Use Tao Decomposition Prover with your AI.

Validate tao decomposition

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. Structured reflection capability 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