Moving support, nearby frequency zeros and the simplex branch

A continuously moving supporting pattern is a legitimate alternative to a frozen spatial profile. The question is whether its equations preserve the selected mass-frequency. The completed radial operator gives enough information to make a precise local test.

Two results follow. First, the small wave frequency can be taken to zero by an exceptionally small allowed complex shape deformation. Second, a positive dynamical model of that deformation admits bounded periodic motion, but its frequency is determined by an additional conserved orbit action. The original simplex coefficient becomes a displacement of the orbit center. This model therefore does not select the particle masses.

The first result concerns the complete infinite normal plane at one fixed surface eigenvalue. The second is an explicitly specified local effective action. Neither is a solution of the full curved particle, a prediction of zero particle inertia, or a derivation of an additional interaction field. All mode coordinates describe patterns of the same proposed field. Mass continues to mean the selected particle frequency in equivalent units.

A complex deformation can close the full normal frequency gap

Use the bounded completed carrier, scaled by its normal frequency unit:

$$\mathcal H(\epsilon,\alpha) =A_g(\alpha)+\epsilon B(\alpha)+\epsilon^2 C_2(\alpha), \qquad \epsilon=\Lambda^{-1}. \tag{1}$$

Here $\alpha=\alpha_d$ is the actual complex centered polynomial deformation. Its two real components include the orientation of the spatial pattern. The scalar, connection and curvature coefficients vary together. A smooth local choice of their component frame is required; the frame audit specifies that requirement.

At $(\epsilon,\alpha)=(0,0)$, let $P$ project onto the two normalized endpoint zero modes, and let $Q=I-P$. The complementary operator is invertible under the already verified gap conditions. Define the exact zero-energy Schur operator using the fixed reference projection:

$$F(\epsilon,\alpha)=P\mathcal HP -P\mathcal HQ(Q\mathcal HQ)^{-1}Q\mathcal HP. \tag{2}$$

Self-adjointness and chiral symmetry give

$$F=\begin{pmatrix}0&\overline{f}\\ f&0\end{pmatrix}. \tag{3}$$

At the radial shape, angular selection and the completed radial chain give

$$f(\epsilon,0)=a_{n,d}\epsilon^d+O(\epsilon^{d+2}), \qquad a_{n,d}=\frac{F_d(g)}{g^{d-1}} \prod_{l=0}^{d-1}(n+l). \tag{4}$$

At the origin of this expansion, $\mathcal HP=0$, so the first shape derivative of the Schur correction vanishes. The complete radial endpoint calculation therefore gives

$$\partial_{\operatorname{Re}\alpha}f(0,0)=g\kappa_d(g), \qquad \partial_{\operatorname{Im}\alpha}f(0,0)=ig\kappa_d(g). \tag{5}$$

The coefficient $\kappa_d(g)$ includes the texture and connection derivatives as well as the curvature derivative. It is nonzero at every maintained depth for the verified $g=0.02,0.05$. Consequently the real Jacobian of $(\operatorname{Re}f,\operatorname{Im}f)$ with respect to the two shape coordinates is invertible.

Assume the regular linked profiles define a smooth bounded-coefficient family in this local chart, with the complementary gap retained. The real implicit-function theorem gives a unique nearby shape $\alpha_*(\epsilon)$ satisfying $f(\epsilon,\alpha_*)=0$. Its leading displacement is

$$\boxed{ \alpha_*=-\frac{F_d(g)}{g^d\kappa_d(g)} \prod_{l=0}^{d-1}(n+l)\,\epsilon^d +O(\epsilon^{d+1}) =-\frac{F_d(g)}{\kappa_d(g)\gamma^d} \prod_{l=0}^{d-1}(n+l)+O(\epsilon^{d+1}), } \tag{6}$$

where $\gamma=g\Lambda$. The expansion fixes $d,g,n$ while $\Lambda$ grows. It is not a uniform statement for arbitrarily large particle index.

At this shape the exact Schur matrix is zero. The two corresponding full normal eigenfields are recovered with their complementary components:

$$\psi=p-(Q\mathcal HQ)^{-1}Q\mathcal HPp, \qquad p\in\operatorname{ran}P. \tag{7}$$

Thus this is a nearby zero of the complete separated spatial operator, rather than only a cancellation in its finite kernel compression. The theorem is local and asymptotic; the numerical coefficients used below are refined approximations, not interval-certified existence radii. Additional frame-transport terms that vanish at the radial reference and enter at order $\epsilon$ leave (5) and the leading displacement (6) unchanged.

This zero is a zero of the candidate wave frequency. The geometric support still has its own energy and stress. It is not a massless particle solution and does not identify a photon.

How small is the required change of shape?

