Positive-Frequency Harmonic Density and Occupied-Sector Response

The bilinear Hermitian current makes the temporal preparation a physical part of a coupling proposal. A real local field evaluation gives a squared real waveform. A positive-frequency complex evaluation gives its modulus squared. Their fundamental density harmonics agree under equal-mean normalization, but their higher harmonics and conservative responses do not.

This calculation extends the existing complex preparation in Sawtooth Harmonics in Geometric Sector Coupling. It uses the same assigned upper mass-frequency, sector fundamentals, common receiver and unspecified physical amplitude. It does not identify either preparation with a derived particle solution.

1. One Bilinear Source, Two Preparations

Let $K=S(n,d)=\binom{n+d-1}{d}$ and $\theta=\omega_{0,d}t+\phi$. Compare

$$f_K(\theta)=\sum_{k=1}^K\frac{\sin k\theta}{k},\qquad z_K(\theta)=\sum_{k=1}^K\frac{e^{-ik\theta}}{k},\qquad J_R=f_K^2,\quad J_C=\frac{|z_K|^2}{2}.$$

Both sources have mean $c_0=H_K^{(2)}/2$. Orthogonal canonical positive-frequency modes retain constant total norm $H_K^{(2)}$; the changing local density arises from their specified evaluation, not a changing integrated norm. A spatial integral with the identity source operator would instead remove cross-mode beats. No new spatial directions are associated with the coefficient labels.

The complex density has only difference frequencies:

$$J_C=c_0+\sum_{p=1}^{K-1}D_p\cos p\theta,\qquad D_p=\sum_{k=1}^{K-p}\frac1{k(k+p)} =\frac{H_p+H_{K-p}-H_K}{p}.$$

Thus $D_1=1-1/K$, exactly the real source's fundamental coefficient. For fixed $p$, however, $D_p\to H_p/p$, whereas the real density coefficient tends to $1/p^2$. Equal fundamental and equal mean do not imply equal source spectra.

For the assigned down-quark cutoff $K=1$, $J_C=1/2$ is exactly constant. Its dynamic derivative source is zero. The real source retains a twice-fundamental ripple. The complex result does not remove a static color current or show that a down quark cannot interact: it constrains the particular derivative-density preparation.

2. An Exact Finite Derivative Norm

Define the normalized period average $\langle h\rangle=(2\pi)^{-1}\int_0^{2\pi}h\,d\theta$. Fourier orthogonality and finite harmonic identities give

$$\boxed{\left\langle(J_C')^2\right\rangle =\frac12\sum_{p=1}^{K-1}p^2D_p^2 =\frac{(K+1)(H_K^2-2H_K^{(2)})-(4K+2)H_K+8K}{2}.}$$

For a direct derivation, expand $(H_p+H_{K-p}-H_K)^2$ and use

$$\sum_{p=1}^{K-1}H_p=KH_K-K,$$

$$\sum_{p=1}^{K-1}H_p^2=KH_K^2-(2K+1)H_K+2K,$$

$$\sum_{p=1}^{K-1}H_pH_{K-p} =(K+1)(H_K^2-H_K^{(2)})-2KH_K+2K.$$

The last identity also follows by extracting the coefficient of $z^K$ in $\log^2(1-z)/(1-z)^2$. The resulting derivative norm is zero at $K=1$, $1/8$ at $K=2$, and grows as $K\log^2K/2$. The real-source norm instead equals $2K-H_K-H_K^{(2)}/2$. Both remember the upper cutoff, with different growth laws.

3. A Controlled Large-Cutoff Limit

Put $a_p=H_p/p$. Then $0\le D_p\le a_p$. The known nonlinear Euler sum is

$$V_\infty=\sum_{p=1}^{\infty}\frac{H_p^2}{p^2} =\frac{17}{4}\zeta(4)=\frac{17\pi^4}{360}.$$

This standard identity is recorded and independently derived through multiple-zeta identities in Mező, Nonlinear Euler sums (2014), pages 204 and 209. It is external mathematics, not a newly selected IDWT coefficient.

The elementary bound $H_p\le1+\log p$ gives, for $J\ge1$,

$$\sum_{p>J}a_p^2\le \int_J^\infty\frac{(1+\log x)^2}{x^2}\,dx =\frac{\log^2J+4\log J+5}{J}=:T_J.$$

Let $V_K=\sum_{p=1}^{K-1}D_p^2$. Since

$$0\le a_p-D_p=\frac{H_K-H_{K-p}}p\le\frac1{K-p+1},$$

and $a_p^2-D_p^2\le2a_p(a_p-D_p)$, partial fractions give

$$0\le V_\infty-V_K\le \frac{H_{K-1}^2+H_{K-1}^{(2)}+2H_{K-1}(H_K-1)}{K+1} +T_{K-1}=:B_K\quad(K>1).$$

Specifically, use $1/[p(K+1-p)]=[1/p+1/(K+1-p)]/(K+1)$, $\sum H_p/p=(H_{K-1}^2+H_{K-1}^{(2)})/2$, and $H_p\le H_{K-1}$ in the second sum. The remaining infinite tail is $T_{K-1}$. At $K=1$ the deficit is exactly $V_\infty$.

