Coherent normal sources and the contact limit

Summary: A canonically normalized coherent normal source approaches the full rank-one contact operator, while exact positive-source moments quantify the required concentration and divergent scalar self-energy cost.

A field realization of a narrow coherent source

The coherent contact construction has the required simplex eigenvalue when a point current on the occupied mode ray is assumed. A direct question is whether the smooth scalar kernel can approach that operator through an actual canonically normalized mode field. It can, with a substantial cost in the same interaction.

Let $D_L=\binom{L+d}{d}$, where $d$ is the physical spatial depth used in the cumulative tangent-mode bundle. Coherent rays live in the coefficient fibre $\mathbb P(\mathbf1\oplus T^*Q_d\otimes\mathbb C)$. They are not new physical spatial dimensions. All ray measures below are normalized coefficient measures; no physical compactification or spatial metric scale is introduced.

Keep the target degree $N$ fixed. A single normal coherent field of degree $M$, canonical norm $q$ and positive scalar polarization has current and symbol

$$C_M=qP_{z,M},\qquad \rho_M(u)=qD_M|u^\dagger z|^{2M},\qquad \int\rho_M\,d\nu=q. \tag{1}$$

This is one occupied normal mode. Its norm is $q$, rather than one unit per basis direction. Its actual physical normal mode, kinetic operator and preparation still require a master-field origin. The source is a mode of the same field, with no independent interaction particle.

The entire projected operator converges

The smooth scalar projection gives

$$V_{M\to N}(C_M)=D_N\int\rho_M(u)P_{u,N}\,d\nu(u). \tag{2}$$

In a frame with $z=e_0$, its transverse shell $j$ has eigenvalue

$$\boxed{\nu_j=q\frac{D_ND_M}{D_{N+M}} \frac{(N)_{\underline j}}{(N+M)_{\underline j}},} \qquad h_{j,d}=\binom{j+d-1}{d-1},\quad 0\le j\le N. \tag{3}$$

This is the beta-integral insertion derived in the complementary Dirac calculation, with source and target degrees interchanged. It includes every target coefficient direction. Its exact trace is $qD_N$ for every finite $M$.

At fixed $N$, $D_M/D_{M+N}\to1$ and every factor in (3) with $j\ge1$ tends to zero. Consequently

$$\boxed{V_{M\to N}(qP_{z,M}) \longrightarrow qD_NP_{z,N}} \quad\text{in operator norm as }M\to\infty. \tag{4}$$

This is the full desired rank-one operator limit, rather than agreement of only its coherent expectation. For $N\ge1$, the exact relative operator error is

$$\frac{\|V_{M\to N}-qD_NP_{z,N}\|_{\rm op}}{qD_N} =1-\frac{D_M}{D_{M+N}} =\frac{dN}{M}+O(M^{-2}). \tag{5}$$

To see why the coherent deficit is the largest error, the other eigenvalues decrease with $j$, and $\nu_1/\nu_0=N/(N+M)$. Meanwhile $D_{M+N}/D_M\ge1+N/(M+1)\ge1+N/(M+N)$. Thus $qD_N-\nu_0\ge\nu_1$. The limit is at fixed target band; it is not uniform over arbitrarily large $N$.

Finite concentration changes the family ratio

For aligned coherent target states, the normal contribution to the source is $g\nu_0$. At two assigned degrees $N_1<N_2$, its ratio is

$$\frac{\nu_0(N_2,M)}{\nu_0(N_1,M)} =\frac{D_{N_2}}{D_{N_1}} \frac{D_{M+N_1}}{D_{M+N_2}}. \tag{6}$$

Every finite positive coherent source gives a strict deficit relative to the simplex ratio. With $\Delta=N_2-N_1$,

$$\delta_M=1-\frac{D_{M+N_1}}{D_{M+N_2}} =\frac{d\Delta}{M}+O(M^{-2}). \tag{7}$$

At depth six, the unchanged assignments are $N_1=12$, $N_2=34$, with target ratio $3515/17$. Exact integer comparisons give the following normal degrees for illustrative mathematical tolerances. These are not fits to measured masses or alternate particle-mode assignments.

