Coupling response of combined coherent bands

Summary: Joint coherent currents expose substantial band dependence hidden by isolated susceptibilities, and exact tensor-current reconstruction identifies the finite-band corrections needed for a different geometric contact.

Test the same action on both mass branches

An isolated mass and an isolated density curvature are insufficient to determine a pair interaction. The two currents must enter one specified action, with the same canonical normalization, and both must be varied.

Use $E_N=\operatorname{Sym}^N W$, $\dim W=d+1$, $D_N=\binom{N+d}{d}$ and the normalized coherent operator

$$\Lambda_{N,z}=D_N|z^{\otimes N}\rangle\langle z^{\otimes N}|, \qquad j_{N,z}=\Psi_N^\dagger(\beta\otimes\Lambda_{N,z})\Psi_N. \tag{1}$$

The rays label field coefficients over an actual physical point. They are not additional physical spatial directions. The independent grade carriers and their canonical time norm are assumptions of the candidate actions below, rather than consequences of nested physical sectors.

For positive coherent fields write $j_{N,z}=D_N A|z^\dagger u|^{2N}$ and $j_{M,z}=D_M B|z^\dagger v|^{2M}$, with unit $u,v$ and $\tau=|u^\dagger v|^2$.

A common square-root action

The single-band homogeneous action has a natural common extension as a norm of the sum of self-normalized current profiles:

$$U_{\rm root}=\mu\left[ \int\left(\sum_N\sqrt{D_{2N}}\,j_{N,z}\right)^2d\nu(z) \right]^{1/2}. \tag{2}$$

Every isolated coherent component retains its specified coefficient $\mu D_N$. Its joint response also depends on the Gram matrix of the normalized profiles. Direct monomial integration gives

$$\begin{aligned} C_{NM}(\tau) ={}&\frac{\sqrt{D_{2N}D_{2M}}}{D_{N+M}}\, \frac{\displaystyle\sum_{j=0}^{\min(N,M)} \binom Nj\binom Mj\tau^j}{\binom{N+M}{N}},\\ U_{\rm root} ={}&\mu\sqrt{(D_NA)^2+(D_MB)^2 +2C_{NM}(\tau)D_ND_MAB}. \end{aligned} \tag{3}$$

The overlap polynomial is the same complete symmetric-tensor contraction derived in coherent pair deformation. Its normalization in (3) makes $C$ a Gram entry, so $0<C\le1$ for these coherent positive profiles. At alignment, $C_{NN}(1)=1$, but $C_{NM}(1)<1$ for different positive degrees.

Put $x=D_NA$, $y=D_MB$ and $R=\sqrt{x^2+y^2+2Cxy}$. The actual reciprocal density Hessian is

$$\frac{\mu(1-C^2)}{R^3} \begin{pmatrix} D_N^2y^2&-D_ND_Mxy\\ -D_ND_Mxy&D_M^2x^2 \end{pmatrix}. \tag{4}$$

At physical depth six and the unchanged degrees $N=12$, $M=34$, $C_{12,34}(1)=0.5151631391$. At $A=B=1$, the occupied source coefficients relative to their isolated values are approximately $0.51870$ and $0.999991$. Their unequal response follows from the common action; it is not removed by the exact isolated mass relationship.

This tests the extension (2). It is not a no-go theorem for every action with the same isolated masses.

The smooth action has a different joint peak width

For the smooth coherent action, a single positive coherent current has the expansion

$$F_N(A)=\mu D_N-\frac{\mu d\sigma}{A}+O(\sigma^2),\qquad F_N'(A)=\frac{\mu d\sigma}{A^2}+O(\sigma^2), \tag{5}$$

at fixed $N,d,A>0$ as $\sigma\to0$. Its isolated leading curvature depends on depth but not on the mass-band count. This is a useful lead, but it does not give the mixed derivative of a combined action.

Put both currents into the same local functional:

$$U_\sigma=\mu\sigma\log\int \exp\left[\frac{j_{N,z}+j_{M,z}}{\sigma}\right]d\nu(z). \tag{6}$$

For aligned rays, let $a=D_NA/\sigma$, $b=D_MB/\sigma$ and $u=|z^\dagger u_0|^2$. Its partition integral is

$$Z=d\int_0^1(1-u)^{d-1}e^{au^N+bu^M}du. \tag{7}$$

Near its unique endpoint peak the width is controlled by

$$H=Na+Mb=\frac{ND_NA+MD_MB}{\sigma}, \tag{8}$$

not by either isolated peak width. With $y=H(1-u)$, the leading integral has the Gamma weight $y^{d-1}e^{-y}$. Thus

$$\begin{aligned} U_\sigma ={}&\mu(D_NA+D_MB)-\mu\sigma d\log H +\mu\sigma\log(d!)+O(\sigma^2),\\ \boxed{\partial_A\partial_B U_\sigma} ={}&\boxed{\mu\sigma d\, \frac{(ND_N)(MD_M)}{(ND_NA+MD_MB)^2}+O(\sigma^2).} \end{aligned} \tag{9}$$

Here the expansion holds with both degrees and positive currents fixed. The complete finite-$\sigma$ Hessian is the covariance matrix of $D_Nu^N$ and $D_Mu^M$, multiplied by $\mu/\sigma$. This supplies an independent numerical calculation of (9) and retains both reciprocal responses.

