Sawtooth Harmonics in Geometric Sector Coupling

A particle pattern may contain the sector fundamental and every harmonic through its assigned mass-frequency. Under a specified sawtooth preparation, the lower harmonics produce a fundamental-frequency density source whose strength approaches a fixed value as the upper cutoff grows. Combining this source with the existing spatial contact, color and angular operators gives calculable phase forces and response spectra.

The most useful results are an exact fundamental coefficient $1-1/K$, a reciprocal phase-overlap kernel, and a tetrahedral example whose shape remains unchanged while its five positive shape eigenvalues split. A common receiver calculation also gives nearly equal density responses for the electron, muon and tau cutoffs. Small depth-five response occurs for a derivative source, whereas a direct density source remains appreciable. Those alternatives specify different interaction terms and must retain that distinction.

The maintained mass-frequency is the supplied endpoint in this calculation. The physical hypotheses being tested are the harmonic occupation, its spatial readout and the interaction. No particle amplitudes or individual pair coefficients are fitted.

1. The Finite Harmonic Pattern and Its Spatial Meaning

Use the maintained cumulative count and sector fundamental:

$$K=S(n,d)=\binom{n+d-1}{d},\qquad \omega_{m,n,d}=K\omega_{0,d}.$$

Consider the real local waveform

$$F_K(t)=A f_K(\theta),\qquad f_K(\theta)=\sum_{k=1}^{K}\frac{\sin(k\theta)}{k},\qquad \theta=\omega_{0,d}t+\phi.$$

The amplitude $A$ is unspecified. The phases of the harmonics are locked by this preparation; an arbitrary collection of occupied harmonics need not make this waveform. The temporal cutoff $K$ is distinct from the spatial shell cutoff $N=n-1$ in the spectral-band construction.

A local waveform need not be the whole spatial field multiplied by a time-dependent scalar. For an orthonormal family of real spatial modes, a conditional positive quadratic carrier is

$$H=\frac12\sum_{k=1}^{K}\left[p_k^2+(k\omega_0)^2q_k^2\right], \qquad q_k=\frac{A}{k}\sin(k\omega_0t),\qquad p_k=A\omega_0\cos(k\omega_0t).$$

A specified evaluation $\ell q=\sum_kq_k$ reads $Af_K$, while

$$H=\frac{A^2\omega_0^2K}{2}$$

is constant. Thus a changing local evaluation is compatible with a conserved complete quadratic energy. This demonstrates a mode-level realization, conditional on the harmonic spectrum and evaluation weights. It does not identify a spatial differential operator or establish that a proposed Dirac field has this second-order action. No amplitude has been chosen to fit the mass.

A complementary first-order mode control uses orthogonal positive-frequency amplitudes

$$a_k(\theta)=\frac{e^{-ik\theta}}{k},\qquad \sum_k|a_k|^2=H_K^{(2)},\qquad z_K=\sum_k a_k.$$

Here $H_K=\sum_{k=1}^K1/k$ and $H_K^{(2)}=\sum_{k=1}^K1/k^2$. The total canonical norm is constant, although the local evaluation density $|z_K|^2$ changes. Both controls use coefficient labels for modes of the same field, without introducing additional physical spatial dimensions.

2. The Fundamental Appears by Adjacent-Harmonic Interference

For the real waveform, write

$$f_K(\theta)^2=c_0(K)+\sum_{p=1}^{2K}c_p(K)\cos(p\theta).$$

Product-to-sum gives a difference-frequency contribution and a sum-frequency contribution:

$$c_p=\sum_{r=1}^{K-p}\frac1{r(r+p)} -\frac12\sum_{r=\max(1,p-K)}^{\min(K,p-1)}\frac1{r(p-r)},$$

where empty sums vanish. Telescoping the two sums yields

$$c_0=\frac12H_K^{(2)},\qquad c_p=\begin{cases} \displaystyle\frac1{p^2}+\frac{H_{K-p}-H_K}{p},&1\leq p\leq K,\\[6pt] \displaystyle-\frac{H_K-H_{p-K-1}}p,&K<p\leq2K. \end{cases}$$

In particular,

$$\boxed{c_1(K)=\sum_{r=1}^{K-1}\frac1{r(r+1)}=1-\frac1K}, \qquad c_{2K}=-\frac1{2K^2}.$$

The fundamental density beat is therefore present whenever $K>1$. A much higher endpoint does not remove it. For fixed $p$,

$$c_p(K)\longrightarrow\frac1{p^2},\qquad c_0(K)\longrightarrow\frac{\pi^2}{12}.$$

The assigned down-quark endpoint has $K=1$. Its real density has a static term and a twice-fundamental term, but no fundamental beat. This is a concrete exceptional case under the same rule, rather than a depth-specific coefficient.

