Tetrahedral geometry from equivalent directions

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. Claim and status

This module contains an exact centred-simplex theorem, an external conventional hybrid-orbital comparison, and a conditional classical angular energy model. IDWT's electron follows a literal deterministic trajectory in six ordinary spatial dimensions. The hybrid-orbital comparison does not replace that trajectory or derive its molecular interactions. Its tetrahedral angle is not independent evidence of electron depth or a solution of the reciprocal orbit problem.

The same result appears in the conventional comparison space

$$\mathcal V_{sp} =\operatorname{span}\{|s\rangle,|p_x\rangle,|p_y\rangle,|p_z\rangle\} \cong\mathbb C^4.$$

Four equivalent orthogonal hybrids in this space point toward the vertices of a regular tetrahedron, giving

$$\boxed{\theta_{\rm tet}=\arccos(-1/3)=109.4712206\ldots^\circ.}$$

This angle follows exactly from equivalence, orthogonality, and the stated observer-space identification.

In the conventional hybrid-orbital comparison, ammonia and water have one or two lone-pair channels. Orthogonality then proves how an unequal $s$-occupation changes the bond angle and proves that greater lone-pair $s$-occupation compresses it. This conditional angle map does not determine the redistribution or supply an IDWT molecular energy.

2. The representation comparison and its physical limits

In the conventional sigma-bonding comparison, one $s$ orbital and three $p$ orbitals span a four-component complex vector space. An orthogonal hybrid basis can be adapted to surrounding nuclear directions. This is a statement within that external model, not an equation for IDWT electron trajectories.

Axial symmetry motivates the bond-directed comparison form. Assuming four orthonormal vectors gives the theorem below. IDWT has not derived this orthogonality from deterministic multi-electron dynamics.

Both derivations express geometry in a copy of $\mathbb C^4$. The pure-spinor bridge supplies a conditional representation correspondence: under $\operatorname{Spin}(3)_{\rm obs}\times\operatorname{Spin}(3)_{\rm hid}$ the chiral four is $(\mathbf2,\mathbf2)$, and diagonal frame locking gives $\mathbf2\otimes\mathbf2=\mathbf1\oplus\mathbf3=s\oplus p$. Until the bound-state action derives that diagonal locking, equality of the two copies of $\mathbb C^4$ is the conditional intertwiner used in this module.

The two-factor representation in this comparison is a model assumption. It does not physically divide the electron's six spatial dimensions: all six are the same kind, including the three shared with the observer. A coordinate grouping alone cannot establish a physical frame lock.

3. The centred-simplex theorem

For a nonzero four-component coefficient vector, define normalized squared weights

$$p_i([z])=\frac{|z_i|^2}{\sum_{j=1}^4|z_j|^2}, \qquad p_i\geq0, \qquad \sum_{i=1}^4p_i=1.$$

Its image is the standard three-simplex $\Delta_3$. Let $e_i$ be its four coordinate vertices and subtract the barycentre:

$$v_i=e_i-\frac14(1,1,1,1).$$

All $v_i$ lie in the three-real-dimensional hyperplane

$$\mathcal H=\left\{x\in\mathbb R^4:\sum_i x_i=0\right\}.$$

Their inner products are

$$v_i\cdot v_i=1-\frac12+\frac14=\frac34,$$

and, for $i\neq j$,

$$v_i\cdot v_j=0-\frac14-\frac14+\frac14=-\frac14.$$

After normalization,

$$n_i=\frac{2}{\sqrt3}v_i,$$

the four directions obey

$$n_i\cdot n_j=-\frac13 \qquad(i\neq j).$$

Therefore

$$\boxed{ \cos\theta_{ij}=-\frac13 \quad\Longrightarrow\quad \theta_{ij}=109.4712206\ldots^\circ }.$$

Any isometry from $\mathcal H$ to the observer's three spatial axes rotates the tetrahedron while preserving this angle.

