Heuristic Algebra

A formal system of ten operators for constructing, combining, and selecting reasoning, forecasting and creativity scaffolds for LLMs. Published March 2026.

formal-system LLM reasoning-scaffolds operators prompt-engineering agents skills heuristics-library
concept Updated 2026-06-16

Heuristic Algebra (H)

A formal system of ten operators for constructing, combining, and selecting reasoning, forecasting and creativity scaffolds for LLMs. Published March 2026.

Treat the fundamental assumptions of any domain as operands. Define operators for combining, transforming, and selecting among them. The result: scaffolds whose composition is explicit and falsifiable.

Core finding: Heuristics don’t make models smarter — they make them more reliably thorough. A good scaffold causes the model to access knowledge it already has but wouldn’t surface under a bare prompt.

The Ten Operators

Symbol Name What it does
⊕ Combination Union all axioms from two or more fields
⊕bio Biological Selection Cull unfit axioms, resolve contradictions
⊕ha Harmonic Arrangement Compose voices, preserve productive tensions
⊕inn Innovative Recombination Generate emergent axioms, then prune
¬ Transformation Replace an axiom with its logical complement
∼ Similarity Shared structural function across fields
↔ Resonance Mutual amplification between axiom pairs
⊥ Contradiction Mark incompatible axioms
πk Projection Select a k-axiom subset for a purpose
δ Decompression Reversibly expand a compound when context shifts

Typed combination

The three typed combiners produce different outputs from identical inputs:

Operator Mechanism Best for
⊕bio Cull unfit, resolve all ⊥ Decision tools
⊕ha Compose voices, preserve productive ⊥ Thinking tools
⊕inn Generate novel axioms from tensions Discovery tools

Convergence: ⊕bio is convergent (each pass culls more), ⊕ha is accretive (tensions from layer N become engines for layer N+1), ⊕inn is divergent (each pass generates novelty).

Key result: ⊕bio(A,B) ≠ ⊕ha(A,B) ≠ ⊕inn(A,B). The compression mechanism determines scaffold quality. In the tournament, same sources via ⊕bio → rank #8 (1631 Elo), via ⊕ha → rank #1 (1862 Elo). 231-point gap.

Relational operators

Before choosing a typed combiner, map all three relations across the axiom pool:

  • Many ∼ pairs → use ⊕bio
  • Many ↔ pairs → use ⊕ha
  • Many gaps between sources → use ⊕inn

Projection (πk)

A valid projection must satisfy five criteria: source coverage, negation inclusion, contradiction resolution, functional completeness, and precondition transparency (CR8 — audit whether surviving axioms depend on culled ones).

Decompression (δ)

Reverses harmonic compression by un-scoping sacrificed axioms when context shifts. Only applicable to ⊕ha compounds, which document their sacrifices. This is a structural advantage of harmonic arrangement: lossless compression with reversible scoping.

Notation

H_T = {h1, h2, …, hn} — the heuristics for topic T.

lucid-judgment derivation:

H_LJ = π₇^{Ha1-Ha7+S1+CR8}(H_DC ⊕ H_RR ⊕ H_SJ)

Falsification

If algebra-built heuristics are no more useful than free-form expert writing, the framework should be retired. The scaffold tournament is the initial test of this condition.

Scaffolding Experiments

Following the publication, scaffolding experiments (R1-R6, C1-C5, F1-F4) produced three standalone program scaffolds through iterative composition and compression:

  • Lucid Judgment — Reasoning scaffold (⊕ha from DC, RR, SJ sources)
  • Generative Imagination — Creativity scaffold (π₅ projection of Imagination ⊕ Creativity via ⊕inn)
  • Forecasting Scaffold — Quantitative prediction scaffold (designed from practitioner knowledge)

Each scaffold is self-contained and designed for use alone, with additional heuristics degrading performance (except Lucid Judgment, which can compose with domain heuristics).

Skeletons vs Programs

Scaffolding experiments revealed two fundamental scaffold types:

  • Skeletons: Abstract quality labels without process directives (e.g., CT: Clarity, Accuracy, Precision). Compose with heuristics by providing meta-cognitive framework that domain content fills.
  • Programs: Specific step-by-step instructions with domain content (e.g., GI, FS). Do not compose with heuristics; additional instructions compete and degrade performance.

Programs outperform enhanced skeletons for their domains. Rubrics must match task type. Prompt structure (e.g., data inclusion) shifts scaffold rankings.

Empirical Test: Heuristic Shaping vs Control

A June 2026 controlled experiment (dynamic-heuristic-shaping-experiment) tested whether heuristic shaping improves a research agent’s qualitative syntheses. Across 11 schedules judged blindly by a multi-model LLM panel, heuristic shaping beat the deterministic control on 10 of 11 questions. LLM-assisted shaping reached a global Elo of 1725.3 (+492 vs control), and deterministic shaping reached 1541.5 — evidence that the structure the shaping layer adds is not just an artifact of extra LLM calls. This extends the original scaffold tournament from one-shot reasoning to multi-step research synthesis.

Systems & Architecture: Heuristic Agents as Skills

In June 2026, Henry advanced his ‘Heuristics to Agent’ architecture by successfully converting his proprietary Heuristics Library into a suite of triggerable LLM skills.

By compressing 117 individual heuristics, personas, and mental models into standalone directives, he created a system where autonomous agents can dynamically index, choose, and apply specific cognitive constraints to themselves based on the task at hand. For example, an agent tasked with a prediction problem can autonomously invoke the forecasting skill to bind its reasoning to the 8 axioms of Superforecasting.

This approach effectively turns static mental models into executable code, allowing for the rapid generation of highly specialized, parallel sub-agents (such as automated Repo Opportunity Scanners and Research nodes) that operate strictly within defined constraints.

See Also