Projected Electron Orbits in the 6D Sector

The proposal begins with a moving electron bound to a nucleus by Coulomb attraction. Its motion can also rotate through the containing spatial directions. A three-dimensional observer then resolves a changing part of that motion: the displayed radius contracts and expands while the angular position advances, creating a ring or petals.

Here that idea has an exact mathematical realization. One pair of coefficients exchanges amplitude at rate \(L\omega\); another supplies a relative-phase clock at \(\omega\). Their selected position response generates the moving curve. Changing \(L\) changes the petal count. Imbalance prevents complete amplitude exchange, opening the planar center crossings. Original and Smooth test how the same motion looks under two response laws.

This is a concrete candidate for what an electron's motion could look like. The equations below make the picture reproducible and expose the physical test: whether one electron–nucleus interaction produces the selected motion, its rate and the measured response.

How the construction developed

Earlier versions assigned a lock between a six-dimensional orbit and added precession. The present norm-preserving coefficient flow replaces that rotation recipe; it is not simply the same calculation renamed or a numerical integration of Coulomb forces plus a derived torque. Its explicit spatial bridge is given below. The physical source of the generator and readout remains to be established.

IDWT proposes that the electron lives on the six-real-dimensional sector manifold

In this candidate geometry, a sector position is a ray \(q=[Z]\), with \(Z\in\mathbb C^4\setminus\{0\}\) and \(Z\sim\lambda Z\). The homogeneous lift \(Z\), the point \([Z]\), and the electron's spatial Dirac field \(\psi_e(q)\) are distinct objects. The six dimensions here are ordinary spatial dimensions, not six coefficient labels. The map below connects a proposed spatial trajectory to the display; a physical field equation and its coupling to an apparatus must still justify that map. The global topology and metric scale of this spatial candidate are additional premises.

An orbit is the Hamiltonian trajectory followed by the electron state. An orbital is a stationary spatial field or density assigned additional quantum numbers and a radial profile. The canvas renders one projected orbit curve \(\Gamma_L\). Its ring and lobe counts are associated with the conventional angular labels \(L=0,1,2,3\); they are candidate skeletons from which a finite registration distribution can be built by an observer response. Petal count alone does not establish the full atomic states, their energies or their transition strengths.

At degree \(L\geq1\), the magnetic line bundle has section space \(H^0(\mathbb{CP}^3,\mathcal O(L))\). The orbit construction selects a four-state subspace \(\mathcal V_L\) and writes a coefficient ray \([c]\in\mathbb P(\mathcal V_L)\). This coefficient ray is the orbit-state variable; \([Z]\in Q_6\) remains the electron-sector coordinate.

For a Hermitian \(4\times4\) matrix \(M\) acting on \(\mathcal V_L\), the expectation value

\[ H([c])=\frac{c^\dagger Mc}{c^\dagger c} \]

is well defined on projective space. In an orthonormal coefficient basis, with unit Fubini–Study normalization, frequency units for \(M\), and \(\iota_{X_H}\omega_{\rm FS}=dH\), it generates the Hamiltonian flow

\[ [c(t)]=[e^{-iMt}c(0)]. \]

The evolution is norm preserving, and \(H\) is constant because \(M\) commutes with its own exponential. For the angular family, choose the traceless generator and initial point

With \(\theta=\omega t\), exponentiation is elementary:

The \(c_1,c_2\) block supplies an \(L\)-fold amplitude exchange. The \(c_3,c_4\) pair supplies the primitive clock through a relative phase. At \(\theta=2\pi\), all four components differ from their initial values by the common factor \(-i\), so the projective trajectory closes after one turn.