Projective geometry supplies the three-dimensional affine coordinate-share simplex; the $p_i$ are not probabilities of electron positions. Identifying its centred hyperplane with molecular bond directions is an additional observer/binding map. The external hybrid-orbital comparison below realizes the same angle under its own premises, without deriving that IDWT map.

4. The equivalent hybrid-state theorem

Write a normalized real sigma hybrid directed along $\mathbf n_i\in\mathbb R^3$ as

$$|h_i\rangle =a_i|s\rangle +\sqrt{1-a_i^2}\, \mathbf n_i\cdot|\mathbf p\rangle.$$

For four occupied orthonormal states spanning $\mathcal V_{sp}$, completeness gives

$$\sum_{i=1}^4|\langle s|h_i\rangle|^2 =\sum_{i=1}^4a_i^2 =1.$$

If the four environments are equivalent, all $s$-shares are equal, so

$$a_i^2=\frac14.$$

Pairwise orthogonality then gives

$$0=\langle h_i|h_j\rangle =\frac14+\frac34\,\mathbf n_i\cdot\mathbf n_j,$$

and hence

$$\mathbf n_i\cdot\mathbf n_j=-\frac13.$$

The hybrid directions are therefore isometric to the normalized centred simplex vertices of the preceding section. Methane is the leading physical example: four equivalent C--H bonding environments approximately realize the regular solution.

The derivation uses:

1. the four-state $s+3p$ space;
2. bond-directed sigma states;
3. orthogonality of the occupied one-electron states;
4. equivalence of the four ligand environments;
5. identification of their direction vectors with observer-space bond axes.

The fifth condition is the assumed direction map in this comparison. None of the five conditions establishes an IDWT orbit law or a probability-cloud description of the electron.

5. Unequal occupations: ammonia and water

For non-equivalent hybrids, orthogonality gives the exact relation

$$\boxed{ \mathbf n_i\cdot\mathbf n_j =-\frac{a_i a_j} {\sqrt{(1-a_i^2)(1-a_j^2)}} }.$$

Let there be $n_b$ equivalent bond states and $n_{\rm lp}=4-n_b$ equivalent lone-pair states. Parameterize their $s$-shares by

$$s_b=a_b^2=\frac14-\delta,$$

$$s_{\rm lp}=a_{\rm lp}^2 =\frac14+\frac{n_b}{n_{\rm lp}}\delta.$$

Then

$$n_b s_b+n_{\rm lp}s_{\rm lp}=1,$$

so completeness is automatic. The bond--bond angle obeys

$$\boxed{ \cos\theta_{bb} =-\frac{s_b}{1-s_b} =-\frac{\tfrac14-\delta}{\tfrac34+\delta} }.$$

Moreover,

$$\frac{d}{d\delta}\cos\theta_{bb} =\frac{1}{(\tfrac34+\delta)^2}>0.$$

Since $\arccos$ is decreasing, positive $\delta$ compresses the visible bond angle below the tetrahedral value. Thus the qualitative ordering follows conditionally:

$$s_{\rm lp}>s_b \quad\Longrightarrow\quad \theta_{bb}<\theta_{\rm tet}.$$

Using gas-phase reference angles gives the occupation parameters

These values invert the measured angles to infer the corresponding occupations in the comparison. They show the redistribution needed by that angle map, with a larger effect in water. They are not predicted IDWT occupations or angles. Section 7 separates the external energy comparison from the deterministic physical calculation still needed.

6. The continuous occupation family

Completeness and orthogonality admit a continuous family of $\delta$ values.

Let $a=(a_1,a_2,a_3,a_4)$ be any positive vector satisfying

$$\sum_i a_i^2=1.$$

The Gram matrix of the remaining $p$ components of four orthonormal hybrids is

$$G_p=I-aa^T.$$

It is positive semidefinite and has rank three. Consequently it always admits a realization by four vectors in $\mathbb R^3$. Dividing the $i$th vector by $\sqrt{1-a_i^2}$ produces unit bond/lone-pair directions satisfying the exact overlap formula of the previous section.

