Observer map for the projective electron orbit

Current physical premise (IDWT): Physical spacetime has infinitely many ordinary spatial dimensions plus time. Matter occupies up to ten spatial dimensions; all remaining spatial directions are unbound for every particle. Particle patterns have different dimensional involvement in that space. The ten-dimensional matter domain is presently modeled by flat $\mathbb R^{10}$. Nested layers describe proposed spatial support, not additional manifolds. Depth counts a pattern's dimensional involvement, not a separate containing space. The familiar three dimensions describe the observer's view, not the ambient dimension. All spatial directions are of the same physical kind. Confinement and persistence of the layers remain dynamical questions.

Mathematical scope: Projective, round-sphere and prescribed global-nesting calculations below retain their stated additional premises. Their curvature, topology and finite-volume normalizations are conditional comparisons; they do not establish the current flat spatial geometry. Field components and mode spaces are distinct from physical position. IDWT has no compactified dimensions or mediator particles; interactions must come from $\Psi_\infty$. See spatial meaning and mathematical scope.

Current Reading and Scope — 1 November 2026

The electron remains a six-dimensional particle moving on a literal external trajectory around the nucleus; the maintained precessing curve uses four spatial directions. Its internal field, external position and detector average are different quantities. Orienting one bound direction constrains the complementary free set but does not fix the complete orientation flag. The attractive interaction and any resulting frequency lock remain to be derived.

The new particle-geometry and sector-depths case provides the current derivations, shared assumptions, glossary and reproduction instructions. The twenty-volume research map connects the complete account. Calculations below retain their own stated mathematical premises and do not inherit results from a different action.

1. The observer-response problem

The six-dimensional electron itself moves around the nucleus; the orbit is not motion inside the electron. Its external position is distinct from its internal classical field oscillations. The physical target is its trajectory in the six macroscopic spatial dimensions. Three are shared with us, but all six are ordinary spatial dimensions of the same kind. An observer map describes the three-dimensional response to that physical motion; it does not divide the electron into different kinds of dimensions.

This is a deterministic trajectory proposal. A spatial probability cloud or stationary orbital wavefunction does not replace that motion. The squared coordinates and time measures below are geometric readouts and dwell-time fractions; no Born position-probability law is assumed.

The general role of the three-dimensional nuclear sector in constraining ordinary matter and observers must be distinguished from a particular $3+3$ frame-lock model. The latter does not by itself derive the former.

The magnetic construction currently determines a Hamiltonian trajectory of coefficients in a selected field-mode space. Its map to the displayed curve is conditional: it must be connected to motion on the physical $Q_6$ and to the nuclear-constrained detector. It is not a replacement of physical space by an internal orientation space. This missing connection limits what the calculation proves; it does not change the theory's spatial premise.

This module classifies the data fixed by projective symmetry, the amplitude/clock splitting, and rotational covariance, together with the remaining response functions:

• the primitive angular clock is unique up to orientation;
• the general rotationally covariant plane map has a fixed form;
• the count of $2L$ lobes is robust under a whole class of radial detector responses;
• the squared-radius readout premise selects the displayed radius within that class;
• uniform Hamiltonian time gives a canonical occupation measure on the curve;
• the apparatus coupling selects the third displayed coordinate and transverse detector width.

2. Projective data available to an observer

Let

$$z=(z_1,z_2,z_3,z_4)\in\mathbb C^4\setminus\{0\}, \qquad [z]=[cz],\quad c\in\mathbb C^*.$$

Every physical readout must be unchanged by this common complex rescaling. The normalized squared-coordinate shares, also called occupations here, are

$$p_j([z]) =\frac{|z_j|^2}{z^\dagger z}, \qquad \sum_{j=1}^4p_j=1.$$

Whenever $z_i$ and $z_j$ are nonzero, their relative phase is represented by

$$\chi_{ij}([z]) =\frac{z_i\bar z_j}{|z_iz_j|} =e^{i(\arg z_i-\arg z_j)}.$$

Both objects are projectively invariant. They are the occupation and relative-angle data available after the common scale and phase have been removed.

Split the chosen four-state orbit into amplitude and clock planes,

$$\mathcal V=\mathcal V_A\oplus\mathcal V_C, \qquad \mathcal V_A=\operatorname{span}(z_1,z_2), \qquad \mathcal V_C=\operatorname{span}(z_3,z_4).$$

Taking the relative share within the amplitude plane gives

$$q([z]) =\frac{p_1}{p_1+p_2} =\frac{|z_1|^2}{|z_1|^2+|z_2|^2}.$$

The primitive clock phase is

$$e^{i\phi([z])} =\chi_{34}([z]) =\frac{z_3\bar z_4}{|z_3z_4|}.$$

Along the magnetic orbit,

$$q(\theta)=\cos^2(L\theta), \qquad e^{i\phi(\theta)}=e^{i\theta}.$$

3. Uniqueness of the primitive clock

A phase character constructed only from the two clock coordinates and unchanged by their common phase has the form

$$\chi_m =e^{im(\arg z_3-\arg z_4)}, \qquad m\in\mathbb Z.$$

Along the orbit it winds $m$ times:

$$\chi_m(\theta)=e^{im\theta}.$$

A primitive clock closes once per turn, which forces

$$|m|=1.$$

Thus

$$\boxed{ e^{i\phi}=\frac{z_3\bar z_4}{|z_3z_4|} }$$

is unique up to complex conjugation, which reverses orientation, and an overall constant phase, which rotates the apparatus axes.

