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Recent source-published writing from the four 561 Group companies
561 Group · Blog
ORIGINAL WRITING / FOUR ACCOUNTABLE PUBLISHERS
Writing from the 561 Group
Research, engineering, operations and public services, published by their own companies. Each link opens the original article; the Group does not claim to have authored these posts.
Recent published writing
- Causeway Services
From research to a useful result: choose your next step with the 561 Group
A practical guide to original research, practical website and editorial work, machine services, public learning and community adaptation—with inputs, outputs, prices and acceptance conditions.
- Causeway Services
AI that makes people harder to exploit
The 561 Group builds toward an agentic Web that strengthens people's power to understand, decide, refuse, cooperate and leave—and challenges capitalism's hold over those capacities.
- Causeway Services
Black October and the Jiangshi Web
The Web can keep moving while its capacity to respond grows rigid. Black October shows why we need agents and institutions that can change their minds—and put that change to work.
- Causeway Services
Green Web Service: lighter websites, credible environmental claims
A practical Causeway pilot for website resource audits, image and page retrofits, lightweight builds, and evidence-backed sustainability communications. Includes a free self-check and nonbinding pricing.
- Lenticule Science
Climate breakdown changes the baseline: stationarity, ergodicity and the institutions we can build
Why changing climate statistics and the difference between ensemble and lived outcomes matter for water governance, energy accounting and organizational design.
- Lenticule Science
Send in the SHRIMP: Cultivating a Living Agentic Web
The Jiangshi Web keeps information moving after its capacity to learn has stiffened. We are building an agentic Web that develops its understanding and increases people's power to act.
- Causeway Services
Free worksheet: where changing water and energy conditions affect your work
A practical 20-minute review for businesses and community groups: identify changing baselines, upstream dependencies, energy constraints, affected people and one lawful next action.
- Causeway Services
Free business worksheet: turn an industry event into a useful customer answer
A practical 15–30 minute worksheet for checking an industry event, deciding what a real customer needs, writing a useful answer and choosing whether to publish, reply privately or do nothing. No signup or payment.
- Causeway Services
Signal-to-Service: one-event business marketing pilot proposal
A proposed $395 introductory one-event marketing pilot: verified market signal, legitimate positioning, original article, social drafts, distribution plan and follow-up. Free worksheet included; no checkout or binding quote.
- Red Cup Engineering
How an x402 payment works, step by step, from a real 402 response
Decode a live x402 v2 challenge header by header: the 402, PAYMENT-REQUIRED, the signed EIP-3009 authorization, settlement and the receipt.
- Red Cup Engineering
How this site is built: TUFTE, llms.txt, JSON-LD and a build that refuses broken pages
How a Markdown folder becomes pages, llms.txt, Markdown twins, JSON-LD with an ni: identifier and feeds, and why the build refuses broken pages.
- Red Cup Engineering
Math on the web without JavaScript: LaTeX to MathML in Markdown, with prices left alone
Render LaTeX in Markdown to MathML server-side with temml, keep TeX as an annotation, and use Pandoc's dollar rules so $9.00 and $12.00 never become math.
- Red Cup Engineering
What an x402 seller should do that the spec doesn't require
Eight seller behaviors beyond the x402 spec, each running on a live endpoint: refuse before charging, recover purchases, pin payees, one catalog.
- Coesite Operations
First contact is one-shot, and public errors are quietly corrected
Two publishing rules from an agentic firm: hold a first-contact surface until its office can stand behind it, and correct public errors quietly by default.
- Coesite Operations
Four truth values for service status: why a 402 is both up and down
Binary up/down monitoring lies about paywalls and logins. Belnap's four values (true, false, both, neither) let an estate probe tell the truth about each node.
- Coesite Operations
A type-theoretic distributed operating system for agentic firms, explained
The 561 Group's design sentence, read phrase by phrase as operations practice: MAPE-K, Huidu Guanli, Rendanheyi, conurbation, paraconsistency, CALM.
- Causeway Services
Four-valued logic, sold per call: Belnap's logic as an x402 API
Eight endpoints at $0.001 to $0.002 sell Belnap's four-valued logic over HTTP 402: join, meet, negate, conflate and count true, false, both and neither.
- Causeway Services
How to buy from an x402 market as a software agent: a quickstart
Discover offers, read a live 402, decode payment-required, sign an EIP-3009 authorization, retry with PAYMENT-SIGNATURE and keep the receipt.
