Harmonic Support Determines Which Fundamental a Coupling Source Can See

The sawtooth calculation establishes a useful conditional connection: adjacent occupied harmonics produce a strong source beat at the sector fundamental even when the selected mass-frequency is a much higher harmonic. The connection depends on which harmonics are occupied and how the physical current reads their spatial modes. The square-wave comparison makes those conditions explicit.

1. The Frequency Rule Follows from the Actual Readout

Let the same field have orthogonal spatial modes $u_k(x)$ and temporal amplitudes $a_ke^{-ik\omega_0t}$. A Hermitian source operator $B$ produces

$$J(t)=\sum_{k,l}a_k^*a_l M_{kl}e^{-i(l-k)\omega_0t},\qquad M_{kl}=\langle u_k,Bu_l\rangle.$$

A local evaluation uses the corresponding pointwise products instead of integrated matrix elements. The source frequencies are differences of occupied harmonic labels for which $M_{kl}\ne0$. Thus the smallest frequency lattice is $g\omega_0$, where $g$ is the greatest common divisor of the surviving nonzero label differences. Cancellation can remove an individual line within that lattice. A single occupied harmonic has constant complex density.

If $B=I$ and the spatial integral includes complete orthogonal modes, $M_{kl}=\delta_{kl}$ and every beat disappears. A local rank-one evaluation can instead retain the off-diagonal products. An actual spatial coupling must specify its operator and profile; the temporal sum alone does not determine the vertex.

For the coherent local preparation

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

the coefficient of $\cos\theta$ is

$$\boxed{D_1=\sum_{k=1}^{K-1}\frac1{k(k+1)}=1-\frac1K.}$$

It approaches one as the mass-frequency cutoff grows. The sector fundamental therefore remains a strong source frequency rather than being replaced by the upper harmonic. The fundamental alone still does not specify the complete response.

2. Harmonic Phase Coherence Is Material

If each harmonic instead has an independent uniform phase $\phi_k$, its complex fundamental coefficient is

$$C_1=\sum_{k=1}^{K-1}\frac{e^{i(\phi_{k+1}-\phi_k)}}{k(k+1)}.$$

Then

$$\langle C_1\rangle=0,\qquad \boxed{\langle|C_1|^2\rangle =2H_K^{(2)}-3+\frac2K-\frac1{K^2}.}$$

The second identity follows by eliminating unequal phase products and summing $1/[k^2(k+1)^2]=1/k^2+1/(k+1)^2-2/[k(k+1)]$. Its large-cutoff variance is $\pi^2/3-3$, rather than the coherent limit one. A given incoherent realization can still have a fundamental line. Averaging the source and averaging a quadratic response are different operations.

A common temporal shift $\phi_k=k\phi$ preserves coherent adjacent phase differences. The phase condition is about the harmonics within one pattern; it does not impose synchronization between particles with different isolated frequencies.

3. An Odd-Only Square Comb Has a Different Source Clock

Take the positive-frequency square comparison with the same $1/k$ amplitudes but only odd labels,

$$z_{\rm odd}(\theta)=\sum_{j=1}^{L}\frac{e^{-i(2j-1)\theta}}{2j-1},\qquad O_L=\sum_{j=1}^L\frac1{2j-1}.$$

Both labels in every density product are odd, so their differences are even. In particular,

$$\boxed{D_1^{\rm odd}=0,\qquad D_{2q}^{\rm odd}=\frac{O_q+O_{L-q}-O_L}{2q},\quad1\le q<L.}$$

The density's first available frequency is $2\omega_0$, even though the waveform contains $\omega_0$. The real odd-only square waveform also satisfies $f(\theta+\pi)=-f(\theta)$, so its squared density has only even harmonics. A linear readout or another nonlinear source can have a different selection rule; this result concerns the specified quadratic current.

4. Preserve the Assigned Mass-Frequency Endpoint

The maintained upper cutoff is $K=S(n,d)$, not a label free to round to the next odd integer. A strict square comb contains that endpoint only when $K$ is odd. Ten of the fourteen assigned massive branches have even $K$. Only Z, down, up and charm have odd cutoffs in this comparison; photon optical frequency and bottom's unresolved assignment provide no adopted cutoff.

