)|K|^a$ for the stated positive fractional exponent. At zero current eigenvalues its differentiability requires a separate completion. On one coherent grade, the inverse binding density is $\lambda_N\|C\|_{\rm HS}+m_c\operatorname{Tr}(J\beta_p)-g\operatorname{Tr}[Jf(K)]$, with $\lambda_N=\sqrt r\,\omega_{0,d}D_N$. The common graded-current prescription below supplies this value.

The unit seed $z$ labels a coherent symmetric coefficient tensor; complex notation packages field quadratures, not physical position. The normalized splitting isometry $T_{Nk}$ embeds its degree-$N$ symmetric tensor into degree-$k$ and degree-$(N-k)$ factors, mapping $z^N$ to $z^k\otimes z^{N-k}$. The order-$k$ current $R_k$ is the partner-current reduction obtained with this map, tracing the $(N-k)$ spectator coefficients and summing $N\ge k$. This is a declared coefficient hierarchy; its actual spatial-harmonic and occupation bridge remains unproved.

Write $\Psi=(\phi,\zeta)^T$. Suppose a spatial matrix $\phi$ satisfies

$$\begin{aligned} \phi^\dagger\phi=\phi\phi^\dagger&=\rho(x)I_r,\\ \mathcal B(\phi):=-i\alpha\cdot\nabla\phi +m_c\beta\phi\beta_p -g\beta\phi f(\phi^\dagger\beta\phi)&=0. \end{aligned} \tag{10}$$

Take $\zeta=0$. Its current $J$ vanishes, while $C=\rho I_r$. The complete upper-spinor mass source on this branch is $\lambda_N\phi/\sqrt r$. The lower component of the full field equation is exactly the second equation in (10). Consequently

$$\boxed{\Psi_N(t,x)=e^{-i\omega_Nt} \begin{pmatrix}\phi(x)\\0\end{pmatrix}\otimes\overline z^{\,N}, \qquad \omega_N=\lambda_N/\sqrt r,\qquad \|z\|=1} \tag{11}$$

solves the complete equation. No residual lower-spinor equation was discarded. The conjugate on $z$ is a matrix-current convention: ordinary column Gram matrices then give $P_{z,k}=|z^k\rangle\langle z^k|$.

For the full common action, split the oscillator indices with the isometries $T_{Nk}$ of the current hierarchy. Trace only the $N-k$ oscillator spectators, retaining the partner indices. Denote the resulting current by $R_k$ after summing all grades $N\ge k$. Set

$$U_{\mathrm{mass}}=\omega_{0,d}\sqrt r \sum_{k\ge0}\binom{k+d-1}{d-1}\|R_k\|_{\mathrm{HS}}. \tag{12}$$

The binding currents $J,K$ instead trace all oscillator coefficients and sum all grades before the stated binding density is varied. These prescriptions define one action for self-interaction and reciprocal coupling.

On (11), $R_k=\rho I_r\otimes P_{z,k}$. Each mass-current order contributes $\omega_{0,d}\phi$ times its weight. Hockey-Stick gives

$$\sum_{k=0}^N\binom{k+d-1}{d-1}=\binom{N+d}{d}, \qquad \omega_N=\omega_{0,d}D_N. \tag{13}$$

The factor $\sqrt r$ in (12) is one stated normalization rule across depths: $\sqrt r\|\rho I_r\|_{\mathrm{HS}}=r\rho$ equals the canonical density. Without it, a common unnormalized coefficient would introduce an artificial rank factor between $r_6=8$ and $r_{10}=32$. The rule preserves the proposed shared charged-lepton frequency unit if $\omega_{0,6}=\omega_{0,10}$ is imposed. It does not derive that equality or remove the documented tau mass residual in the frequency-unit comparison.

The weights, root law, normalized partner trace and coefficient carrier remain inverse premises. Equation (13) proves their consequence, not their origin in the master field. The fixed assignments remain $N=12,34$ at depth six and $N=22$ at depth ten, giving $18564$, $3838380$ and $64512240$.

