Payment Netting Is Easy Algebra and Hard Authority: Our 13-Domain Sweep

Suppose one entity owes a second, the second owes a third, and the third owes the first. A graph solver can identify the directed cycle and produce fewer payment instructions without changing any node’s final net position.

The arithmetic is the easy part. The commercial product exists only when one buyer controls every node, the obligations are mutually enforceable and identical in all relevant dimensions, and a pre-existing setoff right permits cancellation before settlement.

We screened 13 domains against those constraints. The result was return none.

The sweep in numbers

The July 29, 2026 evidence set contained 13 candidate domains and 18 public receipts. The deterministic decision produced:

• 10 rejected domains;

• 3 domains with evidence gaps;

• 0 admitted domains; and

• 0 live payment instructions.

The screened families were corporate treasury, SMB ERP, air transport, rail transport, telecommunications, securities post-trade, online marketplaces, franchising, federal government, healthcare claims, higher education, insurance/reinsurance, and wholesale electricity.

The public solver, verifier, evidence schema, and decision artifacts are available in the Cycle Compression Foundry repository.

Why ten domains failed

Most real-world netting systems span independent counterparties. That is valuable, but it is outside this product boundary because it requires opt-in, representation, clearing, custody, or new contractual terms.

Air transport, rail transport, telecommunications, online marketplaces, franchising, and federal-government examples failed primarily because one buyer did not possess unilateral authority over every legal node. Securities post-trade and regulated clearing are explicitly excluded. Healthcare claims involved independent parties and post-settlement correction rather than a clean pre-settlement cycle.

Mature clearinghouses are evidence that the economic mechanism matters. They are also evidence that a lightweight, self-serve plugin should not impersonate an established multi-party institution.

The three classes that survived only as evidence gaps

Our internal day-60 prepaid-install probabilities were 2% for corporate treasury, 4% for SMB ERP, and 0.5% for higher education. These are research forecasts, not observed conversion rates. The absence of exact receipts dominates the estimate.

The invariant a buyer should demand

For an eligible frozen obligation set, the output must preserve every node’s final position:

net(node) = incoming obligations - outgoing obligations

The optimized instructions must produce the same net(node) for every node as gross settlement. They must not invent an obligation or change currency, unit, due date, tax treatment, priority, permission, exposure, or audit identity.

A legitimate product should replay every proposed instruction against the gross ledger and fail closed on one mismatch. Disputed, contingent, missing, manual, or affiliate-ineligible obligations should never enter the graph.

The authority checklist comes before the solver

A controller evaluating cycle compression should answer these questions in order:

1. Does one organization legally control every node?

2. Can it act unilaterally, without counterparty recruitment or consent?

3. Is there a cited, pre-existing right of setoff for this obligation class?

4. Are unit, currency, settlement window, enforceability, priority, tax treatment, and due date identical?

5. Does the authoritative ERP freeze the obligations before analysis?

6. Can the buyer approve and execute the output without giving the software custody?

7. Can gross-versus-compressed replay be reproduced from the audit trace?

If any answer is “no” or “unknown,” graph compression should not generate executable instructions.

What an exact historical replay would prove

A day-30 replay needs a buyer-authored export from the authoritative system, with disputed and ineligible items already excluded. The verifier should report:

• the gross instruction count and amount;

• every detected directed cycle;

• the sparse proposed instruction set;

• before/after net position for every node;

• transfer-count and avoidable-cost reduction; and

• a machine-readable trace binding every output to source obligations.

This proves algebraic equivalence on that frozen dataset. It does not prove the legal right to set off, the correctness of the source ledger, or future savings.

What buyers can do now

There is no admitted cycle-compression SKU or marketplace listing today. Controllers who own all relevant nodes and can cite an existing setoff right can submit a candidate historical replay. Do not send ledger rows, bank details, confidential contracts, or regulated data through the contact form.

Teams buying general automation can separately review the self-managed platform plans and live Stripe checkout. That checkout sells the automation platform, not payment netting or legal authority.

Bottom line

Cycle detection is not the moat and not the hard risk. The hard problem is proving that one buyer has the authority to cancel obligations while every economic, legal, accounting, and audit invariant survives. Our first sweep found no class meeting the full commercial gate, so the product remains unlisted until an exact replay and stranger-buyer receipt exist.