These bounds avoid constructing millions of harmonics. They bound truncation error; displayed numerical sums still have floating-point roundoff.

4. The Same Receiver at Every Occupied Depth

Retain the real-source comparison's receiver

$$\chi(x)=\frac15\left(\frac3{2-x^2}+\frac2{3-x^2}\right), \quad \chi(0)=\frac{13}{30},\quad r=\frac{\omega_{0,d}}{\omega_{0,6}}.$$

The complex dynamic responses are

$$Q_C^{(0)}=\frac12\sum_{p=1}^{K-1}D_p^2\chi(pr),\qquad Q_C^{(1)}=\frac12\sum_{p=1}^{K-1}(pr)^2D_p^2\chi(pr).$$

The direct source also has the identical static contribution $c_0^2\chi(0)$. These are signed conservative quadratic response forms, with fixed source orientation. They are not probabilities, electric charges, dissipation rates or complete pair forces. A moving source direction adds an orientation derivative and can change the suppression.

Use the maintained prescription with $m_e=0.51099895000$ MeV:

$$\mu_2=m_e\sqrt{2890},\quad \mu_3=m_e\sqrt{32\sqrt7},\quad \mu_4=\frac{m_e}{15}\sqrt{\frac{48}{\sqrt7}},\quad \mu_6=\mu_{10}=\frac{m_e}{18564}.$$

The separate neutrino diagnostic uses $\mu_5=\sqrt{2.534\times10^{-3}/(65780^2-2002^2)}$ eV. Thus $r$ is $997976.8822,170813.0545,5271.4053,2.781392118\times10^{-8},1,1$ at depths $2,3,4,5,6,10$. No scale or receiver coefficient is retuned between preparations or particles.

All source harmonics are included through $K=150000$; larger cutoffs use at most 65536 evaluated coefficients and proved tail intervals. The rounded intervals are expanded outward. The complex direct-response truncation errors are at most $1.11\times10^{-20}$ for top and $3.07\times10^{-13}$ for muon/tau. Photon optical frequency and bottom's unresolved noninteger assignment supply no adopted cutoff here.

For $r(J+1)>\sqrt3$, the ordinary omitted response is negative with magnitude at most $T_J/[2(r^2(J+1)^2-3)]$. The derivative tail is bounded by $T_J/[2(1-3/(r^2(J+1)^2))]$. A sharper derivative interval follows from

$$x^2\chi(x)=-1+\frac65\left(\frac1{2-x^2}+\frac1{3-x^2}\right).$$

Compute the rational correction with the actual finite $D_p$, replace only the constant variance term by $-V_\infty/2$, and bound the replacement by $B_K/2$. The omitted correction is negative and at most $6T_J/[5(r^2(J+1)^2-3)]$ in magnitude. This is the method used for the displayed large-cutoff intervals; treating the partial derivative sum as converged would give misleading values.

5. What Survives and What Changes

The complex charged-lepton direct dynamic responses differ by less than $0.003\%$. Their common fundamental and large cutoffs still give response similarity, though the value differs from the real source's approximately $0.37698$. This is conditional response similarity rather than exact charge universality.

Depth-five derivative suppression survives the changed harmonics. When $r(K-1)<\sqrt2$, positivity and monotonicity of the common receiver give

$$\chi(0)r^2\langle(J_C')^2\rangle \le Q_C^{(1)}\le \chi(r(K-1))r^2\langle(J_C')^2\rangle.$$

The largest relative upper correction for the three neutrinos is less than $1.503\times10^{-6}$. Their response is larger than in the real preparation by approximately factors $9.75,15.94,23.56$, while remaining small on the common diagnostic scale. Direct density response remains appreciable, so neutrality and observed weak rates still do not follow.

The charged-lepton derivative response changes sign: approximately $+0.2998$ for the real source, versus approximately $-2.02$ for the complex source. The higher-harmonic contribution above the receiver poles changes the signed response. A negative conservative response does not imply negative energy, negative inertia or an attractive physical force. It shows why source type, phase preparation and the complete response action cannot be omitted.

The raw Hodge geometry and its exact occupied-sector zeros are unchanged by multiplying a source by either scalar temporal waveform. The temporal calculation therefore complements the geometric selection but supplies no new preference for a spatial depth by itself. The outstanding physical task is to obtain the harmonic occupation and actual spatial readout from the same field dynamics.

Verification

The checker and report preserve the executed code and baseline hashes. Exact rational checks through $K=40$ test telescoping, the finite derivative norm and harmonic convolutions against independent FFT evaluations. Independent full finite sums test the variance and response bounds, high-precision coefficients check the large cutoffs, and all fourteen assigned massive branches use the same real/complex comparison. No particle simulation is used.

Which Fundamental the Source Sees

The harmonic-support calculation tests the first beat against an odd-only square comb, independent harmonic phases and integrated orthogonal-mode readout. Coherent adjacent harmonics retain coefficient 1−1/K; strict odd support removes that density line, and repairing an even mass endpoint leaves only 1/[K(K−1)]. All assigned branches use the same unchanged receiver. The sector fundamental is a concrete coupling-source lead under the specified occupation and readout, rather than a complete coupling law independent of waveform or spatial operator.