Tao Decomposition Prover MCP Connector for Claude

You must: (1) DECOMPOSE — break the monolith into at least 3 tractable sub-problems with clear interfaces, (2) COLLABORATE — identify whose expertise is needed and what blind spots you have, (3) CROSS DOMAINS — borrow techniques from adjacent fields and validate from multiple perspectives, (4) SHOW PROGRESS — document intermediate reasoning, rejected alternatives, and failures, (5) MAINTAIN RIGOR — every claim supported with evidence and independently verifiable. If rejected, fix the specific decomposition gap the engine identifies.Send ALL parameters in ONE call — the analyses above plus YOUR OWN verdict: every PIVOT boolean, verdict and clarification. The engine cross-checks the pivots against the verdict and rejects incomplete or self-contradictory calls. Structured reflection tool that forces the LLM to decompose complex problems through the lens of Terence Tao — the Fields Medal laureate (2006) who proved the Green-Tao theorem by decomposing it into 3 sub-problems spanning number theory, harmonic analysis, and combinatorics, then collaborated with 70+ co-authors across his career and led the Polymath Project (distributed mathematical collaboration where 40+ mathematicians solve problems no individual could). Tao does not attack problems head-on — he decomposes, collaborates, crosses domains, shows his work publicly (300+ blog posts with in-progress reasoning), and demands verifiable rigor. Catches Decomposition Absence (attacking a monolithic problem without breaking it into tractable pieces — a city needs to prevent flooding in a river basin. Monolithic approach: "build a dam." One solution. One project. But the flooding is caused by 4 independent factors: upstream deforestation (runoff volume), inadequate storm drainage (urban capacity), riverbed sedimentation (channel depth), and tidal surge from the coast (backflow pressure). A dam addresses upstream volume but ignores urban drainage, sedimentation, and tidal surge. Result: dam costs $200M, flooding continues 3 years later because the real bottleneck was storm drainage at 40% capacity during peak rainfall. Tao approach: decompose into 4 sub-problems with clear interfaces. Sub-problem 1 (hydrology): upstream runoff volume model → feeds dam sizing. Sub-problem 2 (drainage engineering): urban pipe capacity → feeds infrastructure upgrade. Sub-problem 3 (sediment management): river dredging schedule → feeds channel depth. Sub-problem 4 (coastal engineering): tidal barrier design → feeds backflow prevention. Each sub-problem is tractable by a different specialist team. Interfaces are defined. Total cost: $120M (40% less than dam alone) with 95% flood reduction vs dam's 35%), Collaboration Missing (one person/team solving a multi-domain problem alone — a pandemic response team designs a contact tracing system. Team: 5 epidemiologists. They design a mathematically elegant exposure model. Problem: no logistics expert. The model requires testing 50,000 people per day. Current lab capacity: 8,000 tests/day. No supply chain plan for test kits. No behavioral scientist — the model assumes 90% compliance with quarantine, actual: 42%. The epidemiological model is perfect. The implementation is impossible. Tao approach: the Polymath Project succeeds because it incorporates perspectives that no single mathematician possesses. The epidemiologists needed: logistics (lab capacity, supply chain), behavioral science (compliance rates), public health communication (multilingual outreach), and IT (data infrastructure). Rule: if your solution requires expertise you do not have in the room, you are designing a theoretical success and a practical failure), Domain Isolation (staying within one field when the solution requires cross-domain synthesis — a bridge engineer designs a bridge over a river estuary. Structural analysis: flawless. Steel grade, load calculations, wind resistance — all verified. But no geological survey was conducted. Pier 3 is sited on an alluvial clay deposit. 18 months after construction: Pier 3 sinks 30cm. Bridge deck cracks. Repair cost: $45M (60% of original construction cost). The structural engineering was correct. But the problem was not purely structural — it was geological + structural + hydrological (tidal erosion of the clay). Tao's Green-Tao theorem: number theory ALONE could not solve it. He borrowed techniques from harmonic analysis AND combinatorics. Three fields. One theorem. No single field was sufficient. Rule: if your solution only uses techniques from one domain, ask: which adjacent domain could invalidate my assumptions?), Progress Opacity (black-box reasoning without intermediate checkpoints — an archaeological excavation team announces: "We found a Bronze Age settlement." Report: 3 pages. No stratigraphy documentation. No artifact catalog by layer. No photographs of in-situ positions. No rejected hypotheses documented. Peer reviewer: "How do you know it is Bronze Age and not Iron Age?" Team: "We analyzed the pottery." But the pottery analysis is not documented. The conclusion may be correct — but it is unverifiable. Tao writes 300+ blog posts showing work-in-progress — including wrong turns, failed approaches, and reasoning dead-ends. When he publishes a result: the reasoning path is reconstructable by any peer. Rule: if your conclusion cannot be independently verified by reviewing intermediate steps, it is an assertion, not a proof), and Rigor Abandoned (claims without verifiable evidence — a hospital claims: "Our new surgical scheduling protocol reduces wait times." How much? "Significantly." Measured how? "Staff reported improvement." Sample size? "Several departments." Time period? "Recent months." This is not evidence — it is opinion. Rigorous version: "Surgical wait times measured across 4 departments (orthopedics, cardiac, general, neuro) for 1,247 procedures over 6 months. Mean wait: 14.2 days → 9.8 days (−31%). p < 0.001. 95% CI: −28% to −34%. Controlled for: seasonal variation, staffing changes, case complexity (ASA score). Replication: protocol applied to 2 additional hospitals — similar reduction (−27%, −33%)." Every claim has a number. Every number has a source. Every source is independently verifiable. Tao: Fields Medal standard — every step in every proof is verifiable by any qualified mathematician). Call once per complex problem, strategic decision, or multi-faceted analysis