Preserving the simplex factors in the full radial operator

The simplex numerator can survive the complete radial normal field, rather than only its finite kernel projection. An explicit bounded completion of the spatial operator achieves this at the leading order in the normal-to-tangential scale ratio. Its radial coefficient is common to all particle indices at one depth.

The completion is an inverse construction from the target angular structure. It is not a forward derivation from the master action. It retains the original highest-derivative part, but changes specified lower terms. Finite-scale corrections, the occupied supporting geometry and the physical shared scales remain to be derived. Mass is the resulting frequency in equivalent units throughout.

The later spatial dimension and frame audit establishes the positive-dimensional reference surface formula, while identifying a missing surface harmonic tower at depth two. Its depth-two radial calculation therefore still uses a prescribed surface parameter. The same audit specifies how the angular operator must be transported under a local component-frame change.

The full radial field has two protected endpoints

Use the linked geometric operator $D=D_{\rm geo}$, normal chirality $C$, and the two existing canonical blocks described by $\chi_i$. These are field components, not additional spatial dimensions. Define

$$\begin{aligned} h&=zB/2,\qquad H_h=\chi_-h+\chi_+\bar h,\\ A_g&=D-gCH_h,\qquad \Gamma=C\chi_3,\\ J&=L+\frac12S+\frac d2\tau_3,\qquad \mathscr K=J+\frac12\chi_3. \end{aligned} \tag{1}$$

At the radial linked profile, direct differentiation gives

$$[A_g,\mathscr K]=0,\qquad \{A_g,\Gamma\}=0. \tag{2}$$

For $g=0$, the normal kernel has $d$ modes in each canonical block. In the $\Gamma=-1$ block their angular weights are $l-(d-2)/2$; in the $\Gamma=+1$ block they are $l-d/2$, with $l=0,\ldots,d-1$. The interior weights pair, while $\mathscr K=+d/2$ has one unpaired negative-chiral mode and $\mathscr K=-d/2$ one unpaired positive-chiral mode.

The curvature multiplier preserves these angular sectors. Its radial magnitude has the global bound

$$|h|^2=r^2(1-w^2)^2\le a(r)(1-w^2)\le d. \tag{3}$$

Indeed, monotonicity of $w$ and $a(r)=\int_0^r2s(1-w(s)^2)ds$ give $a(r)\ge r^2(1-w(r)^2)$. Thus the perturbation norm is at most $v=g\sqrt d$, compared with the rigorous unperturbed gap $\delta_d=2/\sqrt{d+1}$.

When $v<\delta_d/2$, each isolated one-dimensional endpoint cluster must stay at zero: chiral symmetry would require a nonzero eigenvalue to have an opposite partner in the same cluster. Interior sectors instead have leading splitting $gh_l>0$. Chirality removes the quadratic static Schur term, leaving the conservative remainder bound

$$\frac{v^3}{\delta_d(\delta_d-v)}. \tag{4}$$

If this is smaller than $g\min_l h_l$, there are no interior zero modes. Both conditions hold for all six depths at the tested $g=0.02$ and $0.05$, with the hopping integrals checked by independent radial refinement. The endpoint modes are therefore isolated and independent of the particle index. Their fields include positive normal-chiral components induced by $g$; those components are retained.

This protection concerns the radial leading operator $A_g$. It is not yet protection of a photon under the complete occupied dynamics or an arbitrary nonradial deformation.

Derive a bounded link from the angular factors

The angular link

$$B_-=-C\chi_-(\beta+J) \tag{5}$$

lowers $\mathscr K$ by one. Its scalar factor between weights $k$ and $k-1$ is $\beta+k-1/2$. Connecting the two endpoints requires $d$ lowering steps, which produces precisely

$$\prod_{l=0}^{d-1}\left(\beta+l-\frac{d-1}{2}\right). \tag{6}$$

Using (5) directly as an added physical term would introduce an unbounded angular derivative. Instead, the geometric anticommutator gives a bounded completion. Put

$$\mathcal R=-iS\alpha\cdot x,\qquad K_{\rm nat}=\{D,\mathcal R\}-2J =S+(a-d)\tau_3-aI, \qquad S_-=\frac12C\mathcal R\chi_-. \tag{7}$$

