Depth-two spectrum, polarization and coherent support

The common curvature-moment calculation now includes all six active depths, $d=2,3,4,5,6,10$. Depth two uses the maintained massive assignments $(n_W,n_Z,n_H)=(76,81,95)$, with the same spatial equations, $h=zB/2$, normalization and hopping strength $\gamma=10000$. No particle receives its own adjusted coefficient.

This extends the normal-plane and finite spectral calculation. The later actual spatial dimension audit shows that the round feature becomes zero-dimensional at depth two and cannot supply the assigned harmonic labels. Its surface parameter is prescribed here; a full two-dimensional spatial realization remains to be constructed.

The spectrum and full centered shape stiffness survive this extension. The coherent support condition changes: with two occupied frequency bands, opposite polarization blocks can cancel every centered beat source exactly. The earlier impossibility result required at least three bands.

These finite polarization blocks are not automatically the observed boson polarizations. Their physical rotation law and interaction with another spatial pattern must also be established. The photon is tested separately as a massless case; a common polarization mechanism and a distinct photon mechanism both remain possible.

The later linked geometric action supplies two protected translation directions with exact null disturbances and positive supporting inertia. Its actual mode classification also retains the distinct centered modes. The bounded full mass operator preserves the leading depth-two simplex factors, but does not yet protect its formal zero at every finite scale or identify that zero with the photon.

The exact two-mode spectrum

The positive surface-sign matrix reduces to

$$T_n=\begin{pmatrix}n&0\\g&n+1\end{pmatrix}, \qquad g=\gamma h_0, \qquad h_0=0.4773956986626318. \tag{1}$$

Both surface signs have the same singular values in this two-mode case. Put $A=g^2+n^2+(n+1)^2$. Then

$$s_+^2=\frac{A+\sqrt{A^2-4n^2(n+1)^2}}2, \qquad \boxed{s_- =\frac{n(n+1)}{s_+}.} \tag{2}$$

The numerator is $2S(n,2)$. Thus the common leading frequency is $2S(n,2)/g$, with its finite correction retained explicitly. For $n>0$,

$$\frac1{\sqrt{1+[n^2+(n+1)^2]/g^2}} \le\frac{s_-}{n(n+1)/g}\le1. \tag{3}$$

This follows from $g\le s_+\le\sqrt A$. It does not treat the leading simplex expression as an exact finite-hopping law.

For the endpoint deformation $T_{01}'=\gamma\kappa_2$, the normalized frequency-response magnitude is exactly

$$\boxed{\frac{|s_-'|}{|\gamma\kappa_2|} =\frac{g s_+}{s_+^2-s_-^2}.} \tag{4}$$

It lies between $g/s_+$ and one. To obtain (4), differentiate $s_+s_-=\det T$ and $s_+^2+s_-^2=\operatorname{Tr}T^\dagger T$. The first derivative of the trace is zero in this direction.

The shared normalized endpoint coefficient is $\kappa_2/\sqrt{M_2}=-0.55216758$. The modest index dependence in the table is calculated, not removed by dividing each response by its particle frequency. These percentages are corrections to this candidate's leading law, not errors relative to measured boson masses.

Full spatial stiffness remains positive

Depth two has one centered complex polynomial deformation, $P(z)=z^2+\alpha_2$. Its real and imaginary coordinates both remain in the calculation. There are no pairs of distinct centered orders at this depth, so the mixed-order list is empty rather than an omitted mode.

The complete fixed-population averaged Hessian includes the actual second geometry, moving spatial basis and explicit coefficient derivatives. At unit canonical charge and with the actual geometric metric, the positive-sign results are:

The negative surface-sign controls are positive as well. Including depth two, all 102 tested real coordinate curvatures of the curvature-moment rule are positive, and the full complex centered matrices are positive across all six depths. The independent spatial refinements include depth two. The separate density-gradient control has 100 negative directions among 102, continuing its earlier failure of this averaged minimum test.

These are constrained energy curvatures. They are not a full particle stability theorem or a calculation of total inertia. The calculation uses the explicitly defined retained action; projection of the full parent's square has an additional complementary term unless an invariant low subspace has been controlled.

Two occupied bands admit exact coherent source cancellation

For $T u_j=s_jv_j$, the two pure positive-oscillator blocks are $(u_j,0)$ and $(0,v_j)$. Choose

$$c(t)=A\left[e^{-is_+t}(u_+,0) +e^{-is_-t}(0,v_-)\right], \qquad A^2=\frac1{2(s_++s_-)}. \tag{5}$$

Every centered source $\partial_a(H_P^2)$ is block diagonal. Its interference matrix element between the two terms in (5) is therefore zero. The diagonal source sum is also zero, by the same weighted trace identity used in the averaged calculation. Thus the centered source vanishes at every time, although the field continues oscillating at both frequencies. Reversing the block assignment works as well.