The source mode is becoming sharply concentrated in coefficient-ray geometry. This statement does not give a physical spatial length scale.

The same scalar interaction charges for that concentration

The normal field's scalar self-energy in the written positive quartic is

$$\boxed{E_{M,\rm self}=\frac g2\int\rho_M^2d\nu =\frac{gq^2}{2}\frac{D_M^2}{D_{2M}} \sim\frac{gq^2M^d}{2^{d+1}d!}.} \tag{8}$$

Its canonical norm remains $q$. For $d=6$ and $q=1$, the three normal degrees in the table have scalar self-energies, in units of $g$, approximately $5.52\times10^{19}$, $5.72\times10^{25}$ and $5.74\times10^{31}$. These are values of the declared normalized interaction, not energies assigned to observed particles.

The contact limit therefore has an unbounded scalar self-energy cost under the same fixed coupling. A physical kinetic contribution or background-energy prescription would have to be calculated in the same master action before any cancellation or subtraction is used. The operator limit alone does not establish a finite-energy particle.

There is also a separate retained self-term. At finite $M$, aligned target and normal fields with norms $J,q$ have

$$\begin{aligned} E_*&=\frac g2\left[J^2\lambda_N+2JqC_{NM}+q^2\lambda_M\right],\\ \omega_N&=g(J\lambda_N+qC_{NM}),& \omega_M&=g(q\lambda_M+JC_{NM}),\\ \lambda_L&=D_L^2/D_{2L},& C_{NM}&=D_ND_M/D_{N+M}. \end{aligned} \tag{9}$$

The rank-one limit makes $C_{NM}\to D_N$ but does not remove $J\lambda_N$. As in the complementary-source calculation, frequency, total energy and a proposed physical rest mass require distinct treatment.

The finite source is a stationary mode preparation

The aligned coherent pair establishes the stationary and stability properties of the corresponding positive-density component action. A simple bound extends its finite energy maximum to the scalar Dirac current when the spatial kinetic operator is zero.

Retain every complex spinor component and both scalar signs. For each family, its scalar ray current $j_L$ and positive norm density $p_L$ obey $|j_L|\le p_L$. At fixed canonical norms $J,q$,

$$\begin{aligned} \frac g2\int(j_N+j_M)^2d\nu &\le\frac g2\int(p_N+p_M)^2d\nu\\ &\le\frac g2\left[J^2\lambda_N+2JqC_{NM}+q^2\lambda_M\right]. \end{aligned} \tag{10}$$

For the last inequality, resolve each positive density into its spinor rows and apply the normalized symmetric-product bound to every pair of coefficient tensors. The single-family sharp coherent bound gives the two self-terms; the cross-product bound gives $C_{NM}Jq$.

Aligned coherent tensors with one common scalar sign attain (10), with arbitrary allowed spinor polarizations in that sign space. The complete scalar gradients preserve both occupied tensors and give (9). Opposite common scalar signs give the same energy maximum; no antiparticle interpretation is assigned to them.

The equality conditions are common coherent coefficient direction and common scalar sign. On the finite product of the fixed-norm spheres, conservation of energy and both norms therefore gives nonlinear orbital stability of this entire maximizing set. This includes the scalar-sign and spinor variations in the declared zero-spatial-kinetic mode system. It establishes neither capture into that set nor stability of omitted spatial modes. In particular, it is not a finite-energy theorem for the singular $M\to\infty$ limit.