In units where time is measured in $R/c$, write the leading endpoint coupling as

$$t_{n,d}+s_d\alpha,\qquad t_{n,d}=\frac{F_d(g)}{\gamma^{d-1}}\prod_{l=0}^{d-1}(n+l), \qquad s_d=\gamma\kappa_d(g). \tag{8}$$

The local frequency is its magnitude. The norm of a small geometric deformation is $\sqrt{M_d}|\alpha|$, with the actual horizontal metric $M_d$ already computed from the spatial fields. At the same illustrative $g=0.02,\gamma=10^7$ used for the spectral bound, the leading norm of (6) is:

These are dimensionless distances in the specified shape coordinate, not particle radii. The values illustrate sensitivity of a very small frequency produced by the angular chain. Improving the large-separation spectral approximation also increases this sensitivity.

The same cavity radius can be retained during these deformations. Consequently common size alone does not protect the internal frequency pattern. A common displacement at one depth changes the leading complex couplings additively by $s_d\alpha$; it does not merely rescale their shared frequency unit. A phase perpendicular to the original coupling removes the first real frequency derivative, but both real deformation directions remain available in (5).

An independent finite-carrier check makes the mechanism transparent. For diagonal entries $n+l$, adjacent entries $\gamma h_l$, and the projected endpoint derivative $\gamma\kappa_{\rm proj}\alpha$,

$$T(\alpha)=T(0)+\gamma\kappa_{\rm proj}\alpha|0\rangle\langle d-1|, \qquad \det T(\alpha)=\prod_l(n+l) +(-1)^{d-1}\gamma^d\kappa_{\rm proj}\alpha\prod_lh_l. \tag{9}$$

Here $\kappa_{\rm proj}$ belongs to the finite normal-kernel compression; it is distinct from the fully dressed $\kappa_d(g)$ in (5). The exact zero and its complete null vector are verified with 110-digit arithmetic for every maintained index and both surface signs. Equation (9) checks a finite operator; the infinite-radial statement follows from (2)-(7), not from a finite numerical cutoff.

A positive moving-support action

Rephase the endpoint and shape coordinates once at a depth so the reference frequency is $t>0$ and the source coefficient is $s>0$. Let $\mathcal M>0$ be a constant local geometric kinetic coefficient, and retain both complex wave polarizations $c\in\mathbb C^2$. An explicit local inverse action is

$$\boxed{ L=\mathcal M|\dot\alpha|^2+\|\dot c\|^2 -|t+s\alpha|^2\|c\|^2-E_0. } \tag{10}$$

The constant $E_0$ denotes the supporting energy in this flat-modulus comparison. Equation (10) retains a positive kinetic term, positive wave energy and arbitrary complex polarization. It assumes constant local metrics and the leading endpoint potential. It is not yet a controlled reduction of the entire moving spatial action. Moving-basis terms, nonlinear metric changes and the complementary field must be included before making that stronger claim.

Set $u=\alpha+t/s$. The complete Euler equations of (10) are

$$\mathcal M\ddot u=-s^2\|c\|^2u, \qquad \ddot c=-s^2|u|^2c. \tag{11}$$

They possess exact solutions

$$u=b e^{i\Omega t_{\rm time}},\qquad c=A\zeta e^{-i\omega t_{\rm time}},\qquad \|\zeta\|=1, \qquad \boxed{\omega=sb,\quad \Omega^2=s^2A^2/\mathcal M.} \tag{12}$$

The symbol $t_{\rm time}$ in (12) is time; $t$ elsewhere in this section is the reference coupling in (10). The wave oscillation and the shape motion are both retained. The arbitrary constant complex vector $\zeta$ includes the full wave polarization, not just a real ray.

The positive wave phase charge and the orbit action are

$$\mathcal Q=2\omega A^2, \qquad \mathcal J=2\mathcal M b^2\Omega. \tag{13}$$

The latter is the conserved phase action for rotation of $u$ in this local model. It has not been identified with the physical angular momentum of the full particle. That identification needs the actual spatial rotation law and its complete supporting field.

Eliminating $A,b,\Omega$ gives

$$\boxed{\omega^3=\frac{s^2\mathcal J^2}{2\mathcal M\mathcal Q}, \qquad E-E_0=\frac32\mathcal Q\omega.} \tag{14}$$

The supporting kinetic energy is $\mathcal Q\omega/2$, in addition to the wave's $\mathcal Q\omega$. No negative inertia was introduced to obtain this motion. A universal numerical factor in energy might be addressed by a derived common normalization; it must not be silently discarded. The constant supporting energy also remains part of the full particle calculation.

