Scalar gap with canonical normalization

Summary: A specified global scalar interaction admits a self-consistent gap and an exact collective response frequency, while canonical normalization determines whether a cumulative mode count enhances either quantity.

One scalar response from the same field

The nonlinear spatial reduction determines how complementary modes alter a proposed linear mass coefficient. A separate constructive question is whether a summed occupied-mode response can itself generate a count-dependent coefficient. The following model makes that question exactly solvable, including its dynamical response.

Use one classical complex field with a fixed physical spatial measure, canonical time-kinetic term and energy

$$\begin{aligned} \mathcal E(\psi)&=\langle\psi,\mathscr D\psi\rangle +\frac{\kappa}{2}S(\psi)^2,& S(\psi)&=\langle\psi,\beta\psi\rangle,\\ \mathscr D^\dagger&=\mathscr D,& \beta^\dagger&=\beta,\quad \beta^2=I,\quad \{\mathscr D,\beta\}=0,\quad \kappa>0. \end{aligned} \tag{1}$$

This is a specified global scalar kernel on that spatial domain. For a probability-normalized spatial mean, its coefficient includes the inverse physical volume. Neither its magnitude nor its equality between different spatial sectors is derived here. The global instantaneous kernel also has the causality limitation of that class of effective models. It is not a completed local master law.

The positive quartic sign is retained. The field is not silently replaced by a fermionic vacuum or a filled quantum sea; the scalar exchange calculation explains why those would be additional assumptions. All modes below are components of the same field, with no additional mediator particle.

The exact equation and conserved positive norm are

$$i\partial_t\psi=(\mathscr D+M(t)\beta)\psi,\qquad M(t)=\kappa S(\psi(t)),\qquad Q=\|\psi\|^2. \tag{2}$$

Immediately, $|S|\le Q$ and hence $|M|\le\kappa Q$. This bound holds for every solution, without choosing a mode basis or a stationary state.

Conserved spectral sectors and the gap equation

First work in a finite invariant collection of nonzero kinetic pairs. In each pair choose a basis with

$$\mathscr D_j=\epsilon_j\sigma_x\otimes I_{r_j},\qquad \beta_j=\sigma_z\otimes I_{r_j},\qquad \epsilon_j>0. \tag{3}$$

The $r_j$ complex polarizations are retained. Their rank counts field components, not extra physical spatial dimensions. The norm $q_j$ of the entire pair is conserved because its projector commutes with both operators in (2). Within a degenerate shell this is a basis-independent total norm. Equal amplitudes in one chosen polarization basis are not an invariant occupation prescription.

For a constant $M>0$, let $v_j^+(M)$ be a unit positive eigenvector of $\epsilon_j\sigma_x+M\sigma_z$, with eigenvalue $E_j=\sqrt{\epsilon_j^2+M^2}$. For arbitrary fixed unit complex vectors $z_j\in\mathbb C^{r_j}$ and phases, the candidate field is

$$\psi_j(t)=\sqrt{q_j}\,e^{-iE_jt} \bigl(v_j^+(M)\otimes z_j\bigr). \tag{4}$$

Its integrated scalar source is time independent even when the $E_j$ differ. Orthogonality of the kinetic pairs and the block structure of $\beta$ remove the integrated cross terms. A local density can still have spatial interference; this exact reduction uses the global kernel.

Since $\langle v_j^+,\sigma_zv_j^+\rangle=M/E_j$, (2) becomes

$$M=\kappa\sum_j\frac{q_jM}{E_j},\qquad \boxed{\frac1\kappa=\sum_j\frac{q_j}{\sqrt{\epsilon_j^2+M^2}}} \quad (M>0). \tag{5}$$

The right side decreases strictly from $\sum q_j/\epsilon_j$ to zero. Thus there is exactly one positive branch if and only if

$$\kappa\sum_j\frac{q_j}{\epsilon_j}>1. \tag{6}$$

There is also the reflected scalar branch at $-M$ and the zero solution of the original self-consistency equation. These are scalar branches; no antiparticle interpretation is assigned to their signs.

Near threshold, write $H_s=\sum q_j/\epsilon_j^s$. Expanding (5) gives

$$M^2=\frac{2(H_1-1/\kappa)}{H_3} +O\bigl((H_1-1/\kappa)^2\bigr). \tag{7}$$

A populated zero-kinetic sector removes the finite threshold: its term $q_0/M$ diverges as $M\downarrow0$. A field supported entirely at zero kinetic energy instead has the exact coefficient $M=\kappa Q$. The nonzero-kinetic response theorem below must not be applied unchanged to that limiting case.

