A whole-depth spatial carrier and physical depth-two polarization

The missing depth-two surface tower can be avoided in a different, already motivated spatial construction. Use the entire proposed $d$-dimensional round geometry, rather than a round feature with two normal directions inside $d$-dimensional Euclidean space.

The critical conformal scalar operator already supplies the even-depth simplex spectrum on this geometry. The local spinor construction below extends its spectral magnitude to odd depths and keeps the actual depth-two harmonic tower. It also gives a first-order causal parent with a positive wave action. Its geometry, retained component chain and supporting field remain assumptions of an inverse construction.

The later homogeneous local current supplies an explicit same-field interaction with exact spatial standing waves of a fully varied positive action. The physical form and stability calculation also realizes the same depth-two component carrier with actual differential forms and distinguishes the stable-component branch from an unstable zero-current competitor. These developments retain the geometric and component assumptions of this note. The complete angular stress constraint and pair response refinements describe the remaining support and coupling limitations.

This does not derive the topology of an IDWT sector or turn its ordinary spatial directions into compactified dimensions. $S_R^d$ is a stated physical geometric candidate with an undetermined scale. Ambient coordinates used to differentiate spherical functions do not add an independent physical coordinate.

The complete spatial operator uses dimension d

Let $D$ be the actual self-adjoint spinor Dirac operator on $S_R^d$, with spinor rank $r=2^{\lfloor d/2\rfloor}$. The positive Killing frame gives the shifted first-order operator

$$B=D-\frac1{2R},\qquad k=\frac{d-1}{2},\qquad \boxed{(RB)^2=R^2\Delta+k^2.} \tag{1}$$

The scalar Laplacian acts on the full coefficient vector in that frame. The spatial frame audit establishes the identity for positive sphere dimensions, including the odd cases after restricting the ambient Clifford representation to one chirality. The current sphere dimension is $d$, rather than $d-2$.

At unit radius, with Hermitian ambient Clifford matrices $\Gamma_i$, the actual differential expression is

$$BF=kF-\Gamma_x\Gamma_{\nabla_S}F. \tag{2}$$

Its tangent symbol is Hermitian and squares to $|p|^2I$. It is an ordinary spatial spinor operator. A complete turn acts as minus the identity on spinors; this does not by itself supply vector-boson polarization.

On scalar coefficient harmonics of degree $n\ge1$, the two shifted Dirac values and their full multiplicities are

$$RB=\pm(n+k),\qquad \dim\Pi_+=r\binom{n+d-1}{d-1},\qquad \dim\Pi_-=r\binom{n+d-2}{d-1}. \tag{3}$$

Their sum is the scalar harmonic multiplicity times $r$. Here the harmonic degree is $n$, as in the original conformal comparison; the previous codimension-two feature used degree $n+1$. The particle index and its simplex count have not been changed.

One local polynomial covers even and odd depths

Define

$$\boxed{\mathcal Q_d(B)= \prod_{j=0}^{d-1}\left[RB+j-\frac{d-1}{2}\right].} \tag{4}$$

On the two spaces in (3),

$$\boxed{ \mathcal Q_d\Pi_+=d!S(n,d)\Pi_+, \qquad \mathcal Q_d\Pi_-=(-1)^d d!S(n,d)\Pi_-. } \tag{5}$$

Thus the squared frequency polynomial is the same on both branches at every depth. The odd-depth sign is a signed operator eigenvalue, not negative particle inertia or an antiparticle assignment.

For even depth this is the existing transferred scalar conformal polynomial. For odd depth $d=2h+1$, it is the local expression

$$\mathcal Q_d=(RB)\prod_{j=1}^{h}[(RB)^2-j^2]. \tag{6}$$

Writing $\mathcal L=R^2\Delta$, useful examples are

$$\begin{aligned} \mathcal Q_2&=\mathcal L,\\ \mathcal Q_3&=(RB)\mathcal L,\\ \mathcal Q_4&=\mathcal L(\mathcal L+2),\\ \mathcal Q_5&=(RB)\mathcal L(\mathcal L+3),\\ \mathcal Q_6&=\mathcal L(\mathcal L+4)(\mathcal L+6),\\ \mathcal Q_{10}&=\mathcal L(\mathcal L+8)(\mathcal L+14) (\mathcal L+18)(\mathcal L+20). \end{aligned} \tag{7}$$

The odd scalar spectral magnitude would contain $\sqrt{\mathcal L+h^2}$. Keeping the actual signed spinor operator $RB$ instead supplies a local polynomial with the required magnitude. This spectral extension is not a proof of a spinorial conformal law on arbitrary deformed geometries. Its full shape-dependent extension and master-field origin remain separate questions.

An independent spatial calculation verifies the two signs before taking their polynomial. Let $u=e_1+i e_2$, $F=(u\cdot x)^n\zeta$, and $G=\Gamma_x\Gamma_u(u\cdot x)^{n-1}\zeta$. Direct derivatives give

$$BF=(n+k)F-nG,\qquad BG=-(n+k)G. \tag{8}$$

Exact products on this verified two-function span give (5), while the complete multiplicities in (3) prevent counting it as two independent copies of every scalar harmonic.

The relation to the integrated density of states is still the exact Hockey-Stick identity

$$S(n,d)=\sum_{q=0}^{n-1}\binom{q+d-1}{d-1} =\frac1{d!}\prod_{j=0}^{d-1}(n+j). \tag{9}$$

The right-hand product is now a spatial spectral magnitude. The identity does not identify all counted modes as occupied or supply their physical population law.