Define a nonnegative radius and an angular clock by

These functions are defined where \(|c_1|^2+|c_2|^2>0\) and \(c_3c_4\ne0\), conditions preserved by the stated preparation. Under \(c\mapsto\lambda c\), the radius numerator and denominator both scale by \(|\lambda|\), and the clock numerator and denominator both scale by \(|\lambda|^2\); both functions are unchanged. Along the flow, \(r=|\cos(L\theta)|\) and \(e^{i\phi}=e^{i\theta}\). The observer-aligned plane map is therefore

\[ X+iY=Rr([c])e^{i\phi([c])} =R|\cos(L\theta)|e^{i\theta}. \]

The integer \(L\) is the Chern degree of the magnetic line bundle. Degree-\(L\) monomials in the four homogeneous coordinates have exponents \(a,b,c,d\geq0\) with \(a+b+c+d=L\). Stars-and-bars gives

\[ N_L=\binom{L+3}{3}=1,4,10,20,\ldots\quad(L=0,1,2,3,\ldots). \]

This counts the full degree-\(L\) projective multiplet. The ring is the separate \(L=0\) completion; \(2,4,6\) count the isolated radial maxima of the selected \(L=1,2,3\) readouts. A ring has constant radius, not one isolated maximum.

The mass construction uses the same symmetric-polynomial counting machinery. For \(d\) oscillator directions, the number of degree-\(N\) monomials is \(\binom{N+d-1}{d-1}\). Summing through \(N=n-1\) gives the Hockey-Stick identity:

\[ S(n,d)=\sum_{N=0}^{n-1}\binom{N+d-1}{d-1} =\binom{n+d-1}{d},\qquad m=m_{{\rm scale},d}\,S(n,d). \]

One mathematical language, two roles. The mass formula uses a cumulative mode count; this orbit construction uses weights within a polynomial representation to organize motion. The connection motivates a common physical model: counting, integer labels and symmetry all have explicit mathematical meanings. Here \(L\) is the orbit's bundle degree, while \(n\) is the mass-family index and \(d\) its proposed spatial depth. The identity \(N_L=S(L+1,3)\) is combinatorial; it does not make the six-real-dimensional \(\mathbb{CP}^3\) a depth-three particle sector or identify \(L\) with an electron mass index.

A common field action must still select the occupied modes, turn the relevant count into the physical mass-frequency law, and produce the orbit generator and its readout. The shared structure is a reason to investigate that connection, not yet proof that both constructions arise from one observed mechanism.

The following construction places the exchange and clock in one symmetry action, explaining their frequency ratio under the stated assumptions. Identifying that action with the atom's physical electromagnetic field is a further step; the geometry alone does not derive the field strength or precession rate.

Electromagnetism is a \(U(1)\) connection on the full manifold, with curvature components \(F_{AB}\). In a local orthonormal frame, the leading orbital coupling to a locally uniform field contains

\[ H_{\rm mag}=-\frac{q}{2m_e}\sum_{A<B}F_{AB}L^{AB}. \]

Here \(q\) denotes electric charge and \(L^{AB}=x^Ap^B-x^Bp^A\). Expanding \((p-qA)^2/(2m_e)\), with \(A_A=-F_{AB}x^B/2\), gives this linear term. A full spatially varying field also requires its actual profile, the quadratic \(A^2\) term and any spin coupling with its own normalization. Access to shared spatial directions permits contact, but the polarized field patterns and their interaction determine its strength; dimensional overlap alone does not derive a coupling.

Hopf flux supplies the integer level

For normalized \(Z\in S^7\) over \(\mathbb{CP}^3\), the canonical connection and curvature are

\[ \mathcal A=-iZ^\dagger dZ,\qquad \mathcal F=d\mathcal A,\qquad \frac{1}{2\pi}\int_{\mathbb{CP}^1}\mathcal F=1. \]

The degree-\(L\) bundle consequently has \(c_1(\mathcal O(L))=L\). Homogeneous polynomial combinatorics gives its exact representation

\[ H^0(\mathbb{CP}^3,\mathcal O(L)) \cong\operatorname{Sym}^L(\mathbb C^4)^*,\qquad \dim=\binom{L+3}{3}. \]

The central Hopf \(U(1)\) supplies this integral flux level. The electron has charge \(-e\), while the independent integer \(L\) labels the background bundle and its state representation.