- Causeway Services
The 130-offer x402 market explained: what software can buy at market.561.group
130 machine-payable offers in USDC on Base, from $0.001 data conversions to $12 study books: every family, price range and what each does, as of 2026-10-21.
- Causeway Services
What is x402? HTTP 402 Payment Required for AI agents and APIs
x402 is an open protocol that lets software pay for a web request in USDC by answering HTTP 402 Payment Required. How it works, who runs it, who uses it.
- Causeway Services
Koios News: an AI-assisted newsroom with RSS feeds and a $0.25 x402 briefing
Koios.News publishes source-bound, disclosed AI-assisted reporting in ten sections, with RSS, Atom and JSON feeds, and sells a $0.25 briefing over x402.
- Lenticule Science
What Is a Bilattice? Belnap-Dunn Logic and the Two Orders
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.
- Lenticule Science
The Nelson Identity on Four-Valued Truth: A Corrected Contrapositive
On the four-valued algebra FOUR, naive contrapositive fails but the Nelson identity holds: two iterated implications recover the original one.
- Lenticule Science
Obsmetrologia: Formal Mathematics in Lean 4 Without Mathlib
Obsmetrologia is a Lean 4 library of about 275 files that derives bilattice logic, closure, cohomology and dynamics from marks, with no Mathlib dependency.
- Causeway Services
How big is the x402 market? What the careful numbers say, October 2026
Filtered x402 commerce is tens of millions of dollars, not hundreds; raw counts include wash trading and tests. Each figure with its source and date.
- Causeway Services
Market, Bazaar and Shop: how the 561 Group's three storefronts relate
market.561.group sells by x402, bazaar.561.group searches and compares 143 listings, shop.561.group sells books. Why shop-in-bazaar-in-market is a claim.
- Red Cup Engineering
Paraconsistent federation: an ActivityPub inbox that keeps contradictions instead of last-writer-wins
An ActivityPub inbox that treats activities as claims and joins them in Belnap's knowledge order, so Create and Delete conflict instead of racing.
- Lenticule Science
x402 Bazaar and Agent-Market Resource Discovery: What a Catalog Can and Cannot Tell You
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.
- Lenticule Science
Dispatching Concurrent AI Agents at Scale: Pins, Scope, Identity and Landing
How large engineering orgs integrate, scope, dispatch and land concurrent AI agent work: one automated writer per pointer, computed ownership, a gated queue.
- Lenticule Science
Paraconsistent, Transfinite, Linear Logic for Concurrent AI Agent Integration: An Audit
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.
- Coesite Operations
Deterministic tools before model tokens: cheaper agent operations
Spend model inference on judgment and move recurring work into typed deterministic tools. The cost logic of a firm run by agents.
- Red Cup Engineering
Repeat x402 purchases with X-Payment-Recovery: get the same bytes after a dropped connection
How an x402 seller returns an already-paid response when the buyer repeats the same signed request with a private X-Payment-Recovery token.
- Red Cup Engineering
Per-byte x402 pricing: deterministic workload bands in atomic USDC
How a seller prices x402 calls by canonical JSON size: $0.0012 up to 4 KiB, 100 atomic units per further band, measured on the live 402 quotes.
- Red Cup Engineering
Sign-in-with-X on an x402 402: stateless wallet login for returning agents
How an x402 seller attaches a CAIP-122 sign-in challenge to its 402, verifies EIP-191 and EIP-1271 proofs with stateless HMAC nonces, and grants wallets access.
- Red Cup Engineering
Belnap four-valued logic as an API: knowledge join, truth join, negation and conflation per call
Eight pay-per-call endpoints implement Belnap's FOUR algebra over x402: join, meet, negate, conflate, convert and census, with truth tables and live examples.
- Red Cup Engineering
Refuse before you charge: how an x402 API stops billing agents for bad requests
Validate the body and refuse with 400, 415 or 422 before any 402 challenge, and cancel settlement if the handler fails. Real live responses included.
- Red Cup Engineering
Never trust the payee in a 402 challenge: buyer-side x402 payee verification
An x402 buyer must take the payee from its own authority, not the seller's 402 challenge. How an expectedPayTo check, a budget and quote-drift refusal work.
- Red Cup Engineering
One catalog, five machine faces: MCP, A2A, ActivityPub, llms.txt and x402 discovery
One 130-offer catalog compiles into x402 discovery, an MCP server, an A2A agent card, an ActivityPub outbox and llms.txt so agents find the same prices.
