Current weights of the coherent scalar projection

Summary: Exact symmetric-index contractions identify the current weights of the smooth scalar kernel, while harmonic reconstruction and the canonical kinetic norm constrain routes to the simplex interaction.

Compare the operators in the same variables

The current self-mass action uses reduced tensor currents with shell weights $h_{k,d}=\binom{k+d-1}{d-1}$. A direct calculation expresses the smooth coherent scalar interaction in those same variables. The comparison identifies exactly what a proposed master reduction must change.

At one physical spatial point let $E_N=\operatorname{Sym}^N\mathbb C^{d+1}$ and $D_N=\binom{N+d}{d}$. The projective rays below describe coefficient directions, not extra physical spatial dimensions. For any Hermitian scalar mode current $C_N$, including the signed current of a full spinor, define

$$j_{N,z}=D_N\langle z^{\otimes N}|C_N|z^{\otimes N}\rangle, \qquad \|z\|=1,\qquad \int d\nu(z)=1. \tag{1}$$

The measure is the normalized Fubini--Study measure on this mode fibre. Write $\rho_{N,k}$ for the exact $k$-index contraction of $C_N$; $\rho_{N,0}=\operatorname{Tr}C_N$.

The full projection identity

The coherent resolution of identity at degree $N+M$ gives

$$\int j_{N,z}j_{M,z}\,d\nu(z) =\frac{D_ND_M}{D_{N+M}} \operatorname{Tr}\!\left[\Pi_{N+M}(C_N\otimes C_M)\right], \tag{2}$$

where $\Pi_{N+M}$ symmetrizes all $N+M$ indices. Both input currents already act on symmetric tensors. In the symmetrizer, permutations with the same number $k$ of indices exchanged between the two inputs give the same contraction, $\operatorname{Tr}(\rho_{N,k}\rho_{M,k})$. Choosing the exchanged indices gives multiplicity $\binom Nk\binom Mk$. Their sum is $\binom{N+M}{N}$. Consequently

$$\boxed{ \int j_{N,z}j_{M,z}\,d\nu(z) =a_{NM}\sum_{k=0}^{\min(N,M)} \binom Nk\binom Mk\, \operatorname{Tr}(\rho_{N,k}\rho_{M,k}),} \tag{3}$$

with

$$a_{NM}=\frac{D_ND_M}{D_{N+M}\binom{N+M}{N}} =\frac{(N+d)!(M+d)!}{d!(N+M+d)!}. \tag{4}$$

This is a linear identity in each matrix current. It applies to indefinite currents and arbitrary complex spinor polarizations; positivity or a coherent preparation is not needed. Varying either current gives its complete reciprocal source, with the same coefficients.

For an isolated coherent current, every reduced purity is $J^2$. Equation (3) therefore gives the familiar smooth self-coefficient

$$\lambda_N=\frac{D_N^2}{D_{2N}}, \qquad \int j_{N,z}^2d\nu=\lambda_NJ^2. \tag{5}$$

The complete quartic variation derives its stationary coherent branch and Hessian. Here (3) locates that result inside the full hierarchy of reduced currents.

What the shell action changes

For equal degrees, normalize the weights in (3) by their sum. The two distributions over current order are

$$q_{N,k}=\frac{\binom Nk^2}{\binom{2N}{N}},\qquad s_{N,k}=\frac{\binom{k+d-1}{d-1}}{D_N}. \tag{6}$$

Vandermonde and Hockey-Stick normalize them, respectively. Their exact mean orders are

$$\sum_k kq_{N,k}=\frac N2,\qquad \sum_k ks_{N,k}=\frac{dN}{d+1}. \tag{7}$$

The first follows from the symmetry $k\leftrightarrow N-k$. For the second, use $k\binom{k+d-1}{d-1}=d\binom{k+d-1}{d}$ and sum. Thus replacing the smooth action by the shell action changes both its overall self-coefficient and the relative weights of its responses.

