Normal stress and translation of the gapped spatial carrier

The gapped carrier produces the simplex frequency under its stated inverse premises, but its partial wave momentum response has a negative direction in the large-coupling regime. This is a limitation of the wave in a prescribed spatial background. The field that generates and sustains that background has not been supplied, so the result is not the inertia of a complete IDWT particle.

This calculation continues the gapped normal carrier. It retains every normal mode forced at first order and does not infer inertia from a truncated kernel momentum matrix. Mass remains the selected frequency in equivalent units. The purpose of the calculation is to test whether the proposed spatial dynamics has the required inertial response.

An exact relation between partial momentum response and wave stress

Fix a surface frequency parameter beta. In dimensionless normal coordinates, the complete trial Hamiltonian has the form

$$H=\Lambda\alpha_i p_i+B(x),\qquad p_i=-i\partial_i,\qquad \{\alpha_i,\alpha_j\}=2\delta_{ij},\qquad \{B,\alpha_i\}=0. \tag{1}$$

Repeated normal indices run over x and y. B includes the prescribed normal matrix texture and all bounded pair terms of the carrier. Lambda converts normal momentum to the common surface frequency unit. Assume a localized normalized eigenstate H psi = omega psi with zero mean momentum, and an isolated eigenvalue for the response under examination. The gap estimates in the carrier note provide one conditional regime where this assumption can be checked.

The anticommutator and position commutator give

$$\{H,\alpha_i\}=2\Lambda p_i,\qquad [H,x_i]=-i\Lambda\alpha_i. \tag{2}$$

Combining them on the stationary state yields an exact response vector:

$$(H-\omega) \left(\frac{\alpha_i\psi}{2} +\frac{i\omega x_i\psi}{\Lambda}\right) =\Lambda p_i\psi. \tag{3}$$

A component along psi can be removed without changing the response to momentum. For translation of the prescribed background, the moving-frame operator is H(w)=H-w_i Lambda p_i, where w is speed divided by the normal wave speed. Second-order stationary perturbation theory gives

$$\omega(w)=\omega(0)-\frac12 M^{\rm wave}_{ij}w_iw_j+O(|w|^3). \tag{4}$$

Substitution of (3), followed by integration by parts, gives

$$\boxed{M^{\rm wave}_{ij}=\omega Q\delta_{ij}+T_{ij},\qquad T_{ij}=\Lambda\operatorname{Re} \langle\psi,\alpha_jp_i\psi\rangle.} \tag{5}$$

Here M^wave names the signed derivative of partial wave momentum with respect to the imposed velocity, not the inertia of the complete particle. Q is the canonical wave norm; Q=1 in the comparisons below. T is integrated wave stress. Equation (5) already includes the response of the complete eigenfunction. It is not a prescription for discarding the normal complement. In physical units, the dimensionless M in (5) has the units of mass times the squared normal wave speed divided by hbar times the surface frequency unit. This dimensional conversion does not identify it with the physical inertia of the complete particle.

In a local reference product, both normal coordinates can be translated. For a round feature embedded in ordinary space, its radial normal coordinate describes a profile displacement; translating that coordinate alone is not a Cartesian translation of the entire feature. The axial normal direction is an ordinary translation. A completed curved spatial construction must respect this distinction.

Independent complete normal control

For winding n=1 the square of D_1 is the two-dimensional oscillator plus a constant Clifford matrix with eigenvalues -2, 0, 0 and 2. With total oscillator degree N, its complete invariant grades are

$$g=N,\quad N+1,\quad N+1,\quad N+2, \qquad D_1^2=2g. \tag{6}$$

Keeping whole grades retains an exactly invariant Dirac space. The bounded spatial multiplier still forces higher grades, so its actual unretained source is calculated in ordinary position space and checked as the grade bound increases. Stress is assembled as the projection of the full product alpha_j p_i; multiplying separately projected factors would omit intermediate components outside the grade bound.

For beta=2 and coupling gamma=2, the largest control gives:

Independent moving-frame eigenfrequencies at speeds 0.001 and 0.002 agree with (5). The unperturbed control, with the additional bounded polarization and hopping removed, has exactly M=omega I. These controls verify the response identity, but do not imply the same limiting stress at higher winding.

