Nuclear Depth, the Four-Direction Orbit and Three-Dimensional Matter

1. Result and Scope

The orbit demonstration uses a literal deterministic electron trajectory through four directions of the surrounding physical space. Its flat candidate law is

$$\ddot q=2\beta J\dot q-kq,\qquad q\in\mathbb R^4,\qquad J=\operatorname{diag}(J_2,J_2),\quad J_2(x,y)=(-y,x).$$

The default $L=2$ example has $\beta=1$ and $k=L^2-1=3$. This is a prescribed restoring and gyroscopic law. It supplies an exact precessing trajectory, but does not derive the attractive interaction between an electron and a nucleus or its directional restriction.

The calculations below establish constant radius and an isotropic second-moment tensor for specified single-centre preparations. They also establish bound motions at every fixed angle in a prescribed two-centre extension. The aligned extension has a free fifth coordinate that must be treated explicitly. The static potential minimum is independent of the angle; the complete potential and mode frequencies are angle-dependent. These results leave open how the coupled field dynamics selects nuclear orientations and the familiar three-dimensional structure of bulk matter.

The electron's position is the deterministic trajectory throughout. A time average of its squared coordinates describes registration along that trajectory, not a probability cloud. Its external motion around nuclei is distinct from its six-dimensional internal classical field structure.

2. Nuclear Depth as a Directional Hypothesis

The maintained down-sector and up-sector assignments are depth three and depth four. One possible nuclear geometry puts their patterns in a common four-direction subspace, with a triple shared by all its quarks. Their depth numbers alone do not establish that alignment.

The force direction does not follow just from the source's dimensional extent. An interaction with a source supported on a subspace can still change when the electron moves transversely to that subspace. For example, an overlap with a localized translated profile can depend on its transverse displacement, giving a transverse derivative of the energy. A projected force must therefore be derived or stated as an additional assumption.

3. What the Single-Centre Trajectory Establishes

The exact trajectory is

$$z(t)=\rho e^{i\beta t}(\cos\delta t,\sin\delta t),\qquad q=(\Re z_1,\Im z_1,\Re z_2,\Im z_2),\qquad \delta^2=\beta^2+k.$$

It has $|q|=\rho$ at every time. The electron maintains a fixed distance from the nucleus while turning at rate $\beta$ and passing between two orthogonal planes at rate $\delta$. The petals and apparent centre crossings in a projection do not mean that the full trajectory reaches the nucleus.

For $\rho=\beta=1$, time averages give:

The integer $L=2,3,4$ second-moment tensors are $I/4$. The irrational example approaches that value in a long-time average; its finite-run error is recorded in the results. $L=1$ has $k=0$ and is a separate preparation.

Isotropic second moments do not make this trajectory or its registration rotationally round. At every time it satisfies

$$q_1q_4-q_2q_3=0.$$

For $L=2$, direct trigonometric averaging further gives

$$\langle q_1^4\rangle=\langle q_2^4\rangle=\frac{11}{64},\qquad \langle q_3^4\rangle=\langle q_4^4\rangle=\frac7{64}.$$

A rotationally uniform unit three-sphere registration in four-dimensional space would have each fourth moment equal to $1/8$. The stated orbit therefore has higher-order directional structure despite its isotropic second moments. How that structure enters an interaction is open; the second moments cannot rule out a shape-dependent effect on bulk geometry.

4. The Empirical Far-Field Constraint

An effective IDWT interaction must reproduce the established electrostatic and gravitational force measurements in their applicable source preparations and distance ranges. For example, Lee et al., Physical Review Letters 124, 101101 (2020) tested the gravitational inverse-square law between 52 micrometres and 3.0 millimetres and found agreement with Newtonian gravity. Electrostatic monopole comparisons refer to appropriate charged sources; arbitrary neutral bodies need not have a monopole electrostatic force.

For an isotropic radial field with conserved outward flux in $n$ spatial dimensions, the sphere area gives

$$\mathcal F(r)\,\operatorname{area}(S^{n-1})r^{n-1} =\text{constant}.$$

The familiar exponent $1-n$ follows under those flux, symmetry and constitutive assumptions. Identifying $\mathcal F$ with a measured force requires the corresponding response law. An observed inverse-square law does not alone determine the full containing dimension of an anisotropic or constrained IDWT field. A construction in the shared higher-dimensional physical space must derive the effective response that measurements test.

