A covariant self-contact mass operator

Current physical premise (IDWT): Physical spacetime has infinitely many ordinary spatial dimensions plus time. Matter occupies up to ten spatial dimensions; all remaining spatial directions are unbound for every particle. Particle patterns have different dimensional involvement in that space. The ten-dimensional matter domain is presently modeled by flat $\mathbb R^{10}$. Nested layers describe proposed spatial support, not additional manifolds. Depth counts a pattern's dimensional involvement, not a separate containing space. The familiar three dimensions describe the observer's view, not the ambient dimension. All spatial directions are of the same physical kind. Confinement and persistence of the layers remain dynamical questions.

Mathematical scope: Projective, round-sphere and prescribed global-nesting calculations below retain their stated additional premises. Their curvature, topology and finite-volume normalizations are conditional comparisons; they do not establish the current flat spatial geometry. Field components and mode spaces are distinct from physical position. IDWT has no compactified dimensions or mediator particles; interactions must come from $\Psi_\infty$. See spatial meaning and mathematical scope.

Current Reading and Scope — 1 November 2026

The title describes a candidate programme, not a completed physical mass operator. The current portion picture has C copies at one fundamental, with average count multiplication under additive registration. The newer compact construction has an exact supported shape, a restricted finite nonlinear energy basin and a proposed square-root cluster Hamiltonian. Its source leaf and count offset are supplied premises; the local field bridge and count selection remain open. The older spectral constructions below retain their distinct actions.

The new particle-geometry and sector-depths case provides the current derivations, shared assumptions, glossary and reproduction instructions. The twenty-volume research map connects the complete account. Calculations below retain their own stated mathematical premises and do not inherit results from a different action.

1. Result

Earlier Conditional Spatial Count Realization

The forward spatial count construction now derives the assigned clocks from two stationary equations of an explicit reconstructed action. A convex dispersion continuation retains those clocks, open affine source charts, nonnegative complete energy and conditional nonlinear spatial persistence at all fourteen maintained assignments under the positive-wave kinetic law. Its finite constitutive window, preferred-time density and assumed ordinary sphere are explicit premises. This is a distinct effective realization of the count spectrum; the covariant contact construction below retains its own premises. Physical localization, complete translational inertia, normalization and the master-field origin remain unresolved.

The spectral and field-mode construction below concerns the particle's proposed same-field structure. Its complex vectors, normalized measures and rank-one operators do not impose a Born position-probability law. For electrons, literal deterministic motion through six physical spatial dimensions remains the orbit premise. A mass-frequency eigenvalue or a stationary classical field profile alone does not derive that bound trajectory; see the actual-orbit correction.

Let $Q_d$ be the real $d$-dimensional manifold on which a particle sector lives. For every positive particle mode $n$, set

$$N=n-1.$$

There is a canonical rank-$S(n,d)$ mode bundle over $Q_d$:

$$\mathcal E_{N,d} = \operatorname{Sym}^{N} \left( \mathbf 1_{\mathbb C}\oplus T^*Q_d\otimes\mathbb C \right).$$

Its rank is

$$\operatorname{rank}_{\mathbb C}\mathcal E_{N,d} = \binom{N+d}{d} = \binom{n+d-1}{d} =S(n,d).$$

The additional trivial line $\mathbf1_{\mathbb C}$ homogenizes the cumulative occupation condition:

$$\operatorname{Sym}^{N}(\mathbb C\oplus V) \cong \bigoplus_{k=0}^{N}\operatorname{Sym}^{k}(V).$$

For $V=T_q^*Q_d\otimes\mathbb C$, the right-hand side contains every $d$-direction oscillator shell from $0$ through $N$. The extra occupation number records the unused portion of the total excitation.

The Hermitian fibre

$$W_{d,q}=\mathbb C\oplus T_q^*Q_d\otimes\mathbb C$$

has ray space

$$\mathbb P(W_{d,q})\cong\mathbb{CP}^{d}.$$

On

$$\mathcal E_{N,d,q}=\operatorname{Sym}^{N}(W_{d,q}),$$

rank-one support on the occupied coherent ray, covariance, and unit normalization uniquely determine a positive localization density

$$\Lambda_{N,d}([z]) =D_{N,d}\, |z;N\rangle\langle z;N|, \qquad D_{N,d}=S(N+1,d),$$

where $|z;N\rangle=z^{\odot N}$ is the normalized symmetric coherent state. This operator obeys

$$\int_{\mathbb{CP}^{d}} \Lambda_{N,d}([z])\,d\nu_{\rm FS}([z]) =I_{\mathcal E_{N,d}},$$

and

$$\Lambda_{N,d}([z])^2 =D_{N,d}\Lambda_{N,d}([z]).$$

It therefore has one nonzero eigenvalue, exactly $D_{N,d}$. A scalar self-contact contribution

$$\mathcal M_{N,d}([z]) =\mu_d\Lambda_{N,d}([z])$$

has the coherent localized mode as an eigenvector with rest mass

$$\boxed{ m(n,d)=\mu_dD_{N,d} =\mu_d\binom{n+d-1}{d}. }$$

This derives the combinatorial multiplier as the eigenvalue of the localized contact density on the cumulative tangent-mode bundle.