A general positive source has the same moment obstruction

The finite coherent family is not the only possible normal preparation. Let $\rho\ge0$ be any fixed integrable normal scalar ray density, and let $t=|u^\dagger z|^2$ relative to a coherent target. Its normalized source coefficient is

$$c_N(z)=\int\rho(u)t^N\,d\nu(u),\qquad \omega_{N,\rm cross}=gD_Nc_N(z). \tag{11}$$

For $N_2>N_1\ge1$,

$$c_{N_1}-c_{N_2} =\int\rho\,t^{N_1}(1-t^{N_2-N_1})\,d\nu. \tag{12}$$

This is strictly positive for every nonzero integrable nonnegative density: $0<t<1$ almost everywhere. Even maximizing $c_N(z)$ over the target ray does not restore equality. Evaluate $c_{N_1}$ at a maximizing ray of $c_{N_2}$; equation (12) still makes it larger.

For a general positive measure, equality at nonzero source requires support on $t=1$ together with possible invisible support on $t=0$. A positive common coefficient therefore requires a point atom at the occupied ray. The exact simplex ratio under this fixed positive-source prescription is not produced by a smooth finite-width normal density. Signed scalar sources and an environment that changes with the target are outside this fixed positive-source statement.

At a common retained canonical norm $J>0$, the complete coherent source expectation adds the self-term:

$$\frac{\omega_{N,\rm coh}}{D_N} =g\left[J\frac{D_N}{D_{2N}}+c_N(z)\right],\qquad \frac{D_N}{D_{2N}}=\prod_{k=1}^d\frac{N+k}{2N+k}. \tag{12a}$$

Every factor in the product decreases strictly with $N$. Thus the complete normalized coefficient decreases at a common ray, or after maximizing over the ray, even in the point-source limit where $c_N$ is constant. For an actual stationary coherent field this expectation is its phase frequency. For a general source, coherence and stationarity must still be checked before giving it that interpretation.

Sharpening one fixed positive normal source therefore cannot remove the retained self-term at common $J$. This is a restriction on the specified complete scalar square and preparation, not on every possible mass operator or on the assigned particle dimensions. A changing normal environment, different selected norms, or a different effective action requires its own dynamical derivation.

A quantitative energy bound for any positive preparation

Let $c=c_{N_1}>0$, $\Delta=N_2-N_1$ and $\delta=1-c_{N_2}/c$. The tilted probability measure $d\pi=\rho t^{N_1}d\nu/c$ obeys

$$\delta=\mathbb E_\pi(1-t^\Delta),\qquad \pi(1-t\ge\eta)\le \frac{\delta}{1-(1-\eta)^\Delta}. \tag{13}$$

For $0<\delta<1/2$, choose $\eta=1-(1-2\delta)^{1/\Delta}$. At least half of the tilted mass then lies in the cap $1-t<\eta$. The unweighted normal density in that cap has mass at least $c/2$. Its normalized ray volume is exactly $\eta^d$, since $t$ has density $d(1-t)^{d-1}$. Cauchy–Schwarz gives

$$\boxed{E_{\rm scalar,self} =\frac g2\int\rho^2d\nu \ge\frac{gc^2}{8\eta^d}, \qquad \eta=1-(1-2\delta)^{1/\Delta}.} \tag{14}$$

At fixed positive $c$ and fixed coupling, this lower bound grows as $\delta^{-d}$ when the family deficit tends to zero. Thus the diverging scalar concentration cost is not peculiar to the coherent source family. Equation (14) is a bound on the positive scalar self-energy term; it does not assert that uncalculated terms in an arbitrary master action cannot contribute to the total energy.

The exact minimum at two fixed moments can also be found in the enlarged class of all nonnegative square-integrable ray densities. Rotational averaging around $z$ preserves the moments and decreases the quadratic norm, so an optimizer depends only on $t$. For moment ratios sufficiently close to one, it has the form

$$\boxed{\rho_*(t)=A\,t^{N_1}(t^\Delta-h)_+,} \qquad A>0,\quad 0<h<1. \tag{14a}$$

Here $(x)_+=\max(x,0)$, and $A,h$ are fixed by $\int\rho_*t^{N_1}=c$ and $\int\rho_*t^{N_2}=c(1-\delta)$. The ratio increases continuously with $h$: the tilted weight $t^{2N_1}(t^\Delta-h)_+$ shifts toward larger $t$. Thus the close-ratio branch determines a unique threshold.