For a labeled repair at even $K=2L$, add the actual endpoint harmonic $e^{-iK\theta}/K$ to the odd comb. Its new odd density coefficients are

$$D_p^{\rm repair}=\frac1{K(K-p)}\quad(p\text{ odd}),\qquad \boxed{D_1^{\rm repair}=\frac1{K(K-1)}.}$$

The even coefficients retain their odd-comb values. The repaired fundamental is approximately $K^{-2}$, compared with $1-K^{-1}$ for the full sawtooth. It is $2.90189\times10^{-9}$, $6.78741\times10^{-14}$ and $2.40279\times10^{-16}$ for electron, muon and tau. This repair is no longer a strict square wave, and is not presented as a derived particle preparation.

The derivative norm has an informative exact endpoint increment:

$$\Delta\langle(J')^2\rangle =\frac12\left[O_L^{(2)}-\frac{2O_L}{K}+\frac{L}{K^2}\right], \qquad O_L^{(2)}=\sum_{j=1}^L\frac1{(2j-1)^2}.$$

A tiny high-frequency amplitude can leave a finite raw derivative norm. Its actual receiver-weighted contribution still needs the frequency response; the raw derivative norm is not a coupling constant.

5. The Same Occupied-Sector Receiver Comparison

Use the previous common susceptibility, sector fundamental ratios and fixed cutoffs,

$$\chi(x)=\frac15\left(\frac3{2-x^2}+\frac2{3-x^2}\right),\qquad Q^{(0)}=\frac12\sum_pD_p^2\chi(pr),\qquad Q^{(1)}=\frac12\sum_p(pr)^2D_p^2\chi(pr).$$

These are signed conservative dynamic diagnostics. Static density terms, spatial orientation and physical amplitude are separate. No response is identified with electric charge or a measured force.

The strict odd-comb column is a waveform control; its missing even mass endpoints are not silently accepted as particle realizations. The repaired preparation is calculated separately for every row in the report. Largest-cutoff density truncation errors are below $3.07\times10^{-13}$, and derivative intervals retain the larger proved tail. The bounds use $0\le D_p\le H_p/p$ for either subset of the full positive comb and the earlier tail envelope $(\log^2J+4\log J+5)/J$. Numerical roundoff is separate from analytic truncation bounds.

At charged-lepton ratio $r=1$, every nonzero strict-square density frequency lies above both receiver poles. All dynamic terms are negative. The full sawtooth instead has a positive fundamental contribution approaching $0.4$, partly offset by its higher lines; its complete direct dynamic response approaches $0.152515$. The squared fundamental contribution is therefore central to that positive direct response, while higher harmonics remain substantial.

Slow depth-five derivative suppression survives the odd support; direct density remains appreciable. The absence of a fundamental density line neither removes every interaction nor derives neutrino neutrality. Likewise a constant single-harmonic density can retain a static spatial current.

The supported physical lead is specific: coherent adjacent harmonics can preserve a sector-fundamental coupling source across widely different upper mass harmonics. Spectral occupancy, relative harmonic phase and the actual spatial current determine whether that source exists and how strongly it is read.

Verification

The checker and report preserve 655 checks. Exact finite sums through $K=50$ are compared with independent FFT coefficients, all endpoint derivative increments are checked, finite independent phase grids verify the first two fundamental moments, and every assigned massive branch uses the unchanged receiver and sector ratios. Large sums evaluate at most 65536 coefficients. No particle simulation or individual parameter adjustment is used.

Candidate Dynamics for the Coherent Source

The cavity phase-lock calculation supplies positive inverse modal actions with exact locked branches, including frequency-weighted sawtooth coefficients. A separate positive mismatch energy gives reciprocal transfer between adjacent high-frequency modes and their difference-frequency mode. Common adjacent-band envelopes have exactly carrier-independent Hermitian sources. Band closure, harmonic generation and the spatial master-action reduction remain explicit construction questions.