The proposed physical identification assumes the following additional occupied-ray prescription for the scalar self-interaction:

The master Dirac interaction is algebraic in the local scalar current and its sector coordinates, with no derivatives of that current. The current is evaluated at the occupied projective mode ray determined by the field's local amplitude and covariant derivatives.

The rank-one formula requires contact at a specified occupied coherent ray in the auxiliary projective fibre, with the stated measure and evaluation map. Locality in the physical base coordinate alone does not imply this. A local identity operator has a delta kernel too, yet band compression gives the identity on the band, not the rank-one operator $\Lambda_{N,d}([z])$. The combinatorial multiplier is therefore derived for the explicitly assumed coherent-ray contact operator. Deriving that operator, its band-independent coefficient $\mu_d$, and the occupied modes from the master action remains open; normalization is not the only remaining problem.

There is also a distinct conformal scalar construction: the critical conformal differential operator on a round even-dimensional spatial sphere has the exact normalized eigenvalues $S(n,d)$. Its conformal recurrence and spatial factorization derive that scalar spectrum without a mode-ray point insertion. The construction uses different geometric and operator assumptions; its realization in the master spinor action, the occupied scalar harmonic degrees and the common sector normalization remain unproved.

The Killing-spinor transfer constructs an exact local spinor version on that specified round geometry. Its opposite Dirac signs carry adjacent simplex values. Direct insertion as a scalar mass fails Hermiticity; the real action gives their average. The scalar-to-spinor transfer is therefore explicit, while the physical master mass term remains unresolved.

Working backward, the inverse polynomial dynamics classify explicit local Hermitian actions with exact simplex frequencies on the complete paired family. They modify both kinetic and scalar terms and retain the complementary spinor equation. The inverse action has a consistent canonical norm. Its full curved commutator now excludes finite propagation when the polynomial acts on every spatial mode. A causal nonlinear origin for selected particle masses remains open.

The inverse scalar-law test retains the original Dirac kinetics and reconstructs an interaction from complete spatial fields. It requires both a scalar full residual and one common function of the actual current. Pure unchanged Jacobi pairs fail the assigned two-frequency target, while dressed nonlinear controls recover the same local law and its coupling derivative.

A dressed excited-branch response now derives a near-resonance from that common scalar action and checks its instability with full spatial derivatives and increasing cutoffs. The interaction that supports a proposed mass state must also give its required spatial stability and reciprocal coupling response.

The scalar-current insertion has also been tested on an exact round-sphere Killing background. Its full spatial response puts the simplex coefficient inside a frequency polynomial and has unstable high harmonics for every nonzero derivative-current coefficient. This is a result for that specified action and background.

An exact local scalar-gap solution is available on the ordinary spinor bundle of a specified round six-space. It has nonzero constant scalar current and complete spatial spectral stability. Its coefficient is fixed by the local self-consistency equation and depends on norm, volume and radius. This closes a conditional spatial field construction, while leaving the simplex enhancement, master geometry and observed mass identification unresolved.

An explicit normal Dirac extension supplies the absolute spatial Dirac operator from a specified half-cylinder of the same field. Its induced kinetic norm cancels the corresponding factor in a constant bulk scalar mass. Local scalar contact retains an inverse-width coefficient but excites complementary normal profiles. This identifies a concrete spatial reduction without yet deriving the complete simplex operator or a physical normal binding law. In the fixed spatial Killing channel, a full-line theorem also excludes a finite-norm stationary normal profile for that local product-space model.

2. The cumulative oscillator bundle

At a point $q\in Q_d$, the complexified cotangent space

$$V_{d,q}=T_q^*Q_d\otimes\mathbb C$$

has complex rank $d$. Its degree-$k$ symmetric power has rank

$$\operatorname{rank}\operatorname{Sym}^{k}(V_{d,q}) =\binom{k+d-1}{d-1}.$$

This is the shell multiplicity of $d$ commuting creation directions. The cumulative band through degree $N$ is

$$\mathcal B_{N,d} =\bigoplus_{k=0}^{N}\operatorname{Sym}^{k}(V_{d,q}).$$

Consequently,

$$\operatorname{rank}\mathcal B_{N,d} =\sum_{k=0}^{N}\binom{k+d-1}{d-1} =\binom{N+d}{d}.$$

The count is independent of coordinates. Changes of orthonormal cotangent frame act through $U(d)$ on $V_{d,q}$ and through its symmetric powers on the band.

