Brock Algebra Field Animator

About Brock Algebra

Brock Algebra is a conceptual framework where numbers are augmented with a history of the operations performed on them. The goal is to make operations invertible, even when they would normally lose information. For example, instead of collapsing (-1) × (-1) to +1, the result might be recorded as a "double negative one," preserving the history of its creation.

What makes it distinct? While a standard complex function f(z) always yields the same output for a given input z, a Brock function's output also depends on an evolving 'history,' represented here by the 'k-rule'. This principle of tracking information for reversibility has deep ties to other fields:

• Abstract & Clifford Algebra: While Clifford algebras generalize complex numbers to model rich geometric structures, they are typically static. Brock Algebra introduces a dynamic, stateful component, akin to a free group that doesn't collapse histories (e.g., not imposing the relation (-1)² = 1).
• Ordinal Analysis: In logic, ordinals measure the proof-theoretic strength of a formal system—a kind of meta-level history. Brock Algebra operationalizes a similar idea, attaching a computational history directly to numbers to change their behavior.
• Category Theory: The framework aligns with making functions reversible (isomorphisms) by augmenting their outputs to carry the necessary history to distinguish between inputs that would otherwise map to the same result.
• Computational & Information Theory: The 'history' acts as the minimal 'witness' or 'advice' needed to reverse a one-way function. This relates to concepts like reversible computing and minimizing conditional entropy (H(Input|Output)) by ensuring no information is lost.

What makes it interesting? It provides a powerful framework for modeling intricate systems that change over time. For a game designer, this is a tool for procedural generation, creating evolving textures, simulating fields of energy, or designing systems that respond to players in complex ways. It's a mathematical engine for exploring potential "timelines" and crafting dynamic, responsive worlds that empower exploration and dignity.