Mixed Spatial-Mode Curvature, Induced Inertia and Spectral Response

A changing complex spatial-mode frame can generate a reciprocal velocity interaction. The same eliminated modes also generate inertia; keeping both gives a different high-frequency response from retaining a constant velocity vertex alone. This is a conditional construction from a stated positive mode action. Its physical mode functions, occupation and gap still need derivation from the common spatial field law.

1. Curvature Is the Relevant Geometric Quantity

Let two actual orthonormal spatial modes of the one field have normalized coefficient frame

$$U(q,\lambda)=\left(\cos\frac{\theta(q)}2, e^{i\phi(\lambda)}\sin\frac{\theta(q)}2\right).$$

The Berry connection and mixed curvature are

$$A=iU^\dagger dU=-\sin^2(\theta/2)d\phi, \qquad F_{q\lambda}=-\frac12\theta'(q)\phi'(\lambda)\sin\theta.$$

Under $U\mapsto e^{i\chi}U$, $A\mapsto A-d\chi$ and $F$ is unchanged. A constant connection alone can be removed locally. A real normalized line has zero Berry curvature. Thus a local expression involving a connection value and changing density is insufficient without its complete phase and continuity dynamics.

The coordinates $q,\lambda$ describe evolving spatial patterns; they add no physical spatial dimensions. In particular the constant total modal action $I$ below is distinct from a changing local density beat.

Locally the curvature gives a gyroscopic term

$$L_B=\frac g2(\lambda\dot q-q\dot\lambda).$$

With positive bare inertias and stiffnesses, the equations are

$$M_q\ddot q+K_qq=-g\dot\lambda,\qquad M_\lambda\ddot\lambda+K_\lambda\lambda=g\dot q.$$

They exchange energy reciprocally and conserve the positive mechanical energy. In canonical inertia units their frequencies obey

$$(\omega_q^2-\Omega^2)(\omega_\lambda^2-\Omega^2) -g_c^2\Omega^2=0,\qquad g_c=g/\sqrt{M_qM_\lambda}.$$

Eliminating one coordinate supplies an extra $\Omega^2$ in the response. A two-component first-order frame alone has only one canonical pair; it does not derive two additional independent inertias. Those must come from a stated reduction such as the following.

2. A Positive Fast-Mode Completion

Take a retained slow action $L_{\rm bare}$ with positive quadratic inertia and restoring energy. Introduce one eliminated complex mode $c$ of the same field with gap $\Delta>0$:

$$L=L_{\rm bare}+\frac i2(c^*\dot c-\dot c^*c) -\Delta|c-\sqrt I\,w|^2,\qquad w=(a q+i b\lambda)/2.$$

This assumes a particular relation between retained and eliminated spatial modes, rather than an independent interaction particle. The full Hamiltonian is positive. With Fourier convention $e^{-i\Omega t}$,

$$i\dot c=\Delta(c-\sqrt Iw),\qquad c(\Omega)=\frac{\Delta\sqrt I}{\Delta-\Omega}w(\Omega).$$

Its exact induced quadratic term is $I\Delta\Omega|w|^2/(\Delta-\Omega)$. At low frequency it yields

$$L_{\rm induced}=\frac{iI}{2}(w^*\dot w-\dot w^*w) +\frac I\Delta|\dot w|^2+\cdots,$$

$$g=\frac{Iab}{2},\qquad \delta M_q=\frac{Ia^2}{2\Delta},\qquad \delta M_\lambda=\frac{Ib^2}{2\Delta},\qquad \boxed{|g|/\sqrt{\delta M_q\delta M_\lambda}=\Delta.}$$

Curvature and induced inertia cannot be tuned independently in this action. If all inertia is induced, a high root of the truncated adiabatic theory can reach the eliminated gap. The complete system must then be retained.

Put $c=(x+iy)/\sqrt2$, $A=\sqrt{I/2}a$, and $B=\sqrt{I/2}b$. The fast energy is

$$H_f=\frac\Delta2[(x-Aq)^2+(y-B\lambda)^2].$$

The six canonical variables $(q,p_q,\lambda,p_\lambda,x,y)$ form three oscillators. Direct elimination gives

$$\delta M_q(\Omega)=\frac{\Delta A^2}{\Delta^2-\Omega^2},\qquad \delta M_\lambda(\Omega)=\frac{\Delta B^2}{\Delta^2-\Omega^2},\qquad g(\Omega)=\frac{\Delta^2AB}{\Delta^2-\Omega^2}.$$

These are exact frequency-dependent Schur terms. Their apparent bare-gap poles must be assessed together in the full response.

3. The Complete Source Response

Prescribe $\lambda$ and eliminate $q,x,y$. Let $K,M$ be the bare receiver stiffness and inertia. The induced source response is

$$\boxed{R(\Omega)= \frac{\Delta B^2\Omega^2(K-M\Omega^2+\Delta A^2)} {(K-M\Omega^2)(\Delta^2-\Omega^2)-\Delta A^2\Omega^2}.}$$

It contains both the direct induced source inertia and receiver-mediated coupling:

$$R=\delta M_\lambda\Omega^2+ \frac{g(\Omega)^2\Omega^2}{K-[M+\delta M_q(\Omega)]\Omega^2}.$$

The limits are

$$\frac{R(\Omega)}{\Omega^2}\longrightarrow \frac{B^2}{\Delta}+\frac{A^2B^2}{K}\quad(\Omega\to0),\qquad R(\Omega)\longrightarrow-\Delta B^2\quad(|\Omega|\to\infty).$$

For $A\ne0$, the apparent bare-gap singularity is removable:

$$R(\Delta)=\frac{B^2(\Delta^2M-K-\Delta A^2)}{A^2}.$$

Actual shifted normal-mode poles remain. The high-frequency constant comes from the complete response; inspecting the decaying $g(\Omega)$ alone misses it. This signed conservative quadratic response is neither electric charge nor a transition rate. No damping or outgoing-wave condition is being inferred.