Therefore a continuous family of $s$-share assignments, including a continuous family of $\delta$ values, is compatible with completeness and orthogonality. The comparison geometry determines the angle once $\delta$ is known. In an energy reduction that actually supplies this family, selection of $\delta$ would belong to that reduction's molecular energy functional:

$$\boxed{ \text{four equivalent projective channels} \longrightarrow109.47^\circ, \qquad \delta=\arg\min_\delta E_{\rm molecule}. }$$

The statement that lone pairs repel more becomes quantitative when the electron interaction energy yields the corresponding occupation inequality.

7. What must select the distortion

In a conventional Born--Oppenheimer comparison, one seeks an angle by minimizing an electronic energy surface at fixed nuclear positions. Its one-parameter reduction has the form

$$E(\delta,R) =E_{\rm one\ centre}(\delta) +E_{\rm two\ centre}(\delta,R) +V_{NN}(\delta,R).$$

Within that comparison, the complete occupied $s+3p$ space makes the trace of every one-electron operator on that space basis-independent. Selection of $\delta$ therefore comes from geometry-dependent multi-centre terms.

A Slater-orbital estimate finds that the one-centre electron--electron contribution changes by less than $0.004$ atomic units over the tested interval. The hydrogen--hydrogen nuclear repulsion increases as the bond angle closes, so that term resists positive $\delta$. The selecting contribution is the geometry-dependent two-centre electronic energy: nuclear attraction, Coulomb repulsion, exchange, and correlation of orbitals extending between the heavy atom and the hydrogen nuclei. These are conventional orbital integrals, not calculated forces between deterministic six-dimensional electron trajectories. Expressing that Hamiltonian in a four-component basis does not turn it into an IDWT orbit Hamiltonian.

The external comparison's stationarity conditions are

$$\frac{\partial E}{\partial\delta}=0, \qquad \frac{\partial E}{\partial R}=0,$$

with measured angles excluded from both equations. These conditions do not close IDWT molecular dynamics. For IDWT the physical target is a coupled action for actual electron positions $q_i(t)\in Q_6$, nuclear positions and the reciprocal same-field response. It must support bounded trajectories and determine nuclear forces and equilibrium directions. A cycle-averaged energy is available only if the dynamics justifies that reduction and its separation of timescales. The geometry here supplies a candidate angle and an angle--share map, not that action or its molecular predictions.

9. A Specified Angular Energy Also Predicts Shape Modes

The angular mode-kernel construction gives an explicit energy for the directed $s\oplus p$ family. The complete scalar–vector band on the observer's unit sphere has normalized overlap

$$K(n_i,n_j)=\frac{1+3n_i\cdot n_j}{4}.$$

With four directions and $E=\kappa\sum_{i<j}|K(n_i,n_j)|^2$, $\kappa>0$, the tetrahedron is the unique zero-energy shape up to spatial rotation. The equal scalar share follows from this band's evaluation normalization. The chosen interaction then selects orthogonal directed states rather than merely permitting them.

For the effective kinetic term $(I/2)\sum_i|\dot n_i|^2$, the constrained Hessian has three rotational zeros, a triplet at $2\kappa$ and a doublet at $3\kappa$. Its ideal angular frequency ratio is therefore

$$\frac{\omega_{\rm doublet}}{\omega_{\rm triplet}}=\sqrt{\frac32}.$$

The result is a conditional angular model with fixed radii and common inertia, not a completed molecular vibrational calculation. The actual electron–nucleus action must supply its band, relative phases, inertia, interaction and radial corrections.

The same derivation compares all angular dimensions. A second feature $w(n)=(n,n\otimes n)/\sqrt2$ gives overlap $(x+x^2)/2$ and selects antipodal orthogonal axes. In three dimensions that is an octahedron, with six lower and three higher shape modes in frequency ratio $\sqrt2$. These new response spectra reuse the geometry that determined the angles.

Reference geometries

• NIST CCCBDB ammonia H--N--H angle
• NIST CCCBDB water H--O--H angle

Companion V2 modules:

• Conditional electron–nucleus frame model
• Observer map for the projective electron orbit
• Bound within, free without