The band can be written as one homogeneous representation. For arbitrary vector spaces $A$ and $B$,

$$\operatorname{Sym}^{N}(A\oplus B) \cong \bigoplus_{k=0}^{N} \operatorname{Sym}^{N-k}(A)\otimes\operatorname{Sym}^{k}(B).$$

Taking $A=\mathbb C$ gives

$$\operatorname{Sym}^{N}(\mathbb C\oplus V_{d,q}) \cong \bigoplus_{k=0}^{N}\operatorname{Sym}^{k}(V_{d,q}),$$

because every symmetric power of a one-dimensional vector space is one-dimensional.

In occupation numbers, a monomial in the cumulative band is labelled by

$$(a_1,\ldots,a_d), \qquad \sum_{j=1}^{d}a_j\leq N.$$

Introduce

$$a_0=N-\sum_{j=1}^{d}a_j.$$

Then

$$a_0+\cdots+a_d=N, \qquad a_j\geq0,$$

and stars-and-bars gives

$$\#\{(a_0,\ldots,a_d):a_0+\cdots+a_d=N\} =\binom{N+d}{d}.$$

Thus a $d$-direction cumulative spectrum has $d+1$ homogeneous slots while its base remains $d$-dimensional.

4. Covariant localization forces the dimension factor

Fix $q\in Q_d$ and write

$$W=W_{d,q}, \qquad \mathcal H_N=\operatorname{Sym}^{N}(W), \qquad D=\dim_{\mathbb C}\mathcal H_N.$$

Give $W$ its metric-induced Hermitian structure. For a unit vector $z\in W$, define

$$|z;N\rangle=z^{\odot N}, \qquad \langle z;N|z;N\rangle=1.$$

Its projector depends only on the ray $[z]$:

$$P_N([z])=|z;N\rangle\langle z;N|.$$

Let $d\nu_{\rm FS}$ be the $U(d+1)$-invariant probability measure on $\mathbb P(W)$. Consider

$$A_N=\int_{\mathbb P(W)}P_N([z])\,d\nu_{\rm FS}([z]).$$

The representation $\operatorname{Sym}^{N}(W)$ of $U(d+1)$ is irreducible. Invariance of the measure gives

$$U A_N U^\dagger=A_N$$

for every $U\in U(d+1)$. Schur's lemma therefore gives

$$A_N=c_NI.$$

Taking the trace fixes the coefficient:

$$1 =\int\operatorname{tr}P_N([z])\,d\nu_{\rm FS} =\operatorname{tr}A_N =c_ND.$$

Hence

$$A_N=\frac{I}{D}.$$

The covariant localization density resolving the identity is consequently

$$\Lambda_N([z])=DP_N([z]).$$

Once the localization operator has rank-one support on the coherent line, covariance, positivity, and resolution of the identity force the factor $D$.

This is the finite-dimensional coherent-state resolution of the identity used in geometric quantization. General coherent-state and reproducing-kernel constructions on compact integral symplectic manifolds are reviewed by Kirwin; explicit $SU(N)$ coherent-state resolutions are constructed by Mathur and Mani.

5. From localization density to rest mass

Let the sector spinor bundle be $\Sigma_d$. For a spin sector this is the ordinary spin bundle. For a $\mathrm{Spin}^c$ sector it includes the required line connection. The mode-$N$ field is a section of

$$\Sigma_d\otimes\mathcal E_{N,d}.$$

The localization density acts only on the mode factor:

$$\widehat\Lambda_{N,d} =I_{\Sigma_d}\otimes\Lambda_{N,d}.$$

It therefore commutes with Clifford multiplication:

$$[\widehat\Lambda_{N,d},\gamma(v)]=0, \qquad v\in TQ_d.$$

This establishes scalarity on the spinor factor. Lorentz covariance of the complete construction also requires transformation laws for the spatial mode tensors, the occupied ray, the measure and the defining frame. A spatial operator can commute with every Clifford matrix without being boost invariant. The rest-current carrier and dynamical rest-frame construction supply conditional completions with responsive frames and induced connections. Their additional dynamics and stability limitations must be retained; they do not derive the mode bundle from the master field. Subject to a complete such specification, the effective projected Dirac equation may take the form

$$\left( i\gamma^a\nabla_a -\mu_d\widehat\Lambda_{N,d}([z]) \right)\psi_{N,d}=0.$$

For a localized coherent mode

$$\psi_{N,d}=\chi\otimes|z;N\rangle,$$

the contact operator gives

$$\widehat\Lambda_{N,d}([z])\psi_{N,d} =D_{N,d}\psi_{N,d}.$$

The rest equation is therefore

$$\left(i\gamma^a\nabla_a-m(n,d)\right)\psi_{N,d}=0,$$

with

$$m(n,d)=\mu_dD_{N,d} =\mu_dS(n,d).$$

The symmetric power acts on the bosonic orbital or derivative directions of one spinor field.

