Spatial dimension and component frames of the mass carrier

The completed radial calculation uses a genuine spatial differential operator, but a physical interpretation also requires its surface and component-frame data. Two independent checks clarify those requirements.

The round-feature surface operator works in each positive tangent dimension used by the construction, with its correct spinor rank. At depth two, however, the surface has dimension zero and cannot supply the assigned harmonic tower. Separately, the bounded angular completion is written in a specified component frame. A change of local phase must transport its angular operator along with the scalar, connection and wave.

These checks concern ordinary spatial dimensions and representations of field components. They introduce no compactified dimensions or new interacting particles.

The surface and normal dimensions add to the physical depth

The round feature is $S_R^m\subset\mathbb R^{m+2}$, with two ordinary normal directions and

$$m=d-2. \tag{1}$$

Its intrinsic spinor rank is $r=2^{\lfloor m/2\rfloor}$. For $m\ge1$, the shifted surface operator is

$$B_K=D_{S^m}-\frac1{2R}. \tag{2}$$

At unit radius, an ambient Clifford representation supplies the complete coordinate formula in a Killing-spinor frame:

$$B_K F=kF-\Gamma_x\Gamma_{\nabla_S}F, \qquad k=\frac{m-1}{2},\qquad \boxed{B_K^2F=\Delta_{S^m}F+k^2F.} \tag{3}$$

For odd $m$, the ambient Clifford representation in $m+1$ dimensions has twice the intrinsic spinor rank. The products in (3) preserve ambient chirality; restricting to one chirality gives the correct intrinsic rank. Both chiral copies must not be counted as independent physical directions. At $m=1$, the induced bounding spin structure gives the usual antiperiodic ordinary spinor; its coefficients in the Killing frame are periodic.

The differential identity can be checked without assuming an eigenvalue. Set $A_i=-\Gamma_x\Gamma_i+x_iI$. Then

$$\partial_iA_j=-\Gamma_i\Gamma_j+\delta_{ij}I, \qquad 2kA_j+\sum_iA_i\partial_iA_j=mx_jI, \tag{4}$$

while the symmetric second-derivative coefficient is $-\delta_{ij}+x_ix_j$ on the unit sphere. These are exactly the coefficients of the positive spherical Laplacian. Independent complex spatial jets verify the identity and its tangent propagation symbol at every maintained positive $m$.

For scalar coefficient harmonic degree $L\ge1$, the two shifted Dirac branches have

$$RB_K=\pm\beta,\qquad \beta=L+\frac{m-1}{2}, \tag{5}$$

with multiplicities

$$r\binom{L+m-1}{m-1},\qquad r\binom{L+m-2}{m-1}. \tag{6}$$

Their sum equals the complete scalar harmonic multiplicity times the intrinsic spinor rank. Taking the maintained $L=n+1$ gives $\beta=n+(d-1)/2$, precisely the value used by the radial carrier. The parity extension therefore introduces no new particle index or separately fitted spectral shift.

This validates the reference surface expression at depths three, four, five, six and ten. It does not supply the full curved normal extension, select the physical geometry, or identify the scalar/vector geometric fluctuations with ordinary spinors. Their different full-turn rotation laws remain explicit in the geometric carrier.

The depth-two limit needs a different spatial realization

For $d=2$, equation (1) gives $m=0$. The round feature $S^0$ consists of two points. It has no tangential derivative and no unbounded sequence of surface harmonics. Its intrinsic Dirac derivative is zero; the shift in (2) supplies only a constant, not a harmonic tower.

The same conclusion follows directly from homogeneous harmonic polynomials on $\mathbb R^1$:

$$\frac{d^2}{dx^2}x^L=L(L-1)x^{L-2}. \tag{7}$$

Only degrees zero and one survive. The assigned boson surface degrees 77, 82 and 96 cannot be obtained by extending this construction to $m=0$.

The earlier depth-two finite matrix, normal profile and polarization checks remain algebraically valid. They prescribe their scalar parameter $\beta$; they do not derive it from this zero-dimensional feature. Consequently those checks should not be described as a complete spatial realization of the depth-two bosons. A two-dimensional cavity or another actual spatial construction must supply the missing operator before it can be combined with that normal model. Its dimension count must remain two; an extra independent spatial coordinate cannot be added merely to restore a harmonic label.

This is a limitation of the codimension-two feature construction. It is not an alternative particle-depth assignment or a failure of the combinatorial mass formula itself. The photon identification is also still separate from the formal zero of the massive matrix.

The whole-depth spherical alternative supplies an explicit different spatial construction with a harmonic tower at every maintained depth, including two. It extends the known conformal spectrum to a local signed spinor polynomial at odd depths and supplies a separate actual tangent-vector polarization control at depth two. Its round geometry and component chain remain physical inputs; it is not a repair of the zero-dimensional feature or a derived supporting state.