Full stress and a direct kinetic derivation of inertia

For the displayed radial cores in (11), both the spatial kinetic expectation and the binding energy vanish pointwise, because they connect the upper and lower blocks. Their variations are nonzero and are precisely what sustain (10). The complete symmetric stress has

$$T^{00}=\omega_N\rho_{\mathrm{tot}},\qquad T^{0a}=T^{ab}=0,\qquad \rho_{\mathrm{tot}}=\operatorname{Tr}(\phi^\dagger\phi). \tag{18}$$

For a constant boost velocity $v\hat n$, use the actual transformed field $S_v\Psi(t',x')$, with $S_v=\exp[\operatorname{artanh}(v)A_{\hat n}/2]$ in natural units. Since $S_v^\dagger B_jS_v=B_j$, every displayed scalar current is invariant. Direct substitution verifies the full boosted field equation. Its stress is

$$T^{\mu\nu}=\omega_N\rho_{\mathrm{tot}}(x')u^\mu u^\nu, \qquad E(v)=\gamma E_0,\qquad P(v)=\gamma E_0v. \tag{19}$$

Lorentz contraction in the spatial integral preserves $Q$. The field continues oscillating in the zero-total-momentum frame.

There is also a direct canonical proof of the inertial coefficient. Let $\Phi=(\phi,0)^T$ and let $\mathcal H'$ be the full real-linear Hessian of the rotating equation. Differentiating the boosted solution at $v=0$ gives

$$\delta_{v_a}\Psi=e^{-i\omega_Nt}(b_a-t\partial_a\Phi),\qquad b_a=\frac12A_a\Phi+i\omega_Nx_a\Phi. \tag{20}$$

The full equations imply

$$\mathcal H'\partial_a\Phi=0,\qquad \mathcal H'b_a=-i\partial_a\Phi. \tag{21}$$

These identities also follow directly from (10). Local phase covariance of its binding source gives $\mathcal B'_\phi(ix_a\phi)=\alpha_a\phi$. Using $\{-i\alpha\cdot\nabla,\alpha_a\}=-2i\partial_a$ and $\{\beta,\alpha_a\}=0$ gives $\mathcal B'_\phi(\alpha_a\phi)=-2i\partial_a\phi$. The root-current Hessian annihilates the local scalar phase and the translation of an isotropic Gram matrix. Combining these terms proves the second identity in (21), retaining the lower component of $b_a$.

The canonical symplectic form from the stated first-order time term is $\Omega(\eta,\xi)=-2\operatorname{Im}\int\operatorname{Tr}(\eta^\dagger\xi)$. The position variation is $-\partial_a\Phi$. Its pairing with the spinor part $A_b\Phi/2$ vanishes by upper/lower orthogonality. Its pairing with the phase part uses

$$\operatorname{Tr}[(\partial_a\phi)^\dagger\phi] =\tfrac12\partial_a\rho_{\mathrm{tot}},\qquad \int x_b\partial_a\rho_{\mathrm{tot}}=-\delta_{ab}Q. \tag{22}$$

The first equality holds for the displayed radial matrix cores; more general cores can contain a spatial trace phase current and need its additional terms retained. Hence

$$\boxed{\Omega(-\partial_a\Phi,b_b)=-\omega_NQ\delta_{ab}.} \tag{23}$$

Restoring the time-term coefficient $\hbar$ and the velocity factor $c^{-2}$ gives (1). This derives the center's inertia from the actual oscillating field's canonical kinetic term. It differs from the two-translation surface pairing: the second variation here is a boost, including its frequency-dependent spatial phase.

What remains physical input

The frequency and inertia identities do not select the shell weights. Replace the binomial weights in (12) by any positive sequence $w_k$, keeping one common action and the same sustaining core. The same contraction and canonical calculation give

$$\omega_N=\omega_{0,d}\sum_{k=0}^Nw_k,\qquad E_0=\hbar\omega_NQ,\qquad M_{ab}=\frac{\hbar\omega_NQ}{c^2}\delta_{ab}. \tag{28}$$

The spatial equation (10) does not contain these weights. This freedom tests what the inverse construction establishes. The Hockey-Stick identity is exact once its weights are supplied, but neither spatial existence nor the boost calculation derives them.

Likewise, $Q=1$ is a common physical normalization condition, not an arbitrary rescaling at fixed nonlinear coefficients. The coefficient $g$ in the unit-norm core was chosen once per depth to implement it. No quantization of $Q$ follows. The relation $m_c=\omega_{0,d}$ is a separate inverse identification. The relativistic kinetic structure of this inverse action is also supplied: the inertia calculation verifies its localized solution's response rather than deriving that kinetic structure.

The displayed boosts treat the oscillator and partner column labels as spectators. This makes the written local spinor action and its boosted solutions consistent. The later same-current rest-frame completion supplies an explicit Lorentz transformation law for the spatial tensors and varies its additional kinetic connection. The stationary frequencies and inertia survive, and zero-rotation-moment rays retain the complete linear response. Complex polarized rays expose a defective principal propagation matrix, however, so this completion does not yet settle a general physical carrier. Likewise, a list of coefficient grades is not a derivation of their underlying spatial harmonics, occupations or selection. Those are distinct carrier problems.

