Contact-kernel audit

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. Depth counts a pattern's bound spatial involvement within that space. “Observed” and “unobserved” describe an observer's access, not different kinds of dimension. Older projective and nested-geometry calculations below retain their additional hypotheses; they do not establish a separate physical manifold or compactified direction. See spatial meaning and mathematical scope.

Summary: Point-contact and smooth-density spectra are compared without identifying their operators.

NOT A PHYSICS MODULE. These are exact finite-band comparison calculations and an action-level audit. The graph-supported point-contact construction remains a live hypothesis; no alternative tested here is silently substituted for it.

1. The source distinction that motivated the test

The self-contact source, Section 6.2, explicitly lifts the scalar current onto the graph of a chosen projective ray:

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

It then writes the scalar interaction $\mathcal L_{\rm sc}=-(\kappa_d/2)J_d^2$, whose scalar Euler--Lagrange term is $-\kappa_dJ_d\Psi_d$. Its diagonal kernel is local multiplication. Later in the same subsection, the self-field insertion is identified with a delta concentrated at the selected ray. Section 6.3 proves the finite-band projection theorem for that point insertion, and Section 6.4 applies it to the proposed nonlinear Dirac term.

The distinction is important: a kernel $f(u)\delta(u,u')$ specifies multiplication by $f$, whereas a coefficient $f(u)=\delta(u,z)$ specifies a point-supported multiplier. Locality of the diagonal kernel alone does not choose that coefficient. The graph lift is explicitly an additional part of the written construction; it must neither be ignored nor mistaken for an already completed derivation of the action's projected dynamics.

Merely integrating $J(q)^2$ against the normalized graph delta returns $J(q)^2$. It does not automatically insert a new finite-band pairing kernel into the action. Establishing that pairing is the remaining dynamical step identified by this audit, not a request for an amplitude measurement.

2. The exact point-contact theorem survives

For

$$\mathcal H_N=H^0(\mathbb{CP}^d,\mathcal O(N)), \qquad D_N=\binom{N+d}{d},$$

use unit Fubini-Study measure and the specified homogeneous Hermitian normalization. The reproducing kernel has constant diagonal $K_N(z,z)=D_N$. Its normalized coherent line is $|z;N\rangle=K_{N,z}/\sqrt{D_N}$.

The point-symbol quadratic form is

$$\langle s,T_{\delta_z}t\rangle=\overline{s(z)}t(z).$$

It gives

$$T_{\delta_z}=P_N\delta_zP_N =|K_{N,z}\rangle\langle K_{N,z}| =D_N|z;N\rangle\langle z;N|.$$

Its nonzero eigenvalue is exactly $D_N$. This result depends on the stated ray, finite band, measure, and covariance assumptions. It is not the claim that every algebraic scalar interaction has this spectrum. The source itself makes isotropic projective completion a necessary premise and separately requires a band-blind normalized coefficient; see Sections 6.3--6.4 and 9.

3. Three different local multiplier prescriptions

In the same finite Hilbert space, compare the following unit-normalized symbols:

All three have projected trace $D_N$. Equal trace therefore does not imply equal spectrum. The smooth case has rank $D_N$, whereas the point case has rank one. These are different operators, not alternative normalizations of the same eigenvector.

The smooth symbol is normalized because

$$\int_{\mathbb{CP}^d}|\langle u,z\rangle|^{2N}d\nu(u)=\frac{1}{D_N}.$$

It is a comparison prescription: the source graph lift selects a point symbol, not automatically this smooth density.

4. Full smooth-symbol spectrum

Choose $z$ along coordinate zero. On normalized degree-$N$ monomials, let $k$ be the occupation of that coordinate. The remaining $d$ occupations sum to $N-k$. Unit-FS monomial moments obey

$$\int \prod_{r=0}^{d}|u_r|^{2m_r}d\nu(u) =\frac{d!\prod_rm_r!}{(d+\sum_rm_r)!}.$$

The normalized monomial of multi-index $\alpha$, $|\alpha|=N$, has squared normalization $(N+d)!/(d!\prod_r\alpha_r!)$. Multiplication by $D_N|u_0|^{2N}$ therefore gives the diagonal eigenvalue

$$\boxed{\lambda_k =D_N\frac{(N+d)!}{(2N+d)!}\frac{(N+k)!}{k!}}, \qquad 0\le k\le N,$$

with multiplicity

$$\boxed{m_k=\binom{N-k+d-1}{d-1}}.$$

Phase invariance makes the off-diagonal monomial integrals vanish. The largest eigenvalue occurs at $k=N$ and simplifies to

$$\boxed{\lambda_{\max}=\frac{D_N^2}{D_{2N}}}.$$

The checks verify positivity, $\sum_km_k=D_N$, and $\sum_km_k\lambda_k=D_N$ for the tested bands, using exact rational arithmetic. They do not require constructing the largest matrices explicitly.

