Canonical contact normalization and finite combinatorial bounds

Counting can constrain a proposed operator without being a derivation of its frequency. This calculation combines canonical normalization with the exact dimension of a space of angular functions. It rules out specific scalar-contact reductions under a shared normal profile, while leaving derivative interactions and dynamically changing normal profiles open.

1. A local quartic has a frequency denominator at fixed charge

Let a normalized spatial eigenmode have frequency $\Omega_n(R)$ and $\int|u_{n,R}|^4=W_nR^{-d}$. For an ordinary quadratic time norm, the unperturbed fixed-charge amplitude satisfies $a^2=Q/(2\kappa\Omega_n)$. A local positive potential $g\int|X|^4$ therefore contributes, at first order in $g$,

$$\Delta E_4=\frac{gQ^2W_nR^{-d}}{4\kappa^2\Omega_n^2}.$$

Replacing this by a constant times $Q^2W_nR^{-d}$ omits canonical normalization. The latter can be studied as a stipulated effective energy, but requires a different derived interaction.

For the particular hypothetical gap law $G=g_0R^{-2}$, the high-gap chain gives $\Omega_n=A P_n R^m$, where $P_n=d!S(n,d)$, $m=d-2$, and $d>2$. With a positive current support cost $\sigma R^{-d}$, the corrected reduced energy is

$$E=A P_n Q R^m+\sigma R^{-d} +B\frac{W_n}{P_n^2}Q^2R^{-q},\qquad q=3d-4.$$

It is an exact algebraic model if stipulated; for a plain local quartic it is only the first-order reduction. At simultaneous radius balance and the additional identification target $E=\omega Q$,

$$B\frac{W_n}{P_n^2}Q^2R^{-q}=\sigma R^{-d},\qquad m A P_n Q R^m=(d+q)\sigma R^{-d}.$$

Thus

$$R^q=\frac{d+q}{mA}\frac{\sqrt{\sigma BW_n}}{P_n^2},\qquad \omega\propto P_n^{d/q}W_n^{m/(2q)}.$$

For mode-independent $\sigma,B$, a linear frequency in $P_n$ requires $W_n\propto P_n^4$. If the charge is also common while the solved support cost may change with branch, linear frequency implies constant radius and requires $\sigma_n\propto P_n$, $W_n\propto P_n^3$. These are consequences of this reduction, not functions that may be fitted into a claimed derivation.

There is a further control on the approximation. At that formal joint balance the quartic energy divided by the linear wave energy is fixed:

$$\frac{\Delta E_4}{Q\Omega_n}=\frac{d-2}{4d-4}.$$

It is $3/16$ at depth five, $1/5$ at six, and $2/9$ at ten. These do not tend to zero by sending $g$ to zero while imposing both balances. Accordingly the formal equality branch is not justified as a controlled weak-quartic solution by this truncation alone. A full nonlinear calculation could change it.

Positive contact also changes the spatial profile. At depth five, $\langle U_l^3,U_j\rangle=j+1$ for $j\le l$ of the same parity, and zero for opposite parity. The nearest same-parity mode therefore has a cubic source of size $l-1$. The prior adjacent-mode stabilization estimates had first-order profile changes around 0.27–0.28 in this channel. They are useful diagnostics, not full-profile stability results.

2. A finite harmonic bound tests every scalar shape in a degree

On $S^m$, the degree-$l$ scalar harmonic dimension is

$$D_{l,m}=\binom{l+m}{m}-\binom{l+m-2}{m},$$

where the second term vanishes below degree two. These count homogeneous polynomials modulo their traces. The diagonal of the orthogonal projector is constant by rotations and equals $D_{l,m}/\mathcal A_m$, with $\mathcal A_m$ the sphere area. Consequently every real or complex normalized harmonic obeys

$$\frac1{\mathcal A_m}\le\int|Y_l|^4 \le\frac{D_{l,m}}{\mathcal A_m}.$$

The lower bound is Cauchy–Schwarz; the upper follows from the projector's pointwise norm. It includes arbitrary mixtures within that degree.

At depth five, $m=3$, the maintained indices $n=10,15,22$ give $l=11,16,23$, products $240240,1395360,7893600$, and dimensions $144,289,576$. The extreme product ratio is exactly $230/7$. A common scalar contact scaling $W_n=cP_n^k$ would require

$$(230/7)^k\le576.$$

Already $(230/7)^2=52900/49>576$. Thus every integer power $k\ge2$ is excluded in this same-degree scalar class with a common normal factor. No numerical shape search is needed.

3. The Hockey-Stick extension includes all lower degrees

For every scalar harmonic through degree $L$, summing the dimension formula gives

$$N_{L,m}=\binom{L+m+1}{m+1}-\binom{L+m-1}{m+1}.$$

This is the Hockey-Stick Identity applied to the two polynomial counts. It counts available coefficients, not occupied particles or a selected frequency. The band projector gives the same fourth-norm bound with $D_{l,m}$ replaced by $N_{L,m}$.

At depth five and $L=23$,

$$N_{23,3}=\sum_{j=1}^{24}j^2=4900, \qquad (230/7)^3=12167000/343>4900.$$

Therefore permitting all lower scalar degrees still cannot supply the $P_n^3$ or $P_n^4$ contact scaling in this model, with one common normal factor. The square ratio is below 4900, so this band bound does not exclude the distinct stipulated-effective $P_n^2$ requirement.

Full Hodge form components, derivative contacts, or a normal profile whose response is derived from the wave change the quantity being bounded. They must be calculated under the same action. Depth two's old scalar $S^0$ feature has no such ladder. These exact finite-mode exclusions do not establish a general prohibition on the physical mass operator.

Verification

Independent verification checks the normalized powers, exact product and dimension ratios, and the complete band sum. An effective quartic model that omits the canonical frequency denominator is a separate stipulated interaction; its results do not verify the plain local-potential normalization.