6. Projection from the master field

Let

$$P_{d,N}$$

project the master spinor onto the particle manifold $Q_d$ and the cumulative mode bundle $\mathcal E_{N,d}$. Separate the master Dirac operator into its kinetic and self-interaction parts:

$$\mathcal D_\infty =i\Gamma^A\nabla_A-\mathcal K_\infty[\Psi_\infty].$$

The projected operator is

$$\mathcal D_{d,N} =P_{d,N}\mathcal D_\infty P_{d,N}.$$

The mass theorem applies if the scalar contact component satisfies

$$P_{d,N}\mathcal K_\infty P_{d,N} \big|_{\rm scalar,contact} =\mu_d\widehat\Lambda_{N,d}([z]).$$

The following subsections specify the additional occupied-ray insertion and derive its exact finite-band projection. The minimal local scalar interaction alone does not supply that insertion.

6.1 The projective ray is local mode-phase data

The fibre

$$W_{d,q} =\mathbb C\oplus T_q^*Q_d\otimes\mathbb C$$

is the homogenized oscillator-symbol fibre. The cotangent factor contains the $d$ complex oscillator amplitudes. In an orthonormal frame centred on the mode, they combine displacement $x_a$ and conjugate momentum $p_a$ as

$$a_a =\frac{1}{\sqrt2} \left( \frac{x_a}{\ell_d} +i\frac{\ell_dp_a}{\hbar} \right), \qquad a=1,\ldots,d.$$

Here $\ell_d$ is the spatial sector length. The connection identifies $p_a$ with the covariant derivative symbol of the field. Both quadratures are dimensionless. Real orthonormal sector-frame changes act through $O(d)$, a subgroup of $U(d)$. The larger unitary oscillator symmetry also mixes canonical quadratures; it is not supplied by spatial frame changes alone.

The additional component $a_0$ records the unused part of the cumulative excitation bound. The completed vector

$$z=(a_0,a_1,\ldots,a_d)\in W_{d,q}$$

defines the projective mode-phase ray $[z]\in\mathbb P(W_{d,q})$. Common complex rescaling leaves the ray invariant. The value of $\ell_d$ changes the physical phase-space embedding while preserving the rank of $\mathcal E_{N,d}$ and the diagonal trace identity.

At fixed homogeneous oscillator action,

$$|a_0|^2+\sum_{a=1}^{d}|a_a|^2=R_N^2,$$

the completed amplitudes lie on $S^{2d+1}$. Quotienting their common phase gives

$$S^{2d+1}/U(1)=\mathbb{CP}^{d}.$$

This is the classical projective completion of the cumulative oscillator band. Quantization at homogeneous degree $N$ gives $\operatorname{Sym}^N(\mathbb C^{d+1})$. The same construction therefore produces both the ray space and its simplex dimension.

Higher oscillator degree fills the degree-$N$ representation of the same spatial coordinates:

$$\operatorname{Sym}^N(W_{d,q}),$$

and the definite ray becomes the coherent line

$$|z;N\rangle=z^{\odot N}.$$

Equivalently, the filtered symbols through derivative order $N$ have associated graded bundle

$$\bigoplus_{k=0}^{N} \operatorname{Sym}^k(T^*Q_d\otimes\mathbb C).$$

The sector connection splits this filtration into the homogenized bundle $\mathcal E_{N,d}$. The ray is the local oscillator phase of the field on $Q_d$. A definite orbit supplies a definite $[z]$ at each instant.

The direct calculation on physical field derivatives keeps the distinction between this carrier identification and an interaction derivation. The written local scalar term acts on Taylor derivatives by a triangular product rule: the count repeats its local source blocks, while their eigenvalues retain the original coefficient. Changing the derivative normalization preserves that result. A physical oscillator projection also needs its induced kinetic metric and control of the next derivatives entering the spatial Dirac equation.

6.2 Conditional occupied-ray insertion

Lift the scalar current to the graph of its occupied projective symbol:

$$\mathcal J_d(q,[u]) =J_d(q)\, \delta_{\rm FS} \left([u],[z_{\Psi_d}(q)]\right), \qquad J_d=\bar\Psi_d\Psi_d.$$

The invariant delta is normalized by

$$\int_{\mathbb P(W_{d,q})} f([u])\delta_{\rm FS}([u],[z])\,d\nu_{\rm FS}([u]) =f([z]).$$

This graph lift assigns a specified ray to the source. Its identification with the field's physical amplitude-gradient data is an additional premise, not a consequence of scalar locality on $Q_d$.

Consider the leading scalar self-interaction of a nonlinear Dirac field:

$$\mathcal L_{\rm sc} =-\frac{\kappa_d}{2}J_d^2.$$

Its Euler--Lagrange contribution is

$$\frac{\partial\mathcal L_{\rm sc}}{\partial\bar\Psi_d} =-\kappa_dJ_d\Psi_d.$$

In integral-kernel language, multiplication by the current has distribution kernel

$$\kappa_dJ_d(q)\, \delta_{Q_d}(q,q')\, \delta_{\rm FS}([u],[u']).$$

The diagonal delta is the kernel representation of multiplication. In the ray variable it represents the identity, not a point insertion at a selected ray. Its band compression is $P_N I P_N=I_{\mathcal H_N}$, which has full rank. General order-zero multiplication can also have a nonconstant coefficient in $[u]$; locality does not determine that coefficient.

