Generalizing polarization cancellation and testing its inertial meaning

The depth-two result supplies an exact cancellation of coherent shape forces. It motivates a stronger test: can phase or polarization also remove the collective inertia, and does that possibility disappear at higher physical depths?

Two calculations separate the relevant effects. Extending the available polarization channels permits force cancellation at higher depths without changing the frequencies. The complete positive quadratic energy of the retained occupied modes nevertheless remains positive and phase independent. Thus the current model supplies a polarization compatibility rule, but does not yet turn that rule into cancellation of inertial mass.

The subsequent actual normal-mode classification finds no independent repeated factor in the existing normal chain. Its geometric modes have different rotation weights. The complete linked stress instead supplies a null traveling disturbance above a background with positive inertia, without cancelling positive kinetic contributions.

The two-channel threshold is not automatically a dimensional law

The actual two-block calculation has two isospectral chiral sectors with different source matrices. At depth two, placing its two occupied frequency bands in opposite blocks removes their beat term. With three or more bands and resolved distinct beat frequencies, the exact endpoint determinant prevents all pair sources from vanishing.

To test whether this depends on retaining only two channels, consider a conditional extension with $r$ identical polarization channels of the same retained field. Each channel has the same squared carrier $T^\dagger T$. No frequency, spatial profile or per-particle coefficient is changed. This extension tests possible component content; it does not derive the physical value of $r$, or identify it with the spatial depth.

Write the normalized polarization of occupied band $j$ as $p_j\in\mathbb C^r$. For a centered shape source,

$$\boxed{\langle u_i\otimes p_i,K_a(u_j\otimes p_j)\rangle =\langle u_i,K_a u_j\rangle\,p_i^\dagger p_j.} \tag{1}$$

The endpoint pair-source vector is nonzero for every distinct pair in the tested finite matrices. With distinct beat frequencies, its cancellation therefore requires $p_i^\dagger p_j=0$. Every beat source can vanish if and only if the occupied polarizations can be mutually orthogonal:

$$\boxed{r\ge N_{\rm occupied}.} \tag{2}$$

Here the filled carrier has $N_{\rm occupied}=d$. Keeping $r=2$ gives the two-band threshold. Allowing $r$ to grow to $d$ permits cancellation at every tested depth. The same orthogonality cancels every centered source, not just the endpoint used to prove necessity. The frequency-weighted diagonal source sum remains zero. The occupied frequencies and their canonical populations are unchanged, so the same averaged energy function and its tested spatial Hessian are preserved. The extension adds polarization content whose physical origin must still be supplied; it does not repair the energy-normalization problem.

This extension is not an assertion that the physical higher-depth field actually contains these independent channels. It shows why that field's polarization content must be derived before a two-only cancellation can be attributed to physical dimensionality.

A quantitative bound when there are too few channels

Let $G_{ij}=p_i^\dagger p_j$. Since $G\ge0$, its trace is $N$ and its rank is at most $r$. Cauchy's inequality on its nonzero eigenvalues gives

$$\operatorname{Tr}G^2\ge\frac{N^2}{r},\qquad \boxed{\sum_{i<j}|p_i^\dagger p_j|^2 \ge\frac{N(N-r)}{2r}.} \tag{3}$$

Use zero as the lower bound when $r\ge N$. If $w_{ij}>0$ is the squared norm of the actual endpoint source for that pair, then

$$\sum_{i<j}w_{ij}|p_i^\dagger p_j|^2 \ge w_{\min}\max\left(0,\frac{N(N-r)}{2r}\right). \tag{4}$$

The six-depth test uses the computed spatial source coefficients in $w_{ij}$. It tests arbitrary complex polarizations, exact binary assignments and orthogonal constructions with enough channels. Neither (3) nor (4) is a new multiplicative factor in the mass formula.

