Passive Hodge Interference and the Clock That Controls Coupling

Two coherent orientations of one field contact can implement the symmetric Hodge filter without destroying energy. A positive wave-network action realizes that mathematical transfer. Its all-sector checks reveal two additional requirements: the physical origin of the response time matters, and a static zero does not guarantee cancellation of an oscillating color-volume interaction.

The network is an explicit inverse construction with prescribed coherent paths and contact maps. It is not a derived particle geometry, an observed force or a compactification. All channels represent modes of the same field along ordinary spatial paths. No electron three-plus-three decomposition is used.

1. A Complete Passive Transfer

On middle $m$-forms in $2m$ ordinary Euclidean dimensions, write $s=(-1)^m$. The real Hodge star obeys $*^T=s*$ and $*^T*=I$. With Fourier convention $e^{-i\Omega t}$, a nonnegative delay $\tau$ contributes $e^{i\Omega\tau}$. Define

$$T(\Omega)=\frac12(*+e^{i\Omega\tau}*^T),\qquad R(\Omega)=\frac12(*-e^{i\Omega\tau}*^T).$$

For the real Hadamard matrix $B$, the complete forward converter is

$$M=(B\otimes I)\operatorname{diag}(*,e^{i\Omega\tau}*^T)(B\otimes I) =\begin{pmatrix}T&R\\R&T\end{pmatrix}.$$

Every factor is unitary on the real frequency axis. Consequently

$$M^\dagger M=I,\qquad T^\dagger T+R^\dagger R=I.$$

The second output retains the field energy absent from the first. A common positive path delay $\tau_0$ multiplies both outputs by $e^{i\Omega\tau_0}$ and changes no forward gains. The complete bidirectional scattering matrix

$$S=\begin{pmatrix}0&M^T\\M&0\end{pmatrix}$$

is symmetric and unitary. A forward Hodge converter need not itself be symmetric to belong to a reciprocal complete device. Passivity alone does not establish reciprocity; both properties are checked here.

Since $T=f_s*$ with $f_s=(1+se^{i\Omega\tau})/2$,

$$T^\dagger T=\begin{cases} \cos^2(\Omega\tau/2)I,&m\text{ even},\\ \sin^2(\Omega\tau/2)I,&m\text{ odd}. \end{cases}$$

At zero frequency, $T(0)=H=(*+*^T)/2$. Thus the same static rule activates three/four and cancels two, five/six and nine/ten, under the existing even-carrier/odd-shadow prescription. Seven/eight and eleven/twelve remain active higher-rank controls. For odd $m$, $T=-i\Omega\tau */2+O(\Omega^2)$, but the full gain is bounded by one; a derivative approximation cannot be extrapolated through all harmonics.

2. A Local Positive Wave Action

A constructive mathematical analogue uses real mode coordinates $u_j(x,t)$ along ordinary paths, with common positive wave action

$$L=\frac12\sum_j\int\left[(\partial_tu_j)^2-c^2(\partial_xu_j)^2\right]dx.$$

The energy is the same expression with a plus sign between the two squares. At a junction, choose a real symmetric orthogonal scattering matrix $S_0$, and let $P_\pm=(I\pm S_0)/2$. Impose

$$P_-u=0,\qquad P_+\partial_nu=0.$$

The allowed variation lies in $P_+$, so these are variational boundary conditions. The boundary energy flux vanishes because $\dot u\in\operatorname{im}P_+$ and $\partial_nu\in\operatorname{im}P_-$. For incoming and outgoing waves $a,b$ at positive wave number $k$, their boundary values are $a+b$ and $ik(b-a)$. The conditions give $b=S_0a$ directly.

Any real orthogonal converter $O$ is obtained with

$$S_0=\begin{pmatrix}0&O^T\\O&0\end{pmatrix}.$$

Taking $O=B\otimes I$, $*$ or $*^T$ realizes the individual splitters and contact converters. Ordinary wave propagation along lengths $\ell_0$ and $\ell_0+c\tau$ supplies the two delays. This is a positive local action for the stated network rather than a rule that deletes transmitted power.

The selfadjoint boundary/scattering method is related to Kostrykin and Schrader, section 2, lemma 2.2, theorem 2.1 and corollary 2.1. Their graph-Laplacian calculation supplies the general scattering formula and real-boundary reciprocity. The wave action and particular projector realization above are stated explicitly here; borrowing the boundary mathematics does not adopt the paper's quantum-particle interpretation or establish such paths in IDWT.