At depth six and the fixed degrees $12,34$, the smooth self-coefficient ratio is $137.029686760779$, while the shell count ratio is $3515/17$. No change of the assigned degrees is made in this comparison.

There is a further constraint on interpreting (3) as independent positive response channels. At a fixed $k$ accessible to two different grades, the grade matrix of its coefficients is not positive semidefinite:

$$a_{NM}^2>a_{NN}a_{MM}\qquad(N\ne M). \tag{8}$$

Strict convexity of $\log\Gamma$ at $2N+d+1$ and $2M+d+1$ proves (8). Multiplication by the two binomial factors does not change the sign of the two-by-two determinant. The complete square $\int(\sum_Nj_{N,z})^2d\nu$ remains nonnegative. Its currents at different orders are constrained contractions of the same fields; they cannot be varied as independent matrix coordinates while retaining that positivity.

A finite-band inverse can change the product

The coherent smoothing operator on the ray fibre is

$$(\mathcal B_Nf)(u)=D_N\int|u^\dagger z|^{2N}f(z)\,d\nu(z). \tag{9}$$

Its harmonic order $\ell$ has multiplicity $g_\ell=D_\ell^2-D_{\ell-1}^2$, with $D_{-1}=0$, and eigenvalue

$$\boxed{b_{N\ell}= \frac{N!(N+d)!}{(N-\ell)!(N+d+\ell)!}},\qquad 0\le\ell\le N. \tag{10}$$

Higher harmonic orders have eigenvalue zero. For an elementary derivation, set $\tau=|u^\dagger z|^2$ and normalize the zonal polynomial as

$$p_\ell(\tau)= \frac{P_\ell^{(d-1,0)}(2\tau-1)}{\binom{\ell+d-1}\ell},\qquad p_\ell(1)=1. \tag{11}$$

The radial measure is $d(1-\tau)^{d-1}d\tau$. Rodrigues' formula, followed by $\ell$ integrations by parts, evaluates $D_Nd\int_0^1\tau^N(1-\tau)^{d-1}p_\ell(\tau)d\tau$ as (10). The boundary terms vanish. The same polynomials have squared norm $1/g_\ell$, giving the expansion

$$D_N\tau^N=\sum_{\ell=0}^Ng_\ell b_{N\ell}p_\ell(\tau). \tag{12}$$

This is the lowest-level coherent smoothing calculation within the broader projective Berezin theory developed by Demni, Mouayn and Yaqine. Only its mathematical operator structure is used here.

On the finite symbol range, the inverse is well defined and satisfies

$$\boxed{\int j_N\,\mathcal B_N^{-1}j_N\,d\nu =D_N\operatorname{Tr}(C_N^2).} \tag{13}$$

For completeness, let $\mathcal S_NC=D_N\langle z^N|C|z^N\rangle$. Then $\mathcal S_N\mathcal S_N^*=D_N\mathcal B_N$ on its range. Inverting there gives $\mathcal S_N^*\mathcal B_N^{-1}\mathcal S_N=D_NI$, proving (13) for every matrix current, not just a coherent one.

Equation (13) is a possible inverse prescription for a different product. It is not the original local scalar product (3). Its highest inverse eigenvalue is $\binom{2N+d}{N}$, and both its finite range and its normalization depend on $N$. A master interpretation must supply that operator and its domain, rather than regarding inversion as cost-free.

A common positive scalar kernel has stronger restrictions

Suppose one fixed isotropic positive operator $K$ acts on the raw scalar symbols at every degree. Let its harmonic coefficients be $k_\ell\ge0$. The self-contact of a normalized coherent symbol is then

$$\langle j_N,Kj_N\rangle =\sum_{\ell=0}^Ng_\ell k_\ell b_{N\ell}^2. \tag{14}$$

Demanding the value $D_N$ for every degree makes this a triangular system. At depth six, imposing the target also at $N=0$ gives

$$(k_0,k_1,k_2,k_3,k_4,k_5) =\left(1,8,\frac{155}{7},\frac17, \frac{3821}{6},-\frac{450440}{21}\right). \tag{15}$$

The negative coefficient is incompatible with the assumed positivity. Leaving $k_0$ free does not rescue the consecutive-degree target: positivity through degree seven requires $k_0\le0.551635757181$, while degree eight requires $k_0\ge0.551638257699$. These displayed decimals summarize exact rational bounds in the diagnostic.

