Built on published mathematics.

The claims on this site are not marketing inventions — they are propositions with proofs. Laoh is the working system that follows from the K3Labs paper.

Algebraic Accounting: A Vector-Algebraic Framework for Immutable, Deterministic, and Branchable Financial Ledgers

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k3labs.ai/papers/algebraic-accounting · Research paper · K3Labs · 2026

Patent pending — U.S. Patent Application No. 63/870,502 (Algebraic Accounting System)

What is algebraic accounting?

Algebraic accounting means books that are provably correct — by mathematics, not trust.

The ledger is a state vector. Every transaction is a balanced change. Every balance is rebuilt from evidence. An unbalanced or tampered book cannot be represented at all.

The idea

The ledger is a vector

The paper unifies three traditions: the algebraic formalisation of double-entry bookkeeping (Pacioli through Ellerman, Mattessich, and Ijiri), event sourcing with cryptographic integrity (Merkle trees, hash-chained records), and isolated work that can be reviewed before it reaches the official ledger. A ledger becomes a state vector; a transaction becomes a balanced change; posting becomes vector addition.

Lₖ₊₁ = Lₖ + ΔTₖ₊₁, ΔT ∈ B, Σᵢ L[aᵢ] = 0

Five centuries of double-entry practice — unchanged in substance, finally implemented in the form the mathematics always suggested. No blockchain, no tokens, no consensus: the integrity techniques transfer directly, at a fraction of the cost.

What is proven

Three properties, not promises

Determinism

Proposition 6.1

Given the same evidence and rules, two independent reconstructions of the ledger produce identical balances — always. There is no "our system says" left to argue about.

Means: two auditors, same data, same result. Every time.

Reproducible history

Proposition 6.2

Any historical state of the books can be rebuilt from source documents and rule versions alone, with no reliance on stored balances.

Means: the evidence is the ledger. Stored balances are a convenience, not the source of truth.

Unbreakable accounting equation

Proposition 6.3 & Theorem 6.4

If every posting is balanced, every reachable state — including every draft that becomes official — satisfies the accounting equation.

Means: balance is not checked after the fact — it is enforced by the algebra. The wrong state is unreachable.

Standing on shoulders

The lineage

Pacioli codified double-entry in 1494. Ellerman proved its group-theoretic structure in 1985. Haber and Stornetta showed how to make sequenced records tamper-evident in 1991. Modern systems made isolated drafts and review routine. Laoh's contribution is the unification — and the working system that follows from it.

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