A bounded normal carrier with linked spatial connection

A linked spatial texture and connection improve the inverse mass operator. Both coefficients are bounded, their radial profiles obey two common first-order equations, and the complete unwanted normal spectrum has an explicit positive gap. A bounded polarization coefficient projects to equally spaced values on the actual localized modes. The resulting small frequency has the same simplex numerator at every assigned particle grade.

These are properties of a specified spatial operator. Its coefficients have not yet been identified with the sustaining motion of the one IDWT field. In particular, the positive geometric functional below is not being assigned as an additional physical field energy. The calculation uses ordinary unbounded spatial coordinates, with no compactification or new particle interaction. Mass remains the selected frequency.

Linked radial coefficients on an ordinary plane

Let z=x+iy, r=|z| and theta=arg(z). For positive integer n, define

$$W=w(r)e^{in\theta},\qquad \boldsymbol A=\frac{a(r)}r\boldsymbol e_\theta, \qquad B=\partial_x A_y-\partial_y A_x=\frac{a'}r. \tag{1}$$

Here W is a complex matrix coefficient and A specifies a connection on field components. Their linked profile rule, in dimensionless normal units, is

$$\boxed{w'=\frac{n-a}{r}w,\qquad a'=2r(1-w^2),} \qquad w(0)=a(0)=0,\quad w(\infty)=1,\quad a(\infty)=n. \tag{2}$$

The regular radial solution has 0<w<1 and 0<a<n away from the origin. Near zero, w=c_n r^n[1+O(r^2)] and a=r^2+O(r^(2n+2)). At infinity, 1-w and n-a decay exponentially. Thus W is smooth and bounded, A is regular at zero and tends to n e_theta/r at infinity, and its curvature has finite integral 2 pi n.

Equation (2) is a rescaled instance of the mathematical first-order vortex profile equations. Their full-plane existence with prescribed zeros is established by Taubes's existence theorem. For the present radial case all n zeros coincide at the origin. The connection and texture are retained as inverse spatial coefficients here; the physical interpretation of that reference is not part of this model.

The linked rule has a finite positive geometric cost

$$\mathcal G=2\pi\int_0^\infty r\left[ w'^2+\frac{(n-a)^2w^2}{r^2} +\frac{a'^2}{4r^2}+(1-w^2)^2\right]dr. \tag{3}$$

Completing squares gives an exact boundary identity:

$$\frac{\mathcal G}{2\pi}=n+\int_0^\infty r\left[ \left(w'-\frac{n-a}{r}w\right)^2+ \left(\frac{a'}{2r}-(1-w^2)\right)^2\right]dr. \tag{4}$$

The boundary term is [a+(n-a)w^2]_0^infinity=n. Consequently (2) minimizes this radial functional and gives G=2 pi n. This supplies an explicit compatibility rule for the coefficients. It does not supply their physical energy units, canonical time evolution or their response to an occupied particle mode. Nonradial configurations with separated zeros also require attention; the radial cost alone does not select coincident zeros against every such deformation.

A complete normal operator with an explicit gap

Use alpha_1=sigma_1 tensor I, alpha_2=sigma_2 tensor I, Gamma_1=sigma_3 tensor tau_1, Gamma_2=sigma_3 tensor tau_2, and chirality C=sigma_3 tensor tau_3. The self-adjoint Dirac operator is

$$D=-i\boldsymbol\alpha\cdot\nabla+ \frac12\boldsymbol\alpha\cdot\boldsymbol A\,\tau_3+ \Gamma_1\operatorname{Re}W+\Gamma_2\operatorname{Im}W. \tag{5}$$

Tensor factors are implicit. It anticommutes with C. All matrix coefficients are bounded; its principal part is the ordinary spatial Dirac derivative. A smooth change of component basis can change the phase of W and the corresponding connection together, but neither coefficient is independently discarded in this calculation.

Let Q be the block from C=+1 to C=-1. In the positive-chiral square, the off-diagonal coefficient is proportional to w'-(n-a)w/r and hence vanishes by (2). The two remaining components have opposite connection signs:

$$\boxed{ Q^\dagger Q= \begin{pmatrix} (-i\nabla+\boldsymbol A/2)^2+B/2+w^2&0\\ 0&(-i\nabla-\boldsymbol A/2)^2+B/2+w^2 \end{pmatrix} =\begin{pmatrix} (-i\nabla+\boldsymbol A/2)^2+1&0\\ 0&(-i\nabla-\boldsymbol A/2)^2+1 \end{pmatrix}.} \tag{6}$$

Each covariant Laplacian is nonnegative on the full plane. Therefore Q has no square-integrable zero mode, and every nonzero frequency of D has absolute value at least one. The gap statement includes the full continuum and every angular mode; it is not a finite-box spectral estimate.

The actual finite kernel

Write q=n-1. Negative-chiral zero modes have the same angular form as in the holomorphic carrier:

$$\Phi_l=(0,e^{il\theta}U_l, i e^{i(l-q)\theta}V_l,0)^T,$$

but their radial equations include the retained connection:

$$U_l'=\frac{l-a/2}{r}U_l-wV_l,\qquad V_l'=\frac{q-l-a/2}{r}V_l-wU_l. \tag{7}$$

For l=0,...,q, the unique decaying radial solution is regular at zero. Both components are asymptotic to a constant times r^(-1/2) exp(-r). They are normalized by 2 pi integral r(U_l^2+V_l^2)dr=1. The ratio V_l/U_l obeys the ungauged ratio equation; the connection adds the same scalar radial weight to both components.

