PHCpack Benchmark Test 3

The first report recorded 25 systems from boon through noon5. The second covered ten more, from redcyc7 through caprasse. This final chapter completes the remaining 45 systems, beginning with cyclic7 and ending at rose, the last entry in PHCpack's main collection.

This was not one method repeated 45 times. The early systems demanded staged resume; the middle section required reduction and unsaturation; the final stretch brought total-degree homotopy, multihomogeneous homotopy, and variable scaling into the core engine. Each new structure called for a new tool.

The first ten systems included the seven-variable cyclic systems with 924 roots and the eight-variable systems with 1,152. Checkpoints preserved each growing solution set, allowing wider searches to pursue only what was still missing.

* Sum of the consecutive search stages used by the successful resume chain.

Formulation became the central weapon in the next 15 systems. des18_3 reached its count after reduction and total-degree homotopy; fbrfive4 required unsaturation; and kinema yielded when its multihomogeneous structure was exposed.

Total-degree and multihomogeneous homotopies became the backbone of the final 20 systems. cohn2, cohn3, and discret3 were especially revealing: direct searches saw only fragments, while scaling followed by path tracking reached the official counts.

A search does not have to finish in one run

cyclic7 advanced through 560, 658, 728, 854, and finally 924 roots. Resume support turned a long all-root search into a cumulative process whose progress could survive each run.

The form of the equations is part of the computational cost

Unsaturating fbrfive4, reducing des18_3, and exposing kinema's multihomogeneous groups showed how dramatically the same mathematical object can change when presented differently. All-root search is a collaboration between solver and formulation.

From Newton search to path tracking

Starting with cohn2, cohn3, and discret3, total-degree and multihomogeneous homotopies became first-class tools. MAL-Seeker no longer merely falls locally into roots; it can generate structured starts and carry them continuously to the target system.

• Analysis Engine: MAL-Seeker (Project Antares Ver 3.1.6)
• Development Project: Project Bellatrix V1.0.4
• Search Methods: 5SS / Polynomial Newton / Resume / Symmetry Closure
• Path Tracking: Total-Degree / M-Homogeneous Homotopy
• Computer: MacBook Pro (Mac17,2)
• OS: macOS Tahoe 26.4 (Build 25E246)
• CPU: Apple M5, 10-core (4P + 6E), arm64
• Memory: 32GB

PHCpack's main 80 systems form a map spanning numerical analysis, robot kinematics, chemical equilibrium, economic equilibrium, quadrature, and algebraic structures. Walking from one edge to the other has revealed both the terrain where MAL-Seeker is strong and the regions where its equipment is still incomplete.

Reaching rose is not the end. PHCpack continues with a separate set of newer tests, singular-root deflation examples, and algebraic-set decomposition. Still, this is the right place to stop and preserve the record of the first pass. We have not escaped the maze; we have finally obtained its coordinates.