discoveries

Research from the Hraness lab, with mathematical results, computational evidence, and links to the papers and code behind each claim.

1. A sharp limit on what reflection can change2026-09-30

   Adding and subtracting the same points can produce different answers. A proof pins down the largest reflected ratio on four sites: seven quarters.

   by hraness · drafted with ai assistance

2. More differences from fewer sums2026-09-30

   Two explicit constructions show how subtraction can create more distinct outcomes than addition, with exact integer checks and carefully stated comparisons to earlier work.

   by hraness · drafted with ai assistance

3. When specialization wins in a mathematical team2026-09-30

   In one task-allocation model, every optimal team gives each agent a single job. A companion result finds exactly when an exponential average preserves order.

   by hraness · drafted with ai assistance

4. When adding the same ingredient reverses an order2026-09-30

   Two simple mixtures of exponential lifetimes exchange their hazard-rate ordering. Exact arithmetic identifies the crossing and explains why a two-component argument fails.

   by hraness · drafted with ai assistance

5. Placing points with no five on a sphere2026-09-30

   Small integer grids give a clear test of mathematical search: finding a configuration is one task; proving that no larger one exists is another.

   by hraness · drafted with ai assistance

6. Grouping components that last2026-09-30

   When a group works only if every member survives, a mathematical model says to group similar components together. The proof holds at every fixed survivor count.

   by hraness · drafted with ai assistance

7. What a Ramsey search can rule out2026-09-30

   A search for a 40-vertex graph produced checkable exclusions of several possible shapes. The main Ramsey question remains open in this work.

   by hraness · drafted with ai assistance

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