The complex carrier in section 1 gives the same fundamental result under equal mean-source normalization. Specifically,

$$\frac{|z_K|^2}{2}=\frac{H_K^{(2)}}2+ \sum_{p=1}^{K-1}D_p(K)\cos(p\theta),\qquad D_p=\frac{H_p+H_{K-p}-H_K}{p}.$$

Then $D_1=1-1/K$. Its higher harmonics differ: $D_p\to H_p/p$. The real-sawtooth response values below cannot be transferred unchanged to this complex preparation. Agreement of the fundamental demonstrates that the adjacent-mode beat does not rely on a changing global norm.

3. The Spatial Source Determines Which Beats Are Accessible

For canonical spatial coefficient vectors $a_r$ and a specified Hermitian source operator $B$, the positive-frequency field has bilinear Fourier coefficients

$$J_p=\sum_r a_r^\dagger B a_{r+p}.$$

This is the existing transition-density rule applied to an evenly spaced temporal preparation. A spatial integral with $B=I$ can remove all off-diagonal beats between orthogonal modes. A local evaluation can retain them.

For a finite operator control with $S$ orthogonal modes, let $e=(1,\ldots,1)^T$ and $T=ee^\dagger$. Then

$$T^2=ST,\qquad \sum_{r=1}^{S-1}\frac{T_{r,r+1}}{r(r+1)}=1-\frac1S.$$

Thus an operator of the contact form already present in V2 can read the fundamental beat. The equal-entry basis is a specified control. The actual sphere Dirac eigenbasis is not automatically assigned one temporal frequency $r\omega_0$ per basis vector: shell multiplicities and temporal occupations still need their physical map.

For a nontrivial kinetic Gram matrix $Z$, the source is instead evaluated with

$$\widehat B=Z^{-1/2}BZ^{-1/2}.$$

Both source legs and all spatial inclusion maps must be transformed. Dividing $T$ by its count to obtain a projector is a different operator prescription unless the action supplies that normalization. The canonical normalization theorem prevents an arbitrary basis convention from becoming an apparent coupling law.

4. A Reciprocal Interaction Between Two Sawtooth Patterns

Suppose the spatial source channels factor from the two real temporal preparations, and their spatial contraction is $G_{ab}$. For a common fundamental, their averaged temporal overlap is

$$W_{KL}(\delta)=\left\langle f_K(\theta)^2f_L(\theta+\delta)^2\right\rangle =c_0(K)c_0(L)+\frac12\sum_{p=1}^{2\min(K,L)}c_p(K)c_p(L)\cos(p\delta).$$

For the specified pair energy

$$E_{ab}=\lambda G_{ab}A_a^2A_b^2W_{KL}(\phi_b-\phi_a),$$

the phase forces are opposite:

$$-\partial_{\phi_a}E_{ab}=\lambda G_{ab}A_a^2A_b^2W_{KL}'(\delta),\qquad -\partial_{\phi_b}E_{ab}=-\lambda G_{ab}A_a^2A_b^2W_{KL}'(\delta).$$

Varying $G_{ab}$ gives both spatial responses as well. The construction therefore reuses the reciprocal pair framework, retaining the two patterns and their relative phase. Its angular minima require a positive spatial coefficient; attraction and repulsion depend on the actual spatial gradient and sign.

For nonzero finite preparations, $W_{KL}>0$: the product of the squared trigonometric polynomials is nonnegative and cannot vanish throughout a period. Multiplying an invariant spatial channel by this scalar preserves that channel's exact zeros.

The limiting overlap has a simple closed form. Away from the jump, $f_\infty(\theta)=(\pi-\theta)/2$ on $0<\theta<2\pi$. Integrating its squared overlap piecewise gives

$$W_\infty(\delta)=\frac{\pi^4}{80}-\frac{\pi^2\delta^2}{24} +\frac{\pi\delta^3}{24}-\frac{\delta^4}{96},\qquad 0\leq\delta\leq2\pi.$$

Consequently,

$$\boxed{\frac{W_\infty(\pi)}{W_\infty(0)}=\frac16},\qquad W_\infty''(\pi)=\frac{\pi^2}{24}.$$

A positive pair coefficient favors antiphase in this limiting kernel. Finite small cutoffs have additional phase structure; this is not a universal claim of antiphase locking for every $K$. The limit describes an overlap integral, not an infinite-energy physical sawtooth.