Integrating the exact derivative

$$\frac{d}{dr}[r^2(U_l^2-V_l^2)] =2r[(l+1)U_l^2-(n-l)V_l^2] -ra(U_l^2-V_l^2) \tag{8}$$

gives, for S=sigma_3 tensor I,

$$\boxed{P[(n+1-a(r))S]P=-2J_z,\qquad (J_z)_{ll}=l-\frac{n-1}{2}.} \tag{9}$$

The weight n+1-a lies strictly between 1 and n+1 away from the origin. Equation (9) excludes every other integer angular label: for l=-1 or l=n the required expectation already has magnitude n+1, which no nonzero square-integrable field can attain; more distant labels require a still larger value. Together with (6) and the regular decaying solutions of (7), this gives dim ker D=n, entirely in negative chirality.

The unweighted operator PSP is generally not equally spaced. For n=6 its diagonal is approximately (0.936811,0.706226,0.267644,-0.267644,-0.706226, -0.936811). The linked bounded coefficient in (9) is essential.

There is also an exact complementary inverse useful for later response calculations. Define the positive-chiral field

$$R_l=r(e^{i(l-n)\theta}V_l,0,0, i e^{i(l+1)\theta}U_l)^T.$$

Direct use of (7) gives

$$\boxed{D R_l=[(n+1-a)S-(n-1-2l)]\Phi_l.} \tag{10}$$

The source is orthogonal to the complete kernel by (9), and the unit gap makes its complementary inverse unambiguous. This is a full spatial identity, not a matrix fit.

The projected mass-frequency branch

Retain the same bounded angular multiplier h=z/sqrt(1+r^2). Its positive adjacent matrix elements are

$$h_l=2\pi\int_0^\infty\frac{r^2}{\sqrt{1+r^2}} (U_{l+1}U_l+V_{l+1}V_l)dr. \tag{11}$$

For n=6 they are approximately (0.707388791,0.822927247,0.852305535,0.822927247,0.707388791). They are fixed spatial integrals, shared by every particle degree at that depth. A canonical two-component pair then gives

$$H_P=\begin{pmatrix}0&T^\dagger\\T&0\end{pmatrix},\qquad T_{ll}=\beta+l-q/2,\qquad T_{l+1,l}=\gamma h_l. \tag{12}$$

No modes in the finite kernel have been dropped. The exact triangular inverse implies the small positive frequency

$$\omega_n(\beta)= \frac{\prod_{l=0}^{n-1}(\beta+l-q/2)} {\gamma^{n-1}\prod_{l=0}^{n-2}h_l} [1+O(\gamma^{-2})]. \tag{13}$$

With the existing n=d and beta=L+(d-3)/2 assignments, the numerator is d! binomial(L+d-2,d). Equivalently it is d! binomial(N+d,d) for N=L-2. The electron, muon, tau and depth-five control indices have not changed. The equality n=d remains a geometric premise of this inverse construction. The finite-gamma correction is present and must not be called an exact simplex law at finite hopping.

For a full first-order parent replace the holomorphic carrier's polarization term by -(n+1-a)S chi_1/2:

$$\mathcal H=\Lambda D-C\chi_1 B_K -\frac{n+1-a}{2}S\chi_1 -C\gamma(\chi_-h+\chi_+\overline h). \tag{14}$$

At a fixed surface eigenvalue beta, the unperturbed square is Lambda^2 D^2+beta^2. The perturbation norm is bounded by v=gamma+(n+1)/2. Its first complementary correction cancels by chirality, just as in the holomorphic construction, and the complete correction obeys

$$\|H_{\rm eff}(\omega)-H_P\| \le\frac{v^2(|\omega|+|\beta|+v)} {\Lambda[\Lambda-|\omega|-|\beta|-v]}. \tag{15}$$

This establishes the controlled normal reduction when the stated denominator is positive and the bound resolves the finite cluster. The actual curved spatial completion and the relation of normal and surface scales still require their own calculation.

Momentum and the remaining physical construction

The distinction between two spatial momenta matters. Equation (7) gives the exact projected covariant identity

$$P\boldsymbol\Pi P=0,\qquad \boldsymbol\Pi=-i\nabla+\tau_3\boldsymbol A/2. \tag{16}$$

It does not give P(-i nabla)P=0. For n=6, the largest absolute adjacent canonical x-momentum entry is approximately 0.254966. The complete physical translation calculation must include the connection's transformation and sustaining motion. Reusing the old unconnected canonical response would give the wrong result. No physical particle inertia has been derived from (16), positive or negative.

The construction now has bounded linked coefficients, finite geometric cost, a complete finite kernel, a rigorous normal gap and the desired projected numerator under one shared rule. It still needs a same-field action for their joint oscillation, its total momentum and stress, and the particle source's reciprocal effect on the geometry. Simply adding the positive functional (3) to the wave energy would change the total energy-frequency accounting and would require a new stationary calculation.

Independent verification uses direct Cartesian derivatives of all four components, a second application of the full Dirac operator to arbitrary positive-chiral test fields, radial and outer-domain refinement, exact projected identities, high-precision complete inverse matrices and the analytic full-complement bound. The report and raw profiles are retained with their source hashes in the research database.