The full normal complement at general winding

Let psi_0 be the canonically normalized projected small-frequency state. At fixed beta and gamma, the first full normal correction is

$$\psi=\psi_0+\Lambda^{-1}\eta+O(\Lambda^{-2}),\qquad \eta=-D_n^{-1}(I-P)V\psi_0. \tag{7}$$

The inverse acts on the complete orthogonal complement. Since psi_0 has negative chirality and eta has positive chirality, the leading wave stress is

$$T_{ij}^{(0)}=2\operatorname{Re} \langle\alpha_jp_i\psi_0,\eta\rangle. \tag{8}$$

The explicit partner factorization reduces the inverse to scalar radial operators for every forced angular integer m:

$$K_m=-\partial_r^2-r^{-1}\partial_r+m^2r^{-2}+r^{2n}. \tag{9}$$

Write x_r=r^(n+1)/(n+1) and nu=|m|/(n+1). The full regular, decaying half-line inverse is

$$\begin{aligned} (K_m^{-1}f)(r)=\frac1{n+1}\bigg[&K_\nu(x_r) \int_0^r I_\nu(x_s)f(s)s\,ds\\ &+I_\nu(x_r)\int_r^\infty K_\nu(x_s)f(s)s\,ds\bigg]. \end{aligned} \tag{10}$$

This expression includes the full radial domain. The numerical evaluation retains all forced Fourier powers and both positive-chiral spinor components. Differentiating (10) gives a separate check of the original first-order equation D_n eta=(omega-B)psi_0. Thus a check of only the squared equation cannot hide an incorrect first-order component.

The constant-polarization source has a useful exact inverse. For the normalized kernel mode with radial functions U_l and V_l,

$$D_n^{-1}(I-P)S\Phi_l =\frac r{n+1} \left(e^{i(l-n)\theta}V_l,0,0, i e^{i(l+1)\theta}U_l\right)^T. \tag{11}$$

The bounded hopping sources are obtained by differentiating h=z/sqrt(1+r^2). Their scalar inverses in (10) have independent adaptive quadrature controls. Tiny canonical chain coefficients are retained with relative accuracy; they are necessary because the frequency is much smaller than individual spatial source terms.

A negative wave-inertia direction survives a large normal gap

At gamma=10000, the leading complete normal result is:

All frequencies use the common dimensionless surface unit of the trial. The depth-ten unit has not been physically related to the depth-six unit. The two normal axes in the table are principal axes of the chosen polarized texture. Rotating the entire geometric data rotates the full stress tensor and preserves its eigenvalues.

Increasing Lambda reduces the size of the complementary field correction eta/Lambda. It does not suppress (8): the momentum in (5) contributes the compensating factor Lambda. Thus the positive normal spectral gap and the identity P p_i P=0 are insufficient to establish acceptable wave inertia.

There is an independent check on the much smaller stress trace. Let h_l' be the derivative of the projected hopping when h(z) is replaced by h(kappa z), at kappa=1:

$$h_l'=2\pi\int_0^\infty \frac{r^2[U_{l+1}U_l+V_{l+1}V_l]}{(1+r^2)^{3/2}}\,dr. \tag{12}$$

Normal dilation and the Hellmann-Feynman identity give, at leading order in the large normal gap,

$$\operatorname{Tr}T^{(0)}= \frac{2\gamma}{n+1}\sum_l h_l' \operatorname{Re}(v_{l+1}^*u_l). \tag{13}$$

Here u and v are the two complete, jointly normalized chain vectors. In the large-gamma simplex regime this becomes

$$\frac{\operatorname{Tr}T^{(0)}}\omega \longrightarrow-\frac1{n+1}\sum_l\frac{h_l'}{h_l}. \tag{14}$$

The mean of the two partial momentum-response eigenvalues divided by frequency is approximately 0.8329 at winding six and 0.8066 at winding ten. This positive mean coexists with a negative individual eigenvalue. The trace was integrated directly before numerical cancellation of the large opposite directional contributions, and agrees with (13).

What a supporting field must supply

The normal coefficient is prescribed in the present trial. A closed same-field solution must instead derive both that coefficient and its momentum and stress. Static stress balance can require a supporting contribution opposite to the wave stress, but asserting such cancellation does not construct the supporting solution or determine its energy. Its canonical norm and boundary conditions also matter.

The earlier matrix-current spatial response provides a concrete warning: the changing norm of a stationary background affects excess energy and inertia. Subtracting its phase-weighted norm without stating the physical ensemble does not establish a complete particle energy. The present normal carrier has not supplied those missing quantities.

The next source test must use the actual spinor currents of (7), retain their angular dependence, and assess whether they generate the assumed texture. No independent scalar interaction or exchanged particle is part of this calculation.

Independent check from every spatial force

The complete stationary equation also gives a force-based check of the integrated wave stress. Taking the expectation of i[H,x_j p_i] and integrating by parts gives

$$\boxed{T_{ij}=\int x_j\,\psi^\dagger(\partial_iB)\psi\,d^2x.} \tag{15}$$

Both coordinate-dependent coefficients must be included: the normal texture W and the bounded hopping h. The constant surface and polarization terms have no normal position derivative in this reference operator.