The master origin of the reconstructed terms, the common canonical norm, the relation of the core scale to $\omega_{0,d}$, and full spatial stability remain necessary for a physical particle interpretation. The complete response calculation gives finite normalized common-probe identities and bounded radial and angular tests. It does not yet identify observed electromagnetic or weak vertices, derive cross-depth coupling, or settle the neutrino control.

The solved-core pair calculation exhibits a specific fractional-current cancellation during an allowed pair preparation. Its smooth binding comparisons remove that particular variation singularity and still have localized depth-six solutions with the exact frequencies and full boosted stress. The smoothing scale is an additional common inverse coefficient; its spatial profiles and stability tests must be kept distinct from the original fractional model.

The trace-current comparison changes the current norm away from the occupied ray while preserving its complete isolated mass source, energy and spatial core. It removes a specific secular high-momentum response of the original norm and gives a positive conserved norm for the constant far-field linearization. This is a further inverse action choice. It does not select the shell weights, provide a physical energy floor, or resolve the polarized carrier's kinetic defect.

Retained source energy

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Stationary identity for a quartic action

Consider canonical Dirac fields $\Phi_j=e^{-i\omega_jt}\phi_j(x)$ with finite integrals and vanishing boundary terms. Write their complete energy as $E=T+U_2+U_4$, where $T$ is the spatial Dirac kinetic energy, $U_2$ is quadratic in the fields, and $U_4$ is quartic. Here the symbols denote integrated quantities. Let $Q_j=\int\phi_j^\dagger\phi_j$.

Multiplying each stationary field equation by its conjugate and summing gives Euler's homogeneous-function identity,

$$\sum_j\omega_jQ_j=T+U_2+2U_4, \qquad \boxed{E=\sum_j\omega_jQ_j-U_4.} \tag{1}$$

Additional binding terms must contribute their actual homogeneous variations; a smoothed or fractional binding term cannot be counted as quartic. Equation (1) is not a formula for particle mass from amplitudes. It is a consistency test on the energy of the specified stationary solution and its canonical normalization.

For a pure cross interaction between target $A$ and retained source $B$, let the target's other energy terms vanish on its solved branch. Then $\omega_aQ_a=U_\times$, while the normal equation gives $\omega_bQ_b=T_b+U_\times$ if it has no other terms. The complete energy is therefore

$$E=\omega_aQ_a+T_b. \tag{2}$$

The algebraic homogeneous comparison has $T_b=0$. A spatial source does not acquire this identity merely by reproducing the same target source matrix. Its kinetic expectation must be evaluated, along with any binding contributions omitted from the hypotheses of (2).

Spatial scaling supplies another necessary condition

Suppose the geometry is ordinary flat $\mathbb R^d$, the coefficients are spatially constant, and all currents and boundary terms allow the norm-preserving variation $\phi_j(x)\mapsto\lambda^{d/2}\phi_j(\lambda x)$. Then $T\mapsto\lambda T$, $U_2\mapsto U_2$, and $U_4\mapsto\lambda^dU_4$. Stationarity at fixed $Q_j$ implies

$$T+dU_4=0,\qquad E=U_2-(d-1)U_4, \quad \sum_j\omega_jQ_j=U_2-(d-2)U_4. \tag{3}$$

In particular, a massless canonical Dirac action with a nonnegative local quartic potential cannot have a nontrivial finite stationary solution of positive energy under these scaling hypotheses when $d>1$. This does not apply to the earlier inverse binding action, to a nonzero background without a justified subtraction, or to a different spatial geometry. It identifies which extra physical structure a proposed localized positive-frequency realization must account for.

Equations (1)–(3) follow directly from the stated action and admissible variations. A numerical stationary calculation should check the full equations, their energy identity and their spatial virial independently.

Scope

The toolkit contains complete conditional boost results as well as unresolved global-support questions. Their assumptions remain attached to each action. Supplied count weights, physical lower-depth confinement, sector scales, common phase-action normalization, occupied cutoffs and reciprocal observed interactions remain separate bridges. A ten-dimensional origin state alone would not establish the full physical mass operator. No mediator particles, compactified directions or probabilistic electron position law are used.

Intrinsic Clock and Contact-Span Energy

The core/span construction distinguishes the core's mass-setting oscillation from a free extension ending in contact. With core action $I$, phase $\theta$, span $L$, and reduced Hamiltonian $H$, the fundamental is $\dot\theta=\partial_I H$. It stays independent of span precisely when $\partial_L\partial_I H=0$ in that family. An additive span energy independent of $I$ can therefore change interaction energy without changing the intrinsic clock. This is a conditional reduced criterion, not a solved endpoint field.