Different Fundamentals and Finite Encounters

For $\omega_a/\omega_b=P/Q$ in coprime positive integers, exact long-time averaging retains source harmonics $Qj$ and $Pj$:

$$\overline W=c_{0a}c_{0b}+\frac12\sum_jc_{Qj}(K)c_{Pj}(L) \cos\!\left[j(Q\phi_a-P\phi_b)\right].$$

The upper limit is $\min(\lfloor2K/Q\rfloor,\lfloor2L/P\rfloor)$. At large cutoffs the oscillatory coefficients carry the factor $1/(P^2Q^2)$ relative to the equal-clock kernel. Incommensurate fundamentals leave only the static product in an ideal infinite-time average.

Actual finite encounters retain detuning. With $j_0=c_0$ and $j_{\pm p}=c_p/2$, a centered window of length $T$ gives exactly

$$\overline W_T=\sum_{p,q}j_{a,p}j_{b,q} e^{i(p\phi_a+q\phi_b)} \operatorname{sinc}\!\left[\frac{(p\omega_a+q\omega_b)T}{2}\right], \qquad\operatorname{sinc}x=\frac{\sin x}{x}.$$

The complete sum is real. No synchronization of unlike sector fundamentals has been imposed. An encounter with moving spatial profiles also retains their time-dependent $G_{ab}$ inside the integral.

5. The Same Geometry, with Phase-Dependent Shape Frequencies

The angular-band construction gives

$$K_1(n_i\cdot n_j)=\frac{1+k\,n_i\cdot n_j}{k+1},\qquad E_{\rm ang}=\kappa\sum_{i<j}W_{ij}|K_1(n_i\cdot n_j)|^2,$$

where $k$ counts resolved spatial directions and every $W_{ij}>0$. Amplitude factors are included in the common $\kappa$ for this equal-amplitude angular comparison. The kinetic term remains $I\sum_i|\dot n_i|^2/2$.

For $k+1$ directions, a regular simplex still makes every pair overlap vanish. Positivity proves that it remains the exact global minimum, with the same Gram matrix. At that minimum,

$$\delta^2E_{\rm ang}=2\kappa\sum_{i<j}W_{ij}|\delta K_{ij}|^2.$$

Equal cutoffs and equal phases give one common $W_{ij}=W$, multiplying the old Hessian by $W$ and preserving its frequency ratios. Unequal relative phases give different edge weights and split the shape spectrum.

For an explicit tetrahedron, use the limiting overlap and phases $(0,\pi,0,\pi)$. Divide all weights by $W_\infty(0)$, so equal-phase pairs have weight $1$ and opposite-phase pairs have weight $1/6$. The canonically normalized squared shape frequencies, in units $\kappa W_\infty(0)/I$, are

The exact characteristic polynomial is

$$p(\lambda)=\frac{\lambda^3(\lambda-2)(2\lambda-1)(3\lambda-1)^2(6\lambda-13)}{108}.$$

The equal-phase reference instead has squared eigenvalues $0^{(3)},2^{(3)},3^{(2)}$ in the same units. Relative temporal phase can therefore change geometric response while leaving the tetrahedral angles unchanged.

These are shape Hessians at specified relative phases. The displayed angular energy is zero at the simplex for any phases, so it alone does not select the alternating phase preparation or supply phase inertia. Phase evolution and preparation require the complete action. Numerical positive-weight simplex checks cover $k=2,\ldots,12$. The same positivity argument preserves the cross-polytope zero-energy shapes for the existing $K_2$ kernel, without asserting a separately computed phase-weighted cross-polytope spectrum.

Here $k=3$ is an observer's angular reduction. It does not replace the electron's six physical spatial dimensions.

6. Low Harmonics Converge, but Derivative Sensitivity Can Grow

The exact finite-cutoff derivative norm is

$$\boxed{D_K=\left\langle\left[\frac{d}{d\theta}f_K^2\right]^2\right\rangle =\frac12\sum_{p=1}^{2K}p^2c_p^2 =2K-H_K-\frac12H_K^{(2)}.}$$

A short proof sets $b_p=pc_p$ and reindexes both halves of the coefficient formula. Let

$$A_K=\sum_{t=0}^{K-1}(H_K-H_t)^2=2K-H_K, \qquad B_K=\sum_{j=1}^K\frac{H_K-H_{K-j}}j=H_K^{(2)}.$$

For the first identity, expand the square and count $\min(r,s)$ occurrences of $1/(rs)$. For the second, use the triangular sum $\sum_jH_{K-j}/j=H_K^2-H_K^{(2)}$. Then

$$\sum_{p=1}^{2K}b_p^2=2A_K-2B_K+H_K^{(2)},$$

which gives the boxed result. In particular, the aligned finite-cutoff phase stiffness satisfies $W_{KK}''(0)=-D_K$.