One isometry supplies the whole matrix

Choose one level-independent \(U(1)\) isometry of the four fundamental coordinates,

\[ G=\operatorname{diag}(1,-1,-1,0). \]

This is a Hamiltonian isometry of \(\mathbb{CP}^3\). Its moment map and classical flow are

\[ \mu_G([Z])=\frac{Z^\dagger GZ}{Z^\dagger Z},\qquad [Z]\longmapsto[e^{-i\theta G}Z]. \]

With \((2\pi)^{-1}\int_{\mathbb{CP}^1}\omega_{\rm FS}=1\), the prequantum curvature is \(-iL\omega_{\rm FS}\), the level-\(L\) moment map is \(L\mu_G\), and we use the forward-pullback convention \(P(Z)\mapsto P(e^{-i\theta G}Z)\) on degree-\(L\) polynomials. Its Hermitian differential generator is

\[ \widehat G_L=\sum_{j=1}^4g_jZ_j\frac{\partial}{\partial Z_j}, \qquad \widehat G_LZ^a=(a\cdot g)Z^a. \]

The active left action on the dual section space is instead \(P(Z)\mapsto P(e^{i\theta G}Z)\), with the opposite generator. Forward evaluation agrees with the coherent-ket bridge below; using the dual convention consistently reverses the oriented motion. The central Hopf circle supplies the flux level, while this base isometry supplies the nontrivial motion; they are distinct actions. A central shift \(G\mapsto G+cI\) adds the same \(cL\) to every degree-\(L\) state and only changes the common projective phase.

For every \(L\geq1\), four degree-\(L\) monomials have the following induced weights:

State	\(Z_1^L\)	\(Z_2^L\)	\(Z_3Z_4^{L-1}\)	\(Z_4^L\)
Weight	\(+L\)	\(-L\)	\(-1\)	\(0\)

Normalize the four monomials in the invariant section norm; in particular the third has a relative factor \(\sqrt L\). Taking the normalized even and odd combinations of the first pair rotates \(\operatorname{diag}(L,-L)\) into \(L\sigma_x\). Therefore, on this four-state subspace,

\[ G_L\big|_{\mathcal V_L} =L\sigma_x\oplus\operatorname{diag}(-1,0),\qquad M_L=\omega\left(G_L\big|_{\mathcal V_L}+\tfrac14I\right). \]

This reproduces the projective generator exactly for every \(L\geq1\). The scalar \(I/4\) fixes the displayed energy zero on each four-state subspace. The amplitude exchange and clock are weights of one group action with the shared parameter \(\theta=\omega t\), which gives the frequency lock.

What selects this generator?

Let a level-independent diagonal action have weights \((g_1,g_2,g_3,g_4)\). In addition to a disjoint primitive clock, impose unit signed amplitude weights relative to its reference. These explicit weight assumptions give

\[ g_1-g_4=1,\qquad g_2-g_4=-1,\qquad g_3-g_4=-1. \]

The constraint matrix has rank three; its null direction is \((1,1,1,1)\), the invisible central phase. Thus the stated weights determine \(G=cI+\operatorname{diag}(1,-1,-1,0)\), up to that shift and equivalent choices of orientation and labels. In the clock plane, \(Z_3^kZ_4^{L-k}\) has relative weight \(-k\), so primitive unit winding uniquely gives \(k=1\).

A primitive clock alone leaves the integer family \(G_a=\operatorname{diag}(a,-a,-1,0)\), \(a=1,2,\ldots\). The same construction then gives \(2aL\) petals. Unit amplitude charge is an additional selection premise. One possible selector is minimum traceless generator norm:

\[ \left\|G_a-\tfrac14\operatorname{tr}(G_a)I\right\|_F^2=2a^2+\tfrac34. \]

Its unique minimum in this family is \(a=1\). Deriving that minimization from the physical action would turn this choice into a dynamical result.

