Null motion and positive inertia of the linked geometry

The linked spatial geometry admits a positive dynamical completion in which a disturbance can keep oscillating and carry null energy–momentum. The same action gives positive inertia to its supporting structure. No negative kinetic contribution is needed for either result.

This closes the time-to-space normalization within a specified effective geometric action. It does not derive that action from the master field, identify its disturbance with the photon, or solve the occupied massive particle. The coefficients below describe proposed modes of the same field. They are not additional particles or a particle-exchange mechanism. All spatial coordinates are ordinary and uncompactified.

Complete the existing geometric cost

Use coordinates $(t,x_1,x_2,y)$, where $(x_1,x_2)$ is the normal plane and $y$ is an available perpendicular direction in the containing space. Set $c=1$, with metric signature $(+---)$. The inverse action is

$$\boxed{ S=Z\int\left[(D_\mu W)^*D^\mu W -\frac18F_{\mu\nu}F^{\mu\nu}-(1-|W|^2)^2\right]d^4x, \quad D_\mu=\partial_\mu-iA_\mu,\quad Z>0.} \tag{1}$$

Its static normal-plane energy is exactly the previously specified cost

$$ZG=Z\int\left(|D_1W|^2+|D_2W|^2+ \frac{B^2}{4}+(1-|W|^2)^2\right)d^2x, \qquad B=F_{12}. \tag{2}$$

Lorentz symmetry in (1) fixes the temporal coefficient of the scalar derivative and the connection's electric term relative to (2). The overall physical coefficient $Z$ is still unspecified. The spatial and temporal derivative coefficients are not separately tuned at each particle index.

Varying the complete action gives

$$D_\mu D^\mu W-2(1-|W|^2)W=0, \qquad \partial_\nu F^{\nu\mu}=4\operatorname{Im}(W^*D^\mu W). \tag{3}$$

The temporal equation in the second expression is the constraint on the connection. Omitting it would give an incomplete moving-field calculation. For a varying static configuration, its solution gives the horizontal metric already obtained from the spatial geometry:

$$Z\int\left(|\delta W-i aW|^2+ \frac14|\delta A-\nabla a|^2\right)d^2x. \tag{4}$$

Here $a$ is the temporal connection associated with that variation. Its equation is

$$(-\Delta+4|W|^2)a =4\operatorname{Im}(W^*\delta W)-\nabla\cdot\delta A. \tag{5}$$

Thus the former positive metric comparison follows from (1), conditional on this explicit choice of action. The overall normalization and its master-field origin remain separate questions.

Keep the connection when the profile travels

Let $(W_0,A^0)$ be any regular solution of

$$(D_1+iD_2)W_0=0,\qquad B_0=2(1-|W_0|^2), \qquad \int B_0\,d^2x=2\pi d. \tag{6}$$

Its tension is $\mathcal T=2\pi Zd$. The maintained carrier uses winding $d$ at depths $2,3,4,5,6,10$; equality of winding and physical depth remains the existing geometric premise.

For any smooth real displacement $X(u)\in\mathbb R^2$, set

$$u=t-y,\qquad \xi_i=x_i-X_i(u),\qquad \boxed{W=W_0(\xi),\quad A=A_i^0(\xi)\,d\xi_i.} \tag{7}$$

The last expression specifies every connection component:

$$A_0=-X_i'A_i^0,\qquad A_y=+X_i'A_i^0, \qquad A_i=A_i^0. \tag{8}$$

Consequently

$$D_0W=-X_i'D_iW_0=-D_yW,\quad F_{0i}=-X_j'F^0_{ji}=-F_{yi},\quad F_{0y}=0. \tag{9}$$

All time and propagation terms cancel in the scalar and transverse connection equations. The remaining longitudinal equations are the static connection equations contracted with $X'$. This proves that (7) solves the full equations, including the temporal constraint and all acceleration terms. It is not merely a projected displacement equation. Replacing $t-y$ by $t+y$ reverses propagation.

The same argument permits a smooth path through the planar static solution family. In that case (5) supplies the temporal connection, and all coordinates of the path must share the same null variable. The construction is conditional on the smooth static family and its regular horizontal lift. The radial linear modes below and finite translated profiles are explicit controls of that statement.

This null-coordinate mechanism has a mathematical precedent in Dziarmaga's traveling multivortex calculation. The normalization, field equations and stress used here are those of (1). The precedent is not evidence that IDWT's particles realize this action.

