The computation and its guarantee
The affine coordinates are α = α₀ + Cu. The solver minimizes Φt = F − tV, with F = −∑ log sin α. It increases t by at most 1 + 1/(20√m), then takes a full Newton step, where m is the corner-angle count.
V* − V(u) ≤ [m + (√m + λ)λ/(1 − λ)]/t ≤ 17m/(12t), when λ ≤ 1/4.
The displayed inequality is the chapter’s exact-arithmetic theorem. Here derivatives, linear solves, decrements and the bound are evaluated in binary64. The reported numbers are numerical estimates, not rigorous floating-point certificates. The independent 80-digit comparisons below test these fixed inputs and do not enclose all rounding errors.
Initial centering work is counted separately. Positivity, conditioning, linear residuals and endpoint centering are checked explicitly; a numerical or budget failure is reported rather than clipped into apparent success. This dense method makes no nearly-linear complexity claim and does not certify an embedding.
Reproduce and inspect
Run from the consolidated repository with its toolkit installed, or from the extracted reproduction archive after installing the supplied requirements:
python experiments/section9/barrier_run.py
A source-tree run can use PYTHONPATH=packages/volume-toolkit. The detached archive contains unchanged maintained numerical modules, the fixed inputs, this driver, the page template, the toolkit license and a payload manifest. It requires only NumPy, SciPy and mpmath; optional monograph files are absent. The single Python inspection downloads do not form a standalone program.
The reference construction distinguishes exact decimal/geometric inputs from high-precision lifts of the actual binary64 angles. This matters especially near a flat boundary maximum, where tiny triangle-sum perturbations change limiting derivatives. Both reference models and their discrepancies are retained in the JSON.