Programs and the references they are checked against

33 programs, 7,254 lines, 228 definitions. Beside each result is the reference to compare it with: a published energy, an independent implementation, or a convergence rate set by theory. A program with tests checks its result itself, and a few are checked by the compiler's own tests; for the rest, marked no test, the comparison is the reader's.

Every program here is loaded into the workbench and the terminal, so you can run, compile, hash and diff any of them without installing anything. In the repo they live in lang/viv/examples/.

Quantum chemistry

Hartree–Fock from the Gaussian integrals up, with the published energies to compare against.

Program	What it computes	Result	Lines
geomopt.viv atlas	Geometry optimization by numerical gradients: relax a diatomic's bond length to its equilibrium by finding where the RHF/STO-3G energy is stationary.	H₂ Rₑ = 1.346 a₀ HF/STO-3G equilibrium checked by a test in lang/viv/tests	344
geomopt_water.viv atlas	Polyatomic geometry optimization: relax water (H2O) to its equilibrium by numerical-gradient Newton on the RHF/STO-3G energy surface.	r 1.870 a₀, ∠ 100.0° HF/STO-3G water checked by a test in lang/viv/tests	376
h2_sto3g.viv atlas	Restricted Hartree-Fock for H2 at fixed geometry in the STO-3G basis.	−1.11671 Eₕ Szabo & Ostlund	382	1 test
hchain_sto3g.viv atlas	Restricted Hartree-Fock for a linear chain of N hydrogen atoms in STO-3G.	H₈ −4.06498, H₂₀ −4.79091 C++ interpreter, exact no test	274
scf_mcmurchie.viv atlas	Restricted Hartree-Fock with a general McMurchie-Davidson integral engine, supporting s AND p Cartesian Gaussians (extends to higher angular momentum).	−1.11671 Eₕ general integral engine no test	378
scf_sto3g.viv atlas	Molecule-driven restricted Hartree-Fock in STO-3G.	12 molecules + MP2 published HF/STO-3G checked by a test in lang/viv/tests	838
scf_water.viv atlas	Restricted Hartree-Fock for WATER (H2O) in STO-3G, with a general McMurchie-Davidson integral engine over s AND p Cartesian Gaussians.	−74.94208 Eₕ Crawford −74.9420787, ~1 µEₕ no test	401
scf_water_631gd.viv atlas	Restricted Hartree-Fock for WATER (H2O) in 6-31G(d) [6-31G*], with a general McMurchie-Davidson integral engine over s, p, AND d Cartesian Gaussians.	−76.0192 Eₕ 6-31G*, 19 basis functions no test	482

Plasma physics

Kinetic and fluid plasma: Landau damping, drift-wave turbulence, tokamak equilibrium, transport coefficients.

Program	What it computes	Result	Lines
debye_scf.viv atlas	Self-consistent field solution of the nonlinear Poisson-Boltzmann equation for Debye screening of a charged wall in a plasma: the classical mean-field cousin of the Hartree-Fock SCF (scf_sto3g.viv).	self-consistent screening Debye length recovered no test	160
grad_shafranov.viv atlas	Grad-Shafranov equilibrium of a tokamak: the self-consistent MHD force balance J x B = grad p, written for the poloidal flux psi(R,Z).	relaxed equilibrium tokamak flux surfaces no test	182
hasegawa_wakatani.viv atlas	Hasegawa-Wakatani (1983): resistive drift-wave turbulence.	enstrophy + energy balance drift-wave turbulence no test	236
oneil_trapping.viv atlas	O'Neil (1965) trapping: the nonlinear sequel to Landau damping.	O'Neil trapping regime nonlinear saturation no test	132
spitzer_harm.viv atlas	Spitzer-Harm (1953) conductivity: the electrical conductivity of a fully ionized plasma from the linearized Fokker-Planck collision operator.	within 2% of NRL Spitzer–Härm 1953 no test	96
two_stream_pic.viv atlas	The two-stream instability by a 1D electrostatic particle-in-cell (PIC) code: the kinetic model in its rawest form -- push real particles, let them make their own field, and watch a beam-plasma instability grow out of noise.	growth rate vs theory particle-in-cell no test	178
vlasov_landau.viv atlas	Landau damping by a 1D1V Vlasov-Poisson solver, written in Viv.	γ matches the dispersion root Landau damping	73	2 tests

Nuclear & astrophysics

Neutron-star structure and nuclear level densities by exact counting.

Program	What it computes	Result	Lines
level_density.viv atlas	Nuclear level density by exact many-fermion state counting.	p(40) = 37338 exact count vs Hardy–Ramanujan no test	49
tov.viv atlas	Neutron-star structure from the Tolman-Oppenheimer-Volkoff equations, with the relativistic free neutron-gas equation of state.	0.7087 M☉ maximum Oppenheimer–Volkoff 1939 no test	109

Numerical methods

Finite elements, molecular dynamics, a spectral Poisson solve, and the records refactor of the FEM solver.