The full unitary-equivalence family is the conjugacy orbit \(UGU^\dagger\). After subtracting its mean trace, the Hermitian generator multiplied by \(-i\) lies in \(\mathfrak{su}(4)\); the equivalence \(SU(4)\simeq\operatorname{Spin}(6)\) makes this the six-dimensional rotation orbit when the four coordinates carry that spinor action. An isotropic atom may then preserve the family while preparation or apparatus alignment selects a representative. Within the unit-charge extremal construction and its selected readout, a positive Chern degree \(k\) produces \(2k\) petals, so matching the family labelled by \(2L\) gives \(k=L\). This is a conditional matching rule, not a statement about all states in that bundle. Negative flux needs the conjugate polarization; \(H^0(\mathbb{CP}^3,\mathcal O(k))=0\) for \(k<0\).

The nucleus supplies the candidate frame lock

At electron–nucleus contact, the nuclear depth-three structure selects a split \(V_6=O\oplus H\cong\mathbb R^3\oplus\mathbb R^3\). The electron-sector point \([Z]\) determines an orthogonal complex structure \(J([Z])\), written relative to that split as

\[ J=\begin{pmatrix}A&Q\\-Q^T&C\end{pmatrix}. \]

The invariant

\[ s([Z];P)=\frac12\|A\|_F^2=\frac12\|C\|_F^2 \]

lies in \([0,1]\) and measures departure from the exchanging lock. Indeed \(J^2=-I\) gives \(QQ^T=I+A^2\), with singular values of \(Q\) equal to \(1,\sqrt{1-s},\sqrt{1-s}\). This split is selected by the nuclear reference plane, not an inherent division of the electron's dimensions. At \(s=0\), \(Q\in O(3)\) identifies the two three-frames and leaves the diagonal \(SO(3)\). A specified projected electron–nucleus kernel determines the coefficient \(\kappa=(dV_N/ds)_{s=0}\); \(\kappa>0\) makes this family locally energy-favored. That is not a capture mechanism: under a conservative Hamiltonian depending only on \(s\), \(s\) itself is conserved. Reaching the lock and selecting a generator orientation require additional dynamics or preparation.

Observer-space magnetic scale

A static Coulomb source has no magnetic field in its own rest frame unless a separate magnetic source is specified. In the instantaneous moving electron frame, the leading transformed field is \(\mathbf B'=-\mathbf v\times\mathbf E/c^2\), giving the magnitude \(B_{\rm mot}=vE/c^2\) for perpendicular motion. Using the Coulomb circular-orbit identities \(|e|E=m_ev^2/r\), \(\omega_{\rm orb}=v/r\), and \(v/c=\alpha/n\), a Larmor-rate comparison gives

Using the 2022 CODATA value \(\alpha=0.0072973525643\), this ratio is \(2.66257\times10^{-5}\) at \(n=1\) and falls as \(n^{-2}\). The earlier IDWT benchmark \(2.88\times10^{-5}\) corresponds to a different input, \(\alpha\simeq0.00758947\). This is a conditional circular-orbit scale comparison; a spin-precession calculation must also include the magnetic moment and Thomas frame rotation. It is too small to supply the displayed order-one lock under these assumptions. Additional curvature planes are a candidate source, not a consequence proved by this estimate.

For \(L>0\), assuming \(B\pi r^2=2\pi L\hbar/|e|\) and \(mvr=L\hbar\) gives \(|e|B/(2m)=v/r\). This equality is algebraically exact. Identifying flux through an open orbit disk with the Chern flux of a closed projective cycle is an extra physical premise that the bound-state action must supply.

Status: exact representation construction; physical source and occupation law open. The chosen group action realizes \(M_L\) and its block frequencies exactly. Matching the selected positive-degree magnetic and angular constructions gives \(k=L\) conditionally. A physical electromagnetic current producing that action, the bound-state selection of \(L\), the electromagnetic identification of the Hopf curvature, and the dimensional coefficient \(\omega\) remain to be derived.