The harmonic restoring force used in this note is not an inverse-square force. Its bound trajectories neither reproduce that empirical law nor constitute a failure of a higher-dimensional theory: they test a separate, explicit candidate for the orbit. The link between the candidate orbit and the measured long-range interaction remains missing.

5. Orientation and Polarization Have Different Assumptions

A pattern's bound directions and its extension through other directions can motivate a geometric model. Its orientation is a continuous spatial subspace, not a selection among named coordinate axes. The space of $d$-planes in $\mathbb R^n$ has dimension $d(n-d)$. Changing that orientation is distinct from changing polarization within a fixed subspace.

The working photon picture is a structure spanning source and target with phase accumulating along its extent. It is not a point object travelling along an initially empty line. Its proposed two bound directions supply a useful transverse comparison, but this picture does not by itself derive its polarization constraints or their massive counterparts.

For an explicitly external mathematical comparison, Maxwell and Proca vector equations give the following counts:

For the massive case, the rest-frame constraint removes the temporal component and leaves $n$ spatial vector components. In the massless case, the transverse constraint and residual gauge equivalence leave $n-1$. These are statements about specified vector fields. See Tong's Maxwell derivation and Ueda and Ishibashi's Proca analysis for the corresponding field assumptions and the three massive-vector degrees of freedom in four spacetime dimensions.

There is no general rule that every massive spatial pattern gains one polarization or that its polarization spans the containing space. Scalars, spinors and constrained vector patterns need their own equations. A depth-three IDWT pattern consequently has no polarization count derived here. In particular, the W/Z depth-two assignments and their high mode indices do not by themselves produce a longitudinal polarization; a same-field mechanism must establish that connection.

Some geometric facts remain useful under explicit assumptions. Two linear three-planes through one origin in $\mathbb R^4$ share at least two directions, whereas generic such planes in $\mathbb R^{10}$ intersect only at the origin. Translated affine supports can be disjoint. These statements do not set the contact acceptance of finite or extended field patterns.

If nuclei already share three directions, and their fourth unit axes are independent and uniform in the seven-dimensional perpendicular complement, then

$$\mathbb E|\cos\theta|= \frac{\Gamma(n/2)}{\sqrt\pi\,\Gamma((n+1)/2)},\qquad \mathbb E\cos^2\theta=\frac1n,$$

giving $5/16=0.3125$ and $1/7$ for $n=7$. These are conditional orientation statistics, not a stochastic electron-position law. They describe one possible configuration ensemble; they do not derive the shared triple, independent axes, mechanical rigidity or three-dimensional bulk matter. Testing that picture requires the interaction between actual nuclear and electron patterns, including their relative phases and orientations.

6. Two Nuclei with Different Fourth Directions

Choose $e_1,e_2,e_3$ shared, $e_4$ the fourth axis of nucleus A, and $e_4^B=\cos\phi\,e_4+\sin\phi\,e_5$ that of nucleus B. The nuclear positions are $a=-(d/2)e_1$ and $b=(d/2)e_1$, with $d=2$. As an explicit model assumption, each nucleus pulls and gyrates only in its assigned four directions:

$$\ddot q=2\beta(J_A+J_B)\dot q -k[P_A(q-a)+P_B(q-b)].$$

$P_A,P_B$ are the projectors, and each $J$ rotates the shared $(e_1,e_2)$ plane and the plane pairing $e_3$ with its own fourth axis. The prescribed nuclei do not themselves move or rotate in this equation. The equilibrium is $q=0$, with a force-free spare direction when $\phi=0$.

Conserved Energy and Bounded Motion

Put $G=J_A+J_B$ and $K_\phi=k(P_A+P_B)$. Since $G^T=-G$,

$$E_{\rm rel}=\frac12|\dot q|^2+\frac12q^TK_\phi q, \qquad \frac{dE_{\rm rel}}{dt}=0.$$

For $0<\phi\le\pi/2$, the stiffness eigenvalues are

$$2k,\ 2k,\ 2k,\ k(1+\cos\phi),\ k(1-\cos\phi).$$

They are positive, so every solution at each such fixed angle is bounded by its conserved energy. This is an all-time statement for the supplied linear law; the bound is not uniform as $\phi$ approaches zero. At exact alignment, $q_5''=0$. Restricting to $q_5=\dot q_5=0$ gives the bounded four-direction problem. A nonzero constant $q_5$ with zero velocity is also bounded, but a nonzero $\dot q_5$ gives linear drift. Imaginary or zero eigenvalues alone do not exclude that Jordan-block drift.