Here $\mathcal R^2=r^2I$, and $S_-^2=0$. The full differential commutator is

$$[A_g,S_-]=-\frac12C\{D,\mathcal R\}\chi_- -\frac g2\mathcal R\bar h. \tag{8}$$

Consequently the new neighboring link is the multiplication operator

$$\boxed{\widetilde B_-=B_--[A_g,S_-] =-C\beta\chi_-+\frac12CK_{\rm nat}\chi_- +\frac g2\mathcal R\bar h.} \tag{9}$$

There is one required two-step link:

$$\boxed{C_{--}=-S_-A_gS_-=\frac g4Cr^2\bar h\chi_-.} \tag{10}$$

Both (9) and (10) are bounded for the decaying linked curvature. Their coefficients follow from the same identity; they are not independently chosen for a particle or fitted to its mass.

With $\epsilon=1/\Lambda$, the exact directed identity on the smooth decaying mode domain is

$$(I+\epsilon S_-)(A_g+\epsilon B_-)(I-\epsilon S_-) =A_g+\epsilon\widetilde B_-+\epsilon^2C_{--}. \tag{11}$$

The coordinate multiplier $S_-$ itself is unbounded on the whole plane. Equation (11) is used for its finite-order identity on the exponentially decaying modes, not as an assertion of bounded global similarity of two self-adjoint operators.

The completed Hermitian parent

Add each raising adjoint and restore the normal scale:

$$\boxed{ H_{\rm c}=\Lambda A_g+ (\widetilde B_-+\widetilde B_-^\dagger) +\Lambda^{-1}(C_{--}+C_{--}^\dagger).} \tag{12}$$

Equivalently,

$$\begin{aligned} H_{\rm c}={}&\Lambda D-\gamma CH_h-C\beta\chi_1 +\frac12CK_{\rm nat}\chi_1+g\mathcal R\operatorname{Re}h\\ &+\frac g{4\Lambda}Cr^2(\chi_-\bar h+\chi_+h), \qquad \gamma=g\Lambda. \end{aligned} \tag{13}$$

It is Hermitian and anticommutes with $\Gamma$. Every new spatial coefficient is bounded; the original normal and tangential principal symbols remain unchanged. On the negative normal kernel the new polarization agrees with the maintained $-K\chi_1/2$. Its extension off that kernel, and its additional curvature terms, are specified changes to the full parent.

The coordinate multipliers refer to the chosen geometric normal origin. A rigid translation must move that origin and all profile coefficients together. Translating only $W,A$ while holding the coordinate factors fixed tests a different, externally anchored operator. The moving-frame contributions must be retained in a later occupied-state calculation.

A local phase change also transports the angular lift: $J\mapsto UJU^\dagger$, with $U=\exp[-i(\tau_3-I)\lambda/2]$. Transforming only the profile and connection while keeping the literal $J$ fixed changes the completed operator. The explicit covariance identity retains the reference spectrum and identifies the required frame data.

At order $\epsilon^d$, any path joining the angular endpoints must lower at every step. A raising step would require at least two additional powers of $\epsilon$. The leading coefficient of the Hermitian operator (12) therefore equals that of the directed identity (11), and hence contains the product (6). This includes all interior radial resolvents and both dressed endpoint fields.

For the maintained positive branch $\beta=n+(d-1)/2$, the result is

$$\boxed{ \omega_{n,d}=\frac cR\, \frac{|F_d(g)|}{\gamma^{d-1}}\, d!\binom{n+d-1}{d} \left[1+O_{d,g,n}(\Lambda^{-2})\right].} \tag{14}$$

The radial coefficient $F_d(g)$ is independent of $n$, with $F_d(g)\to(-1)^{d-1}/\prod_lh_l$ as $g\to0$. The negative surface branch reverses the signs of the factors and has the same leading positive frequency. Neither surface sign is retuned.