The periodic motion has a positive stability bound

Use dimensionless time $\tau=t\,t_{\rm time}$, displacement $u=(t/s)z$, and $c=\sqrt{\mathcal Q/(2t)}v$. Dividing the nonconstant Lagrangian by $\mathcal Q t/2$ gives

$$\widehat L=\mu|z'|^2+\|v'\|^2-|z|^2\|v\|^2, \qquad \mu=\frac{2\mathcal M t^3}{\mathcal Qs^2}. \tag{15}$$

The corresponding action is $(\mathcal Q/2)\int\widehat L\,d\tau$. The conserved dimensionless charges are $q=-\operatorname{Im}(v^\dagger v')$ and $j=\mu\operatorname{Im}(\bar z z')$. The chosen positive-frequency wave normalization gives $q=1$.

For arbitrary complex polarization, Cauchy's inequality and the two conserved charges give, with $a=\|v\|$, $b=|z|$,

$$\widehat E\ge\frac{q^2}{a^2}+\frac{j^2}{\mu b^2}+a^2b^2 \ge\boxed{3(q^2j^2/\mu)^{1/3}}. \tag{16}$$

The second inequality is the arithmetic-geometric mean inequality. Equality occurs at the circular solutions. For nonzero $q,j$, this effective potential also bounds both amplitudes away from zero and infinity at finite energy.

At $q=1$, equilibrium has $b=(j^2/\mu)^{1/3}$ and $a=b^{-1/2}$. The radial Hessian is

$$\begin{pmatrix}8b^2&4\sqrt b\\4\sqrt b&8/b\end{pmatrix}, \qquad \det=48b>0. \tag{17}$$

Thus the orbit minimizes energy at its fixed charges, modulo its phase and polarization symmetries. Independent time integrations retain both polarizations, recover the exact solutions and conserve their charges. Nearby amplitude perturbations stay near the orbit in the tested local system. This is a stability result for (10), not for unspecified spatial or complementary modes.

The bare moving model loses the particle index

The translation $u=\alpha+t/s$ removes $t$ from (11) exactly. At one common $s,\mathcal M,\mathcal Q,\mathcal J$, every particle index therefore has the same frequency (14). To reproduce the prescribed simplex frequency by these orbits, their actions would have to obey

$$\boxed{\frac{\mathcal J_n}{\mathcal J_{n_0}} =\left[\frac{S(n,d)}{S(n_0,d)}\right]^{3/2}.} \tag{18}$$

No equation in (10) selects (18). For example, the required muon/electron orbit-action ratio is about $2.97\times10^3$; the top/up ratio is about $2.31\times10^7$. These are required inputs to match the frequencies, not derived predictions. The wave amplitudes in (12) merely implement the declared common phase normalization.

On the full coordinate plane of this local model, the corresponding Hamiltonians at different $t$ are related by a unitary translation of $\alpha$. Their quantum spectra would therefore also be identical under translated domains. Quantizing the same flat local action does not recover the missing index. Physical restrictions on the shape domain or additional metric and energy terms can break that equivalence, but must be derived rather than inserted as a separate frequency fit.

Starting at the radial reference with the geometry momentarily still also fails to produce the circular solution automatically. The direct integrations retain the wave's initial oscillation and show substantial subsequent changes of the instantaneous endpoint gap. An instantaneous gap during that motion is not automatically the frequency of a persistent particle state.

The available curvature stiffness is insufficient in its tested regime

The ordinary tube calculation supplies an optimistic harmonic pinning term for this centered deformation. In the same $R/c$ time units, let $x=\sqrt{M_d}\alpha$ and $E_0$ be the bare geometric energy. Its leading coefficient is

$$E_{\rm pin}=K_x|x|^2, \qquad \frac{K_x}{E_0}=\frac{A_d}{2\pi d\Lambda^2}, \qquad A_d=\frac{d(d-2)(d-3)}2. \tag{19}$$

This follows by integrating the same geometric density and kinetic metric over the round feature. It is not an independently chosen spring. At depths two and three this leading coefficient vanishes. At higher depths the other curved shape forces remain; ignoring them here makes the single-direction test favorable to this pinning mechanism.

Write $s_x=\gamma|\kappa_d|/\sqrt{M_d}$. In the positive branch, minimizing $K_x|x|^2+\mathcal Q|t+s_xx|$ gives

$$x_*=-\frac{\mathcal Qs_x}{2K_x},\qquad \omega=t-\frac{\mathcal Qs_x^2}{2K_x}, \tag{20}$$

provided this frequency stays positive. The shift is additive and common at a depth, rather than proportional to the simplex count. Requiring a fractional shift no larger than $\delta$ yields the necessary bound

$$\boxed{ \frac{E_0}{\mathcal Q t}\ge \frac{\pi d\Lambda^2\gamma^2|\kappa_d|^2} {M_d A_d\delta t^2}.} \tag{21}$$

For $g=0.02,\gamma=10^7,\delta=10^{-3}$, the leading bound is approximately $1.57\times10^{71}$ for the up reference, $4.21\times10^{79}$ for the first neutrino, $3.12\times10^{90}$ for the electron and $5.03\times10^{134}$ for the tau. These are conditional ratios in this particular large-separation pinning model, not physical energies. The calculation also checks a smaller separation. It does not establish that every possible same-field support has this cost.