To obtain the point insertion $\kappa_d\delta_{[z]}$, separately assume that the interaction evaluates against the graph-supported source at the occupied ray. Its finite-band projection is computed in the next section. A physical derivation must specify the lifted action, its measure, and its pairing; formally squaring the distributional graph current in a quartic action is not a defined replacement for those data.

A defined graph-measure pairing with one finite-band kernel retains the simplex coefficient. Its complete self-variation is a scalar-current variation: the moving ray contributes no self-orientation stiffness. This candidate action still needs its physical origin and normalized spatial solutions.

6.3 Exact projection of a point contact

Let

$$\mathcal H_N =\operatorname{Sym}^{N}(W_{d,q})$$

and let $P_N$ be the orthogonal projector onto its holomorphic realization over $\mathbb P(W_{d,q})$. Denote its reproducing kernel by $K_N$. For every $s\in\mathcal H_N$,

$$s([z])=\langle K_{N,z},s\rangle, \qquad K_{N,z}(\,\cdot\,)=K_N(\,\cdot\,,[z]).$$

Define the Toeplitz projection of the point contact by its quadratic form:

$$\langle s,T_{N,[z]}t\rangle = \int \overline{s([u])}\, \delta_{\rm FS}([u],[z])\, t([u])\,d\nu_{\rm FS}([u]).$$

Evaluation gives

$$\langle s,T_{N,[z]}t\rangle =\overline{s([z])}t([z]) =\langle s,K_{N,z}\rangle \langle K_{N,z},t\rangle.$$

Therefore

$$T_{N,[z]} =P_N\delta_{[z]}P_N =|K_{N,z}\rangle\langle K_{N,z}|.$$

Its nonzero eigenvalue is

$$\|K_{N,z}\|^2=K_N([z],[z]).$$

The diagonal is constant because $U(d+1)$ acts transitively on $\mathbb{CP}^{d}$ and preserves both $P_N$ and $d\nu_{\rm FS}$. Its value is fixed by the trace:

$$\int_{\mathbb{CP}^{d}} K_N([u],[u])\,d\nu_{\rm FS}([u]) =\operatorname{tr}P_N =\dim\mathcal H_N.$$

The measure has total volume one, hence

$$K_N([z],[z]) =\dim\mathcal H_N =\binom{N+d}{d} =S(N+1,d).$$

Writing

$$|z;N\rangle =\frac{K_{N,z}}{\sqrt{K_N([z],[z])}}$$

gives the exact identity

$$\boxed{ P_N\delta_{[z]}P_N =S(N+1,d)|z;N\rangle\langle z;N| =\Lambda_{N,d}([z]). }$$

Here $U(d+1)$ is the covariance of the homogenized cumulative mode fibre. The original $U(d)$ rotates the $d$ derivative directions. Projective completion adds the homogeneous slack component and extends the finite band to the irreducible $U(d+1)$ representation $\operatorname{Sym}^N(\mathbb C^{d+1})$. Treating that completion isotropically is a necessary mode-space premise of the exact simplex law.

This is the required projection theorem. Toeplitz operators with singular measure and distribution symbols are part of the general reproducing-kernel framework described by Rozenblum and Vasilevski. Here the finite-dimensional projective case is an exact consequence of the reproducing property.

6.4 Conditional projected nonlinear Dirac equation

Assume the occupied-ray interaction of Section 6.2 supplies the following term, with the ray held fixed during this operator evaluation, and project it onto the mode band:

$$P_{d,N} \left(\kappa_dJ_d\delta_{[z]}\right) P_{d,N} = \mu_dP_N\delta_{[z]}P_N,$$

where $\mu_d$ is the sector-normalized scalar contact strength after the $Q_d$ profile and spinor normalization have been taken. The projection theorem gives

$$P_{d,N}\mathcal K_\infty P_{d,N} \big|_{\rm scalar,contact} = \mu_dS(N+1,d) |z;N\rangle\langle z;N|.$$

Acting on the occupied coherent line,

$$P_{d,N}\mathcal K_\infty P_{d,N}|z;N\rangle =\mu_dS(N+1,d)|z;N\rangle.$$

With $n=N+1$, the projected Dirac equation therefore contains the rest mass

$$\boxed{ m(n,d)=\mu_dS(n,d). }$$

All explicit $N$-dependence comes from the diagonal of the finite-band reproducing kernel. The resulting $\mu_d$ is shared by all ordinary particle modes in sector $d$ provided the normalized one-particle scalar-current contraction is the same in every band. This is automatic for a universal unit-normalized point current, but must be checked for the normalized spinor convention used by the projected master action.

The action-level derivation has four necessary ingredients:

1. Locality in the mode ray. The self-interaction evaluates the wave against its own phase-space ray $[z]$.
2. Covariance. Rotating the mode frame sends $\Lambda([z])$ to $\Lambda([Uz])$ by conjugation.
3. Resolution normalization. Integrating the mutually overlapping localized rays over invariant measure gives the identity.
4. Band-blind contact strength. The coefficient $\mu_d$ depends on the sector and is independent of $N$.

