Following the log-sine barrier path

The computation and its guarantee

The affine coordinates are α = α₀ + Cu. The solver minimizes Φ_t = 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.

A triangle approaching the regular ideal tetrahedron

On the exact triangle angle simplex, strict concavity and symmetry give the unique maximum at three angles pi/3.

18/18 checks passed. The independent initial decrement is 1.16001; the exact-space and lifted-input reference maxima differ by 1.28e-16. All values, trace rows, methods and tolerances are in the JSON download.

An asymmetric geometric disk with a moving central path

The exact geometric angles close at u=0. The positive disk Hessian gives their unique maximum. F'(0)=-28/15, so the barrier center is different.

18/18 checks passed. The independent initial decrement is 0.881695; the exact-space and lifted-input reference maxima differ by 2.94e-17. All values, trace rows, methods and tolerances are in the JSON download.

A closed maximum with one flat triangle

For 0<s<0.1, angles are (pi-2s,s,s) and (0.1-s,1+s,pi-1.1), with u=s-0.05. Strict concavity and V'(0+)=2 log 2+log(sin(0.1)/sin(1))<0 give the unique closed maximum at s=0. No geometric realization is asserted.

18/18 checks passed. The independent initial decrement is 0.968172; the exact-space and lifted-input reference maxima differ by 7.14e-15. All values, trace rows, methods and tolerances are in the JSON download.

The exact-space maximum has a flat triangle. Rounded triangle sums can change the endpoint singularity; the lifted Clausen model has its own nearby stationary reference. Neither is an interval certificate for machine arithmetic.

Reproduce and inspect

All fixed-fixture numerical checks passed. Source commit: c1f100b957166b93eca0d214586322d5af71b9cf; checkout dirty at start: False. Captured Python 3.12.11; NumPy 2.5.4, SciPy 1.18.1, mpmath 1.3.0.

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.