Heuristic Algebra: A Formal System for Composing Reasoning Scaffolds for Large Language Models
Author: Henry Carstens Date: March 2026 Pages: 25 (including appendices) Project: heuristicchat — Version 1.0, Campaign H1 Full paper: PDF
Abstract
Presents heuristic-algebra (H), ten operators for constructing, combining, and selecting reasoning scaffolds injected into LLM system prompts. Validates the framework through a 26-condition Elo-rated scaffold tournament across four frontier providers. Every scaffold outperforms the baseline (+160 to +799 Elo). The top result — lucid-judgment, a three-generation harmonic composite — achieves 1862 avg Elo vs 1063 baseline. Same three sources combined via different operators produce a 231-point gap, demonstrating that the compression mechanism determines scaffold quality.
Three central results
- Scaffolding works. Every heuristic condition outperforms the unscaffolded baseline. Minimum gain: +160 Elo (Strategy). Maximum: +799 (Lucid Judgment).
- Algebraic composition works. Top 5 are all composites. Median composite: 1637 Elo. Median primitive: 1464.
- The operator matters. ⊕bio(DC, RR, SJ) → Sovereign Judgment, rank #8, 1631 Elo. ⊕ha(DC, RR, SJ) → Lucid Judgment, rank #1, 1862 Elo. 231-point gap, same sources.
Tournament design
- 26 conditions (25 scaffolds + 1 baseline)
- 4 providers: Grok-4, Claude Opus 4.6, GPT-5.4, Gemini 2.5 Pro
- 5 strategic decision queries per condition per provider
- 3-judge Elo panel (xAI, Anthropic, OpenAI), K=32, initial 1500
- 185 total experiments
- Kendall’s W = 0.560
Top 10 results
| # | Condition | Avg Elo | Derivation |
|---|---|---|---|
| 1 | Lucid Judgment | 1862 | ⊕ha(DC, RR, SJ) |
| 2 | Actionable Reasoning | 1784 | ⊕bio(CR, DC, AR) |
| 3 | Superior Judgment | 1767 | ⊕bio(6 sources) |
| 4 | Decisive Calibration | 1734 | ⊕bio(CR, MR) |
| 5 | Reflexive Reasoning | 1725 | ⊕bio(AR, MR, CR) |
| 6 | Forecasting | 1647 | Primitive |
| 7 | Master Reasoning | 1637 | ⊕bio(CT, F, SM) |
| 8 | Sovereign Judgment | 1631 | ⊕bio(DC, RR, SJ) |
| 9 | Calibrated Reasoning | 1625 | ⊕bio(CT, F) |
| 10 | Grounded Sovereignty | 1623 | ⊕inn(SV, CR) |
Baseline (no scaffold): 1063 Elo.
Key findings from analysis (§6)
- LJ on Claude (1983) is the single highest cell in the tournament. +740 delta over Claude’s baseline (1243).
- Meta-reasoning sources + ⊕ha is the strongest derivation method tested. Ranking: ⊕ha(meta) > ⊕bio(meta) > ⊕inn(meta) > ⊕ha(domain).
- Preserved tensions outperform resolved tensions. Sovereign Judgment resolved all contradictions. Lucid Judgment preserved two. The scaffold with productive tensions won by 231 points.
- Generation predicts rank. 3rd-gen > 2nd-gen > 1st-gen > primitives.
- LJ beats all three of its sources individually by 95–137 points (source avg: 1742).
Caveats (§6.8)
- All rounds use the same 5 strategic decision queries
- Elo values are relative to this pool
- LJ shares deep lineage with its sources — meaningful test is whether combination exceeds sources
- Kendall’s W = 0.560 is moderate, not high
Paper structure
| Section | Content |
|---|---|
| §1 | Motivation — scaffolds make LLMs reliably thorough |
| §2 | 10 operators defined |
| §3 | Tournament design and condition pool |
| §4 | Provider-condition heatmap (26 × 4) |
| §5 | Full results and structural observations |
| §6 | LJ derivation, operator comparison, key findings |
| §7 | Conclusion and reproduction instructions |
| App A | Full algebra specification with per-operator falsification |
| App B | Heuristics of Lucid Judgment (7 axioms, tensions, dependencies) |
| App C | Five strategic decision queries |
External resources
See Also
- heuristic-algebra — the formal system
- lucid-judgment — the top scaffold
- heuristicchat — the software