Force cancellation leaves the retained positive energy intact

For the explicit action

$$L=\|\partial_t(Bc)\|^2-c^\dagger H_P^2c,$$

hold the geometry fixed and use normalized positive-frequency modes $\phi_j$. Their orthogonality gives

$$c(t)=\sum_j A_je^{i\theta_j-is_jt}\phi_j, \qquad \boxed{E=2\sum_js_j^2|A_j|^2 =\sum_jq_js_j,\quad q_j=2s_j|A_j|^2>0.} \tag{5}$$

No relative phase occurs in the complete integrated energy. The kinetic Gram matrix is the identity on these independent modes. A positive spatial Gram form can vanish only when the represented field variation vanishes; an algebraic null direction is not automatically a moving massless particle.

For the source-balancing populations $q_j=s_j/\sum_ks_k$, total canonical charge is one and

$$\boxed{E_0=\frac{\sum_js_j^2}{\sum_js_j}>s_{\min}.} \tag{6}$$

This also holds for the exact depth-two opposite-block state. Its shape force cancels, while its positive kinetic and potential energies remain. Direct evaluation of the complete wave and its time derivative verifies (5) for arbitrary phases and polarizations across all six depths. Amplitudes set canonical populations here; they are not particle mass labels.

The energy–momentum test is also dimension independent

A specified common free propagation extension gives $\omega_j^2=c^2k_j^2+s_j^2$. In units with wave speed one, its canonical contributions obey

$$E_j=q_j\omega_j,\qquad P_j=q_jk_j, \qquad E_j^2-|P_j|^2=(q_js_j)^2. \tag{7}$$

For positive-energy contributions, the total invariant is

$$\boxed{E^2-|P|^2 =\sum_jm_j^2+2\sum_{i<j}(E_iE_j-P_i\cdot P_j), \quad m_j=q_js_j.} \tag{8}$$

Each pair term is at least $m_im_j$: align the momenta to minimize it, then write their energies and magnitudes as $m_j\cosh\eta_j$ and $m_j\sinh\eta_j$. The difference is $m_im_j\cosh(\eta_i-\eta_j)\ge m_im_j$. Consequently

$$\boxed{\sqrt{E^2-|P|^2}\ge\sum_jm_j.} \tag{9}$$

Positive massive contributions in this additive retained action therefore do not cancel their energy–momentum mass through phase choice or added spatial dimensions. Under a common boost the bound is saturated, and the invariant equals the positive $E_0$ in (6).

For massless contributions the pair term becomes $E_iE_j(1-\widehat P_i\cdot\widehat P_j)$. It vanishes when every occupied component propagates in the same direction, for any number of components. Opposing directions increase the invariant instead of removing it. This zero is compatible with positive energy and continued motion; it does not require negative local inertia.

These conclusions are conditional on the stated retained action and free propagation extension. They do not calculate the full master field's binding energy or supporting stress. Those contributions must be obtained before identifying this retained energy with a complete physical particle.

What a physical cancellation mechanism must additionally provide

The current construction needs a physical polarization module, rather than an assumed channel count. It also needs the complete same-field stress and kinetic normalization. A viable masslessness mechanism must give null total energy–momentum for the photon while preserving its physical motion, and retain the intended nonzero mass-frequencies in the other sectors. The formal depth-two zero is still lifted by an allowed shape deformation, so its protection is a separate unresolved test.

The shared and distinct photon-mechanism possibilities remain open. Neither the force-cancellation threshold nor the positive-energy calculation alone settles the complete physical theory.

Verification

polarized_energy_momentum_checks.py checks the actual retained kinetic Gram matrix, phase-independent energy, canonical charge and common-boost invariant at every maintained massive assignment and both surface signs.

polarization_dimension_cancellation_checks.py uses the actual endpoint pair coefficients to test the varying-channel extension, Gram bound, exact orthogonal cancellations and positive energy–momentum inequalities across all six depths. Their 588-check energy report and 484-check channel report record the executed source hashes and exact input spatial report. Both scripts accept that report filename with --input-report.