For illustration, the following seeded trajectories use the same start except that both spare components are zeroed at alignment. They are propagated to $t=400$ with $dt=0.004$. On the positive-energy block the transformation by the square root of the energy metric makes the generator skew-symmetric; NumPy's Hermitian eigensolver evaluates its exponential. At alignment the oscillatory invariant block is treated separately and the spare coordinate follows its exact linear-drift formula. Energy conservation and an analytical aligned solution check numerical accuracy.

“Seen by A” means the time-averaged value of $|P_Aq|^2$ divided by that of $|q|^2$, and likewise for B. This geometric quantity is not a measured interaction strength. These numbers concern one initial preparation. An orbit confined to the invariant shared $(e_1,e_2)$ plane has fractions one for both nuclei at every angle.

At $\phi=\pi/2$, the bare mode with frequency $\sqrt{k}$ has vector $(e_4-e_5)/\sqrt2$. The two gyroscopic contributions cancel on this mixed direction, and each projector sees half its squared norm. Its fractions are therefore $1/2$, rather than the seeded values near $2/3$.

Static Degeneracy Is Not Full Orientation Indifference

The full potential is

$$V(q,\phi)=\frac{kd^2}{4}+\frac{k}{2}q^T(P_A+P_B(\phi))q.$$

Its minimum over $q$ is $kd^2/4=3$ at every angle, but its value at a fixed nonzero displacement generally changes with the angle. For example,

$$V(e_4,\phi)=3+\frac32(1+\cos^2\phi),\qquad -\left.\frac{\partial V(e_4,\phi)}{\partial\phi}\right|_{\phi=\pi/4} =\frac32.$$

The quadratic restoring energy can therefore distinguish orientations. The velocity-linear gyroscopic term contributes no separate term to the mechanical energy at fixed position and velocity; that fact does not remove this dependence. The mode frequencies also change with $\phi$, so comparisons at fixed mode actions need not have equal energies.

No nuclear orientation equation or its kinetic term is included here. The nonzero potential derivative is consequently an orientation-sensitive contribution to a prospective coupled action, not a demonstrated alignment or misalignment history. Existence of bound electron trajectories at every prescribed angle shows that this law does not require exact fourth-axis alignment for binding. It does not show that all angles are energetically equivalent or select three-dimensional bulk matter.

7. What Is Established and What Remains

• Under the stated preparation and force law, the single-centre trajectory has constant radius and, for the specified nonexceptional ratios, isotropic second moments. Higher moments retain directional structure.
• Under the projected two-centre law, conserved positive energy gives bound motions at every fixed nonzero relative angle; the aligned free direction has the explicit zero-velocity qualification above.
• The static potential minimum is angle-independent. The complete potential, normal frequencies and preparation-dependent spread are angle-dependent.
• Independently oriented fourth axes above a shared triple are a conditional geometric proposal. A reciprocal same-field interaction, nuclear orientation dynamics and the observer response must determine whether it produces the observed bulk geometry and force laws.

8. Reproduction

experiments/nuclear_depth_orbit_dimension_checks.py writes experiments/results/nuclear-depth-orbit-dimension-20261022.json. The corrected experiment has eighteen scoped checks, retains the original seeded comparison, and records the executed code hash. Its propagator uses the installed NumPy Hermitian eigensolver in conserved-energy coordinates, with the aligned invariant block reduced and its free drift added analytically. It does not invert the singular aligned eigenvector matrix.

The checks include the fourth-moment counterexample, full potential variation, mixed-axis bare mode, analytical aligned motion, conserved energy and propagator composition. The all-angle boundedness conclusion uses the explicit energy proof above, rather than extrapolating from four sampled trajectories. These calculations test the stated classical model; they do not establish the missing nuclear interaction or polarization law.

Alpha Separation Through the Fourth Direction

The real fourth-direction alpha proposal uses the maintained up4/down3 assignments. The full emitted alpha has six up and six down valence quarks. A specified static scalar equation in four real spatial directions is exactly the elliptic equation obtained by Euclideanizing the matching three-space scalar operator. This motivates a real spatial separation geometry. The field must still supply the composite path, barrier and outgoing energy flux. The supplied orbit-tail control and its L3 cubic correction are recorded in that note.