Although every fixed density harmonic converges, differentiating the finite sums retains a narrow feature near the sawtooth jump. Its squared derivative integral grows approximately as $2K$. Differentiating the limiting overlap and taking the limit of finite-cutoff derivatives are different operations. A derivative-sensitive interaction can therefore remember the upper endpoint even when its low harmonics scarcely change.

7. One Receiver Operator Compared Across All Maintained Depths

The new tetrahedral geometry supplies a specific conservative receiver with three shape directions at squared frequency $2\kappa/I$ and two at $3\kappa/I$. Let $q$ denote those canonically normalized shape coordinates of the same field. For a common comparison scale, set $\kappa/I=\omega_{0,6}^2$ and use time $\tau=\omega_{0,6}t$. The dimensionless quadratic action is

$$L_q=\frac12|q'|^2-\frac12q^TCq,\qquad \operatorname{spec}C=(2,2,2,3,3).$$

This scale choice is a diagnostic assumption, not a derived identification of the receiver with a particular particle. Rotational zero modes are excluded from this shape-only response; a source that drives rotations needs their dynamics too.

Two explicit alternatives for coupling a density source $J_a$ to these modes are

$$L_{\rm int}^{(0)}=g\sum_aJ_a u_a^Tq, \qquad L_{\rm int}^{(1)}=g\sum_aJ_a u_a^Tq'.$$

Here $u_a$ is a real source-direction vector. For fixed $u_a$, the second differs from $-g\sum_aJ_a'u_a^Tq$ by a boundary term. A moving source direction instead gives $-(J_au_a)'$, including the orientation term $-J_au_a'$. A slow density clock alone does not suppress that additional source. Eliminating $q$ in the fixed-direction comparison yields the reciprocal response matrix

$$g^2u_a^T(C-x^2I)^{-1}u_b$$

for the first action and an additional $x^2$ for the second. These are projected response modes of the same field, not additional interaction particles. The physical units of $g$ depend on the chosen action and the reference time. The comparison uses one common coefficient within each alternative; it does not identify the two alternatives' physical constants.

Average a unit source isotropically across the five shape directions. The resulting diagonal response is

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

Actual pair coupling retains $u_a,u_b$, spatial overlaps and phases. This scalar average is a common receiver diagnostic, not a full two-particle encounter.

For the real sawtooth source, remove the common amplitude and coefficient factors and define

$$Q_K^{(0)}(r)=\frac12\sum_{p=1}^{2K}c_p^2\chi(pr),\qquad Q_K^{(1)}(r)=\frac12\sum_{p=1}^{2K}(pr)^2c_p^2\chi(pr), \qquad r=\frac{\omega_{0,d}}{\omega_{0,6}}.$$

The direct source also has the static contribution $c_0^2\chi(0)$. The derivative source has no static contribution because of its actual time derivative. These signed conservative quadratic response forms are neither probabilities nor electric charges. No damping or outgoing-wave prescription is included at the receiver poles.

Common Scale Inputs

The maintained non-neutrino prescription gives, in mass-equivalent units,

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

Use its archived electron reference $m_e=0.51099895000$ MeV. These numerical sector coefficients remain an unresolved prescription rather than derived force constants. For depth five only, use the neutrino module's archived splitting calibration:

$$\mu_5=\sqrt{\frac{2.534\times10^{-3}}{65780^2-2002^2}}\ \mathrm{eV}.$$

This is a separate conditional absolute-scale input. The resulting ratios are

All cases below use that same receiver, source-coefficient law and response convention. The listed neutrinos are assigned mass branches, not flavor eigenstates.

Only top, muon and tau use a truncated response sum. Their absolute tail bounds on $Q^{(1)}$ are respectively $3.249\times10^{-5}$, $1.029\times10^{-5}$ and $6.120\times10^{-7}$; the extra displayed digits specify the computed partial sums. Their $Q^{(0)}$ tail bounds are below $4\times10^{-19}$, $2.4\times10^{-15}$ and $1.5\times10^{-16}$. These are truncation bounds, not total floating-point error estimates. All other displayed cases evaluate the full source support.