The graph-supported contact is the first premise; it is not forced by the local scalar current. The Bergman projection proves the second and third within the chosen isotropic mode space and produces the factor $S(n,d)$. Absence of a band label in a bare coupling does not by itself establish the fourth: normalized projected profiles and scalar-current matrix elements can depend on the band. Their independence must also be demonstrated.

If the occupied ray is itself a function of the varied field or its derivatives, variation of the lifted action must include that dependence. The fixed-ray operator calculation does not already derive those additional Euler--Lagrange terms or prove a self-consistent occupied-ray solution. The family-ratio theorem is conditional on this interaction and normalization.

This makes the shared scale experimentally meaningful: the successful family-ratio tests are evidence that the projected contact strength is band-blind.

At one fixed ray, $\Lambda_{N,d}([z])$ has a $(D_{N,d}-1)$-dimensional nullspace. The coherent rays form an overcomplete family and

$$\int\Lambda_{N,d}([z])\,d\nu_{\rm FS}=I,$$

so every nonzero state is detected by some contact density. At a given $[z]$, the occupied localized particle is the one nonzero eigenline.

7. Kinetic and contact operators

A spatial Dirac or Laplace operator on $Q_d$ supplies kinetic eigenvalues and their multiplicities through

$$\slashed D_{Q_d}$$

and

$$\Delta_{Q_d}$$

respectively. The localized contact operator supplies the rest-energy endomorphism whose eigenvalue grows as $S(n,d)\sim n^d$.

The kinetic operator constructs the mode eigenspace. The contact density then maps its rank $D$ to the eigenvalue of a normalized localized operator. This is the two-stage origin of the mass multiplier.

7.1 Actual Spatial Evaluation and Its Kinetic Correction

The spatial spectral-projector construction now supplies a common reference-normalized version on actual spatial bands. If $E_N$ evaluates the band at a physical point and $B_N=E_NE_N^\dagger=S(n,d)B_0$, then

$$T_N=E_N^\dagger B_0^{-1}E_N,\qquad T_N^2=S(n,d)T_N.$$

The round-sphere positive Dirac bands satisfy this at every $d\ge2$; the fixed-$\mathcal O(m)$ projective zero-form bands satisfy it at $d=2m$. The sphere contact has one nonzero channel per spinor component. The projective line branch has one. Their physical volume and reference multiplicities enter $B_0$ explicitly.

This connects cumulative spatial counting to a specified contact operator. For a sphere band with the kinetic term retained, the coupled block is $H=\operatorname{diag}[(l+d/2)/R]+\mu vv^\dagger$, $v_l=\sqrt{\binom{l+d-1}{d-1}}$. Its frequencies satisfy

$$1=\mu\sum_{l=0}^{n-1} \frac{\binom{l+d-1}{d-1}}{z-(l+d/2)/R}.$$

The exact count therefore also supplies a computable kinetic dressing, including a variance bound for the occupied bright channel. The normalized evaluation image has an induced connection and definite overlap vertices, so its frequency and transport can be tested on the same carrier. The contact's physical source, occupied branch and compatible global geometry remain the requirements of this module.

8. Boundaries of the theorem

8.1 The photon

The construction begins at $n=1$, where $N=0$ and

$$S(1,d)=1.$$

The proposed photon assignment has $n=0$. Extending the count by $S(0,2)=0$ records a vanishing contribution from this mode family. The formal empty bundle $\mathcal E_{-1,d}=0$ has no nonzero state and therefore supplies neither a zero eigenvalue nor a propagating photon.

A physical photon requires a nonempty transverse state space and a contact operator satisfying

$$\mathcal M\psi_\gamma=0,\qquad \psi_\gamma\ne0.$$

Its kinetic propagation, complete energy–momentum response and stability must follow from the same field action. The zero-count boundary alone does not prove these conditions or the protection of a massless branch.

8.2 The bottom beat

The ordinary theorem gives the single-band masses

$$m(n,3)=\mu_3S(n,3).$$

The bottom formula

$$m_b=\mu_3\sqrt{S(16,3)S(17,3)}$$

requires a two-band coupling. Its closure condition is that the off-diagonal contact operator between $\mathcal E_{15,3}$ and $\mathcal E_{16,3}$ have a singular value equal to the geometric mean of their diagonal contact eigenvalues.

8.3 Sector scales

For a band-blind master contact coefficient, one $\mu_d$ multiplies every ordinary occupied band in a sector. Numerical projection must determine

$$\mu_2,\mu_3,\mu_4,\mu_5,\mu_6,\mu_{10}$$

and their cross-sector relations from the normalized master kernel and its overlap between the corresponding $Q_d$ sectors. Observed masses provide the empirical test of those projected values.