For W, its coefficient Lambda multiplies the first current j^(1)/Lambda, so the contribution remains finite in the large-gap limit. For h, the leading kernel pair already gives the required source. Write k(x)=sum phi_(1)^dagger phi_(0), summing every normal spinor component. The two contributions are

$$\begin{aligned} T^W_{ij}&=\operatorname{Re}\int x_j (j_1^{(1)}+i j_2^{(1)})^*\partial_iW\,d^2x,\\ T^h_{ij}&=2\operatorname{Re}\int x_j\, \gamma e^{-i\vartheta}k\,\partial_i h\,d^2x. \end{aligned} \tag{16}$$

Here vartheta is the retained normal orientation. The sum agrees independently with the kinetic calculation (8), including the full off-diagonal tensor entries and their rotation. The trace additionally obeys Tr T^W=-n Tr T and Tr T^h=(n+1) Tr T in this limit. Keeping only one coefficient's force would therefore give the wrong stress.

Lorentz contraction does not remove the partial wave stress

There is a useful distinction between a rigidly translated prescribed coefficient and the explicitly contracted coefficient appropriate to the normal Lorentz transformation of this trial equation. It does not separate mass from frequency; it concerns which part of the field and its supporting structure is included in momentum.

For a boost along normal direction i, put gamma_w=(1-w^2)^(-1/2). The contracted moving-frame operator uses B(gamma_w x_i,x_perp) and subtracts w Lambda p_i. The complete boosted eigenfunction can be written

$$\varphi_w= \exp\!\left(\frac{\operatorname{arctanh}w}{2}\alpha_i\right) \exp\!\left(\frac{i\omega\gamma_w w x_i}{\Lambda}\right) \psi(\gamma_w x_i,x_\perp). \tag{17}$$

The anticommutation relations of (1) verify directly that its moving-frame phase frequency is omega/gamma_w. The stationary internal oscillation has not stopped. At first order, (17) has exactly the response vector (3). Its wave momentum derivative is therefore still omega+T_ii at unit canonical norm.

The second derivative of the contracted coefficient contributes x_i partial_i B to the frequency curvature. Combining (15) with the rigid-coefficient result gives

$$\left.\frac{d^2\omega_{\rm contracted}}{dw^2}\right|_0 =-M^{\rm wave}_{ii}+T_{ii}=-\omega. \tag{18}$$

Thus the expected contracted phase frequency and a nonzero partial wave stress coexist. Reading only the curvature of the contracted phase would miss the explicit velocity dependence of the supporting coefficient. It would not compute the momentum of the complete supporting field.

In fact, (17) also gives an exact finite-velocity momentum identity. For a localized stationary state, the commutator with x_i gives the integrated current <alpha_i>=0, so the boost preserves its canonical norm Q. Define Pi_i=Lambda <p_i>. Changing variables to x_i'=gamma_w x_i and using the square of the spin boost gives

$$\Pi_i(w)=\gamma_w\Pi_i(0) +\gamma_w w\bigl(\omega Q+T_{ii}\bigr). \tag{19}$$

In a reference with Pi_i(0)=0 and Q=1 this is gamma_w w (omega+T_ii). The wave stress therefore persists in the full boost formula, not only in a first-order expansion. This identity concerns the same prescribed coefficient and does not supply the missing momentum of its physical source.

The complete-grade n=1 control verifies the small-velocity limits directly. At Lambda=30, its base frequency is about 1.99715209. The contracted phase has that curvature, while the x momentum response approaches 1.99841825, including the independently calculated stress 0.00126616. Both normal directions and increasing complete grades agree with (18).

For a closed localized same-field particle, the sustaining degrees of freedom must supply their own energy, momentum and stress. The present prescribed-background calculation does not determine them. The negative wave-only direction found above is therefore a source requirement to resolve, rather than a demonstrated negative inertia of a complete IDWT particle.

Reproducible evidence

The complete-grade normal control report linear-normal-grades-20260914T053528_685814Z.json passes 78 checks. holomorphic-components-20260914T054737_589091Z.json passes 18 independent high-precision checks of every small canonical component. The full normal stress report holomorphic-normal-stress-20260914T055018_172524Z.json passes 126 checks, including radial refinement, independent Green integrals, the first-order field residual, physical rotations, and the dilation trace. Source versions and raw arrays accompany these reports.

holomorphic-force-virial-20260914T072132_349807Z.json passes 90 independent full spatial-force checks. The contracted normal control normal-contraction-20260914T072306_274477Z.json passes 28 checks of both phase curvature and wave momentum, including normal-grade refinement.

The later algebraic audit holomorphic-source-compatibility-20260914T074340_237342Z.json also compares all eight stored contracted momentum results with (19). Their maximum absolute discrepancy is about 4.10e-10. The identity itself follows from (17), beyond those sampled velocities.

These establish the stated conditional calculations and their resolved signs. They do not establish a physical normal texture, a complete particle inertia, or a derived IDWT mass operator.

Related research

• Spatial mass operator: results and remaining dynamics — connects this calculation to its premises, later corrections and the complete source question.