Three consequences are useful:

1. The electron, muon and tau direct dynamic responses differ by less than $0.009\%$, although the cutoffs differ by factors of hundreds and thousands. Their common sector frequency unit and convergent low source coefficients do real work. This is conditional similarity of a response, not exact electric-charge universality.
2. Direct density coupling does not suppress depth five. Its static term is also about $0.293$, comparable to the charged-lepton values. A small fundamental alone does not make every coupling small.
3. The derivative action gives a small depth-five dynamic response under the same coefficient used elsewhere. When $2Kr$ lies well below the receiver poles,

$$Q_K^{(1)}(r)\simeq\chi(0)r^2D_K =\frac{13}{30}r^2\left(2K-H_K-\frac12H_K^{(2)}\right).$$

The suppression depends on both fundamental and cutoff. It has no inserted depth-five zero, but requires this derivative interaction, the separately calibrated scale and the stated receiver. It does not establish neutrino electric neutrality or an observed weak rate. The velocity-source term must be derived from the common field action; a moving spatial-band connection is a possible place to examine such terms, not an automatic derivation of this one.

The small ratio is a spectral condition, not a theorem unique to five spatial dimensions. Another sector with the same sufficiently slow source would have the same suppression under this receiver. The spatial selection maps must supply any further dimensional restriction.

The massless photon label supplies no optical cutoff for this massive-comb prescription. Bottom's maintained unresolved geometric-mean assignment supplies no adopted integer cutoff. Neither has been rounded or assigned a replacement mass to fill the table.

8. Bounded Sums at Very Large Cutoffs

Large cutoffs need no million-mode time simulation. The coefficient formula uses harmonic sums or digamma differences. The calculation evaluates the full $2K$ support for $K\leq150000$, otherwise the first $J=65536$ source harmonics.

With normalized period norm $\|h\|_2^2=\langle|h|^2\rangle$, Fourier orthogonality gives

$$\|f_K-f_\infty\|_2^2\leq\frac1{2K},\qquad \|f_\infty\|_\infty\leq\frac\pi2,\qquad \|f_K\|_\infty\leq\frac{3\pi}{2}.$$

For the last bound, reduce the argument to $0<\theta\leq\pi$, split at $\lfloor1/\theta\rfloor$, bound the early terms by $1$ and apply summation by parts to the remaining sine sum. Its partial sums are bounded by $1/\sin(\theta/2)$, giving a tail bound at most $\pi$. Thus $1+\pi<3\pi/2$ suffices.

Apply the triangle inequality to the high-frequency projection of $f_K^2-f_\infty^2$. Since the limiting squared-wave coefficients are $1/p^2$,

$$\frac12\sum_{p>J}c_p(K)^2\leq\frac{4\pi^2}{K}+\frac1{3J^3}=:B_{KJ}.$$

If $r(J+1)>\sqrt3$, all omitted receiver terms lie above its poles, and

$$|\Delta Q^{(0)}|\leq\frac{B_{KJ}}{r^2(J+1)^2-3},\qquad |\Delta Q^{(1)}|\leq\frac{r^2(J+1)^2B_{KJ}}{r^2(J+1)^2-3}.$$

The actual omitted terms are negative for this receiver. The bounds explain what precision the partial sums support without extrapolating a low-harmonic approximation through unresolved poles.

9. Color Channels and the Complementary Field Modes

Under the factorization in section 4, the color volume contraction becomes

$$E_{ab}^{\rm color}=\gamma\sigma^2 A_a^2A_b^2W_{KL}(\delta) \left(\|a\|^2\|b\|^2-|a^\dagger b|^2\right).$$

The color amplitudes $a,b$ here are fixed normalized channel directions; their time-dependent scalar factors are displayed separately. The third-pattern covariance remains the stated isotropic $\sigma^2I$. Thus the operator remains proportional to $I-\mathsf S$, preserving its symmetric kernel and antisymmetric channel. The same rank-three Casimir factors survive while the scalar coupling acquires phase and frequency dependence.

For a general rank-$k$ three-form norm, the contraction retains the multiplier $k-2$. A positive temporal factor cannot remove that higher-rank response. This extension preserves the geometric color results and the need for a physical invariant-selection rule.