For example, at $(N,d)=(3,3)$ the complete spectrum is $5/21$, $20/21$, $50/21$, $100/21$, with multiplicities $10,6,3,1$, respectively.

5. A same-sector shape test, not a scale fit

Using the already specified electron and muon bands, $N_e=12$, $N_\mu=34$, $d=6$, a common band-independent coefficient cancels in either prescription.

The point prescription gives

$$\frac{D_{34}}{D_{12}}=\frac{3515}{17} \approx206.764705882353.$$

The smooth-symbol occupied-line prescription gives

$$\frac{D_{34}^2/D_{68}}{D_{12}^2/D_{24}} =\frac{9441729375}{68902802} \approx137.029686760779.$$

These are calculated spectral ratios under two different prescriptions, not new experimental measurements. An overall electron unit conversion cannot make the prescriptions identical. This demonstrates the importance of the operator choice. It does not disprove the graph-supported point construction by imposing the smooth construction in its place. No experimental precision or significance claim is made in this note.

6. Single contact evaluation versus double smearing

The finite-band coherent pairing kernel is

$$B_N(z,w)=\langle z;N|T_{\delta_w}|z;N\rangle =D_N|\langle z,w\rangle|^{2N}.$$

It satisfies

$$\int B_N(z,w)d\nu(w)=1, \qquad B_N(z,z)=D_N.$$

A point source evaluated at its own ray uses the coincident value of this one kernel. Replacing the source by its smooth coherent density and averaging the test coherent density against it instead gives

$$\int B_N(z,w)^2d\nu(w)=\frac{D_N^2}{D_{2N}}.$$

This is the previously undocumented explanation of the two eigenvalues. A graph-source interaction with one such finite-band pairing can retain the linear simplex count; two smoothings yield a different quantity. Neither operation may be silently substituted for the other.

What remains open is the action-level origin and ordering of the pairing: the proposed graph current, the nonlinear variation, and the finite-band projection have to be joined in one specified construction. The test establishes a consistent mathematical distinction, not that the full master action has already supplied this joining.

7. Moving self-consistent ray: the limited cancellation that is exact

Differentiate $P^2=P$ for a differentiable orthogonal projector:

$$(\delta P)P+P(\delta P)=\delta P.$$

Multiplying by $P$ on both sides gives

$$\boxed{P(\delta P)P=0.}$$

For a normalized $u$ on the occupied line,

$$\langle u,\delta P\,u\rangle=0.$$

This was checked with a complex ray $u=(3/5,4i/5)^T$ and a Hermitian generator of projector motion. It removes the automatic objection that every change of the self-consistent ray must shift the occupied eigenvalue at first order.

It does not imply $(\delta P)u=0$, erase off-diagonal variations, or prove the full nonlinear Euler--Lagrange equation. In particular, derivative dependence of the ray, variation of the contact coefficient, kinetic contributions, and stability of the coherent branch remain separate questions. No claim of complete variational closure follows from the diagonal projector identity.

8. Findings to carry forward

Graph contact variation supplies a defined measure-pairing action with one finite-band kernel. Its self-contraction retains $D_N$, while its full ray-dependent variation cancels self-orientation stiffness. The remaining scalar coefficient still needs a normalized spatial solution before it can be interpreted as a rest mass.

Coherent quartic variation gives the complete action-level calculation for the smooth prescription. Coherent tensors are its sharp fourth-moment extrema, and its full conjugate-linear Hessian and finite-mode orbital stability can be derived. Its nonlinear frequency still contains $D_N^2/D_{2N}$. Thus the existence of a coherent stationary branch does not by itself select the point-contact mass sequence.

Checked: the point-contact identity; the constant, point, and smooth comparison spectra; the exact smooth-symbol family ratio; normalization and coincidence of the single pairing kernel; its squared-kernel integral; and the diagonal moving-projector cancellation.

Not established: that ordinary diagonal locality alone forces the graph symbol; that the graph lift by itself supplies the required action pairing; that double smearing is the correct V2 prescription; that the chosen coherent branch solves and is selected by the full projected dynamics; or that these calculations determine a baryon frequency.

No current magnitude, field amplitude, or profile parameter was fitted. The unknowns are operator identification and the derivation or cancellation of their physical coefficients. The regression script reproduces the finite tests; the source passages are self-contact Sections 6.1--6.4 and 9.

Related

• Research status — current proof boundaries and correction guidance.

Band consistency and finite resolution

Consistent band contacts tests how the coherent family changes under the specified cumulative oscillator inclusion. A fixed bare kernel has one compatibility law; an effective kernel obtained by eliminating same-field modes has a different Schur consistency law. Finite ray resolution independently varies the symbol width and retained band, giving the complete spectrum and an exact operator-norm error relative to the point contact.