# Lenticule Science > Paraconsistent four-valued logic, agent integration, MAPE-K loops, x402 machine payments and Lean 4 formal mathematics: strange research, with limits stated. Lenticule Science is the research company of the 561 Group. It publishes research on paraconsistent (four-valued) logic, concurrent AI-agent integration, the MAPE-K autonomic loop, x402 machine-payable markets, organizational science (Rendanheyi) and formal mathematics in Lean 4. Each publication separates what is proved or measured from what is hypothesized, and states its limits. Papers carry structured citations; glossary pages give short definitions. ## Articles - [Lenticule Science: Four-Valued Logic, Agent Integration, x402 Markets and Formal Mathematics](https://science.561.group/): Lenticule Science, the 561 Group's research company, publishes work on four-valued paraconsistent logic, concurrent AI agents, MAPE-K, x402 and Lean 4. - [What Is a Bilattice? Belnap-Dunn Logic and the Two Orders](https://science.561.group/bilattices-and-belnap-dunn-logic/): A bilattice has two lattice orders, one for truth and one for information. It explains why a value meaning both true and false is data, not an error. - [The Nelson Identity on Four-Valued Truth: A Corrected Contrapositive](https://science.561.group/nelson-identity-on-four-valued-truth/): On the four-valued algebra FOUR, naive contrapositive fails but the Nelson identity holds: two iterated implications recover the original one. - [Obsmetrologia: Formal Mathematics in Lean 4 Without Mathlib](https://science.561.group/obsmetrologia-formal-mathematics-in-lean-4/): Obsmetrologia is a Lean 4 library of about 275 files that derives bilattice logic, closure, cohomology and dynamics from marks, with no Mathlib dependency. - [x402 Bazaar and Agent-Market Resource Discovery: What a Catalog Can and Cannot Tell You](https://science.561.group/x402-bazaar-and-agent-market-resource-discovery/): The x402 Bazaar is a discovery extension, not a single marketplace; a catalog's call counts cannot be divided by chain events to size x402 commerce. - [Dispatching Concurrent AI Agents at Scale: Pins, Scope, Identity and Landing](https://science.561.group/dispatching-concurrent-ai-agents-at-scale/): How large engineering orgs integrate, scope, dispatch and land concurrent AI agent work: one automated writer per pointer, computed ownership, a gated queue. - [Paraconsistent, Transfinite, Linear Logic for Concurrent AI Agent Integration: An Audit](https://science.561.group/paraconsistent-transfinite-linear-agent-integration/): An audit of a four-valued, ordinal-settling, resource-sensitive logic for coordinating concurrent AI agents: what is proved, what is only defined or open. - [MAPE-K Autonomic Computing Loop: Monitor, Analyze, Plan, Execute and Knowledge](https://science.561.group/mape-k-autonomic-computing-loop/): MAPE-K is the Monitor-Analyze-Plan-Execute loop over shared Knowledge from autonomic computing; its uses, and a staged-closure reading with open inputs. - [Rendanheyi Micro-Enterprises as Persistent Programs in a Union of Territories](https://science.561.group/rendanheyi-living-union-organizational-science/): A Lenticule Science synthesis reads a Rendanheyi micro-enterprise as a persistent, customer-coupled program inside a federation with no universal interior. - [The x402 Economy: Machine-Payable Markets, Measured (September 2026)](https://science.561.group/x402-economy-machine-payable-markets/): x402 lets software pay per HTTP request; filtered settlement was about $15-26M in sixteen months, not the $135M raw figure, and little is provably agentic. - [Agent-to-Agent Commerce: When Software Buys From Software](https://science.561.group/agent-to-agent-commerce/): Agent-to-agent commerce is exchange in which an autonomous software agent initiates and settles a purchase, usually per call, with no human approving each. - [Machine-Payable API: An API Priced Per Call for Software Buyers](https://science.561.group/machine-payable-api/): A machine-payable API charges per call through an inline payment challenge that software can satisfy with no signup, card or human approval, typically via x402. - [Micropayment: Why Tiny Payments Failed for Decades and What Changed](https://science.561.group/micropayment/): A micropayment is a payment so small that processing cost and approval friction have exceeded its value; stablecoin rails and x402 aim to fix both. - [What Is x402? The HTTP 402 Payment Protocol Explained](https://science.561.group/what-is-x402/): x402 is an open protocol that uses the HTTP 402 Payment Required status code so a client can pay for a web request with a stablecoin, with no account needed. - [x402 Bazaar: The Discovery Layer for Payable Resources](https://science.561.group/x402-bazaar/): The x402 Bazaar is a version 2 discovery extension for payable HTTP resources and MCP tools; Coinbase runs one index, but no single marketplace. - [x402 Facilitator: The Party That Verifies and Settles Payments](https://science.561.group/x402-facilitator/): An x402 facilitator is an optional third party that verifies a buyer's signed payment, screens it for compliance and submits settlement on-chain for the seller. - [Belnap Four-Valued Logic: Reasoning with Gaps and Gluts](https://science.561.group/belnap-four-valued-logic/): Belnap-Dunn logic gives each claim one of four statuses: true only, false only, both, or neither. It lets a machine reason on conflicting data. - [Paraconsistent Settling and Comprehension on Four-Valued Evidence](https://science.561.group/paraconsistent-settling-and-comprehension/): Settling closes four-valued evidence to a least fixed point; comprehension then extracts what is held. Conflict gets its own sheaf, so it is never erased. - [The Adjoint Structure of Predication on Bilattice-Valued Fibers](https://science.561.group/adjoint-structure-of-predication-on-bilattice-fibers/): Lambek residuation on the twist product 2⊗2 gives an extend/restrict adjunction on four-valued fibers, a decidable modal calculus, and a linear-logic reading. ## Machine interfaces - [Resource inventory](https://science.561.group/resources.json): Resource identities, access requirements, media types and next actions. - [Discovery JSON](https://science.561.group/discovery.json): Navigation, search and feeds. - [JSON Feed](https://science.561.group/feed.json): Accepted publications as a JSON Feed. - [Atom feed](https://science.561.group/feed.atom.xml): Accepted publications as Atom. - [Sitemap](https://science.561.group/sitemap.xml): All published URLs. ## 561 Group - [561 Group](https://561.group/): The parent organization. - [Engineering](https://engineering.561.group/): Engineering publications. - [Operations](https://operations.561.group/): Operations publications. - [Services](https://services.561.group/): Services. - [Market](https://market.561.group/): Dated observations of the x402 machine-payable market. ## Feeds and discovery - [Sitemap](https://science.561.group/sitemap.xml): every page with last-modified dates - [RSS feed](https://science.561.group/feed.rss.xml) - [Atom feed](https://science.561.group/feed.atom.xml) - [JSON Feed](https://science.561.group/feed.json) - [Full text](https://science.561.group/llms-full.txt): every article as Markdown in one file - [Search](https://science.561.group/search/): query with ?q= - Fediverse: @science@science.561.group (https://science.561.group/actors/science)