Program	What it computes	Result	Lines
annotated.viv atlas	The worked example for the native-annotation syntax: the Boys function and the primitive Gaussian overlap from the SCF programs, annotated at all three levels, with two labeled regions and a variable glossary.	—	59	1 test
fem_poisson.viv atlas	Finite-element solution of the 2D Poisson equation -Laplacian(u) = f on the unit square with u = 0 on the boundary, using linear (P1) triangular elements.	error ratio → 4 O(h²), the P1 rate checked by a test in lang/viv/tests	166
fem_poisson_records.viv atlas	The 2D Poisson FEM (see fem_poisson.viv), refactored with records to show the named-field bundling that motivated them: the mesh is a single Mesh value passed to functions (instead of three loose global arrays), and each element's geometry is a Tri value passed to helpers (instead of nine loose scalars).	identical to the above records are zero-cost no test	178
fft_poisson.viv atlas	Poisson's equation, solved spectrally: -u'' = f on a periodic domain.	error 1.7e-16 exact solution — spectral, to rounding	118	4 tests 3 fig
fused_views.viv atlas	Elementwise expressions and reductions over slices, as the compiled engines fuse them (lang/viv/src/fuse.rs): every result is compared to the element loop that writes the same arithmetic in the same order, to the bit.	—	130	1 test
md_polymer.viv atlas	Molecular dynamics of a Lennard-Jones bead-spring polymer.	energy conserved ~0.5% R_g 11.5 → 6.0 checked by a test in lang/viv/tests	180
xyz_atoms.viv atlas	Read a molecule written as XYZ text into atoms: symbol, atomic number, position (the parser is lib/xyz.viv).	—	16

Automatic differentiation

Forward mode by dual numbers, reverse mode by tape, and deriv() — AD as a compiler transform.

Program	What it computes	Result	Lines
autodiff_deriv.viv atlas	Automatic forward-mode AD: deriv(fn(x) -> expr, x0) returns the EXACT derivative d/dx expr at x0 -- the compiler dualizes the expression (chain rule over + - * /, exp, sqrt, pow), so no dual arithmetic is written by hand.	f′(2) = 9.926737 AD as an IR transform no test	25
autodiff_forward.viv atlas	Forward-mode automatic differentiation via dual numbers.	f′(2) = 9.926737 dual numbers, exact no test	74
autodiff_reverse.viv atlas	Reverse-mode automatic differentiation via a tape ("micrograd in Viv").	∇g matches forward one backward pass no test	134

Symbolic computation & discovery

A computer-algebra system on flat arrays, symbolic regression, Meijer-G metamodeling, symmetry discovery.

Program	What it computes	Result	Lines
meijer_g.viv atlas	Symbolic metamodeling by continuous optimization: the Meijer G-function trick of Alaa & van der Schaar (NeurIPS 2019), in Viv.	log ✓ rational ✓ arctan ✗ identification by snapping	248	4 tests
mini_cas.viv atlas	A mini computer-algebra system in Viv: symbolic computation written with no pointers, no strings and no Int values, only arrays of numbers.	11 asserts pass symbolic diff, simplify to fixpoint	492	5 tests
nn_symmetry.viv atlas	NN-guided symmetry discovery: the AI-Feynman outer loop, in Viv.	exact after the substitution the oracle finds x−y	447	1 test
symbolic_regression.viv atlas	Symbolic regression in Viv: recover a formula from data by brute force.	4 / 4 recovered exactly AI-Feynman inner loop no test	240

Language features

The minimal demos: arithmetic and functions; control flow and recursion.

Program	What it computes	Result	Lines
area.viv atlas	A few numeric functions, in real Viv syntax.	—	22
control.viv atlas	Control flow: recursion, while, for, break/continue, if/else.	—	35

The basis-set ladder for water

At one fixed water geometry, the variational principle requires each larger basis to lower the Hartree–Fock energy toward the basis-set limit, and it does:

The STO-3G energy the program prints, −74.94208, is within about a microhartree of the value in T. D. Crawford's published programming projects (−74.9420787); no test checks it. On top of the converged reference, scf_sto3g.viv computes the MP2 correlation energy through four O(N⁵) quarter-transforms: it prints −0.0491496 Eₕ for water, against the reference −0.049149636. Helium and neon give exactly zero, as they should, because a minimal basis leaves them no virtual orbitals.

A convergence-rate check

fem_poisson.viv solves −∇²u = f on the unit square with P1 triangles against a manufactured solution. P1 elements are second-order accurate, so the error has to quarter each time the mesh halves. Measured ratios as the mesh refines: 3.45, 3.74, 3.87, approaching 4. The check needs only the rate, not a table of reference values.

The program runs one mesh and prints its rms error. The test lang/viv/tests/fem_poisson.rs recompiles it at N = 8, 16, 32… and checks the ratios, so cargo test checks the rate, not the program's own output.

fem_poisson_records.viv is the same solver rewritten with records — a Mesh value and a per-element Tri instead of nine loose scalar arguments. It produces identical numbers. Passing a record does not copy its array fields, so passing a mesh around copies no array data.