8.4 Mode selection

The operator supplies a collective mass eigenbranch for every positive integer $n$. A separate stability or resonance condition must select which branches occur as particles. The accuracy of the mass-depth scan is conditional on those occupied indices.

8.5 Occupied-ray and ray-averaged quartic densities

The distinction between contact at the occupied ray and averaging over all rays is mathematically observable.

For a normalized coherent state $|w;N\rangle$, define its normalized phase-ray density

$$\rho_w([z]) =D_N|\langle z;N|w;N\rangle|^2.$$

It obeys

$$\int\rho_w\,d\nu_{\rm FS}=1.$$

Because

$$|\langle z;N|w;N\rangle|^2 =|\langle z|w\rangle|^{2N},$$

the invariant moment formula gives

$$\int_{\mathbb{CP}^{d}} |\langle z|w\rangle|^{4N}\,d\nu_{\rm FS} =\frac{1}{D_{2N}}, \qquad D_{2N}=\binom{2N+d}{d}.$$

A quartic energy averaged over the whole ray space is

$$\int\rho_w([z])^2\,d\nu_{\rm FS} =\frac{D_N^2}{D_{2N}},$$

while occupied-ray self-contact has eigenvalue $D_N$.

The two interactions are

$$\text{occupied-ray self-contact} \quad\longrightarrow\quad D_N,$$

$$\text{ray-averaged quartic density} \quad\longrightarrow\quad\frac{D_N^2}{D_{2N}}.$$

Of these two specified prescriptions, only the point contact has the simplex multiplier with a band-independent coefficient. This comparison does not establish uniqueness among nonlinear actions.

The full smooth quartic variation establishes coherent stationary solutions, a complete mode Hessian and finite-mode orbital stability for the second action. Its nonlinear frequency retains $D_N^2/D_{2N}$. Deriving a coherent branch and deriving the point-contact multiplier are therefore separate requirements.

These conditional field calculations do not yet select a stable physical particle, its occupied index, or its mass.

8.6 Coordinate harmonic kernel and phase-ray contact

A scalar coordinate kernel whose isotropic mean field is

$$V_d(q)=\lambda_d|q|^2$$

derives a harmonic well and hence the shell spaces $\operatorname{Sym}^k(T_q^*Q_d\otimes\mathbb C)$. That is enough to derive the rank $D_N$ of the cumulative band.

The corresponding band projection is

$$P_{N,d}|q|^2P_{N,d}$$

and the phase-ray contact density is

$$D_N|z;N\rangle\langle z;N|.$$

Their relation requires either

1. a master interaction written locally on the homogenized phase-ray bundle; or
2. a derivation showing that the coordinate-space nonlinear kernel transforms into the occupied-ray contact density after the oscillator and sector projections.

This step completes the map from coordinate confinement to phase-ray self-contact.

The actual spatial oscillator point form provides a separate explicit comparison: its eigenvalue is the spatial reproducing kernel on the diagonal, with a weighted Hockey-Stick central value and fixed-width growth of degree $d/2$. The simplex dimension is its integrated trace, so spatial point evaluation alone does not supply the projective coefficient at every point.

Consistency across cumulative bands adds a test for one underlying kernel, distinguishing fixed bare compression from a dynamical Schur reduction. The specified band embeddings and any eliminated modes are part of that bridge.

9. Sector normalization

The universal combinatorial part of the scalar contact projection is now fixed:

$$P_N\delta_{[z]}P_N =S(N+1,d)|z;N\rangle\langle z;N|.$$

Write the scalar coefficient left after sector projection as

$$\mu_d=\kappa_dR_dB_d.$$

Here $\kappa_d$ is the projected master-contact coefficient, $R_d$ is the normalized spatial-profile overlap on $Q_d$, and $B_d$ is the scalar spinor contraction. Suppose the order-zero vertex is a scalar identity on the normalized non-mode factors, its coefficient is constant on the occupied contact graph, and its coefficient is independent of the band projector. Under these band-blind hypotheses, all three factors are independent of $N$. The contact theorem then gives

$$m(n,d)=\kappa_dR_dB_d\,S(n,d).$$

For the stated band-blind scalar vertex, $\mu_d=\kappa_dR_dB_d$ is independent of $N$. Derivative vertices, nonconstant order-zero coefficients, finite contact width, spectral functions, mode-dependent spinor polarization, and band-dependent state normalization generate testable modifications of the mass sequence.

For two sectors with equal self-couplings, the ratio of mass units is

$$\frac{\mu_a}{\mu_b}=\frac{R_aB_a}{R_bB_b}.$$

Thus the proposed $\mu_6=\mu_{10}$ symmetry requires both $g_{66}=g_{10,10}$ and equality of the normalized projected overlaps.

The local scalar profile calculation shows explicitly how normalization can change the frequency produced by a contact coefficient. Its virial and exact massive benchmark concern a flat local comparison; they do not replace the physical geometry and bundle terms needed for the occupied particle solutions.

