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. Depth counts a pattern's bound spatial involvement within that space. “Observed” and “unobserved” describe an observer's access, not different kinds of dimension. Older projective and nested-geometry calculations below retain their additional hypotheses; they do not establish a separate physical manifold or compactified direction. See spatial meaning and mathematical scope.

Summary: A finite supported-space nuclear profile extended cylindrically through unsupported directions gives restoring cycle-averaged radii for both trigonal and tetrahedral 6D orbit-axis geometries.

Correction being tested

Under the selected cylindrical bound-within/free-without premise, a depth-three nucleus is localized only in its supported three-plane and extends through the other electron coordinates. The electron--nucleus separation entering the nuclear profile is therefore

ra=∣Pa(q−Ra)∣, r_a=|P_a(q-R_a)|,

not the full six-dimensional distance to (Ra,0)(R_a,0). An ideal −Z/ra-Z/r_a potential is singular because the candidate orbits genuinely encounter the cylindrical nuclear support. A finite supported-space charge profile is therefore required before force balance is meaningful.

The earlier embedded-point calculation belongs to the mutually exclusive unobserved-localized ontology. Its unstable trigonal root and missing tetrahedral root do not apply here.

Finite profile

Use the potential of a uniformly charged three-dimensional ball of supported radius aa, extended constantly through unsupported directions:

Va(r)={−Z/r,r≥a,−Z(3a2−r2)/(2a3),r<a. V_a(r)= \begin{cases} -Z/r,&r\ge a,\\[2mm] -Z(3a^2-r^2)/(2a^3),&r<a. \end{cases}

The attractive force on the nucleus per unit electron--nuclear charge is

Fa(d)=dmax⁡(∣d∣,a)3,d=P(q−Ra). F_a(d)=\frac{d}{\max(|d|,a)^3}, \qquad d=P(q-R_a).

Potential and force are continuous at r=ar=a. This uniform-ball choice is a finite-profile diagnostic, not yet the derived IDWT nuclear density.

Cycle-averaged symmetric force balance

For each prescribed electron orbit, place equal ligand nuclei on its exact trigonal or tetrahedral axes and solve

F‾R(R)=1T∫0TFR(q(t),R) dt+FNN(R)=0. \overline F_R(R) =\frac1T\int_0^T F_R(q(t),R)\,dt+F_{NN}(R)=0.

The central and ligand effective charges remain the prior dimensionless diagnostic values: (Z0,ZL)=(1,0.1)(Z_0,Z_L)=(1,0.1) for trigonal and (0.1,0.1)(0.1,0.1) for tetrahedral. Nuclear repulsion is independently modeled as a point-centre Coulomb force. It equals the mutual force of nonoverlapping spherical profiles with a common support plane, but not of overlapping profiles. In particular, at a=0.6a=0.6 both central--ligand roots below have R∗<2aR_*<2a and the profiles overlap. Those rows are hybrid-model diagnostics, not a consistent application of the same uniform-ball density to every pair interaction.

supported radius aa trigonal R∗R_* trigonal F‾R′\overline F_R' tetrahedral R∗R_* tetrahedral F‾R′\overline F_R'
0.01 0.998986407 -9437.74 0.999588141 -5837.76
0.03 0.993867571 -1012.91 0.997253187 -642.783
0.10 0.963036065 -76.5450 0.980348565 -55.4964
0.30 0.937606436 -5.99580 0.901811745 -5.62379
0.60 1.191686445 -1.28256 0.805967852 -1.59962

Every listed slope is negative, so every root is restoring in the symmetric radial coordinate with the electron route frozen. This is not a derived Born--Oppenheimer energy surface. This statement alone does not imply full shape stationarity: the later Cartesian test confirms a strict trigonal minimum but finds transverse force at the nominal regular-tetrahedral configuration. The latter relaxes to a stable disphenoid. The trigonal calculation also has one outer root near R=3.459R=3.459 with positive slope; it is rejected as unstable. The tetrahedral calculation has no second root in the tested interval.

Doubling midpoint quadrature from 4,800 to 9,600 orbit samples changes the representative a=0.03a=0.03 roots by at most 3.57×10−73.57\times10^{-7}. The restoring roots are therefore not single-sample encounter artifacts.

Interpretation and remaining test

This is a restoring multi-nucleus result within the selected cylindrical ontology; an observed distribution does not select that ontology, nor does projection alone spread an instantaneous point electron. The stable radii lie near the orbit-axis encounters, showing how a finite nucleus converts an ideal singularity into a restoring impulse region. It supports the proposed link between near-straight electron motion, repeated nuclear encounters, impulse closure, and molecular angle.

It is not yet a complete molecular solution. The electron paths were held at their exact trigonal and tetrahedral candidates while nuclear force was averaged. The next calculation must evolve the electron, all nuclear centres, and their finite-inertia projectors reciprocally, then solve recurrence and the full Floquet spectrum. The result must also be repeated with a nuclear density derived from V2 rather than the uniform-ball control.

Reproduction

cylindrical_nuclear_profile_checks.py checks profile continuity, both five-radius root families, restoring slopes, root multiplicities, and doubled-quadrature convergence.

  • Multicentre orbit axes — prerequisite and correction target: supplies the orbit axes and distinguishes the rejected unobserved-localized force test.
  • Reciprocal cylindrical molecule — continuation: inserts this finite profile into moving-centre, flexible-plane electron dynamics.
  • Cylindrical molecular shape stability — correction and extension: tests every ligand coordinate and resolves the regular-tetrahedral candidate into a stable disphenoid.