Projecting the field kinetic term onto a surface

Summary: The canonical field kinetic term gives a boundary expression for the two-normal surface coefficient; common background circulation can make it profile-independent, while a fully localized normal field has a cancelling outer contribution.

A kinetic coefficient with a stated field origin

The surface-bending frequency calculation uses a geometric kinetic coefficient $\Gamma$. It is possible to test its origin directly from the canonical field action, independently of the curvature energy.

This addresses a normalization coefficient, not a new way of solving for mass. The selected frequency already specifies mass in equivalent units. A field's fixed-point equation may determine the common background and the profile used in the kinetic projection. Their existence as solutions of the same spatial action remains necessary.

Take the canonical time term

$$L_{\mathrm{time}}=\frac{i\hbar}{2}\int \left(\Psi^\dagger\dot\Psi-\dot\Psi^\dagger\Psi\right)d^dx. \tag{1}$$

All complex spinor components are retained. Their number does not add physical dimensions. The following comparison singles out two ordinary coordinates normal to a proposed field feature; it does not distinguish two fundamental directions in the containing sector.

Let $y=(y_1,y_2)$ denote those normal coordinates. Locally approximate a translated core by $\Psi(y-X)$, with slow dependence along the surface. The two displacement variations are $\delta_a\Psi=-\partial_a\Psi$. The exterior derivative of the kinetic one-form in (1) gives the oriented coefficient per tangential area

$$\boxed{ \Gamma_{12}=-2\hbar\operatorname{Im}\int_{\mathbb R^2} (\partial_1\Psi)^\dagger\partial_2\Psi\,d^2y. } \tag{2}$$

The sign follows the order of the normal coordinates and the convention that the symplectic form is the exterior derivative of the kinetic one-form. Reversing their orientation reverses the sign. The positive $\Gamma$ in the bending calculation is its magnitude after that orientation choice. Neither sign supplies an antiparticle interpretation.

Equation (2) is a pullback of the existing kinetic term, rather than an independently chosen field or pair coupling. The approximation that a curved feature is represented by these translated normal profiles, and the contributions of its other field variations, still require checking.

The coefficient is a boundary circulation

Define the normal current one-form

$$j_a=\operatorname{Im}(\Psi^\dagger\partial_a\Psi). \tag{3}$$

This is a canonical phase-current expression, not an assigned ordinary electric charge. Commuting the two spatial derivatives gives

$$\partial_1j_2-\partial_2j_1 =2\operatorname{Im}\big[(\partial_1\Psi)^\dagger\partial_2\Psi\big]. \tag{4}$$

For a smooth core and a normal disk $B_L$, Stokes' theorem therefore yields

$$\boxed{ \Gamma_{12}(L)=-\hbar\oint_{\partial B_L}j_a\,dy_a. } \tag{5}$$

The outer field matters even when the core is narrow. Restricting the integral to a convenient inner region can change the kinetic coefficient. This is an instance of the need to retain the complete canonical normalization.

A common background makes the result independent of core shape

As a declared spatial control, take

$$\Psi(r,\theta)=f(r)e^{ik\theta}\chi,\qquad \chi^\dagger\chi=1, \qquad f(0)=0, \tag{6}$$

where $k$ is a nonzero integer and $\chi$ is a full complex spinor. This is a common-core ansatz for its components, not a generic zero set of an arbitrary spinor and not a solved Dirac field. Equation (5) becomes

$$\Gamma_{12}(L)=-2\pi\hbar k f(L)^2. \tag{7}$$

Equivalently, the bulk integrand in (2) is $-2\hbar k f f'/r$. Its radial integral gives the same boundary value. If $f(r)^2\to\rho_\infty>0$, then

$$\boxed{|\Gamma_{12}|=2\pi\hbar|k|\rho_\infty.} \tag{8}$$

The detailed core width and radial shape have disappeared. A common winding and background density therefore supply a coefficient independent of the occupied surface harmonic, under these assumptions. The background density could be set by one common fixed-point equation; it is not a separate value to retune for each mass.

The transverse core width in this calculation is different from the round surface radius or a particle's proposed common cavity scale. Equation (8) establishes no equality between those spatial lengths.

For a general multicomponent boundary pattern, the correct expression remains $-\hbar\oint\operatorname{Im}(\Psi^\dagger d\Psi)$. Different component windings or spatial polarization textures cannot silently be replaced by (8).

A localized outer field cancels the coefficient

If the complete normal field decays so that its outer current circulation vanishes, (5) gives $\Gamma_{12}=0$. For example, a smooth localized profile $f(r)=r^{|k|}\exp[-r^2/(2a^2)]$ has positive and negative radial contributions that cancel. Keeping only the rising core would miss the returning outer contribution.

Thus a nonzero coefficient of the form (8) needs the stated nonvanishing surrounding field or a different complete kinetic reduction. A nonzero background also requires its own finite relative energy, spatial stress and global consistency analysis; (5) by itself does not provide them.

What this supplies for the frequency operator

With the stated positive coefficient, the bending frequency would read

$$\omega_{n,d}= \frac{\kappa d!}{mR^d\,2\pi\hbar|k|\rho_\infty}S(n,d), \qquad m=d-2. \tag{9}$$

This conditionally supplies the origin of one common normalization factor. It does not derive the bending energy coefficient $\kappa$, the particle indices, the common radius or the full inertial response. In particular, $\Gamma$ is a first-order geometric kinetic coefficient with different units from mass; it must not be relabelled as inertia.

The next dynamical requirement is a same-field fixed point and reduction that produce the needed curvature energy and its supporting terms together with (2). The unrestricted bending evolution's propagation limitation also remains. A boundary identity cannot resolve that limitation.

Verification

The localized oscillation calculation now supplies a different complete kinetic reduction. It pairs a spatial translation with an actual boost of a solved field, including the boost's frequency-dependent spatial phase, and obtains inertia $\hbar\omega Q/c^2$ in every ordinary spatial direction. This leaves the two-translation boundary cancellation in (5) intact. The new result belongs to an explicitly reconstructed local matrix-current action; it does not derive the bending energy or its surface coefficient from the original scalar quartic term.

surface_kinetic_projection_checks.py verifies (5) with independent Cartesian differences of the actual complex fields and Fourier differentiation of their boundary currents. It retains all eight or thirty-two spinor components in the six- and ten-dimensional controls, and checks covariance under a complete unitary basis change.

Different core widths with the same surrounding field give the same coefficient. Localized controls retain the outer contribution and give zero. Spatial-step and quadrature refinements check these conclusions. The preserved radial densities display where the contributions enter; the prescribed profiles do not establish a spatial master-field solution.