At depth six, $ND_N$ for degrees $12$ and $34$ has ratio $3515/6=585.8333333$. With $A=B=\mu=1$ and $\sigma=0.01$, resolved quadrature gives

The mixed-band value is about $0.00680463$ times the equal-band leading value. Therefore the isolated curvature (5) did not establish a universal coupling group for the common action (6).

These are local current controls. They are not measured lepton vertices, particle encounter rates or alternative assignments of physical depth.

Exact linear mass and amplitude curvature

There is also a simple general constraint. If a differentiable stationary branch has $E_N(Q)=m_NQ$ on an open interval of canonical norms, then

$$\omega_N=\frac{dE_N}{dQ}=m_N, \qquad \frac{d\omega_N}{dQ}=0. \tag{10}$$

A nonzero density curvature cannot coexist with that exact linear branch on the same interval. The constraint does not exclude polarization couplings, a physically selected isolated norm or a different stationary branch. It does explain why smoothing the peak generates both a response and a mass correction. The exact entropy energy-frequency defect of (6) must also be retained.

Reconstructing a fixed-order current from a coherent symbol

The adjacent-current contact uses exact one- and two-index contractions of the field. A raw coherent symbol does not already equal either contraction.

Write $r=d+1$ and, for one grade, define the raw moments

$$M_k=\int|z^{\otimes k}\rangle\langle z^{\otimes k}|\, j_{N,z}\,d\nu(z),\qquad \operatorname{Tr}M_k=J. \tag{13}$$

Let $\rho_1,\rho_2$ be its actual reduced tensor currents, and let $\mathcal S(A)$ be $A\otimes I$ compressed to the symmetric square. Direct integration of normalized monomials gives

$$\begin{aligned} M_1&=\frac{N\rho_1+JI}{N+r},\\ M_2&=\frac{N(N-1)\rho_2+4N\mathcal S(\rho_1)+2JI_2} {(N+r)(N+r+1)}. \end{aligned} \tag{14}$$

For example, at $\rho_1=JP_{e_0}$ and $\rho_2=J|e_0^2\rangle\langle e_0^2|$, the three degree sectors of $M_2/J$ have numerators $(N+1)(N+2)$, $2(N+1)$ and $2$ over the common denominator in (14). These are the independent Dirichlet monomial integrals. Covariance and polarization extend the identity to every matrix current, including the full signed scalar current.

The trace factors can be checked without a large band:

$$\operatorname{Tr}\mathcal S(A)=\frac{r+1}{2}\operatorname{Tr}A, \qquad \operatorname{Tr}[\mathcal S(A)\mathcal S(B)] =\frac{(r+2)\operatorname{Tr}(AB)+\operatorname{Tr}A\operatorname{Tr}B}{4}. \tag{15}$$

Equations (14) are the first two cases of the symmetric-tensor coherent moment identities discussed by Lewin, Nam and Rougerie, Theorem 2.2. Only the tensor identities are used here; that paper's many-particle interpretation is not a premise about constituent particles in IDWT.

For $N\ge2$, the exact inverse is

$$\boxed{ \begin{aligned} \rho_1&=\frac{(N+r)M_1-JI}{N},\\ \rho_2&=\frac{(N+r)(N+r+1)M_2 -4(N+r)\mathcal S(M_1)+2JI_2}{N(N-1)}. \end{aligned} } \tag{16}$$

These coefficients are fixed by the projection and its normalization. They are not a freedom to fit a separate coupling to each mass band. They nevertheless have to be produced by the proposed master reduction; knowing their inverse formula does not establish that the master action implements it.

In particular, applying a fixed raw ray kernel $|z^\dagger w|^2-|z^\dagger w|^4$ to $j_{N,z}j_{M,w}$ gives $\operatorname{Tr}(M_{N,1}M_{M,1})- \operatorname{Tr}(M_{N,2}M_{M,2})$. It does not give the adjacent reduced-current action. A finite-band coherent symbol is spread over rays, and this raw action has a nonzero isolated coherent self contact. The corrected contractions (16) cancel it exactly.

This identifies a concrete remaining operator problem: derive the normalized low-order current from the same physical master field, with its actual kinetic metric. Agreement of the isolated mass formula cannot replace that derivation.

Verification

Separate complex-ray quadrature reconstructs the first two complete tensor currents in (16). Exact small-index trace algebra and assigned large-degree controls check the coefficients without allocating the full large mass carriers.