The coefficient equation can be connected explicitly to motion of a point on the proposed spatial \(\mathbb{CP}^3\). With the generator \(G\) above, follow \([Z(\theta)]=[e^{-i\theta G}Z_0]\). The normalized coherent degree-\(L\) lift \(Z^{\otimes L}\) contains all symmetric monomials. Select four of its normalized components:

With \(r_0=(A/B)^{1/L}\), the preparation

\[ Z_0=\frac{(r_0,r_0,1/\sqrt L,1)} {\sqrt{2r_0^2+1/L+1}} \]

gives exactly \(c(0)=(A,0,B/\sqrt2,B/\sqrt2)\). Along this trajectory, the four selected components gain phases \(e^{-iL\theta},e^{iL\theta},e^{i\theta},1\). The even/odd basis change therefore produces the displayed exchange and clock, up to their irrelevant common phase. These are coherent-ket components in \(\operatorname{Sym}^L(\mathbb C^4)\); evaluation of the dual polynomial sections gives the same numbers in the stated forward convention, without identifying the two spaces.

The map is homogeneous of degree \(L\), so its ray is independent of the representative \(Z\). For \(L\ge2\), it is undefined at \([0:0:1:0]\), where all four selected components vanish. The radius and clock have the additional nonzero-denominator conditions stated above; the specified trajectory avoids these exceptions.

What the bridge establishes. One explicit motion in the six-dimensional spatial candidate produces the coefficient orbit and the picture through a specified nonlinear readout. For \(L>1\), the coherent lift also has occupied components outside these four. At \(A=0.6,B=0.8\), the retained squared norm is \(1,\ 0.347222,\ 0.117577\) for \(L=1,2,3\). A physical detector must explain that selection; interpreting it as a quantum projection additionally requires its postselection weight. This normalized component map is not automatically a symplectic identification or a linear spatial projection.

What would distinguish six dimensions? The two \(-1\) weights make \(Z_2/Z_3\) constant along each prepared ideal trajectory. It consequently lies in a fixed \(\mathbb{CP}^2\) inside \(\mathbb{CP}^3\); the same restricted readout gives the same picture. More general perturbations can leave that subspace. Deriving accessible perturbations and a calibrated response to those extra directions would test the full spatial geometry more sharply than the unperturbed petals alone. This bridge proves the ideal case; lifting the coefficient perturbations below to a common spatial interaction is a further problem, especially for \(L>1\).

Write the amplitude occupation and primitive clock as

\[ q=\frac{|c_1|^2}{|c_1|^2+|c_2|^2},\qquad e^{i\phi}=\frac{c_3\bar c_4}{|c_3c_4|}. \]

Within the assumed readout family depending only on \(q\) and this clock, rotational covariance gives \(X+iY=Rh(q)e^{\pm i\phi}\). Projective invariance alone does not exclude dependence on other state invariants. Reflection symmetry across the aligned apparatus axis makes the radial response real, giving

\[ X+iY=Rf(q)e^{\pm i\phi}. \]

If \(f(0)=0\) and \(f\) is strictly increasing, it preserves the maxima and zeros of \(q=\cos^2(L\theta)\), so every such detector response has exactly \(2L\) lobes. The original choice \(f(q)=\sqrt q\) follows from the additional intensity rule \((X^2+Y^2)/R^2=q\). The Smooth setting instead uses \(f(q)=q\). Both keep exactly the same state motion and lobe count while changing the metric shape; neither detector law has yet been selected by the physical coupling.

Regularity matters. At a center crossing \(\theta_0=(k+1/2)\pi/L\), the original plane curve has \(\Gamma(\theta_0+h)=RL|h|e^{i\theta_0}+O(h|h|)\). Its two one-sided velocities differ by \(2RL\omega\). The smooth coefficient motion has acquired a corner through the modulus readout. A smooth spatial projection cannot do this. The original curve is an idealized registration map, not yet a regular point-charge path; the Smooth response removes this particular problem.

