Algorithmic Verification of Combinatorial Flips

The problem

Flips are a concept from birational geometry used in the Minimal Model Program, and whether a given integer sequence represents a valid flip depends on three conditions grounded in group theory, ring theory, and modular arithmetic, not on anything directly computable by eye. The task was to translate genuinely abstract graduate-level algebra, coprimality of coordinate subsets, positivity constraints, and a modular sum inequality, into a program that could verify any sequence in three or higher dimensions and explain exactly which condition it failed, not just whether it passed.

My approach

• Architecture: three independent condition checks, each mapped directly to a distinct piece of theory, GCD-based coprimality across every (n-1)-sized subset, a simple positivity sum, and a modular inequality checked per positive coordinate. Keeping them as separate, independently testable functions made it possible to diagnose exactly which mathematical constraint a given sequence violated, rather than returning a single pass or fail.
• Technical decision: Condition 1's coprimality check generates every (n-1)-element subset via backtracking and computes each subset's GCD independently, giving O(n² log M) time. This was the direct, provably correct translation of the mathematical definition, prioritizing correctness first before optimizing.
• Validation methodology: rather than testing only clear pass or fail cases, nine test cases were deliberately constructed to isolate every combination of which conditions pass and fail, single failures, double failures, all three failing, and invalid input entirely, to confirm each condition was being checked independently and correctly rather than one condition's logic accidentally masking another's.
• Complexity analysis performed after implementation, not assumed beforehand: profiling the three conditions showed Condition 3's modular inequality, not the combinatorial subset generation in Condition 1, is the actual runtime bottleneck as the positive coordinates grow large, which matches the theoretical expectation that this condition represents the most delicate constraint in the geometry of flips.

Results

• Correctly classified all nine constructed test cases, including sequences designed to fail exactly one, two, or all three conditions, confirming each condition operates independently and correctly.
• Identified Condition 3's modular sum inequality as the true computational bottleneck at scale, consistent with the mathematical theory that this condition encodes the most delicate balance constraint on a flip.
• Delivered a full mathematical bridge from graduate algebra, group actions, ideals, quotient rings, to a working, testable computational verification tool.

What I would do differently

Condition 1's coprimality check currently regenerates every (n-1)-sized subset from scratch and computes each one's GCD independently, an O(n squared log M) approach. My own complexity analysis identified a concrete fix I did not implement, precomputed prefix and suffix GCD arrays would let 'GCD of everything except index i' be answered in constant time per index, cutting the whole condition down to O(n log M). I prioritized proving correctness first and treated the optimization as a documented next step rather than something to build before the deadline, which was the right call for a dissertation, but it is the first thing I would implement given more time.