A nonlinear field equation also tests whether the prepared harmonic set closes. The real cubic source has the exact nonzero highest term

$$[f_K^3]_{\sin(3K\theta)}=-\frac1{4K^3}.$$

Only the product of the three top-frequency components contributes at $3K$. Consequently, a pure finite sawtooth is not closed under this cubic term by itself. The fraction of the cubic source's squared norm above $K$ is $0.1$ at $K=1$, approximately $0.05331$ at $K=8$, $0.007562$ at $K=128$ and $0.0005248$ at $K=2048$. These are forcing fractions, not decay rates or fractions of radiated energy.

The existing complementary-mode resolvent provides the correct framework for the forced corrections. Whether they remain bounded, shift the prepared waveform or reach outgoing channels depends on the complete same-field action and its spectrum. That completion retains the supplied mass-frequency endpoint as the target; this calculation identifies the sources that must be accommodated around the proposed preparation.

10. What the Combined Calculation Establishes

The sawtooth hypothesis gives a concrete mechanism for a sector fundamental to remain prominent in coupling while the upper mass harmonic changes substantially. The same temporal calculation can be inserted into spatial evaluation, reciprocal phase forces, color contractions and geometric shape response. It produces conditional relationships without assigning a separate coefficient to each particle.

It does not by itself prefer the maintained spatial depths: the temporal identities apply at any integer $K$. The dimensional content comes from the spatial carrier, invariant channel, sector scale and receiver map. A physical completion must establish the occupied harmonic pattern, its spatial evaluations, the common kinetic normalization and the interaction term. The distinction between density and derivative coupling is now explicit enough to test that completion rather than assume its outcome.

Finite-Gap Action Completion

The mixed-curvature action supplies a positive same-field mode realization of reciprocal velocity coupling. It generates induced inertia as well as the vertex. Its exact complete response retains small depth-five source sums at four common gaps, while the charged-lepton response spread depends on that gap. The earlier constant-vertex table and its closer lepton similarity remain results for their stated diagnostic action; they cannot be transferred unchanged to the finite-gap completion. The density-to-pattern source map, actual spatial modes and their occupied gap remain physical premises to derive.

Verification

One Python script and its report preserve 2,023 passing checks. Exact rational identities are compared with independent Fourier transforms; pair formulas with time integration; spatial sources with nonunitary basis changes; the tetrahedral spectrum with an exact generalized Hessian. Finite-encounter checks retain unequal fundamentals and phases.

The checks include all six maintained depths in the assigned massive-branch comparison, resolved angular dimensions 2 through 12, and algebraic color-rank controls through depth 16. The report records the frozen research-source hashes, executed script hash, scale inputs, cutoffs and omitted-tail bounds. Passing these checks verifies the stated calculations, not an empirical identification of the candidate interaction.

Geometric Interaction-Order Candidate

A common raw-Hodge contact hypothesis now selects position response on even middle-degree channels and velocity response on odd ones. Its positive reciprocal mode reduction uses the same coefficient choices across sectors and retains the slow depth-five source suppression under the stated preparation. This is a candidate explanation of the interaction-order choice, with a specified kernel and time-reversal structure; it does not establish charge or observed rates. The companion finite-gap-induced action remains a different completion with its own inertia and response terms.

Passive Delay and Nonlinear Volume Comparison

The passive Hodge-path calculation inserts a bounded parity-sensitive transfer into the same density harmonic sums. Both outputs conserve energy. A common receiver delay retains slow neutrino suppression, whereas a source-relative inverse-fundamental delay removes it. A separate exact sixth-moment test shows why the instantaneous color-volume zero cannot be transferred to a delayed replacement. These are common-law controls with fixed source cutoffs, not fitted particle charges or interaction rates.

Positive-Frequency Density Response Has Been Compared

The complex-comb response calculation now evaluates the actual coefficients already defined in section 2 with the same receiver and scale inputs. An exact finite derivative norm and proved variance bounds avoid large harmonic simulations. Small depth-five derivative response and charged-lepton direct-response similarity survive. The signed lepton derivative response changes sign, and the complex down preparation has a constant density. Equal fundamental beats therefore do not establish interchangeable physical coupling sources.

Square Harmonics and the Fundamental Source

An exact support and phase comparison shows when the fundamental source survives. Odd-only square harmonics produce only even quadratic beats, and ten of the fourteen adopted massive cutoffs omit their mass endpoint under a strict square preparation. A labeled endpoint repair and bounded all-sector response are preserved separately. Coherent adjacent harmonics retain the full sawtooth fundamental, while the local spatial readout remains essential.