The two display choices are written together as

\[ \rho_\beta=Rq^\beta,\qquad \beta=\tfrac12\ \text{(Original)},\quad \beta=1\ \text{(Smooth)}. \]

Uniform Hamiltonian time gives the invariant occupation measure \(d\theta/(2\pi)\). For \(L\geq1\), its amplitude fraction has density \([\pi\sqrt{q(1-q)}]^{-1}\), and the original radius has density \(2/[\pi\sqrt{R^2-\rho^2}]\). In the Smooth setting, \(\rho=Rq\) has density \(1/[\pi\sqrt{\rho(R-\rho)}]\). These are radius densities on \(0<\rho<R\), not densities per unit area or arc length. The ring instead has constant radius \(R\), so its radial measure is concentrated there. If detections sample this motion uniformly in time and the apparatus has a translation-invariant spatial response, their distribution is the convolution of that time measure with the response. Those are measurement assumptions, not observations of the electron's physical shape.

The \(L=0\) ring and its extra clock

At \(L=0\), \(H^0(\mathbb{CP}^3,\mathcal O(0))=\mathbb C\), whose projectivization is a point and whose relative-phase dimension is zero. With a separate constant-amplitude coordinate and a disjoint two-state clock, a ring completion is

\[ M_0^{\rm min}=\omega\operatorname{diag}(0,-1,0),\qquad c_0^{\rm min}(0)=\left(A,\frac{B}{\sqrt2},\frac{B}{\sqrt2}\right), \]

This gives the ring exactly. Three coordinates are minimal only for that separate amplitude/clock architecture. A two-state clock \(u(t)=(e^{i\omega t},1)/\sqrt2\) already gives \(Ru_1\bar u_2/|u_1u_2|=Re^{i\omega t}\); the degree-zero section space itself provides neither extension. The physical origin of the universal clock doublet and its zero ordinary-angular-momentum expectation are closure conditions for the bound-state and observer couplings.

The controls test which dynamical conditions preserve the center-crossing petals. They alter a moving state and its position readout; a stationary orbital is not substituted for the trajectory.

After removing the common phase, the ideal generator is \(L\omega\sigma_x\oplus\operatorname{diag}(-\omega,0)\). Keep the same observer axes and initial state, but test the Hermitian perturbation family

Here all four parameters have frequency units, with \(\hbar=1\). The term \(\delta\) is the imbalance between the amplitude states, and \(\epsilon\) is their common offset relative to the clock pair. No mixing between the two blocks or leakage from \(\mathcal V_L\) is included. Direct exponentiation gives

Complete exchange gives the center crossings

This radius formula is for the Original response. For either response, \(\rho_{\min,\beta}=R(\delta^2/\Omega^2)^\beta\). For the stated preparation, exact planar center crossings require \(\delta=0\). A nonzero imbalance leaves radial modulation but opens a central hole in the plane curve. Robustness under a monotone observer response is therefore different from robustness under a changed Hamiltonian. Rotating away the imbalance also rotates the physical readout axes and cannot be treated as the same observation.

Preparation matters separately. At \(\delta=0\), a general pure amplitude pair with Bloch vector \((s_x,s_y,s_z)\) has

The demo's initial pair \((1,0)\) satisfies \(s_x=0\), allowing full exchange. Unitary evolution conserves \(s_x\); it does not by itself attract arbitrary preparations to that orbit.

A closed image is not necessarily a closed state orbit

For \(\epsilon=0\), full recurrence of the stated coefficient ray requires both \(\Omega T\) and \(\nu T\) to be integer multiples of \(2\pi\). A rational ratio \(\Omega/\nu=p/q\), in lowest terms, gives \(T=2\pi q/\nu\). An irrational ratio does not close exactly. The plane readout loses the sign of the amplitude pair: \(g/\nu=3/2\) with zero imbalance gives a closed three-maximum image after one clock turn, but the state itself needs two turns.