The general combinatorial bound agrees: two bands can have opposite colors with no same-color pair, whereas three or more occupied bands cannot. The depth-two minimum force variance is exactly zero, verified by direct projection of the complete source.

This removes the beat-force obstruction for this particular two-band trial. It does not remove the supporting state's energy cost. The existing positive-energy lower bounds remain about 3896, 3432 and 2500 times the small frequency for W, Z and Higgs controls respectively. Nor are these two different occupied frequencies the two equal-frequency components of an ordinary polarized light wave.

Physical polarization is a further spatial test

For a complex spatial vector $\varepsilon=a+ib$, the real harmonic motion is

$$\operatorname{Re}(\varepsilon e^{-i\omega t}) =a\cos\omega t+b\sin\omega t. \tag{6}$$

Its trajectory lies in the real plane spanned by $a,b$. Collinear vectors give linear motion; orthogonal equal lengths give circular motion; the general case is elliptical. This trajectory can have a component along the propagation direction unless transversality is imposed or derived. The ellipse's rank does not determine the spatial support dimension of the complete particle field.

The physical photon plane in photon transversality is perpendicular to propagation. Its two circular components have spatial rotation weights $\pm1$. The existing two-form coupling gives the analyzer overlap $|e^\dagger\varepsilon|^2$: a cosine-squared law for linear preparations and one half for a circular preparation with a linear analyzer. Direct real-space contractions verify these comparisons.

Massive bosons need not inherit that transverse-plane restriction. Longitudinal W/Z polarization is part of the measured comparison, including joint longitudinal states in the ATLAS WZ measurement. The Higgs mass assignment also does not prescribe a vector polarization: the ATLAS spin study found evidence for spin zero against its tested alternatives. The depth assignment is an IDWT premise, while these measurements constrain the required mode content.

Within each nonzero singular-frequency doublet, the current retained shape-source matrix is proportional to the identity. Its self-shape response therefore does not itself supply a polarization analyzer. The spatial plane identification, vector versus scalar state content and the massive longitudinal channel remain to be obtained from the full field. Higher-depth patterns can be compared with massive-boson polarization without assuming that the photon is either universal or exceptional.

The formal photon zero exists, but is not protected yet

Extending (1) formally to $n=0$ gives

$$T_0=\begin{pmatrix}0&0\\g&1\end{pmatrix},\qquad u_0=\frac{(1,-g)^T}{\sqrt{1+g^2}},\qquad v_0=(1,0)^T. \tag{7}$$

The full chiral mass matrix has the two zero vectors $(u_0,0)$ and $(0,v_0)$. A separately specified free propagation term leaves both mass eigenvalues zero and allows nonzero wave frequency. The normal mass matrix alone does not derive that propagation law or establish the availability of the required physical surface mode.

For the allowed centered deformation $T_{01}=\gamma\kappa_2\alpha_2$, the leading small mass-frequency becomes

$$\boxed{s_-(\alpha_2)= \frac{|g\gamma\kappa_2|}{\sqrt{1+g^2}} |\alpha_2|+O(|\alpha_2|^2).} \tag{8}$$

Consequently the reference zero alone is insufficient. The physical action still needs a constraint or identity that protects the photon zero under the permitted dynamics. The common kernel may eventually provide it; the current calculation has not done so.

Reproducible checks

The six-depth rerun uses mass_operator_sectors.py, full_spatial_stiffness_checks.py, flux_moment_checks.py and curvature_moment_mixed_checks.py. The coherent and pure-polarization searches explicitly retain the depth-two zero-bound case. The high-precision two-mode frequency and response identities are checked separately from the spatial integration accuracy.

depth_two_polarization_checks.py tests the actual zero-mode and shape-source matrices, the deformation lifting, and the separate physical-plane interface. Its polarization tests are conditional on that interface and do not silently identify algebraic components with physical helicities.

The reviewed outputs are the 549-check spatial report, 176-check curvature report, 151-check full Hessian report, 788-check coherent report, 134-check polarization report, 89-check physical-interface report and the unchanged six/ten scale comparison's 50-check rerun. Dependent scripts take the curvature report filename with --input-report; original executed source versions and raw spatial arrays are preserved with their reports.

The generalized cancellation and inertia test extends the polarization channel count and evaluates the retained positive energy and complete energy–momentum invariant.