Increasing the coefficient of this bare positive geometric action can hold the shape more tightly, but its energy must then be counted. It cannot be used as cost-free rigidity while mass is read from the small wave frequency. The complete common-radius pressure test remains an additional condition.

The same positive action also constrains the motion's time scale

The geometric kinetic coefficient in (10) is not independent of its background energy if both come from the declared positive geometric action. Restore normal length $L$, feature radius $R$, and $\Lambda=R/L$. Static normal gradients scale as $L^{-1}$, while derivatives with respect to the chosen time coordinate scale as $R^{-1}$. The same horizontal metric therefore gives, to leading order in the tube expansion,

$$\boxed{\mathcal M= \frac{E_0M_d}{2\pi d\Lambda^2}.} \tag{22}$$

The common geometric coefficient and feature volume cancel in this ratio. The temporal connection constraint is already included in $M_d$. Equation (22) applies to this specified geometric action and reference profile, not to an arbitrary unknown master-field reduction.

Consider the circular orbit that would retain $\omega=t$, and set $e_0=E_0/(\mathcal Q t)$. Equations (12), (13) and (22) give

$$\Omega=\frac{\gamma|\kappa_d|\Lambda}{t} \sqrt{\frac{\pi d}{e_0M_d}}. \tag{23}$$

The leading normal operator has interior modes with frequency at most $\Lambda g\sqrt d$: they descend from the original normal kernel, and the perturbation has norm at most $g\sqrt d$. The gap conditions ensure that these modes remain in the isolated cluster. Thus a local moving-profile approximation cannot obtain separation from every retained normal frequency unless, at minimum,

$$\frac{\Omega}{\Lambda g\sqrt d}\le\eta \quad\Longrightarrow\quad \boxed{\frac{E_0}{\mathcal Q t}\ge \frac{\pi\Lambda^2|\kappa_d|^2} {M_d\eta^2t^2}.} \tag{24}$$

Here $\eta<1$ is the chosen separation tolerance. This is a necessary condition relative to the leading normal carrier, not a sufficient adiabatic estimate or a full finite-scale evolution theorem. With $g=0.02,\gamma=10^7,\eta=0.1$, the required ratio already exceeds $10^{25}$ for the depth-two reference and grows much larger at higher depth. Both couplings and both separations used above are checked.

Consequently the stable local orbits cannot currently be promoted to controlled moving spatial solutions at a supporting energy comparable to the selected wave frequency. Making the geometric inertia sufficiently large also increases its positive energy. An exact rapid solution could evade the need for an adiabatic approximation, as the separately derived null pulses illustrate, but that would require solving its complementary spatial fields. Equation (24) does not exclude such solutions.

What the calculation establishes

The later periodic endpoint and temporal-frame calculation identifies a different moving possibility. A perpendicular periodic deformation preserves the leading simplex ratios through one common multiplier, with a uniform analytic error bound. Its reciprocal force still has a nonzero static component and higher temporal harmonics. Exact temporal transport can preserve all frequencies in a stated positive inverse action, but its physical connection and supporting energy remain to be derived. These results extend the moving test without turning the flat local action above into a frequency-selecting particle solution.

The completed radial mass polynomial remains a valid conditional spectral result. Its nearby allowed shape zero is now explicit. Positive periodic support is possible in a stated local action, but does not by itself retain that polynomial as a selected spectrum. Ordinary curvature pinning has a quantified energy limitation in the same regime.

The next supporting law must therefore preserve the relevant spatial spectrum under its own motion, with the actual component frame, complete energy and common normalization retained. This is a requirement on the joint same-field action, beyond merely choosing a cavity radius.

Verification

moving_endpoint_support_checks.py passes 222 checks: symbolic Euler equations, arbitrary complex-polarization energy bounds, exact and perturbed time integrations, conserved charges, and the maintained cross-depth coefficients. Its moving-support report retains the trajectories and their exact source references.

endpoint_gap_closure_checks.py passes 484 checks of the exact finite determinant, complete null vectors, the complex Jacobian and the curvature-pinning estimate. Its gap-closure report identifies the complete radial input report and raw spatial coefficients.

support_time_scale_checks.py passes 129 independent symbolic and numerical checks of (22)-(24), including all maintained particle indices. Its time-scale report preserves the positive-energy normalization and the restricted scope of the frequency-separation condition.