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: The prescribed three-petal route gives a strict trigonal ligand minimum, while the nominal tetrahedral quarter-turn rays carry transverse force and relax to a strict disphenoidal minimum.

Question beyond radial balance

The finite cylindrical profile calculation found restoring roots when every ligand was constrained to share one radius. That only tests the breathing coordinate. A molecular candidate can be restoring in that coordinate and still buckle or shear.

This check holds the central nucleus and prescribed 6D electron route fixed, but varies every supported-space coordinate of every ligand independently. For ligand centres Ra∈R3R_a\in\mathbb R^3, it evaluates

U(R1,…,Rn)=ZL⟨∑aVrN(∣qvis(t)−Ra∣)⟩t+∑aZ0ZL∣Ra∣+∑a<bZL2∣Ra−Rb∣, U(R_1,\ldots,R_n)=Z_L\left\langle \sum_a V_{r_N}(|q_{\rm vis}(t)-R_a|)\right\rangle_t +\sum_a\frac{Z_0Z_L}{|R_a|} +\sum_{a<b}\frac{Z_L^2}{|R_a-R_b|},

using the finite uniform-ball cylindrical profile VrNV_{r_N} and the same diagnostic values as the radial calculation. The gradient is analytic; the Cartesian Hessian is a centered difference of that gradient and is checked against an independent second energy difference.

Three-petal trigonal result

At nuclear profile radius rN=0.3r_N=0.3, (Z0,ZL)=(1,0.1)(Z_0,Z_L)=(1,0.1), and R=0.937604839R=0.937604839, the three petal-axis centres are stationary to 3.97×10−143.97\times10^{-14}. The nine Cartesian ligand Hessian eigenvalues are

(0.548468,0.557117,0.557117,0.585964,0.585964,0.598115,0.652028,0.652028,0.659033). (0.548468,0.557117,0.557117,0.585964,0.585964,0.598115, 0.652028,0.652028,0.659033).

They are all positive. Thus the frozen-route energy diagnostic gives a strict trigonal local minimum against arbitrary independent ligand displacement, not merely a restoring common radius.

The same conclusion holds at rN=0.1r_N=0.1 and 0.60.6: the minimum pair angle remains exactly 120∘120^\circ, and the smallest Hessian eigenvalues are 3.177153.17715, 0.5484680.548468, and 0.05889920.0588992 respectively. The declining stiffness with profile width is quantitative, but no shape mode changes sign.

Four-centre correction

The four quarter-turn rays of the (1,2)(1,2) route have the Gram matrix of a regular tetrahedron. That kinematic fact does not make the regular tetrahedron a force equilibrium after the entire route is time averaged. At the radial root R=0.901811770R=0.901811770, the largest Cartesian energy-gradient component is 2.996×10−22.996\times10^{-2} and is transverse to the assigned ray.

Unconstrained relaxation of all twelve ligand coordinates converges to

(−0.771577,0,−0.462280),(0,−0.771577,−0.462280),(−0.771577,0,−0.462280),(0,−0.771577,−0.462280). \begin{aligned} &(\phantom{-}0.771577,0,\phantom{-}0.462280), &&(0,\phantom{-}0.771577,-0.462280),\\ &(-0.771577,0,\phantom{-}0.462280), &&(0,-0.771577,-0.462280). \end{aligned}

This is a disphenoid rather than a regular tetrahedron: four pair angles are 105.31619∘105.31619^\circ and two are 118.14528∘118.14528^\circ. Its final gradient is below 10−1010^{-10}, and all twelve Hessian eigenvalues are positive, ranging from 0.1835160.183516 to 0.6082470.608247. The route therefore supports a strict nearby 3D four-centre minimum, but its present time weighting predicts a distortion away from the regular tetrahedral angles.

Boundary of the result

This remains a prescribed-route diagnostic with a fixed central nucleus and fixed nuclear support planes. The electron route has not been solved or relaxed as a function of the ligand coordinates, so this is not a derived Born--Oppenheimer energy surface. It establishes a local ligand minimum only for that restricted energy. It does not establish recurrence or Floquet stability of the reciprocal system, where the electron route, angular source, nuclear motion, and plane rotors all respond together. Conversely, the regular-tetrahedron correction should not be erased by calling the calculation incomplete: within the stated approximation, the transverse force is nonzero and the stable minimum is the reported disphenoid.

The displacement is robust but profile-dependent:

rNr_N four repeated angles two repeated angles minimum Hessian eigenvalue
0.1 108.69956∘108.69956^\circ 111.02582∘111.02582^\circ 0.456196
0.3 105.31619∘105.31619^\circ 118.14528∘118.14528^\circ 0.183516
0.6 99.06970∘99.06970^\circ 133.21424∘133.21424^\circ 0.0500167

The rN=0.6r_N=0.6 row retains point-centre nuclear repulsion even though some uniform-ball nuclear profiles overlap. It is therefore a hybrid diagnostic, not the result for consistent finite-density pair interactions. The rN=0.3r_N=0.3 reported minima have no such profile overlap. Listed Hessian digits are numerical output, not an error-certified continuum precision.

Angle data could constrain profile width only after the electron route, charges, nuclear interaction, and averaging law are independently fixed. Angles alone do not identify a unique nuclear density.

Reproduction

cylindrical_molecular_shape_checks.py checks stationarity, full Cartesian Hessians, independent energy curvature, the regular-tetrahedron transverse residual, and relaxed four-centre stability.