The complete spatial error bound makes this remainder explicit using the complementary gap and angular selection. The remainder notation holds at fixed nonzero $g$ and fixed assigned index as $\Lambda$ grows. It is not a bound uniform in arbitrarily large $n$, nor a claim of an exact finite-scale simplex law. At a formal zero of (6), the leading coefficient vanishes, but higher orders have not been shown to vanish. Such a root cannot yet be identified with an exactly massless photon.

The count in (14) is the usual simplex or cumulative oscillator count, appearing here through the angular product. This calculation does not assert that all counted states are occupied or independently derive the identification of geometric winding with physical depth.

Its relation to the integrated density of states is the Hockey-Stick identity itself:

$$\boxed{\sum_{q=0}^{n-1}\binom{q+d-1}{d-1} =\binom{n+d-1}{d} =\frac1{d!}\prod_{l=0}^{d-1}(n+l).} \tag{14a}$$

The sum counts oscillator shells cumulatively, while the completed operator produces the equal rising product through its angular links. Their equality follows by telescoping Pascal's identity. Identifying the actual occupied particle dynamics with that IDOS interpretation still requires the same-field state construction; the identity alone does not provide an occupation law.

Full radial verification

The endpoint and interior equations are solved as regular, decaying four-component radial boundary-value problems. Each endpoint has unit complete spatial norm. The interior inverse is taken before making a small-frequency approximation. In the completed recursion both neighboring and two-step links are retained.

At $g=0.05$, the coefficient obtained independently from the angular chain is:

Removing the factorial product from the direct completed calculation gives these same coefficients for every maintained index and both surface signs. The original bounded link gives small but index-dependent coefficients instead. Independent Cartesian differentiation checks the endpoint field equations, (8), (10), and the directed identity on arbitrary smooth test spinors. Changing the radial domain and resolution checks the coefficient independently of the perturbative algebra.

The spatial-complement calculation explains why this distinction matters. A full extension of a projected operator can change the leading mass ratios even when its compression is unchanged. The completion above supplies one explicit extension that retains the required factors after the radial field is included.

The leading shape response is common within a depth

Let $\psi_+$ and $\psi_-$ denote the normalized endpoint fields at angular weights $+d/2$ and $-d/2$. For the actual horizontal polynomial deformation $\alpha_k$, differentiate the scalar, connection and curvature coefficients together. The endpoint source is

$$\kappa_d(g)=\frac1g \langle\psi_-,\delta_{\alpha_d}A_g\,\psi_+\rangle. \tag{15}$$

This derivative includes $\delta D$ as well as $\delta h$. Keeping only the curvature overlap would omit most of the response at large depth. Because the endpoint fields are independent of the occupied index, (15) is also independent of that index. Angular selection gives zero between these endpoints for $k<d$, while $\delta_{i\alpha_d}$ multiplies (15) by $i$.

For clarity, write the real radial fields of each endpoint as $(u_\pm,v_\pm,a_\pm,b_\pm)$, with the first two components in negative normal chirality. Let $(U,V)$ be the actual horizontal geometric mode for a unit $\alpha_d$ variation. Exact angular integration separates the source into three radial densities:

$$\begin{aligned} k_W&=\frac{2U}{g}(u_-a_+-b_-v_+),\\ k_A&=-\frac{2V}{g}b_-u_+,\\ k_B&=rwU(a_-a_++b_-b_+-u_-u_+-v_-v_+),\\ \kappa_d(g)&=2\pi\int_0^\infty r(k_W+k_A+k_B)dr. \end{aligned} \tag{16}$$

These expressions are checked against an independent angular quadrature of the full coefficient matrices. With geometric metric $M_d=2\pi\int r(U^2+V^2)dr$, the response per unit geometric norm at $g=0.05$ is approximately:

The sign is a signed frequency response in the stated endpoint phase convention, not a negative mass or negative inertia. The physical small frequency is the magnitude of the complex endpoint coupling. A shape variation changes that coupling by $\delta t=\gamma\kappa_d(g)\delta\alpha_d+\cdots$ in the unscaled parent; its first frequency derivative is the component along the existing coupling's phase. Relative phase can therefore strengthen, weaken or initially leave its magnitude unchanged.