A constrained energy maximum and its full response

At fixed pair norms the phase space is a product of finite spheres. Introduce a real variational parameter $m$ through the exact identity

$$\frac\kappa2 S^2=\max_m\left(mS-\frac{m^2}{2\kappa}\right).$$

Maximizing first over the field at fixed $m$ gives

$$\max_{\|\psi_j\|^2=q_j}\mathcal E(\psi) =\max_m F(m),\qquad F(m)=\sum_j q_j\sqrt{\epsilon_j^2+m^2}-\frac{m^2}{2\kappa}. \tag{8}$$

The parameter $m$ is only an algebraic device for this extremization. For (6), $F$ has maxima at $\pm M$. Its maximizing fields are (4), including the full degenerate polarization and phase families. Conservation of energy gives orbital stability of each maximizing family on this finite fixed-norm manifold. It is an energy maximum, and therefore does not establish a passive ground state. Stability under arbitrary omitted spatial interactions or changes to the kernel remains a separate problem.

For an occupied ray, let $a_j=x_j+iy_j$ describe the normalized perturbation into its opposite eigenspinor. Choose that eigenspinor's sign so that the scalar transition matrix element is $b_j=\epsilon_j/E_j$. The quadratic energy at fixed $q_j$ is

$$\delta^2\mathcal E =-2\sum_j q_jE_j|a_j|^2 +2\kappa\left(\sum_j q_jb_jx_j\right)^2. \tag{9}$$

Cauchy--Schwarz and (5) give

$$\kappa\sum_j\frac{q_j\epsilon_j^2}{E_j^3} =1-\kappa M^2\sum_j\frac{q_j}{E_j^3}<1, \tag{10}$$

so (9) is strictly negative on these ray directions. The phase and positive-band polarization directions remain flat. Opposite-band polarizations orthogonal to $z_j$ are also retained: they have frequency $2E_j$ and do not drive the scalar source to first order.

The coupled ray equations are

$$\dot x_j=-2E_jy_j,\qquad \dot y_j=2E_jx_j-2\kappa b_j\sum_kq_kb_kx_k. \tag{11}$$

After a positive diagonal change of coordinates, their squared-frequency matrix is the symmetric matrix

$$\mathcal K=4\left[\operatorname{diag}(E_j^2)-\kappa ww^T\right], \qquad w_j=\frac{\sqrt{q_j}\epsilon_j}{\sqrt{E_j}}. \tag{12}$$

Subtract $4M^2I$ and use (5). The remaining rank-one perturbation of $4\operatorname{diag}(\epsilon_j^2)$ is nonnegative and has a one-dimensional null space, since $\kappa\sum w_j^2/\epsilon_j^2=1$. Therefore

$$\boxed{\Omega_{\mathrm{scalar}}=2M} \tag{13}$$

is an exact lowest ray response frequency, with all other ray frequencies larger. In the original coordinates its eigenvector has $x_j\propto 1/\epsilon_j$, so its integrated scalar variation is nonzero.

The complete field linearization verifies that no polarization was lost. In the rotating frame of (4), set $v=\beta\phi$ and $W=\bigoplus_j E_j I_{2r_j}$. For arbitrary complex perturbations,

$$i\dot\eta=A\eta+B\bar\eta,\qquad A=\mathscr D+M\beta-W+\kappa vv^\dagger,\qquad B=\kappa vv^T. \tag{14}$$

This retains changes of each pair norm as well as all phase and polarization directions. Such norm changes can produce zero-frequency Jordan motion in the rotating frame. The fixed-norm orbital-stability statement does not mistake these for a strictly positive full Hessian.

The scalar pole, its residue and the field energy

An applied scalar coefficient $f(t)\beta$ gives $\delta M=\kappa\delta S+f$. Solving (11) with time convention $e^{-i\Omega t}$ yields

$$\delta S=\chi(\Omega)f,\qquad \chi(\Omega)=\frac{\chi_0(\Omega)}{1-\kappa\chi_0(\Omega)},\qquad \chi_0(\Omega)=4\sum_j \frac{q_j\epsilon_j^2}{E_j(4E_j^2-\Omega^2)}. \tag{15}$$

At $\Omega=2M$, (5) makes $\chi_0=1/\kappa$. The pole is simple, with

$$\chi(\Omega)=\frac{\mathcal R}{4M^2-\Omega^2}+O(1),\qquad \mathcal R=\frac{4}{\kappa^2\displaystyle\sum_jq_j/(E_j\epsilon_j^2)}>0. \tag{16}$$

The diagnostic obtains the same response by solving the complete real linearization (14), independently of the ray reduction. For a purely zero-kinetic field the scalar source is conserved and its driven response vanishes; a formal opposite-chirality frequency $2M$ is then invisible to this scalar probe. Equation (16) explicitly assumes every $\epsilon_j>0$.

The total energy of (4) is

$$\boxed{\mathcal E_*= \sum_j q_jE_j-\frac{M^2}{2\kappa},\qquad \frac{\partial\mathcal E_*}{\partial q_j}=E_j.} \tag{17}$$

The derivative follows by stationary elimination of $M$ in (8). Thus $M$, the phase frequencies $E_j$, the collective frequency $2M$, and the full energy $\mathcal E_*$ have distinct meanings within this same model. The local scalar normalization gives the corresponding energy-frequency distinction for a spatially local action. None of these quantities has been identified here as the observed rest mass of an isolated particle.