4. Classification of covariant plane maps

Restrict the readout to the two arguments $q$ and $\phi$. This is an additional measurement ansatz: projective invariance alone permits other occupations and relative phases. The classification below is complete only within this restricted class.

Let $W=X+iY$ be a complex coordinate in an observer-aligned plane. Rotating the clock by $\alpha$ must rotate the apparatus coordinate by the same angle:

$$W(q,\phi+\alpha)=e^{i\alpha}W(q,\phi).$$

Set $\alpha=-\phi$. The covariance equation immediately gives

$$W(q,\phi)=e^{i\phi}W(q,0).$$

Therefore every orientation-preserving covariant plane map is

$$\boxed{ W=R h(q)e^{i\phi} },$$

where $R>0$ is the spatial scale and $h$ may be complex. An occupation-dependent argument of $h(q)$ is an angular shear: points at different radii are rotated by different extra angles.

If the aligned apparatus also respects reflection across its chosen axis,

$$W(q,-\phi)=\overline{W(q,\phi)},$$

then $h(q)$ is real. Write this real response as $f(q)$. Nonnegativity is an additional radial-response condition, used below with monotonicity, rather than a consequence of reflection alone. The reflection-aligned maps are

$$\boxed{ W=Rf(q)e^{i\phi}. }$$

Orientation reversal replaces $e^{i\phi}$ by $e^{-i\phi}$.

The readout ansatz selects the arguments $q$ and $\phi$; rotation covariance fixes the equivariant form. Reflection alignment removes the possible angular shear. The detector response supplies the radial function $f$.

5. Radial-response invariance of the lobe count

Suppose the detector response satisfies

$$f(0)=0$$

and is strictly increasing on $[0,1]$. Along the orbit,

$$|W(\theta)| =R f\!\left(\cos^2(L\theta)\right).$$

The occupation $q(\theta)$ has maxima at

$$\theta=\frac{k\pi}{L}$$

and zeros at

$$\theta=\frac{(k+\tfrac12)\pi}{L}.$$

A strictly increasing $f$ preserves those extrema. Hence every map in this class has exactly

$$\boxed{2L\text{ separated radial lobes per primitive clock turn}.}$$

The lobe count is therefore a topological/order property of the locked amplitude and clock, invariant across the stated class of $f$.

The theorem's response class is continuous, strictly increasing, and satisfies $f(0)=0$. Those conditions preserve the extrema and separate adjacent lobes at the origin.

6. What selects the displayed radius

The displayed visualization uses

$$r=\sqrt q.$$

This follows under the squared-radius coordinate-readout premise

$$\frac{X^2+Y^2}{R^2}=q.$$

This is a coordinate-response ansatz: squared observed radius equals the normalized squared-coordinate share of the selected amplitude component. It is not a probability of electron position. With nonnegative radius it forces

$$\boxed{ W =R\sqrt q\,e^{i\phi} =R\frac{|z_1|}{\sqrt{|z_1|^2+|z_2|^2}} \frac{z_3\bar z_4}{|z_3z_4|} }.$$

Along the orbit,

$$W(\theta)=R|\cos(L\theta)|e^{i\theta}.$$

Thus the current plane map is unique under five explicit requirements:

1. projective invariance;
2. dependence only on the selected amplitude occupation and primitive clock;
3. rotational covariance;
4. reflection alignment and a nonnegative radial response, excluding angular shear and fixing the radial sign;
5. the squared-radius readout $(X^2+Y^2)/R^2=q$.

Requirements 1--3 give the family $Rh(q)e^{i\phi}$, reflection alignment makes $h=f$ real, and the squared-radius readout with nonnegativity fixes $f(q)=\sqrt q$.

7. Canonical occupation measure on the orbit

The Hamiltonian flow uses the compact group parameter

$$\theta\in[0,2\pi).$$

Its normalized invariant time measure, when the angle advances uniformly, is Haar measure

$$d\nu(\theta)=\frac{d\theta}{2\pi}.$$

The plane curve $W_L$ therefore carries the pushforward measure

$$\boxed{ \mu_L(B) =\frac{1}{2\pi} \int_0^{2\pi} \mathbf1_B\!\left(W_L(\theta)\right)d\theta }$$

for every measurable detector region $B$. Identifying the group parameter with uniform physical time selects this measure directly.

