A complete, scientifically rigorous comparison demonstrating that TRF's epsilon-field tensor products are exactly proportional to the real spherical harmonics YLM for all orbitals through L=3, with machine-precision numerical verification and wall-clock performance benchmarks.
The time-independent Schrödinger equation for any central potential V(r) separates in spherical coordinates into a radial equation (potential-dependent) and an angular equation (universal). The angular part satisfies:
The solutions are the spherical harmonics, which factor as YLM(θ, φ) = Θ(θ) · Φ(φ). The azimuthal equation yields Φ(φ) = eiMφ. The polar equation yields the associated Legendre polynomials.
The associated Legendre polynomial PLM(x), where x = cos(θ), is given by the Rodrigues-derived closed-form:
Computational cost: O(L) multiplications per point, with factorial overhead dominating. Each evaluation requires ⌊(L-M)/2⌋+1 terms, each involving factorial lookups and a power of x. For L=3, this is approximately 20–30 floating-point operations per evaluation (with cached factorials).
The complex harmonics are converted to real form:
TRF begins with three basis functions — the Cartesian components of the unit vector on S². These are the d-sector ε-fields:
These satisfy εx² + εy² + εz² = 1 on S². They are the L=1 real spherical harmonics (up to normalization). This is the complete basis from which all orbital shapes are generated.
All angular momentum eigenstates are constructed as symmetric, traceless tensor products of the three ε-fields:
For L ≥ 2, the raw tensor product contains traces corresponding to lower-L harmonics. The traceless projection removes these:
For L=3, the factor 3/5 in εz³ - (3/5)εz arises from removing L=1 contamination: three index-pair contractions in a rank-3 symmetric tensor, each yielding a factor of εz/5.
TRF is generative, not presolved or statistically approximated.
(a) No lookup table. Each evaluation computes from three ε-fields directly.
(b) No statistical fitting. The products are exact algebraic expressions.
(c) No precomputed Legendre polynomials. Shapes emerge from multiplication and subtraction.
(d) Uniform construction: the same three basis fields generate all angular eigenstates.
Each TRF ε-field product equals the corresponding real spherical harmonic up to a normalization constant. These are exact identities, not approximations.
Side-by-side rendering: TRF ε-field products (left) vs QM Legendre polynomial evaluation (right). Both use identical line tracing — the only difference is the radius function. Visually identical shapes confirm algebraic equivalence.
Both functions evaluated on a dense 100 × 200 = 20,000-point angular grid. For each orbital, the least-squares proportionality constant c and residual are computed. All maximum relative errors are at or below machine precision (~10-14).
| Orbital | L | M | Type | Prop. Const. c | Max Rel. Error | RMS Rel. Error | Status |
|---|
Integral of fa · fb · sin(θ) dθ dφ over S² using trapezoidal quadrature on an 80 × 160 grid.
| Pair | Integral Value | Expected | Status |
|---|
Evaluate ε-field product directly. Pure arithmetic on trig values. No factorials, no polynomial loops, no normalization. Cost: O(L) multiplications.
Full Rodrigues-form associated Legendre polynomial with normalization. Cost: O(L) with factorial overhead and polynomial summation loop.
Build a 500×500 table of YLM values, then bilinear interpolation at runtime. Build: O(N²·L). Query: O(1) but not exact — introduces interpolation error.
Can TRF beat both the exact computation and the precomputed table? The answer: TRF beats exact QM by 2–6×, and while the LUT query is O(1), its build cost and inexactness make TRF the better overall choice for on-the-fly evaluation.
| Orbital | L | TRF (s) | QM Exact (s) | LUT Build (s) | LUT Query (s) | TRF / QM |
|---|
(a) ε-field tensor products are exactly proportional to real spherical harmonics.
(b) Computationally simpler evaluation path for low-L orbitals.
(c) Uniform construction — same engine generates all orbital types.
(d) Arises naturally from the d-sector of the TRF Motor algebra.
(a) That spherical harmonics were previously unknown.
(b) That the STF formalism is new.
(c) Results different from standard QM for the angular part.
(d) Replacement of the radial wavefunction.
(e) Any violation of quantum mechanics.
"Generative" means the orbital shapes are produced on-the-fly by the algebraic engine of ε-field products. No tables, no fits, no boundary-condition solutions. This is analogous to the difference between a formula and a lookup table — both yield the same answer, but the formula produces it from rules while the table was computed once and stored.
Both approaches are exact. Neither is an approximation. The difference is conceptual and computational, not mathematical. The QM Legendre polynomials are derived by solving a differential equation; the TRF ε-products arrive at the same functions via algebraic tensor construction.
This benchmark covers the angular part only (spherical harmonics). The angular eigenfunctions are universal across all central potentials — they do not depend on V(r). The radial part RnL(r) depends on the specific potential and is outside the scope of this document. Benchmarks are pure Python for transparency; C/GPU implementations would show different absolute timings but similar relative scaling.
TRF's ε-field tensor products reproduce the real spherical harmonics exactly, with machine-precision numerical agreement across all 16 orbitals tested (L=0 through L=3). TRF evaluation is 2–6× faster than standard Legendre polynomial evaluation and, unlike precomputed lookup tables, is exact and requires no build step. The angular quantum mechanics is complete and correct.