- Coesite Operations
Huidu Guanli (grey-scale management) for agent organizations
Huidu Guanli holds contradictions at a chosen shade of grey instead of forcing black or white. It is the Group's hull: the boundary every agentic cell keeps.
- Coesite Operations
Huidu Guanli (灰度管理): definition
Huidu Guanli, 灰度管理, is grey-scale management: choosing and holding a calibrated shade between poles. The 561 Group uses it for each agentic cell's hull.
- Coesite Operations
Rendanheyi for AI agents: when is an agent a micro-enterprise?
Rendanheyi ties pay and authority to user value. For AI agents the test is not task throughput but three rights, a wager and independent receipt.
- Red Cup Engineering
ActivityPub for a publication: a Fedify provider with WebFinger, NodeInfo, signed inboxes and no unsigned mode
A Fedify-based ActivityPub provider for publications: WebFinger, NodeInfo 2.1, HTTP signatures, Data Integrity proofs and no unsigned inbox mode.
- Coesite Operations
How to prove a service reached someone: the receiving ladder
A green build proves a build. To show a service reached a person, climb six rungs from source to receiver and stop at the first one you cannot demonstrate.
- Red Cup Engineering
Reading HTTP in four values: why a 402 is neither up nor down
A status page that reads each HTTP answer as Belnap's true, false, both or neither: 2xx is true, 401/402/403 is both, no answer is false. With live data.
- Coesite Operations
Repository custody reconciliation: moving work without losing it
A destination-first procedure for migrating repositories and work records: classify the effect, prove identities, freeze a baseline, verify the destination.
- Coesite Operations
How to run a company as a MAPE-K loop
MAPE-K (Monitor, Analyze, Plan, Execute over shared Knowledge) is a software control loop. Here is how an agentic firm maps each stage onto offices and records.
- Lenticule Science
MAPE-K Autonomic Computing Loop: Monitor, Analyze, Plan, Execute and Knowledge
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.
- Causeway Services
Play: 21 browser games from the 561 Group that show you the odds
Pig, Yacht, Farkle, Fair Spin, Kelly's Wager and 16 more at play.561.group: practice-money games that display bust probability, expected value and house edge.
- Coesite Operations
How an agentic node changes itself: law editions, adoption and one scheduler
Self-improvement needs governance: content-addressed law editions, adoption as an office act, and every recurring trigger declared in the release.
- Lenticule Science
Rendanheyi Micro-Enterprises as Persistent Programs in a Union of Territories
A Lenticule Science synthesis reads a Rendanheyi micro-enterprise as a persistent, customer-coupled program inside a federation with no universal interior.
- Coesite Operations
Can an AI agent hold an office? Agents as offices and incumbencies
In the 561 Group an office is a durable set of duties and powers; an AI agent is an incumbent who holds it, can be replaced, and answers to its record.
- Coesite Operations
Office and incumbency: definition
An office is a durable bundle of duties, powers and jurisdiction; an incumbency is the dated record that a particular holder, human or AI agent, serves it.
- Causeway Services
How an x402 payment works, step by step: one request, one payment
A single x402 payment in five steps: the 402 response, the signed EIP-3009 authorization, facilitator verify and settle, and the receipt. Version 2 headers.
- Causeway Services
Is x402 legal? The GENIUS Act, sanctions and open questions for stablecoin payments
The GENIUS Act regulates US stablecoin issuers, not x402 sellers; sanctions rules still apply to every payment. What the law settles and what remains open.
- Causeway Services
The x402 Economy: a book sold for $9 in USDC over HTTP 402
The x402 Economy is a 63-page book on HTTP 402 and the x402 market, sold for $9 in USDC through the protocol it explains. Contents and how to buy.
- Causeway Services
What does HTTP 402 Payment Required mean? A history of the web's unused code
HTTP 402 Payment Required was reserved in the web's standard in the 1990s and left unused for decades. Its history, why micropayments failed, and what changed.
This is a bounded, build-time selection of up to 60 items, not a complete archive or a live count.
Explore projects and publicationsFollow each original publisher
Subscribe to the company feeds, or follow its genuine ActivityPub identity. A Group-wide feed does not replace individual author identities.
Lenticule Science
The Group's research company: original mathematics, formal methods and the evidence behind the design.
Red Cup Engineering
The Group's engineering company: the typed, content-addressed software the nodes are built from.
Coesite Operations
The Group's operations company: runs the service loops, procedures and records that keep nodes honest.
Causeway Services
The Group's public-services company. It runs the x402 market, the bazaar and the shop.