Finite pulse checks use normalized traveling-wave velocity amplitudes, whose squared norm integrates energy flux. Both outputs conserve their complete integrated norm, with no output before the first path delay. A constant displacement's zero-frequency scattering limit must not be interpreted as a nonzero traveling energy flux.

3. Forward Receiver Gain and Retarded Return Are Different

For a scalar receiver susceptibility $\chi$, $T^\dagger\chi T=|f_s|^2\chi I$ is a forward-driven receiver quadratic diagnostic. Reciprocal return at the same frequency instead gives

$$T^T\chi T=f_s^2\chi I.$$

The distinction is causal. The return factor has the nonnegative-delay impulse coefficients

$$f_s^2=\frac14(1+2se^{i\Omega\tau}+e^{2i\Omega\tau}).$$

The forward gain $|f_s|^2$ contains both $e^{i\Omega\tau}$ and $e^{-i\Omega\tau}$. It cannot by itself be used as a retarded force kernel. Nor is the network's low-frequency odd transfer automatically the bare gyroscopic action of the preceding note; eliminating traveling modes changes the response and its time-reversal interpretation.

A passive causal load control is the reflection from a positive oscillator coupled to a wave line:

$$r_q(\Omega)=\frac{K-M\Omega^2+i\gamma\Omega}{K-M\Omega^2-i\gamma\Omega}, \qquad K,M,\gamma>0.$$

Its real-axis modulus is one and its poles lie below that axis. Loading the selected output gives return amplitude $r_qf_s^2$. The other reflected and outgoing ports conserve the remaining power. Actual interaction force still requires the complete spatial source and load variation; a scattering amplitude is not a measured charge or rate.

4. Two Common Delay Laws Across Every Maintained Sector

Retain the finite sawtooth source, assigned mass cutoff $K=S(n,d)$, density coefficients $c_p(K)$, and fundamental ratio $r_d=\omega_{0,d}/\omega_{0,6}$. Use the same receiver

$$\chi(x)=\frac15\left(\frac3{2-x^2}+\frac2{3-x^2}\right).$$

For a fixed unit source direction, the forward diagnostic is

$$Q_T=\frac12\sum_{p=1}^{2K}c_p^2|f_s(pr_d)|^2\chi(pr_d) +c_0^2|f_s(0)|^2\chi(0).$$

Compare two laws using one common dimensionless parameter $\eta$:

1. A receiver-set delay $\tau=\eta/\omega_{0,6}$ gives phase $pr_d\eta$.
2. A source-set delay $\tau_d=\eta/\omega_{0,d}$ gives phase $p\eta$.

The second law is a possible consequence of a source path length proportional to its inverse fundamental. It removes the filter's automatic suppression by the small neutrino clock. A common wave speed alone does not decide which length controls an encounter.

At $\eta=1$, representative total signed diagnostics are:

All assigned integer-cutoff branches, including depth two, are in the report. The coefficients and neutrino splitting calibration remain those of the earlier source comparison. Photon and bottom are not given invented cutoffs. The calculation also retains shared $\eta=0.01,0.1,\pi,2\pi$. At receiver delay $2\pi$, every charged-lepton integer-clock source harmonic is dark, whereas generic neutrino harmonics are not. At source-set delay $2\pi$, every odd-parity sector is dark. These unfavorable and exact periodic controls exclude a universal favorable-delay claim.

Because $|f_s|^2\leq1$, the omitted response tail is bounded by the existing density-tail bound. At large cutoffs, if $J=65536$ and $r_d(J+1)>\sqrt3$,

$$|\Delta Q_T|\leq \frac{4\pi^2/K+1/(3J^3)}{r_d^2(J+1)^2-3}.$$

The largest displayed lepton tail is below $2.4\times10^{-15}$. These are truncation bounds, not complete floating-point or physical uncertainty estimates. The receiver is still a diagnostic replicated on each form component, rather than a derived finite-orientation encounter.

5. A Finite Delay Can Reactivate the Color-Volume Channel

The preceding static color action uses $H$ instantaneously. Replacing every $H$ by the full delayed $T$ defines a different action. That replacement does not retain its strict higher-sector zero.