The nonlinear spatial reduction now derives the effective stationary action and induced norm after complementary Dirac modes respond. For a normalized chiral zero-mode branch, the frequency is $gA-4g^3C+O(g^5)$ while the total energy is $gA/2-g^3C+O(g^5)$. Under the stated positive-potential hypotheses, $C$ vanishes exactly when the scalar source preserves the selected zero-mode line. A common graph coefficient $g_N=\kappa D_N$ must therefore also control these spatial corrections before it supplies a common physical mass scale. The graph coefficient itself remains to be derived from the master action.

A covariant kernel preserving the coherent branch then makes every holomorphic shape in that degree degenerate. The genuine mean kernel gives an exact, nonlinearly stable phase orbit, while a finite same-field static response retains a bounded nonzero spatial correction. Exact linearity therefore does not by itself select the occupied shape.

The scalar self-consistency comparison constructs a nonzero scalar coefficient and an exact collective response frequency from one specified global kernel. Its leading coefficient is $M\simeq\kappa Q$, where $Q$ is the full canonical norm. A cumulative count enhances it only when that norm grows with the number of modes; at unit total norm, $|M|\le\kappa$ for every band. This locates a normalization requirement that a physical mass derivation must supply.

9.1 The conditional common-anchor relation

There is a compact way to state the scale relations that have direct mass tests. Let $n_d^\star$ be a specified reference band in sector $d$ and define its contact mass

$$M_d^\star=\mu_dS(n_d^\star,d).$$

Under the projected fixed-point condition

$$\frac{(M_d^\star)^2}{g_{dd}} = \frac{m_e^2}{g_{66}}.$$

the sector scale is

$$\boxed{ \mu_d = \frac{m_e}{S(n_d^\star,d)} \sqrt{\frac{g_{dd}}{g_{66}}} }.$$

The square root is the direct algebraic consequence of this quadratic fixed-point equation. The master equation of motion must supply that equation.

For the proposed reference choices, the formula reads

$$\begin{aligned} \mu_6&=\frac{m_e}{S(13,6)},\\ \mu_3&=m_e\sqrt{\frac{g_{33}}{g_{66}}},\\ \mu_4&=\frac{m_e}{S(3,4)} \sqrt{\frac{g_{44}}{g_{66}}}. \end{aligned}$$

The electron fixes the unit. The $d=3$ and $d=4$ expressions then make conditional predictions to compare with inferred quark-mass parameters. A quantitative comparison must specify the mass definition, renormalization scheme, scale and matching to measured hadronic or scattering observables. These parameters are not directly measured free-particle masses; see the PDG quark-mass review. No common-scheme refit is performed here. The proposed $d=2$ formula

$$\mu_2=m_e\sqrt{\frac{g_{22}}{g_{66}}}$$

uses a unit-count back-reaction reference. It is testable because it predicts the absolute $W$, $Z$, and Higgs masses after their mode indices are specified. Likewise, $\mu_{10}=\mu_6$ is testable against the tau mass, but still requires the normalized $6\leftrightarrow10$ overlap equality just identified.

These are the direct empirical checks of the conditional fixed-point equation and coupling counts.

9.2 The neutrino scale is unresolved

The proposed relation

$$\mu_5 = \frac34\frac{\mu_6^3}{\mu_4^2}$$

is a phenomenological normalization calibrated to the observed neutrino order of magnitude. V2 treats $\mu_5$ as an undetermined projected normalization. An operator equation must select the factor $3/4$ and powers $(3,-2)$ before this expression becomes a derived scale law.

Neutrino oscillations measure mass-squared differences. Given the proposed mode indices, they constrain a common $\mu_5$. An absolute neutrino-mass measurement will provide the direct scale test.

The current status is

$$\boxed{\mu_5\text{ remains an undetermined projected normalization.}}$$

A derivation requires evaluating $\kappa_5R_5B_5$ from the master action.

9.3 Remaining numerical calculation

The outstanding work is to evaluate

$$(\kappa_d,R_d,B_d), \qquad d\in\{2,3,4,5,6,10\},$$

from one normalized projected action and recover the cross-sector relations. The simplex multiplier is derived, the non-neutrino scale relations are conditional and empirically testable, and the neutrino expression is a phenomenological ansatz.

The construction also makes sharp deformation predictions:

• a finite ray-contact width replaces the constant diagonal evaluation with a convolution of the Bergman kernel and changes the mass sequence;
• derivatives of the contact delta insert powers of the representation generators and generate additional $N$-dependence;
• a band-dependent scalar coefficient spoils the scale-free family ratios;
• a mixed state on several projective rays splits or averages the rank-one mass eigenvalue.

Order-zero self-contact at a definite projective mode ray is one operator realization of the simplex spectrum. The homogeneous coherent actions provide other exact occupied eigenvalues with different complete mode responses. Their nonconstant spatial solutions have canonical energy and full linear stability for the specified independent kinetic bundle. The functional normalization and master-field carrier remain assumptions; the direct derivative identification fails finite propagation in its stated flat control. The spectrum therefore constrains a master interaction without determining it uniquely.