This conclusion concerns one positive raw-symbol kernel required to work at every consecutive degree shown. It does not exclude a kernel adjusted only to the physical assignments, a nonlinear reduction, a degree-dependent reconstruction, or a frequency-dependent effective operator.

The common-probe requirement also distinguishes raw symbols from reduced currents. A nonzero harmonic in a raw pair kernel has a factor $b_{N\ell}b_{M\ell}$. At fixed probe degree $M$ and $\ell>0$, $b_{N\ell}$ increases strictly with target degree $N$. Thus this fixed kernel cannot have an exactly unchanged nonconstant angular response on two target grades. The corrected contractions in combined-band response remove that raw-symbol dependence by an explicit reconstruction.

Counting field copies also counts their kinetic norm

A separate normalization test applies to a seemingly simple explanation of the shell weights. Suppose each of the $h_{k,d}$ channels is an independent copy of the actual field $T_{Nk}\psi$, with its canonical kinetic term. The direct-sum lift obeys

$$L_N=\bigoplus_{k=0}^N\bigoplus_{a=1}^{h_{k,d}}T_{Nk},\qquad L_N^\dagger L_N=D_NI. \tag{16}$$

These natural bundle maps preserve the connection, so the spatial kinetic term is multiplied by the same factor. The canonical field is $f=\sqrt{D_N}\psi$, and its scalar density is $J_f=D_NJ_\psi$. The coherent potentials become

$$\begin{array}{c|c} \text{Potential before canonical normalization}& \text{Potential in }f\\ \hline \kappa D_NJ_\psi^2/2&\kappa J_f^2/(2D_N)\\[2pt] \mu D_N|J_\psi|&\mu|J_f|\\[2pt] \mu D_N(\sqrt{J_\psi^2+\sigma^2}-\sigma)& \mu(\sqrt{J_f^2+(D_N\sigma)^2}-D_N\sigma) \end{array} \tag{17}$$

The desired hierarchy is absent in this direct-copy explanation. In particular, the root source is $\mu$ at positive current, and the quartic source scales as $\kappa J_f/D_N$. Potential multiplicity without its kinetic contribution would give the wrong physical equation.

This leaves a concrete effective-operator problem. A dynamical elimination of actual complementary modes of $\Psi_\infty$ can have a different normalization from (16), but it must include its frequency dependence and induced norm. Those requirements are derived in consistent band contacts and nonlinear spatial reduction. No independent interaction particle is needed or introduced.

Verification

The coherent normal-source limit now obtains the entire rank-one target operator as a limit of this smooth projection at fixed normal canonical norm. Its exact finite-width spectrum and positive-source moment inequalities quantify the accompanying scalar self-energy cost. The operator limit does not remove the retained self-interaction or supply a physical normal preparation.

The complementary-mode calculation uses this exact insertion in a canonical normal Dirac reduction, including the complete scalar square. Its balanced normal response gives a common leading geometric vertex by dynamical spectral cancellation, with the finite kinetic gap and normal self-interaction retained.

smooth_current_projection_checks.py compares (3) with independent fully symmetrized tensor products, varies both full complex spinors, and checks exact weight identities and the canonical lift. coherent_kernel_inverse_checks.py uses independent Jacobi quadrature, exact rational reconstruction and positivity intervals to check (10)–(15). The bounds have the consecutive- degree scope stated above; they are not scans of alternative particle dimensionalities.