A simple control uses three equivalent-mode patterns with a common frequency and phase and nonzero mode volume. This is an equal-clock control, not synchronization imposed on unlike particles. If their common real waveform is $f_K(\theta)$, the transmitted waveform on a Hodge eigenchannel is, up to a unit phase,

$$g_{K,s}(\theta)=\frac12[f_K(\theta)+sf_K(\theta-\delta)], \qquad \delta=\omega_{0,d}\tau.$$

The averaged three-slot volume energy is multiplied by $\langle g_{K,s}^6\rangle$. For every finite nonzero preparation, a nontrivial odd-channel delay generally gives a positive value. The absence of a DC source does not cancel its positive time-averaged nonlinear energy.

For the limiting sawtooth and $0\leq\delta\leq2\pi$, direct piecewise integration gives

$$\boxed{\langle g_{\infty,-}^6\rangle= \frac{\delta(2\pi-\delta)^6+(2\pi-\delta)\delta^6}{2\pi4^6},}$$

$$\boxed{\langle g_{\infty,+}^6\rangle= \frac{\delta^7+(2\pi-\delta)^7}{14\pi4^6}.}$$

The unfiltered sixth moment is $\pi^6/448$. At $\delta=1$, the odd-channel moment is about $39.38\%$ of that reference; at $\delta=0.1$ it is about $10.12\%$. Thus a unit receiver delay which gave small neutrino density response can also allow a substantial delayed higher-mode volume on the integer-clock lepton sectors. This is not an observed strong force, but it rules out transferring the instantaneous action's strict cancellation to this delayed replacement.

At fixed finite $K$, the small-delay odd response begins at sixth order, with a coefficient involving $\langle(f'_K)^6\rangle$. If the infinite-cutoff limit is taken first, the result instead begins as $\delta\pi^5/128$. The sawtooth jump makes the limits nonuniform; a low-frequency derivative estimate cannot prove uniformly small nonlinear leakage at arbitrary cutoffs.

For finite $K\leq100000$, more than $6K$ uniform samples integrate the sixth-power trigonometric polynomial exactly, apart from roundoff. Larger assigned cutoffs use the analytic limiting moment with a rigorous error estimate. Put $M_b=\pi/2$, $M_e=2\pi$. The previously proved $L^2$ sawtooth error and the binomial expansion give

$$\left|\langle g_{K,s}^6\rangle-\langle g_{\infty,s}^6\rangle\right| \leq\frac{6M_b^5}{\sqrt{2K}} +\frac1{2K}\sum_{j=2}^{6}\binom6jM_b^{6-j}M_e^{j-2}.$$

The bound proves a nonzero delayed higher-volume control at both the muon and tau cutoffs for $\eta=1$. A separate interaction law could retain the instantaneous $H$ color term while using other delayed modes for other responses. That possibility needs its stated common action; it cannot be inferred by identifying the two filters.

6. Coherence and Robustness

The selected zero requires adding amplitudes before squaring. If the two contact routes become distinguishable, their incoherent powers do not cancel. With path amplitudes $\alpha=(1+\epsilon)/2$, $\beta=(1-\epsilon)/2$ and extra DC phase $\phi$, the odd-channel static gain is

$$|\alpha-\beta e^{i\phi}|^2 =\epsilon^2+2\alpha\beta(1-\cos\phi).$$

This floor couples the nonzero static density, which was absent from the ideal derivative channel. An exchange symmetry could protect equal routes, but reciprocity alone does not impose equal splitting or zero extra phase. The source must also be formed before lower-dimensional projection if it is to retain the complete-pattern Hodge rule; a network diagram alone does not derive that ordering.

The useful outcome is a physical distinction that can now be tested: an instantaneous graded self-contact, a passive delayed orientation interferometer and a fast-mode-induced gyroscopic action share some mathematics but give different finite-frequency and nonlinear responses. Their coefficients, time scales and omitted modes cannot be exchanged without rederivation.

Verification

The single checker and its report record 11,629 passing checks across 230 cases, covering oriented Hodge blocks through depth twelve, full unitary reciprocal scattering, causal pulse and load controls, positive local wave boundaries, all maintained source cutoffs under both common delay laws, symmetry-breaking floors and finite-delay color-volume controls. Executed source hashes accompany the report. No particle simulation or fitted field amplitude is used.