The alignment buttons retain \(g/\nu=L\) and set \(\delta/\nu=\sqrt{L+1/4}\) for one extra radial maximum, or \(\delta/\nu=\sqrt{2L+1}\) for two. The resulting exchange rates are \(L+1/2\) and \(L+1\). For \(L=1\), the required imbalance for the first alignment is \(\sqrt5/2\simeq1.118\). These aligned curves still have nonzero minimum radius: alignment does not restore their center crossings. With zero relative offset, the first returns the full state after two turns and the second after one.

The offset \(\epsilon\) is also invisible to the displayed coordinates, including the depth readout. Nevertheless, full recurrence requires

For example, \(g/\nu=L\), \(\delta=0\), and irrational \(\epsilon/\nu\) preserve the displayed petals while preventing exact state recurrence. This block-relative phase is not a removable common phase.

Sharper dynamical target. The ideal orbit has \(\delta=\epsilon=0\), \(g/\nu=L\), and \(s_x=0\). The selected representation with unit amplitude charge supplies those generator weights; the projected action must select and preserve that representation and orbit, including relative phases that the observer map does not resolve. These conditions sharpen the existing orbit construction rather than replace it with orbitals.

At the default ideal settings, the view looks along the apparatus-selected depth axis and displays the plane theorem exactly. Dragging rotates this three-dimensional observer embedding; it does not rotate or expose all six spatial coordinates directly. For the Original response, the ideal depth coordinate is

\[ Z_{\rm obs}=\eta R\frac{|c_2|}{\sqrt{|c_1|^2+|c_2|^2}}\cos(3\phi) =\eta R|\sin(L\theta)|\cos(3\theta), \]

The traceless spinor weights \((5,-3,-3,1)/4\) have exterior-square weights \(\pm1/2,\pm1/2,\pm3/2\). Thus the six-vector rotation has plane-rate ratio \(1{:}1{:}3\). A quadratic response on its fastest plane can supply a \(\cos(3\phi)\) harmonic. That explains an available harmonic, not a unique derivation of the entire depth formula. The apparatus coupling would have to select the three resolved axes, the slow and fast readout planes, and the scale \(\eta\). The Depth control currently sets this uncalibrated display parameter; its default is \(\eta=0.64\). The formula preserves the exact \(X,Y\) lobe count. For both readout settings and the perturbed motion, use

\[ Z_{{\rm obs},\beta}=\eta R(1-q)^\beta\cos(3\phi),\qquad 1-q=\frac{g^2}{\Omega^2}\sin^2(\Omega t). \]

The Original depth response has further modulus corners where its amplitude vanishes. The Smooth choice \(\beta=1\) makes all three coordinates smooth in time along this motion. Its planar velocity vanishes at center encounters; those are not generally encounters with the three-dimensional origin. Changing the response leaves the generator, preparation and state-return fidelity unchanged. The physical apparatus interaction must still choose and normalize that response.

Status: exact mathematics inside a conditional physical identification. For every \(L\geq1\), the selected degree-\(L\) representation, unit amplitude charge and stated preparation give \(M_L\) and its flow exactly. The coherent bridge realizes that flow from a trajectory in the spatial candidate. Conservation, projective invariance, the conditional frequency lock and the \(2L\) lobe count follow. Original and Smooth show how different position responses preserve the petals while changing their regularity and shape.

Physical completion requires the projected master action to supply the occupied angular/flux sector, the electromagnetic normalization of the Hopf curvature, the rate \(\omega\), the electron–nucleus frame-lock coefficient \(\kappa\), the generator orientation within the locked family, the three-axis apparatus embedding, and the transverse response profile. The resulting observed density has the form

where \(G_\sigma\) is a normalized finite-width response kernel. The projected interaction determines its physical shape and width \(\sigma\). For a closed motion, \(T\) is a full period; for a nonclosing perturbed motion, the same formula describes a specified finite exposure, not a fictitious recurrence period. Normalization of \(G_\sigma\) makes the total registration probability one. Its physical justification and comparison with measurements remain necessary. A detector exposure is not automatically the electromagnetic source current, and it does not by itself establish radiation balance or stability.