A causal parent retains the whole spatial spectrum

Using an unrestricted order-$d$ operator directly as a first-order time generator would have the previously identified finite-propagation problem. Instead, consider the explicit inverse first-order chain

$$T=\begin{pmatrix} cB-ck/R&0&\cdots&0\\ G&cB+c(1-k)/R&\ddots&\vdots\\ 0&G&\ddots&0\\ \vdots&\ddots&G&cB+ck/R \end{pmatrix},\qquad \boxed{H=\begin{pmatrix}0&T^\dagger\\T&0\end{pmatrix}.} \tag{10}$$

Every adjacent coefficient is the same $G>0$. No winding, normal profile or independently chosen hopping function is used here. The $d$-component chain is an explicit assumption about retained patterns of the same field. Its occurrence in that field must still be derived; matrix rank is not being identified with physical spatial dimension.

The operator is self-adjoint on the ordinary first-order domain. Its principal symbol is $c\chi_1\otimes I_d\otimes\sigma_B(p)$, whose square is $c^2|p|^2I$. Bounded lower-order coefficients do not alter that propagation cone. The full positive wave action

$$L_{\rm wave}=\|\dot\Psi\|^2-\|H\Psi\|^2 \tag{11}$$

has conserved nonnegative energy and frequencies equal to the singular values of $T$. It retains the entire spatial operator and component chain. It is not obtained by discarding the complement before squaring. The geometry and its support are still prescribed in (11).

On the positive branch in (3), let $\gamma=GR/c$. The dimensionless chain has diagonal $n,n+1,\ldots,n+d-1$ and subdiagonal $\gamma$. Its determinant is the exact rising product. The negative branch reverses these diagonal entries and has the same singular values. The lowest positive frequency is therefore

$$\boxed{\omega_{n,d}= \frac cR\frac{d!}{\gamma^{d-1}}S(n,d) [1+O_{d,n}(\gamma^{-2})].} \tag{12}$$

For a finite $\gamma$, all entries of the complete inverse are

$$(T_n^{-1})_{ji}= \frac{(-\gamma)^{j-i}}{\prod_{l=i}^{j}(n+l)} \quad(j\ge i), \qquad \frac1{\|T_n^{-1}\|_F}\le\sigma_{\min}(T_n) \le\frac{\prod_{l=0}^{d-1}(n+l)}{\gamma^{d-1}}. \tag{13}$$

These bounds retain the complete finite matrix. High-precision inversion checks every maintained index and both signs at $\gamma=1000,10000$. There is no hidden normal-plane approximation in this different carrier; its unresolved input is the physical component-chain realization and supporting geometry. Equation (12) remains an asymptotic mass pattern, not an exact finite-gap law or a determination of the sector scales.

Actual tangent-vector polarization at depth two

The spinor calculation above should not be relabeled as observed boson polarization. A separate vector control is available on the actual two-dimensional sphere. Let $\Delta_1=d\delta+\delta d$ be the Hodge Laplacian on tangent one-forms. For a scalar harmonic $\Delta_0Y=n(n+1)Y/R^2$,

$$\boxed{ \Delta_1(dY)=\frac{n(n+1)}{R^2}dY, \qquad \Delta_1(*dY)=\frac{n(n+1)}{R^2}*dY. } \tag{14}$$

Here the star rotates a real tangent one-form through a right angle. For a real standing harmonic $Y$, the two nonzero vectors $E=\nabla Y$ and $B=x\mathbin{\times}\nabla Y$ are pointwise orthogonal and have equal length. Thus

$$V(t_{\rm time},x)= \operatorname{Re}\{e^{-i\omega t_{\rm time}}[aE(x)+bB(x)]\} \tag{15}$$

gives actual tangent-plane linear, circular or elliptical polarization according to the relative magnitude and phase of $a,b$. The physical vectors and their entire spatial texture transform under a common ordinary rotation. A complete turn returns the vector to itself. At nodes of $\nabla Y$, the field vanishes and a polarization axis need not be assigned.

The normalized operator $R^2\Delta_1/2$ has spectral value $S(n,2)$ on both branches. This is a spatial vector-polarization control, not a derivation of all observed massive-boson degrees of freedom, their charges or their interactions. The scalar Higgs assignment remains a scalar spectral control; a vector test at its numerical index is not an identification of the Higgs with that vector field.

The scalar constant at $n=0$ has $dY=*dY=0$. Moreover, the positive identity

$$\langle V,\Delta_1V\rangle =\|\nabla V\|^2+R^{-2}\|V\|^2 \tag{16}$$

shows that the round two-sphere has no nonzero harmonic tangent one-form. The scalar zero is therefore not a polarized photon in this construction. A different photon realization remains possible and must supply its own massless propagation and transverse spatial field.

What this alternative improves and what it still needs

The common local polynomial now uses all $d$ physical dimensions, has the maintained spectral magnitude at odd and even depths, and avoids the zero-dimensional surface at depth two. The inverse causal parent uses fixed geometric shifts and one common gap within each depth.

This does not derive a self-supported particle. The round geometry, the origin of the retained chain, the physical scales, occupied-state selection and complete particle inertia remain open. The periodic-support test also shows why a frequency-preserving mode frame is insufficient without its temporal connection and physical supporting energy. Neither construction has yet derived the observed coupling filters or neutrino suppression.

Verification

whole_depth_sphere_checks.py passes 275 checks of actual spatial derivatives, complete spinor ranks, the exact polynomial, positive squared frequencies, finite-chain bounds, both signed branches and actual depth-two vector polarization. Its whole-depth report retains the original particle indices and explicitly labels the proposed geometry and the separate vector representation control.