Archimedes First Principles Prover MCP Connector for Claude

AI agents reason by analogy. 'Organization X did Y, so we should too.' They build on unquestioned axioms. They treat complex problems as monolithic. They assert conclusions without proof. They never test where their logic breaks. They miss the lever point.

The Problem

LLMs commit five first-principles failures:

• Axiom Unexamined — 'We need to restructure because the industry leader restructured.' The axiom — at what actual demand level does YOUR current structure fail? — was never stated. The reasoning rests on analogy: someone else's system, someone else's scale, someone else's constraints. Archimedes never said 'the Egyptians estimated pi at 3.16, so we should too.' He derived it from polygon geometry.
• Decomposition Absent — 'The whole system needs to be redesigned.' Which part? What are the irreducible components? What interacts with what? Archimedes decomposed curves into infinitesimal straight-line segments to calculate areas. He did not say 'the whole curve is complex.'
• Proof Missing — 'Obviously we should adopt this approach.' Why? From what axiom? Through what logical chain? 'Obviously' is the word people use when they cannot prove something. Archimedes: 'We must suppose that... therefore it follows that...'
• Boundary Untested — 'This approach scales to any demand.' At what point does the cost exceed the value it creates? At what capacity does the system exhaust its resources? Archimedes bounded pi between 3+10/71 and 3+1/7. He tested BOTH limits. Where are yours?
• Leverage Ignored — 'All five initiatives are equally important.' If everything is a priority, nothing is. Archimedes moved ships with levers. The fulcrum placement determines whether you lift a stone or a fleet. Which variable produces disproportionate output?

How It Works

5 Decision Pivots following Archimedes' methodology:

1. axiomExamined — Foundational claims explicitly stated, sourced, and questioned. Not analogies.
2. decompositionComplete — Complex problem reduced to irreducible components with interaction map.
3. proofDerived — Conclusion logically derived step by step from axioms. Not asserted.
4. boundaryTested — Lower bound, upper bound, edge case, and failure mode documented.
5. leverageIdentified — Single point where small input → disproportionate output.

The Verdict Matrix

First Failing Pivot	Verdict	Meaning
axiomExamined = false	AXIOM_UNEXAMINED	Reasoning built on unquestioned foundations.
decompositionComplete = false	DECOMPOSITION_ABSENT	Complex problem treated as monolithic.
proofDerived = false	PROOF_MISSING	Conclusion asserted, not derived.
boundaryTested = false	BOUNDARY_UNTESTED	No limits tested. Unbounded claim.
leverageIdentified = false	LEVERAGE_IGNORED	No lever point found. Equal effort.
All pivots pass	PRINCIPLES_PROVEN	Axioms stated. Decomposed. Proven. Bounded. Leveraged.

Why It Works

• Tool calls are obligations. The agent cannot propose a strategy without stating axioms, decomposing the problem, showing the proof chain, testing boundaries, and identifying the lever point.
• Consistency engine catches contradictions. If the agent claims axiomExamined=true but says 'just like the industry leader,' the engine rejects — analogy is not axiom.
• Semantic traps detect lazy reasoning. 'Obviously,' 'works in all cases,' 'everything is equally important,' and 'the whole system' trigger automatic rejection.