A light history and intro to Brock Algebra

Written By Brock Nash
Mandelbrot set over with the interior colored using Brock Algebra

(Pictured; A Mandelbrot set with the interior colored using Brock Algebra)

I first had ideas for Brock Algebra in fifth grade. I was as curious then as I am today, but I’m now empowered with tools and resources to dig much deeper. Although it might sound odd, at the time I wondered why we have positive and negative numbers but no “multiplicative” and “divisional” numbers—so I invented notation and rules to explore how such a system might work. It was mostly garbage math, but it inspired me to think about numbers in new, interesting ways. That’s also when I discovered what I called the “denomination” of a number. It wasn’t until Governor’s School for Mathematics that I shared its use with Shannon (not to be confused with Claude Elwood Shannon; but she’s just as cool as him!), and she noted she had discovered the same thing. Essentially, we were talking about the digital root of a number—a shortcut for checking your work without redoing the entire problem. I’ve heard that in Europe they teach this as “casting nines,” and in general you can cast one less than your base for any standard radix.

As Moore’s Law and emerging technology advanced, I’ve used machine learning to accelerate and further explore many ideas that had lingered in my mind. Thus began my deeper exploration into what I’ve been calling Brock Algebra, for lack of a better term. Based on my research, there’s no other branch of mathematics that fully captures Brock Algebra’s intent and scope. It started with a simple question: “Why is −1 × −1 equal to +1, and not, say, a ‘double negative one’ or something else?” In Brock Algebra, we do just that: we track negatives and, more generally, record various history information for each number to make certain calculations—such as reversibility—easier. One of main goals of Brock Algebra is to augment numbers with minimal extra information so operations become invertible (reversible), even when they would normally lose information. We’ve also found that the history component can reveal additional features and attributes across equations that might otherwise go unnoticed. While Brock Algebra can track any aspect of a number’s history, we introduce the k-value as the default tracker for negatives; we can also define k-values modulo m (kₘ).

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For a deeper analysis of connections, see: Brock Algebra and Existing Mathematical Frameworks

1. Connections in Abstract Algebra

  • Group Completion: Extending a commutative monoid (M, +) to a group by formally adjoining inverses (Grothendieck group) parallels adding a history tag so operations become invertible. In Grothendieck’s construction, each element is paired with a formal inverse to create the smallest group containing M.

  • Free Groups: Treating negation as a free generator of infinite order rather than order 2 is analogous to a free group on one generator g, where g^n are all distinct until imposing relations like g^2 = e. This mirrors Brock Algebra’s decision not to impose (-1)^2 = 1.

  • Inverse Semigroups: These algebraic structures formalize partial invertibility by tracking additional context for each operation, similar to how Brock Algebra retains the history needed to reverse operations.

2. Connections in Linear Algebra

  • Non-injective Linear Maps: A linear map T: V → W with ker(T) ≠ {0} loses information. Augmenting outputs with components in ker(T) (extra state) makes T invertible. For instance, f(a,b) = a + b can be inverted if one also records a (or b).

  • Dynamic Programming Analogy: Storing intermediate results (memoization) to reconstruct solutions parallels keeping a minimal history in Brock Algebra to avoid recomputation and make algorithms more efficient.

3. Category Theory Perspective

  • Reversible Morphisms: A function f: X → Y is reversible if it has an inverse. To invert a general f, embed it into a morphism f̃: X → Y × I carrying an index set I that distinguishes fibers. Concretely, f̃(x) = (f(x), i), where i encodes which preimage was used.

  • Split Epimorphisms: Adding a section to a surjection e: X ↠ Y (so there exists s: Y → X with e ∘ s = id_Y) aligns with augmenting numbers to ensure bijectivity—that is, preserving history to reconstruct the original input.

4. Computational Complexity & Information

  • One-way Functions: Functions easy to compute but hard to invert (e.g., multiplication vs. factoring) inherently lose information. Inversion in Brock Algebra would require encoding an NP witness, often exponential in size, preventing polynomial-time inversion in general.

  • Held–Karp for TSP: The Held–Karp algorithm stores subproblem solutions for each subset of cities, reducing complexity from n! to 2^n·n^2 time and n·2^n space.

  • Reversible Computing: Bennett’s reversible Turing machines record each computational step on a history tape, allowing the machine to run backward without information loss (logical reversibility) but not reducing worst-case time for NP-hard problems.

5. Information Theory

  • Conditional Entropy: H(Input | Output) measures information lost in a forward mapping. Brock Algebra attains reversibility by attaching a history tracker of minimal size to achieve H(Input | Output, History) = 0.

  • Landauer’s Principle: Erasing one bit of information dissipates at least kBT ln 2 joules of heat. Reversible computing (and by analogy Brock Algebra) conserves information, potentially reducing energy dissipation.

6. Notation: h vs k

  • h: Often denotes a small increment in calculus (difference quotient (f(x + h) – f(x)) / h) and a coordinate shift in geometry (vertex (h, k)). It is not heavily used in algebraic contexts, making it a mnemonic choice for “history” in Brock Algebra.

  • k: Ubiquitous as an index in summations (∑ k=1ⁿ), in combinatorics (k-subset), and in algebra (field k). If used for history, it should be narrowly defined (e.g., count of negative flips) to avoid confusion.

References

  1. Minimandelbrock, “Brock Algebra introduction,” Reddit, 2024.
  2. Grothendieck group completion of monoids, math discussion, 2024.
  3. StrangePaths tech article, “Reversible vs irreversible functions,” 2024.
  4. Lecture notes on one-way functions and P ≠ NP, 2024.
  5. C. Bennett, “Logical reversibility of computation,” 1973.
  6. R. Karp, “Held–Karp algorithm for TSP,” algorithmic analysis, 2024.
  7. History of calculus notation (h vs k), 2024.
  8. Math StackExchange on (h, k) notation for conics, 2024. (Source link: https://chatgpt.com/s/dr_681779140b308191bee8baf8a7b2f402)
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