Winding and spatial depth are distinct inputs

The normal profile has winding $w$, which fixes its localized kernel dimension. The construction chooses $w=d$. Spatial depth determines the surface dimension in (1); it does not by itself force that winding.

Keeping the two integers distinct gives the leading angular product

$$\prod_{l=0}^{w-1} \left[n+l+\frac{d-w}{2}\right]. \tag{8}$$

The required degree-$d$ polynomial is recovered when $w=d$. This makes the role of that choice explicit. It is not a new unrestricted mode search or evidence that another dimensionality reproduces the maintained mass construction. A same-field geometric selection of the winding remains to be derived.

The completed angular operator has specified frame data

The linked geometric action has the local phase transformation

$$W'=e^{i\lambda}W,\qquad A'=A+\nabla\lambda. \tag{9}$$

Let $Q_A=(\tau_3-I)/2$ and $U=e^{-iQ_A\lambda}$, extended over the canonical pair. The full normal Dirac operator obeys $D'=UDU^\dagger$. The curvature $B$, and therefore $h=zB/2$, is unchanged. Independent Cartesian differentiation confirms that the geometric energy density is unchanged.

The completed mass operator also uses

$$J=L+\frac12S+\frac d2\tau_3, \qquad L=-i(x_1\partial_2-x_2\partial_1), \qquad K_{\rm nat}=\{D,\mathcal R\}-2J. \tag{10}$$

Write $\xi\lambda=x_1\partial_2\lambda-x_2\partial_1\lambda$. Since the angular derivative acts on the local phase,

$$\boxed{J'=UJU^\dagger=J+Q_A\xi\lambda.} \tag{11}$$

If $W,A$ and the wave are changed but the literal angular derivative in (10) is held fixed, the raw completed formula instead has the defect

$$\boxed{ H_{\rm raw}[W',A']U-UH_{\rm raw}[W,A] =C\chi_1Q_A(\xi\lambda)U. } \tag{12}$$

The operator here is in the unscaled parent units. The defect is multiplied by $\epsilon$ in the scaled operator (1) of the moving-support calculation. It is a bounded multiplication term for a smooth localized phase change.

Transporting $J$ according to (11) removes (12) exactly. This keeps the radial reference operator and its simplex calculation unchanged. It does not establish a master-field origin for the selected angular lift, or turn a component-frame change into an observed force.

Equivalently one can describe the lift with a reference connection $\mathcal A$, whose angular component transforms by $a_{\rm ref}'=a_{\rm ref}+\xi\lambda$. Then

$$J_{\rm ref}=L+a_{\rm ref}Q_A+\frac12S+\frac d2\tau_3, \qquad K_{\rm ref}=S-d\tau_3+2(a-a_{\rm ref})Q_A. \tag{13}$$

In the original reference frame $a_{\rm ref}=0$, this is the maintained coefficient. Equation (13) records the operator's frame data; it does not posit a new independently interacting field. Whether that lift is supplied by the ambient geometry, a chosen gauge description, or retained same-field modes must be stated in a complete physical action.

The fixed-frame spectral calculations are meaningful within their stated frame. The correction is that an arbitrary phase change cannot be applied to only some coefficients and then interpreted as the same physical operator. Conversely, a claimed gauge redundancy of a proposed joint action needs the complete transformation law, including (11).

The frame term can reach the small-frequency endpoints

The defect is not necessarily invisible after radial projection. For $\lambda=f(r)\sin[(d-1)\theta]$, the leading endpoint contraction in the stated real radial convention is

$$\boxed{ \langle\psi_-,C\chi_1Q_A(\xi\lambda)\psi_+\rangle =\pi(d-1)\int_0^\infty r f(r)u_-(r)u_+(r)dr>0, } \tag{14}$$

when $f$ is positive away from the origin and decays appropriately. Both endpoint fields include their complete radial dressing. The other component term has angular weight $d+1$ and drops out of (14). Direct angular quadrature agrees with this independent radial integral at every maintained depth.

Thus correct frame transport matters even for the small-frequency subspace. Equation (14) is a leading endpoint matrix element; it is not being used as an unverified complete finite-scale eigenvalue derivative. The normal leading operator is already covariant, so the leading physical shape derivative used for the nearby-zero theorem is unaffected by this order-$\epsilon$ bookkeeping correction.

Verification

surface_dimension_checks.py passes 194 checks of actual Clifford representations, intrinsic spinor ranks, the complete differential square, tangent symbols, full-turn rotations, multiplicities and the depth-two boundary case. Its surface-dimension report keeps that missing tower explicit.

angular_frame_covariance_checks.py passes 156 checks of independent Cartesian derivatives, unchanged geometric energy, the exact raw frame defect and its transported cancellation. The frame-covariance report also retains full radial endpoint contractions and their raw fields.