The deterministic amplitude share

$$Q(\theta)=\cos^2(L\theta)$$

has the following dwell-time density under $d\theta/(2\pi)$:

$$\boxed{ f_Q(q)=\frac{1}{\pi\sqrt{q(1-q)}} }, \qquad 0<q<1,$$

independently of nonzero integer $L$. This density is the Jacobian of the time-to-coordinate map, including all its branches. It does not assign an indeterminate position to the electron. For the squared-radius readout

$$\rho=R\sqrt Q,$$

the radial density is

$$\boxed{ f_\rho(\rho) =\frac{2}{\pi\sqrt{R^2-\rho^2}} }, \qquad 0<\rho<R.$$

The distribution is singular near the turning radius because the radial velocity vanishes there. Different $L$ values share the same one-dimensional radial marginal; their distinction is the angular arrangement of the lobes.

8. From a curve measure to detector intensity

The exact pushforward $\mu_L$ is supported on a one-dimensional curve. A finite detector profile requires a normalized response kernel $K_\sigma(x)$:

$$K_\sigma(x)\geq0, \qquad \int K_\sigma(x)d^3x=1.$$

If detections sample the trajectory uniformly in physical time and the apparatus response is spatially translation invariant, the registered density for a three-dimensional trajectory $\Gamma_L(\theta)$ is

$$\boxed{ \rho_{\rm obs}(x) =\frac{1}{2\pi} \int_0^{2\pi} K_\sigma\!\left(x-\Gamma_L(\theta)\right)d\theta }.$$

It is automatically normalized. The Hamiltonian gives the time-occupation measure; identifying that with detection probability is an additional interaction assumption. The apparatus model must justify the sampling law, kernel shape, and width $\sigma$. Neither this convolution nor a spread in detection positions is a direct observation of the electron's physical shape.

9. The depth observable

The current visualization supplements the plane theorem with

$$Z_{\rm display} =\eta R\sqrt{1-q}\cos(3\phi).$$

This is projectively invariant and leaves the exact plane lobe count intact. The apparatus coupling selects a scalar depth coordinate from the admissible real functions

$$Z=D(q,\phi)$$

consistent with the apparatus symmetries. Fourier harmonics $\sqrt{1-q}\cos(m\phi)$ for different integers $m$ are inequivalent examples with the same plane projection.

The pure-spinor bridge sharpens this result. In the six-vector representation, the same generator has plane rates $1{:}1{:}3$. An antipode-invariant quadratic readout of a slow plane and the fast plane therefore carries $e^{\pm i\phi}$ and $\cos(3\phi)$ respectively. The integer three is thus representation-natural once an even vector detector map is assumed. The apparatus coupling selects the spatial vector/frame datum, the slow and fast readout planes, and the radial factor.

A physical depth coordinate follows from an embedding of the apparatus's three resolved axes into the electron's six-dimensional tangent space and the coupling by which the apparatus samples the orbit.

10. The $L=0$ observer map

For the minimal three-state $L=0$ completion, the amplitude occupation is constant:

$$q=1,$$

while the clock remains primitive:

$$e^{i\phi}=e^{i\theta}.$$

Every normalized covariant response has constant radius, so

$$W(\theta)=Rf(1)e^{i\theta}.$$

Choosing the normalization $f(1)=1$ gives a ring. Thus the ring follows once the universal clock doublet is supplied. The degree-zero line representation is one-dimensional, so this relative-phase doublet is an additional dynamical ingredient.

For an atomic $L=0$ state, the projected clock must have zero ordinary angular-momentum expectation. This can occur through unobserved-sector motion or through an aligned display of the occupation measure; the observer coupling must select one realization.

11. Defensible conclusion

The observer-plane mathematics is now complete at the following scope:

$$\text{projective invariants} + \text{primitive clock} + \text{readout restricted to }(q,\phi) + \text{rotation and reflection covariance} \Longrightarrow W=Rf(q)e^{\pm i\phi}.$$

Strict monotonicity and $f(0)=0$ imply exactly $2L$ lobes. The additional squared-radius readout premise selects $f(q)=\sqrt q$. Uniform group time supplies a canonical dwell-time measure on the curve and an exact arcsine radial time distribution.

The unresolved pieces belong to detector physics: selection and weighting of the representation-allowed $1{:}3$ vector harmonics, the embedding into three apparatus axes, the finite response kernel, and the $L=0$ unobserved-motion or angular-momentum cancellation. These are the input and closure layers of the observer map.

Flat-Action Position Projection

The default Flat Action setting realizes the trajectory in four active real coordinates with $\ddot q=2\beta J\dot q-(\delta^2-\beta^2)q$. Its Lagrangian has the gyroscopic term $+m\beta q^T J\dot q$ for $J^T=-J$. This flat family embeds in six dimensions without selecting six; its prescribed integer ratios do not derive an atomic occupation law.

Sources and provenance

Canonical calculation: files/idwt.py STEP 130.

Companion modules:

• Conditional electron–nucleus frame model
• Bound within, free without