The complete stress separates disturbance and support

For the Lagrangian density $\mathcal L$ in (1), the stress tensor is

$$T_{\mu\nu}=2Z\operatorname{Re}[(D_\mu W)^*D_\nu W] -\frac Z2F_{\mu\rho}F_\nu{}^\rho-g_{\mu\nu}\mathcal L. \tag{10}$$

Let $e$ denote the static density in (2) without $Z$, and let $v=X'(u)$. The self-dual equations imply $\operatorname{Re}[(D_iW)^*D_jW]=|D_1W|^2\delta_{ij}$. Direct substitution into the complete stress gives

$$\begin{aligned} T^{00}&=Ze(1+|v|^2),&T^{0y}&=Ze|v|^2,\\ T^{yy}&=Ze(|v|^2-1),&T^{0i}=T^{yi}&=Ze\,v_i,\\ T^{ij}&=0,&&i,j\in\{1,2\}. \end{aligned} \tag{11}$$

The negative static $T^{yy}$ is tension, a spatial stress. It is not negative inertia. For a pulse returning to the same center at both ends, with square-integrable $X'$, integration gives

$$\boxed{E_{\rm pulse}=\mathcal T\int|X'|^2du, \quad P_y=E_{\rm pulse},\quad P_i=\mathcal T[X_i(+\infty)-X_i(-\infty)]=0.} \tag{12}$$

These are finite excess quantities over the same static background. They include every momentum component. The background itself has positive energy per unit propagation length and extends along that direction. Equation (12) does not make its complete energy vanish.

Independently, boost the entire static profile in a transverse direction with speed $V$, retaining its contraction and temporal connection. Per unit propagation length, its complete stress gives

$$E=\gamma_V\mathcal T,\qquad P_x=\gamma_V\mathcal T V,\qquad \boxed{M_{\rm support}=\mathcal T>0.} \tag{13}$$

Restoring units gives $M_{\rm support}=\mathcal T/c^2$. Its quadratic translation metric is $\mathcal T\delta_{ij}/2$, in agreement with the independently derived $M_1=\pi/d$ for the polynomial center coordinate. Positive support inertia and null excess propagation therefore coexist in one specified action.

An exact bound retains arbitrary phase and field variation

Write $\mathcal E(y)$ and $\mathcal P_y(y)$ for the energy and longitudinal momentum integrated over the normal plane. For any smooth field data with the same fixed winding and finite transverse energy, completion of the full stress gives, for $s=\pm1$,

$$\begin{aligned} \frac{\mathcal E-s\mathcal P_y-\mathcal T}{Z} =\int\bigg[&|D_0W+sD_yW|^2+|(D_1+iD_2)W|^2\\ &+\left(\frac B2-1+|W|^2\right)^2+\frac14F_{0y}^2\\ &+\frac14\sum_{i=1}^2(F_{0i}+sF_{yi})^2\bigg]d^2x\ge0. \end{aligned} \tag{14}$$

Only the ordinary finite-energy boundary term of the static completion has been integrated out. The inequality holds before imposing the field equations, so it retains arbitrary phases, source disturbances and connection motion. The traveling solution saturates it in its propagation direction. The bound explains precisely which cancellation occurs: the time and propagation derivatives satisfy a null relation while the positive topological tension remains.

Physical polarizations and the higher-depth control

The displacement $X=a\cos\omega u+b\sin\omega u$ describes a line, ellipse or circle in the actual normal plane. Its two directions are normalizable translation modes, protected by translation symmetry. This protection survives a nonradial change of the static profile. It differs from the formal $n=0$ zero of the retained massive matrix, which an allowed shape perturbation lifts.

The full geometric kernel contains more than the center motion:

$$P(z)=z^d+\sum_{k=1}^d\alpha_kz^{d-k},\qquad \alpha_k\longmapsto e^{ik\theta}\alpha_k. \tag{15}$$

There are $2d$ real zero modes, with one complex pair at each normal rotation weight $k=1,\ldots,d$. Only $k=1$ translates the whole profile. All have positive kinetic metrics. A flat critical geometric action permits their linear null waves; occupation or curvature can change the centered modes. Their different weights do not make them identical spectator channels of the mass operator. The actual multiplicity calculation tests that distinction.

This particular construction has two normal translation directions at every tested depth because its feature has codimension two. It therefore does not select depth two as the only possible null sector. A physical photon identification still requires the appropriate global background, state and rotation content, localization in the available directions, and its coupling to other particles. The massive carrier's simplex frequencies and sustaining source are additional constraints on that same-field realization.

Verification

linked_geometric_dynamics.py implements the full fields and uncompleted stress. linked_null_stress_checks.py passes 942 conditions at all six depths, including independent Cartesian field derivatives, both propagation directions, three polarizations, complete boosts and all geometric mode weights.

linked_topological_stress_checks.py passes 263 conditions. It checks (14) on arbitrary smooth field data and solves nonradial configurations at two independent spatial resolutions. The latter retain their positive translation metric and fixed tension. Both reports preserve exact code hashes and separate raw spatial arrays. The complete results are the null-stress report and topological-stress report.