4. One Common Comparison Across the Sectors

Use the finite sawtooth source, including its exact coefficients $c_p(K)$, maintained count $K=S(n,d)$ and ratio $r=\omega_{0,d}/\omega_{0,6}$. The source map $\lambda=f_K^2$ is a conditional preparation, not a derived identity between density and pattern coordinate. Hold its orientation fixed and average the same receiver stiffnesses $2,3$ with weights $3/5,2/5$:

$$Q_K(r)=\frac12\sum_{p=1}^{2K}c_p(K)^2 \left[\frac35R(pr;K_q=2)+\frac25R(pr;K_q=3)\right].$$

The notation $K_q$ in this expression is receiver stiffness, distinct from the harmonic cutoff $K$. All cases use $M=1$, $A=B=1/10$ and common gaps $13/10,37/10,10,30$. These dimensionless diagnostic parameters are not particle fits. The source note defines the electron-based non-neutrino scale prescription and separate neutrino splitting calibration retained here.

Representative complete-response sums at common gap $\Delta=10$ are:

All three depth-five branches remain small at all four tested common gaps. Their sums agree within one percent with the low-frequency estimate

$$Q_K(r)\simeq\left(\frac{B^2}{\Delta}+\frac{13A^2B^2}{30}\right) r^2\left(2K-H_K-\frac12H_K^{(2)}\right).$$

This survives the full fast-mode completion without a depth-five coefficient. It depends on the calibrated slow fundamentals and the specified source and receiver, rather than proving a unique five-dimensional effect.

Charged-lepton similarity depends on the common gap. The computed tau/electron response spread is about $0.0245\%$, $0.238\%$, $1.16\%$ and $12.1\%$ at the four gaps respectively. Partial-sum tail bounds are recorded in the report; these spreads are not exact charge ratios or empirical predictions. The closer similarity in the earlier constant-vertex diagnostic cannot be transferred unchanged to this action. The shifted spectral poles and direct induced term matter.

For large cutoffs, only the first $J=65536$ source harmonics are evaluated. If the omitted frequencies satisfy $\Omega^2\ge2(K_q/M+\Delta^2+\Delta A^2/M)$, the rational response obeys $|R|\le3\Delta B^2$. The existing coefficient-tail estimate therefore gives

$$|\Delta Q_K|\le3\Delta B^2 \left(\frac{4\pi^2}{K}+\frac1{3J^3}\right).$$

This is a truncation bound, not a complete floating-point uncertainty estimate.

5. Paired Responses and a Related Spectral Method

The paired-mode continuation checks two equal positive fast bands with opposite pattern maps. Their gyro cross-response cancels at every frequency, while their induced inertias and direct source responses add. Unequal gaps or occupations leave residuals. The same continuation shows why zero Berry curvature does not establish a color singlet.

A related general adiabatic method derives both Berry forces and an inverse-gap-weighted inertia tensor; see D’Alessio and Polkovnikov, equations 15–27. The independently checked finite-mode completion here is a conditional application of that kind of geometry, with its exact complement retained. Its collective inertia is not the maintained particle mass-frequency spectrum. The attributed external formula is preserved separately in sources/external/spectral-inertia-geometric-forces-2014.md.

6. Physical Completion and Reproducibility

The same complex spatial-mode geometry generates curvature, velocity interaction, induced inertia and a calculable spectral response. Its existence at every sufficiently rich spatial depth means that dimensional selection must enter through actual mode functions, their sector embeddings, occupation and source maps. Rotating source directions adds orientation derivatives absent from the fixed-direction comparison. Their complete dynamics is still required.

The single checker passes 281 conditions across 71 cases. Its report records exact executed hashes, full-versus-reduced Hamiltonian comparisons, phase-gauge covariance, all maintained sectors, common-gap controls and tail bounds. These checks establish the stated conditional action results; they do not derive the action from the IDWT master law.

The actual spherical evaluation-band calculation now supplies explicit spatial mode functions, their curvature and quantum tensor. It is a concrete spatial control for the abstract two-mode geometry above. Its natural full-spinor connection does not directly supply an independent color triplet or an electric-charge line, and applying it to inertia requires the actual complement gaps.

Testing the Generated Clock at Actual Carrier Frequencies

The ordinary gyroscopic-clock continuation retains this positive action's three exact branches at every supplied nonzero-wave-number endpoint. A constant low-frequency gyro coefficient does not imply a constant carrier-branch gap above the eliminated band. A common far-band ratio approaches the intended clock, while a ratio of one hundred fails for high carriers. Full symplectic residues and complete mode energy are checked; a nearly universal partial vertex alone does not establish the required fundamental frequency.