Complete scalar response and null directions

Summary: The complete local scalar-source derivative is invertible at nonzero scalar density, while a scalar-null field retains one real response direction whose kinetic protection and projected channel meaning must be tested separately.

Source response is distinct from a projected vertex

At a physical spatial point, consider the written scalar contribution

$$\mathcal N(\phi)=g s\beta\phi,\qquad s=\phi^\dagger\beta\phi,\qquad \rho=\phi^\dagger\phi, \tag{1}$$

with real $g$, Hermitian $\beta^2=I$, and the full spinor fiber retained. The invariants obey $|s|\le\rho$. They are scalar and positive densities over ordinary physical space; spinor components do not introduce more spatial dimensions.

The complete derivative is

$$\boxed{J_\phi\eta=D\mathcal N(\phi)\eta =g\left[s\beta\eta+2\beta\phi\, \operatorname{Re}(\phi^\dagger\beta\eta)\right].} \tag{2}$$

This note determines its entire pointwise response spectrum. It does not assert that every projected particle vertex is nonzero. A vertex includes spatial wavefunctions, an output projection, kinetic normalization and the appropriate current channel. Also, $J_\phi$ is only the nonlinear source derivative: the complete rotating-frame operator includes $D-\mu$. Its symmetry zero modes or cancellations are not excluded by an invertible $J_\phi$ alone.

Exact real response spectrum

Let $\beta$ have $p\ge1$ positive and $q\ge1$ negative complex directions, with $n=p+q$. Regard the complex fiber as a real $2n$-dimensional Hilbert space with inner product $\operatorname{Re}(\eta^\dagger\xi)$. Writing $v=\beta\phi$, equation (2) is the real symmetric operator

$$\boxed{J_\phi=g\left(s\beta+2v\otimes_{\mathbb R}v\right).} \tag{3}$$

The real Hessian of the energy density $g s^2/2$ is $2J_\phi$; the factor two follows from $\delta E=2\operatorname{Re} (\mathcal N^\dagger\delta\phi)$. The rank-one term in (3) is real-linear, not a complex-linear rank-one operator acting alone.

Diagonalize $\beta$ and use unitary changes within its two eigenspaces to put $\phi=(a,b)$ in one real direction of each, where

$$a^2=\frac{\rho+s}{2},\qquad b^2=\frac{\rho-s}{2}.$$

The remaining real directions give $g s$ with multiplicity $2p-1$ and $-g s$ with multiplicity $2q-1$. On the two displayed real directions, $J_\phi/g$ is

$$\begin{pmatrix} 3a^2-b^2&-2ab\\-2ab&-a^2+3b^2 \end{pmatrix}. \tag{4}$$

Its trace is $2\rho$ and determinant is $-3s^2$. Thus the full spectrum is

$$\boxed{g s\ (2p-1),\quad -g s\ (2q-1),\quad g\left(\rho+\sqrt{\rho^2+3s^2}\right),\quad g\left(\rho-\sqrt{\rho^2+3s^2}\right).} \tag{5}$$

The multiplicities in parentheses count real directions. Formula (5) also has the correct pure-chirality limits $|s|=\rho$. No spatial-depth counting identity changes this pointwise spectrum.

Nonzero scalar density leaves no full source kernel

If $g\ne0$ and $s\ne0$, every eigenvalue in (5) is nonzero. In particular,

$$\boxed{\det_{\mathbb R}J_\phi=3g^{2n}s^{2n},} \tag{6}$$

and the exact real-linear inverse is

$$\boxed{J_\phi^{-1}\xi=\frac1{g s} \left[\beta\xi-\frac{2\phi}{3s} \operatorname{Re}(\phi^\dagger\xi)\right].} \tag{7}$$

Substitution in (2) verifies (7). Equivalently, apply the rank-one inverse identity to $sI+2\phi\operatorname{Re}(\phi^\dagger\beta\,\cdot)$, whose rank-one functional takes value $2s$ on $\phi$.

For example, suppose $\phi^\dagger\beta\eta=0$. The scalar itself has zero first variation in that direction, but the full source still changes by $g s\beta\eta$. Freezing the final field factor would erase this term and create an artificial nullspace.

The nonlinear map has a related exact invariant:

$$\omega=\mathcal N(\phi)\quad\Longrightarrow\quad \omega^\dagger\beta\omega=g^2s^3. \tag{8}$$

For nonzero output scalar $t=\omega^\dagger\beta\omega$, take the real cube root $s=(t/g^2)^{1/3}$ and recover $\phi=\beta\omega/(g s)$. The scalar-null input cone maps to zero. A nonzero scalar-null output does not occur in the image of this local cubic map.

A small scalar does not suppress every response

The exact smallest singular value in the canonical real metric is

$$\boxed{s_{\min}(J_\phi)=|g| \left(\sqrt{\rho^2+3s^2}-\rho\right) =\frac{3|g|s^2}{\sqrt{\rho^2+3s^2}+\rho}.} \tag{9}$$

The second expression avoids subtractive numerical cancellation. It is no greater than $|g s|$, because $|s|\le\rho$. For $0<|s|\ll\rho$, the weakest response is approximately $3|g|s^2/(2\rho)$, but the largest is approximately $2|g|\rho$.

Near cancellation of the background scalar can therefore leave a very weak direction and a strong direction simultaneously. It is insufficient to explain universal rare interaction without specifying the actual incident and observed spatial channels.

