Bounded mass operator and its joint dynamics

The inverse spatial mass operator can now use smooth bounded coefficients. Its complete normal spectrum has a proven positive gap, and its actual localized modes give the equally spaced factors needed by the simplex frequency construction. There is also a derived common geometric response within each depth. The construction is still conditional: its occupied wave drives a deformation of the proposed supporting geometry, so their joint stationary dynamics has not yet been obtained.

Mass is the selected frequency, mc^2=hbar omega. These calculations retain the assigned particle indices and read frequency from the operator. Field magnitude supplies canonical normalization where required; it is not substituted for mass. All spatial coordinates remain ordinary and uncompactified. The normal matrices describe field components, and the sectors remain nested rather than the particles.

The spatial operator is more constrained

The radial identity proves that equally spaced polarization survives arbitrary admissible radial changes to the winding profile. Thus the earlier W=z^d coefficient is not required. A bounded radial example retains the complete finite kernel and a rigorously positive gap to its full complement.

The linked connection construction goes further. Its common radial rule is

$$w'=\frac{n-a}{r}w,\qquad a'=2r(1-w^2).$$

It gives bounded coefficients W=w exp(i n theta) and A=a e_theta/r, a finite positive geometric functional, and the full factorization

$$Q^\dagger Q= \operatorname{diag}\left[ (-i\nabla+\boldsymbol A/2)^2+1, (-i\nabla-\boldsymbol A/2)^2+1\right].$$

Consequently the complete normal kernel has n modes and every other normal frequency has absolute value at least one in the stated units. The actual projected polarization satisfies P[(n+1-a)S]P=-2J_z. With n=d and the maintained grade assignment, the resulting small frequency is

$$\omega_{N,d}=C_d\binom{N+d}{d} \left[1+O(\gamma_d^{-2})\right],\qquad C_d=\frac{d!}{\gamma_d^{d-1}\prod_l h_l} \quad\text{in the chosen surface frequency unit}.$$

The spatial integrals h_l and the scale are shared by all grades at the same depth. The relation to physical cavity size across depths, the choice n=d, the full curved geometry and physical occupation remain explicit model questions. At finite hopping the frequency has a definite grade-dependent correction; it is not an exact simplex spectrum.

The same calculation gives a common shape response

The complete reciprocal source retains the wave's force on W, the connection and the polarization term. It agrees with an independently recomputed frequency derivative. Common translations cancel when every coefficient is moved together.

A deformation that changes the constant coefficient of the geometric polynomial affects only the opposite endpoint entry of the projected operator. Its frequency derivative is

$$\frac{\partial\omega}{\partial\epsilon} =2\gamma\kappa_d\operatorname{Re}(v_0^*u_{d-1}),$$

where kappa_d is one derived spatial integral. At large hopping the endpoint product approaches a grade-independent value. At depth six, the electron and muon shape derivatives are approximately 339.0613 and 339.0534 in the common test units, despite their different frequencies. This is a concrete route to grouping responses by depth. It is still a geometric source derivative, not a derived observed force or encounter rate. No physical neutrino suppression follows from this comparison.

The same deformation preserves the proposed geometry's finite functional. Its nonzero wave force therefore exposes what the joint model must handle: the supporting geometry cannot be solved independently and then held fixed when the particle mode is occupied.

The localized action supplies a different piece

A separate polarization-ratio current action has an explicit localized profile with the same shape for all grades at one depth. Its radial ratio is

$$\frac{h}{f}= \frac{I_{d/2}((\kappa-m_c)r)} {I_{d/2-1}((\kappa-m_c)r)},\qquad \kappa>m_c>0.$$

The common magnitude is absent from this equation. Under its declared current weights and canonical norm, the full action gives E=hbar omega Q and P=gamma hbar omega Q v/c^2. Thus it realizes positive total inertia on the displayed oscillating branch.

Its stability test is limiting. Exact full-multiplet algebra and a translated-packet argument show that the formal linearized spatial evolution is not uniformly bounded. This action therefore supplies an existence and inertia example, not a stable physical mass operator. Its binomial current weights also remain premises. It has not been identified with the bounded normal carrier above.

The phase calculation also establishes when a signed partial momentum response is opposite in phase to an imposed oscillating velocity. Such a response must not be called negative physical particle inertia. Computing the complete particle still requires its sustaining motion, momentum and stress.

The next useful model is a joint same-field action whose occupied wave and supporting geometry solve the same stationary and dynamical equations. The endpoint source and the full linear response now provide concrete tests for it. A specified common physical premise is acceptable; its predicted frequencies, geometry and stability must then be checked together.

Controls across physical depths

The depth-three and depth-four comparison retains the actual intrinsic spinor representations, fixed quark assignments and complete canonical partner phases. The same spatial carrier passes, while odd-depth response reversal, the core's persistent unbounded linear response and the bottom two-grade mixing constraint sharpen the joint-dynamics test.