What the cumulative count does at fixed canonical norm

Let $Q=\sum q_j$, $\Lambda=\kappa Q$, and $a_r=\sum q_j\epsilon_j^r/Q$. For a fixed finite spectrum and $\Lambda$ large compared with its kinetic frequencies, (5) gives

$$M=\Lambda-\frac{a_2}{2\Lambda} +\frac{3a_4-4a_2^2}{8\Lambda^3} +O(\Lambda^{-5}). \tag{18}$$

The omitted coefficient contains sixth-order spectral moments. Convexity of $(x+M^2)^{-1/2}$ also gives the exact bounds, on an existing branch,

$$\sqrt{\max(0,\Lambda^2-a_2)}\le M \le\sqrt{\Lambda^2-\epsilon_{\min}^2}<\Lambda. \tag{19}$$

If every counted mode carries one unit of canonical norm, $Q$ equals the number of modes. The leading coefficient $M\simeq\kappa Q$ then contains that count without inserting it into $\kappa$. This is a result about an extensive field norm. Its leading total energy is $\mathcal E_*\simeq\kappa Q^2/2$, rather than $\kappa Q$.

If the total canonical norm is one, the occupation weights instead sum to one and $M\le\kappa$ for every band. In the strong-response limit $M\simeq\kappa$, independently of the mode count. Both the scalar coefficient and its collective frequency inherit this normalization.

More sharply, suppose shell energies increase with their label and occupation is uniform per counted mode at fixed total norm. Adding the next shell decreases the weighted average of $(\epsilon^2+M^2)^{-1/2}$ at every trial $M$. Its intersection with $1/\kappa$ therefore moves to a smaller $M$, or the positive branch ceases to exist. This is an exact comparison at unchanged $\kappa$. It does not change the committed particle mode selections.

A field-coordinate rescaling cannot turn the extensive case into the normalized case while preserving their apparent coefficients. Substituting $\psi=\sqrt D\,\varphi$ multiplies the original kinetic term by $D$ and its quartic by $D^2$. Restoring the canonical field restores the original physical norm. Dividing the entire action by a band-dependent $D$ would be an additional physical prescription.

Hockey-Stick moments and a specified spatial spectrum

For cumulative oscillator shell weights

$$w_k=\binom{k+d-1}{d-1},\qquad D_N=\sum_{k=0}^Nw_k=\binom{N+d}{d}, \tag{20}$$

Hockey-Stick also supplies all factorial moments:

$$\frac1{D_N}\sum_{k=0}^Nw_k(k)_{\underline r} =\frac{d}{d+r}(N)_{\underline r}. \tag{21}$$

Indeed, multiplying $w_k$ by the falling factorial gives $d(d+1)\cdots(d+r-1)\binom{k+d-1}{d+r-1}$; summing and dividing by $D_N$ proves (21). Consequently $\langle k\rangle=dN/(d+1)$ and $\langle k^2\rangle=dN(N-1)/(d+2)+dN/(d+1)$. These identities calculate the kinetic moments in (18) once a kinetic spectrum has actually been specified. The count alone does not specify it.

One geometric comparison is the round even-dimensional spatial sphere of physical radius $R$. Its Dirac frequencies and positive multiplicities are

$$\epsilon_k=\frac{k+d/2}{R},\qquad r_k=2^{d/2}\binom{k+d-1}{d-1},\qquad \sum_{k=0}^Nr_k=2^{d/2}D_N. \tag{22}$$

These are the standard spherical Dirac eigenvalues and degeneracies; see Camporesi and Higuchi, equations (3.41)–(3.44). Its directions remain ordinary macroscopic spatial directions, with no compactification or chosen small radius. Equation (22) supplies a conditional spectral comparison, not a selection of physical topology or a complete nested geometry. The spin factor is eight at depth six and thirty-two at depth ten, and cannot silently be discarded in a comparison of extensive norms.

The diagnostic uses the assigned cumulative bands $N=12,34$ at depth six and $N=22$ at depth ten, retaining those spin factors. It solves (5) for both extensive and unit total norms with the same stated coefficients, measured in units of $R^{-1}$. No radius or coefficient is fitted to observed masses. The fixed-norm bound and decreasing-gap result survive.

Verification and physical consequence

The complementary Dirac source extends this normalization question to the actual coherent scalar projection. Its finite-gap shell equations retain the complete normal self-field, and its full canonical energy derivative gives the retained source without dropping the populated band's norm.

scalar_gap_normalization_checks.py checks the exact shell sums and six factorial moments, gap thresholds, bounds and asymptotic orders. Complex controls retain one to four kinetic pairs, up to three degenerate polarizations per pair, all conjugate variations, and common coordinate covariance. Independent energy derivatives, the complete field spectrum, driven scalar response and nonlinear time evolution verify the finite model's equations.

The calculation provides a self-consistent scalar coefficient and a collective response pole from one specified interaction. It also locates the normalization needed for a count enhancement. A physical simplex mass law still requires a master-derived kernel, a justified occupation and canonical norm, viable spatial particle states, and a rest-mass identification. This comparison neither establishes nor overturns the assigned particle dimensionalities.