If a spatial perturbation is supported where $|s(x)|\ge s_*>0$ and $\rho(x)\le\rho_{\max}$, integrating (9) gives

$$\|J_\phi\eta\|_{L^2}\ge |g|\left(\sqrt{\rho_{\max}^2+3s_*^2}-\rho_{\max}\right) \|\eta\|_{L^2}. \tag{10}$$

This is a full-source estimate on the stated support. A localized field generally has $s\to0$ in its tails, so (10) must not be promoted to a uniform gap over all space. For a complete orthonormal output family $f_a$, Parseval gives $\sum_a|\langle f_a,J_\phi\eta\rangle|^2=\|J_\phi\eta\|^2$. A selected channel can still vanish, and a finite output band can miss the response altogether. Counting that band is not a substitute for evaluating its spatial projection.

What the rank-one null response actually means

At $s=0$ and $\rho>0$,

$$\boxed{\mathcal N(\phi)=0,\qquad J_\phi=2g(\beta\phi)\otimes_{\mathbb R}(\beta\phi).} \tag{11}$$

The complete source derivative has one real responsive direction and $2n-1$ real null directions. In particular, $J_\phi(\beta\phi)=2g\rho\beta\phi$. A zero background self-source therefore does not mean an absence of interaction with a probe.

This rank-one operator also does not provide an occupied-ray self-contact. Let $P_\phi=\phi\phi^\dagger/\rho$. At scalar nullity,

$$\phi^\dagger\beta\phi=0\quad\Longrightarrow\quad P_\phi J_\phi=0. \tag{12}$$

Its responsive direction is orthogonal to the occupied complex ray. Projecting onto that ray removes the response. The rank-one source derivative cannot be identified with a nonzero mass operator on the occupied ray. The graph-contact variation likewise distinguishes an occupied coefficient from the complete field-dependent response; its graph kernel and normalization are separate specified ingredients.

Spatial kinetics can activate an initially null scalar

The exact zero crossings of local polarized pulses use their actual traveling solutions and full spatial compatibility. Nullity of the scalar at one time alone does not impose that dynamics.

A simple control uses two ordinary spatial directions $x,y$, with anticommuting $\alpha_x,\alpha_y,\beta$. Choose a unit initial spinor $\phi_0$ satisfying

$$\alpha_x\phi_0=\phi_0,\qquad \alpha_y\phi_0=-i\beta\phi_0. \tag{13}$$

The full plane field $\Psi(t,y)=e^{iky}\phi(t)$ obeys

$$i\dot\phi=(k\alpha_y+g s\beta)\phi,\qquad s=\phi^\dagger\beta\phi, \tag{14}$$

and begins with $s(0)=0$, $\rho=1$. Direct differentiation gives $\dot s(0)=-2k$, independently of $g$. The actual transverse kinetic term immediately produces a scalar source.

This is an unbound plane control per unit volume. The preparation in (13) mixes both signs of the free energy associated with its actual momentum along $y$; it is not proposed as a positive-energy particle state. An eigenplane with $\alpha_y\phi_0=\phi_0$ instead has an invariant scalar zero and solves the full equation as a free positive-frequency wave. The difference is the occupied kinetic mode, not scalar nullity alone.

For completeness, the nonlinear scalar evolution in (14) closes exactly. On the invariant two-component plane use $\alpha_x=\sigma_z$, $\alpha_y=-\sigma_y$, $\beta=\sigma_x$. Writing $v=\langle\sigma_y\rangle$, $w=\langle\sigma_z\rangle$, the conserved energy is $-kv+g s^2/2=0$ and $\dot s=-2kw$. The Bloch equations therefore imply

$$\boxed{\ddot s+4k^2s+2g^2s^3=0,\qquad s(0)=0,\quad\dot s(0)=-2k,} \tag{15}$$

$$\dot s^2+4k^2s^2+g^2s^4=4k^2. \tag{16}$$

For $k>0$, its amplitude is

$$\boxed{|s|_{\max}=\sqrt{\frac{2k}{\sqrt{k^2+g^2}+k}}.} \tag{17}$$

At $g=0$, $s(t)=-\sin(2kt)$. This exact source activation occurs with conserved positive norm and conserved total energy; the scalar term has not introduced an external driving field or a mediator particle.

Consequence for physical coupling tests

The classification of scalar-null fields identifies constant linear preparations whose zero source survives every spatial derivative. The full null-plane response separately determines whether perturbations outside an exact zero-source background stay bounded. Its stability threshold shows why invariance alone is insufficient.

The local scalar term admits real source nullspaces at particular fields and exact mismatched traveling solutions. Its nonzero-scalar response is nevertheless full rank before projection, and its null cone is not invariant under arbitrary spatial motion. A protected physical suppression therefore requires the actual occupied solution, its full kinetic evolution and the derived channel map.

These statements do not establish electric neutrality or eliminate weak vertices. The neutrino channel control keeps those physical requirements separate. No special depth-five zero, band-independent mass coefficient or observed rate follows from (5) or (11).

Verification

local_scalar_response_rank_checks.py runs 2,116 checks. Independent finite differences of the cubic source test the derivative; full real spectra, singular values, multiplicities, determinants and inverse maps are checked in complex rotated Clifford frames. Null-source and occupied-ray projections are tested separately from selected output channels.

Full complex spatial-plane evolution in dimensions three, six and ten is compared with the independently integrated scalar equation (15), its first integral and the exact free limit. These tests concern the specified local equation and its source geometry, not measured particle interactions.