Copying the vibrating core along a finite span is a different realization: its norm and energy can scale with span, and a fixed-norm constraint can shift its clock. That calculation does not impose the same behavior on a free part lacking the mass-setting vibration. The complete system still requires energy and momentum accounting. The independently assessed pair and frame diagnostics retain their restricted variations, the failed convergence criterion and the interrupted execution where applicable; none supplies a completed general inertia theorem.

The tension of a free extension

A pattern is localized in its bound directions and already spread along its free ones, as far as the things it touches. Let $\Omega=E-\omega N$, of which every stationary pattern is a critical point. $\Omega$ is unchanged by rotating ordinary space with the spinor turned along, so a straight pattern tilted between a bound direction $x$ and a free direction $w$ is again an exact solution. A slowly bent pattern with bound position $X(w)$ has

$$\Omega[X]=T\int\sqrt{1+X'^2}\,dw,\qquad T=\frac{p}{p+1}\,g\int\rho^{p+1},\qquad p=\tfrac19 .$$

$T$ is positive. With an outside energy $U(X)$ per unit length, a static shape obeys $T X''=U'(X)$ in the small-slope limit. This is Newton's equation in the inverted potential, with $w$ in the place of time: the equation of the imaginary-time path, on a real direction, with the tension in the place of the mass. The cost of a shape running from $X_1$ to $X_2$ is $\int\sqrt{2T(U-U_0)}\,dX$, the same form as a barrier exponent.

Sixteen of sixteen checks hold at depth two, for both seeds at $\omega/m_0=0.9$ and $0.7$, with bending cost over $TX'^2/2$ equal to 1.000 at the longest wave. The next term of the arc length also appears at the predicted size. The law's own energy on covariantly turned trial fields follows the reduced functional to five figures, 0.99592 at $\lambda=1$, and without the spinor's turn the ratio is 0.50, so the tension comes from the spinor turning with the axis.

The ends carry this tension. $T$ is $dE/dL$ at fixed norm and is positive, so a span pulls on whatever holds its ends. Contact does real work: a free extension is not self-supporting, and its ends keep it open. In direct evolution of the copied profile with soft edges touching nothing, a span shortens, a long span breaks into humps and a short one does not. The barrier is nearly the same all along $w$, which is why $w$ plays the part of imaginary time in alpha decay (volume 18).

A massive pattern's clock does not depend on how far it reaches: at fixed density, the same profile has the same clock at any span, and only its norm and energy grow with length. A ray of light is the opposite case. Its energy does not depend on how far it reaches, so its norm is fixed and its density falls with the span.

Pushing a pattern

A weak prescribed force applied to a solved depth-two pattern of the density law gives direct readings of inertia, energy and clock. At $\omega/m_0=0.80,\,0.75,\,0.70$:

Phase rate changes are in units of $Mv^2/(2N)$. The inertia is positive and close to the bare rate $m_0$ times the norm at every clock tried, and it differs from the energy. The energy at fixed norm rises by $Pv/2$ to four parts in ten thousand, so a slowly pushed pattern stays on the family of moving patterns and sheds nothing measurable. Pushing a pattern deforms it and raises its energy per unit norm and its lab phase rate by half the inertia per norm times the speed squared. Its own co-moving clock slows by the same amount. The slowing is 1.31 to 1.53 times $v^2/2$, more than a boost-covariant law gives, because the density term of the law is not boost covariant. The pattern contracts along its motion. The value at depth ten and $\omega/m_0=0.99$ is $M/(m_0N)=1.012$. The behaviour of two patterns side by side is in volume 15.

Identifying the Conserved Functional in Numerical Controls

In the new copied-envelope runs, the reported energy diagnostic is actually the rotating grand potential $\Omega=E-\omega N$. At fixed $\omega$, its drift and the physical energy drift obey $\Delta E=\Delta\Omega+\omega\Delta N$. A small relative grand-potential drift is not interchangeable with a physical-energy bound, especially when its own stated criterion fails. The line-density width diagnostic also uses staggered cell averaging rather than exactly the canonical node sum.

For the covariantly turned translated trial family, by contrast, the shifts and unitary turns preserve total norm. Energy differences then equal grand-potential differences, while the coefficient $T$ still means grand potential per projected length. These are concrete bookkeeping distinctions in the specialist control account, not assumptions that every physical field has the same normalization or free-part realization.