For a direct optimality proof, put $f=A(t^{N_2}-ht^{N_1})$, so $\rho_*=f_+$. If another nonnegative density $s$ has the same two moments, $\int f(s-\rho_*)=0$. On the complement of the positive cap, $f\le0$ and $s\ge0$. Consequently

$$\|s\|_2^2-\|\rho_*\|_2^2 =\|s-\rho_*\|_2^2+2\int\rho_*(s-\rho_*) \ge\|s-\rho_*\|_2^2. \tag{14b}$$

This proves uniqueness of the minimum in that enlarged source class. Its scalar norm $\int\rho_*$ is calculated, rather than prescribed. If a separate canonical norm is required, the result remains a lower bound for that smaller admissible class. The cap profile is not itself claimed to be a finite-band field or a spatial particle solution.

For small $\delta$, let $L=1-h^{1/\Delta}$. The cap profile becomes linear in $L-(1-t)$ at leading order. Its moment constraint gives $L\sim(d+2)\delta/(d\Delta)$, and its optimal scalar self-energy is

$$\boxed{E_{\min}\sim gc^2\frac{d+1}{d+2} \left[\frac{d\Delta}{(d+2)\delta}\right]^d.} \tag{14c}$$

Independent Gaussian orders and nontrivial positive moment-preserving variations verify the minimum numerically. At the two moments of the $d=6$, $M=131908$ coherent source, the minimum is approximately $1.76027\times10^{25}g$, with scalar norm $0.9999999453$. The original coherent source has norm one and energy $5.71726\times10^{25}g$. Its ratio to the optimal energy tends to $8^7/(2^7\,6!\,7)\simeq3.25079$ in the sharp limit. Optimizing the positive source shape therefore improves the coefficient but retains the concentration divergence.

A simpler bound applies without source positivity:

$$\left|\int\rho\,D_Nt^N d\nu\right|^2 \le\left(\int|\rho|^2d\nu\right)\frac{D_N^2}{D_{2N}}. \tag{15}$$

One fixed finite quadratic source norm therefore cannot produce an unbounded exact $D_N$ sequence at all degrees. Its scope is a fixed source acting through the stated smooth scalar product, rather than every nonlinear or spatial master reduction.

A populated background also needs a selection rule

For a fixed collection of isotropic normal currents $C_M=q_MI_M$, their scalar ray symbol is the constant $Q_f=\sum_Mq_MD_M$. The cross source at every retained grade is consequently

$$\boxed{\mathcal N_{N,\rm cross}=gQ_f\beta\Psi_N.} \tag{16}$$

The sum is finite, or explicitly convergent. It is independent of $N$. Populating a different normal band $M=N$ for each target supplies a different preparation; a master derivation must explain that matching. The concentrated coherent source (1) is a distinct possibility with fixed canonical norm and an explicit contact limit, but it incurs the self-energy cost just calculated.

These results sharpen the physical operator problem. A viable reduction must supply its normal preparation, account for the complete energy, retain the target self-term and solve the spatial field equations. The rank-one projection itself now has an explicit finite-source approximation and quantitative bounds, without establishing a physical mass operator or changing any assigned particle dimensionality.

Verification

coherent_normal_contact_checks.py checks the complete target operator against independent signed tensor projection, its exact trace and norm error, finite integer concentration thresholds, the scalar self-energy cost, full scalar Dirac stationary gradients and the fixed-norm maximum bound. Positive-density quadrature checks the source moments and concentration inequalities. The program also verifies that one fixed isotropic normal population gives the same scalar source at distinct target degrees.