# Validation Protocol

Concrete, reproducible steps to verify the mathematical correctness of
conformallab++.  Every check below has a deterministic expected outcome.

---

## Prerequisites

```bash
cmake -S code -B build -DWITH_CGAL=ON -DCMAKE_BUILD_TYPE=Release
cmake --build build --target conformallab_cgal_tests -j$(nproc)
```

---

## Check 0 — All tests pass

```bash
ctest --test-dir build -R "^cgal\." --output-on-failure
```

**Expected output (last lines):**
```
100% tests passed, 0 tests failed out of 170

The following tests did not run:
    206 - cgal.HomologyGenerators.Genus2_FourGeneratorPaths_BLOCKED (Skipped)
```

If any test fails, stop — the implementation is broken.

---

## Check 1 — Gauss–Bonnet (topological identity, error < 1e-10)

```bash
./build/conformallab_cgal_tests --gtest_filter="GaussBonnet.*" -v
```

**Expected:** all 8 tests `[  PASSED  ]`

What is verified: for each test mesh (tetrahedron χ=2, torus χ=0, open mesh χ=1):

```
Σᵥ (2π − Θᵥ)   =   2π · χ(M)   ± 1e-10
```

This is a **pure topology check** — it fails only if vertex/face counts or
property-map assignments are wrong.  It does not depend on the Newton solver.

---

## Check 2 — Euclidean gradient consistency (FD vs. analytic, error < 1e-6)

```bash
./build/conformallab_cgal_tests --gtest_filter="EuclideanFunctional.GradientCheck*" -v
```

**Expected:** all `GradientCheck_*` tests `[  PASSED  ]`

What is verified: for ε = 1e-5,

```
|G(u)ᵢ  −  (E(u + εeᵢ) − E(u − εeᵢ)) / (2ε)| < 1e-6
```

This check **proves that the energy and its gradient are mathematically consistent**.
A failing FD check means Newton will converge to the wrong point — it is the most
important correctness check for any new functional.

Also run for Spherical and HyperIdeal:
```bash
./build/conformallab_cgal_tests --gtest_filter="SphericalFunctional.GradientCheck*" -v
./build/conformallab_cgal_tests --gtest_filter="HyperIdealFunctional.GradientCheck*" -v
```

---

## Check 3 — Newton convergence on canonical test meshes

```bash
./build/conformallab_cgal_tests --gtest_filter="NewtonSolver.*" -v
```

**Expected:** all 11 tests `[  PASSED  ]`

Each test verifies:
- `res.converged == true`
- `res.grad_inf_norm < 1e-8`
- `res.iterations < 50`  (typically 5–20 for the small test meshes)

---

## Check 4 — Period matrix: SL(2,ℤ)-reduction invariants

```bash
./build/conformallab_cgal_tests --gtest_filter="PeriodMatrix.*" -v
```

**Expected:** all 7 tests `[  PASSED  ]`

The three mathematical invariants checked for **any** genus-1 output τ:

| Property | Condition | Why |
|---|---|---|
| Upper half-plane | `Im(τ) > 0` | τ encodes a positive-area lattice |
| Outside unit disk | `|τ| ≥ 1 − 1e-10` | SL(2,ℤ) reduction step |
| Vertical strip | `|Re(τ)| ≤ 0.5 + 1e-10` | SL(2,ℤ) reduction step |

These hold for **any** well-formed genus-1 mesh — they are topology, not geometry.

---

## Check 5 — Möbius arithmetic (complex analysis correctness)

```bash
./build/conformallab_cgal_tests --gtest_filter="MobiusMap.*" -v
```

**Expected:** all 8 tests `[  PASSED  ]`

What is verified:
- `T ∘ T⁻¹ = Id` (inverse is correct)
- `(T₁ ∘ T₂)(z) = T₁(T₂(z))` (composition is associative)
- `from_three(z₁, z₂, z₃)` maps z₁→0, z₂→1, z₃→∞ (unique Möbius transformation)

A bug here would corrupt **all** hyperbolic holonomy computation.

---

## Check 6 — End-to-end pipeline (build + solve + layout)

```bash
./build/conformallab_cgal_tests --gtest_filter="Pipeline.*" -v
```

**Expected:** all 5 tests `[  PASSED  ]`

What is verified: starting from a mesh file, the full pipeline
(setup → Gauss-Bonnet → Newton → layout → serialise → reload)
produces a consistent result.

---

## Check 7 — Manual torus τ verification

This check requires adding a small program (or modifying an existing test).
It validates that `torus_4x4.off` produces τ in the fundamental domain:

```cpp
#include "conformal_mesh.hpp"
#include "euclidean_functional.hpp"
#include "newton_solver.hpp"
#include "cut_graph.hpp"
#include "layout.hpp"
#include "period_matrix.hpp"
#include "mesh_io.hpp"
#include <iostream>

int main() {
    conformallab::ConformalMesh mesh;
    conformallab::load_mesh(mesh, "code/data/off/torus_4x4.off");

    auto maps = conformallab::setup_euclidean_maps(mesh);
    conformallab::compute_euclidean_lambda0_from_mesh(mesh, maps);
    conformallab::enforce_gauss_bonnet(mesh, maps);

    auto res = conformallab::newton_euclidean(mesh, std::vector<double>(maps.n_dof, 0.0), maps);
    std::cout << "Converged: " << res.converged
              << "  iterations: " << res.iterations
              << "  |G|∞: " << res.grad_inf_norm << "\n";

    auto cg  = conformallab::compute_cut_graph(mesh);
    conformallab::HolonomyData hol;
    conformallab::euclidean_layout(mesh, res.x, maps, &cg, &hol, true);
    auto pd  = conformallab::compute_period_matrix(hol);

    std::cout << "τ = " << pd.tau_reduced.real()
              << " + " << pd.tau_reduced.imag() << "i\n";
    std::cout << "|τ| = " << std::abs(pd.tau_reduced) << "\n";
    std::cout << "|Re(τ)| = " << std::abs(pd.tau_reduced.real()) << "\n";
}
```

**Expected output (torus_4x4.off, R=2, r=1 torus of revolution):**
```
Converged: 1   iterations: <30   |G|∞: <1e-8
τ = [small]  + [positive]i        (Re close to 0 by 4-fold symmetry)
|τ| ≥ 1.0                          (fundamental domain)
|Re(τ)| ≤ 0.5                      (fundamental domain)
```

The exact value of Im(τ) depends on the 3D embedding (R=2, r=1 gives unequal
inner/outer edge lengths).  Use `torus_8x8.off` for a finer approximation.

---

## Summary checklist

```
[ ] Check 0: 176 tests pass, 0 skipped
[ ] Check 1: Gauss–Bonnet exact (1e-10)
[ ] Check 2: FD gradient < 1e-6 for all 3 geometries
[ ] Check 3: Newton convergence < 50 iterations
[ ] Check 4: τ in SL(2,ℤ) fundamental domain
[ ] Check 5: Möbius arithmetic (inverse, compose, from_three)
[ ] Check 6: End-to-end pipeline
[ ] Check 7: Torus τ in upper half-plane with correct symmetry
```

All checks are deterministic and do not depend on random initialization or
floating-point non-determinism beyond standard IEEE-754.