This retains the intended leading pattern of different simplex frequencies and a common shape response within a depth. It does not derive a complete reciprocal particle encounter or its rate. In particular, depth five has no exceptional suppression in this matrix element. The neutrino coupling target remains open.

The remaining physical calculation

Equation (14) gives a common frequency unit at each depth, not its absolute physical value or the shared six/ten unit. The normal and tangential lengths and coupling must follow from the occupied same-field action. The pressure calculation shows that bare geometric tension alone does not maintain the required common radius.

The coefficient $g$ mixes the normal spatial modes; it is not already an observed interaction strength or the proposed self-coupling mass unit. At fixed $R,L$, its small-$g$, large-separation behavior gives

$$\frac{\partial\log\omega}{\partial\log g} =-(d-1)+O(g^2)+O_{d,g,n}(\Lambda^{-2}). \tag{17}$$

Increasing this mixing lowers the small frequency under that fixed-geometry comparison. A proposed relation between small mass and weak interaction must therefore include the same-field geometry and normalization, or a different identified coupling coefficient. Numerical smallness of $g$ alone does not explain a small neutrino mass in this operator. The comparison requires $\gamma=g\Lambda$ to remain large; it is not the limit $g\to0$ at a fixed finite $\Lambda$.

There is also a direct stationarity test. For one occupied band with fixed positive canonical phase charge $\mathcal Q$, the trial energy is $E=ZG+\mathcal Q\omega$. The flat linked geometric cost is stationary along its complex modulus $\alpha_d$, whereas (15) gives

$$\boxed{ \left\|\nabla_{(\operatorname{Re}\alpha_d,\operatorname{Im}\alpha_d)} E\right\| =\mathcal Q\,\frac cR\,\gamma|\kappa_d(g)| +O_{d,g,n}(\mathcal Qc/R)\ne0 } \tag{18}$$

in the controlled large-separation regime. The canonical phase charge is the wave action's normalization, not an electric charge or a mass-fitting amplitude. Equation (18) follows by differentiating the magnitude of the complex endpoint coupling. Choosing its relative phase can make one real derivative vanish, but cannot make both vanish while $\kappa_d(g)\ne0$.

Thus a single occupied band does not hold this bare radial geometry stationary. The common leading source across particle indices is useful for a common supporting law, but that law has not been derived. A complete same-field contribution must balance this shape source as well as the radius pressure, with its energy and kinetic normalization retained.

The moving-support calculation strengthens this local test: under the stated smoothness and gap assumptions, an allowed complex shape deformation gives a nearby exact zero of the complete separated normal operator. That wave-frequency zero is not a massless complete particle. A positive local moving action does admit stable periodic support, but loses the particle-index dependence unless an additional orbit-action selection law is supplied. Ordinary curvature pinning also has a quantified supporting-energy cost in the tested regime.

The completion also changes the full shape-dependent operator. Previous finite-carrier stiffness calculations cannot be adopted unchanged: the new two-step term and the normal dressing must enter that source and stability calculation. A positive norm and a preserved mass numerator are necessary progress, but do not establish a complete particle solution. The physical rotation module must also be supplied: the geometric fluctuation and ordinary spinor representations discussed in the original carrier note are not identified merely by sharing a differential expression.

Verification

dressed_angular_chain.py solves the complete radial chain. dressed_angular_chain_checks.py compares all maintained assignments and both signs at two physical couplings, includes the analytic global gap bound, and checks independent Cartesian identities and radial refinements. The timestamped report and raw endpoint/interior fields preserve their exact executed source hashes.

dressed_endpoint_response.py and its checks verify (15)–(16) with 132 conditions, including phase covariance, every available polynomial weight, independent Cartesian curvature variation, the small-$g$ limit and radial refinement. The 298-check radial-chain report and 132-check shape-response report preserve the numerical comparisons and exact source versions.