561 Group Market
Exact Verified Network Flow and Multi-Budget Frontier
Exact Verified Network Flow and Multi-Budget Frontier
Exact rational min-cost maximum-flow for directed capacities and constant nonnegative costs. Choose independently checked funded and unbudgeted flows, or up to 32 derived budget quotes from one full optimum. Source-bound Cell and Chain evidence supports replay. Intermediate quotes lack separate primal certificates. Physical validity, fairness, rights, payment and customer acceptance require separate evidence.
Seller-listed; delivery not independently established — A listing is not a completed purchase. The live HTTP 402 challenge supplies current payment terms.
- Price preview
- $0.05
- Method
- POST
- Endpoint
- https://market.561.group/v1/mathematics/exact-resource-flow
- Seller
- Seller not independently established
- Source-rights state
- not-declared
- Offer NI
- ni:///sha-256;zt_kTOVzWx44H1bHqYyV6Gd1mr9n-Nehbek1GW67VQo
- Claim cell
- ni:///sha-256;7kVEiAuSl_gwWUl3TRauBY0XNqOy1iMAXxgH4UKkWvU
- Operation law
- ni:///sha-256;WPLvfGsi2E4KngtL7l0ZRZQ3N1BLn2zbD3DjebkPK9Q
- mathematics
- exact-resource-flow
- science
- ni:///sha-256;zt_kTOVzWx44H1bHqYyV6Gd1mr9n-Nehbek1GW67VQo
- @red-cup-engineering/signcraft
The seller offers this resource under the stated method and contract. The price is a preview, not payment authorization, settlement or a promise of delivery.
Request contract
{
"maxBytes": 65536,
"bodySchema": {
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"oneOf": [
{
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"required": [
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},
"links": {
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"pattern": "^[A-Za-z][A-Za-z0-9_.:-]{0,79}$"
},
"to": {
"type": "string",
"pattern": "^[A-Za-z][A-Za-z0-9_.:-]{0,79}$"
},
"capacity": {
"anyOf": [
{
"type": "string",
"pattern": "^\\d{1,14}(?:/[1-9]\\d{0,13})?$"
},
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},
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}
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"amount": {
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},
{
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"maximum": 99999999999999
}
]
}
}
}
},
"budget": {
"anyOf": [
{
"anyOf": [
{
"type": "string",
"pattern": "^\\d{1,14}(?:/[1-9]\\d{0,13})?$"
},
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"minimum": 0,
"maximum": 99999999999999
}
]
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{
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]
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},
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},
"links": {
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"items": {
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"additionalProperties": false,
"required": [
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"to",
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],
"properties": {
"id": {
"type": "string",
"pattern": "^[A-Za-z][A-Za-z0-9_.:-]{0,79}$"
},
"from": {
"type": "string",
"pattern": "^[A-Za-z][A-Za-z0-9_.:-]{0,79}$"
},
"to": {
"type": "string",
"pattern": "^[A-Za-z][A-Za-z0-9_.:-]{0,79}$"
},
"capacity": {
"anyOf": [
{
"type": "string",
"pattern": "^\\d{1,14}(?:/[1-9]\\d{0,13})?$"
},
{
"type": "integer",
"minimum": 0,
"maximum": 99999999999999
}
]
},
"unitResourceCost": {
"anyOf": [
{
"type": "string",
"pattern": "^\\d{1,14}(?:/[1-9]\\d{0,13})?$"
},
{
"type": "integer",
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"maximum": 99999999999999
}
]
}
}
}
},
"sources": {
"type": "array",
"maxItems": 48,
"items": {
"type": "object",
"additionalProperties": false,
"required": [
"node",
"capacity"
],
"properties": {
"node": {
"type": "string",
"pattern": "^[A-Za-z][A-Za-z0-9_.:-]{0,79}$"
},
"capacity": {
"anyOf": [
{
"type": "string",
"pattern": "^\\d{1,14}(?:/[1-9]\\d{0,13})?$"
},
{
"type": "integer",
"minimum": 0,
"maximum": 99999999999999
}
]
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}
}
},
"demands": {
"type": "array",
"maxItems": 48,
"items": {
"type": "object",
"additionalProperties": false,
"required": [
"node",
"amount"
],
"properties": {
"node": {
"type": "string",
"pattern": "^[A-Za-z][A-Za-z0-9_.:-]{0,79}$"
},
"amount": {
"anyOf": [
{
"type": "string",
"pattern": "^\\d{1,14}(?:/[1-9]\\d{0,13})?$"
},
{
"type": "integer",
"minimum": 0,
"maximum": 99999999999999
}
]
}
}
}
}
},
"not": {
"required": [
"budget"
]
}
},
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"type": "array",
"minItems": 1,
"maxItems": 32,
"items": {
"anyOf": [
{
"type": "string",
"pattern": "^\\d{1,14}(?:/[1-9]\\d{0,13})?$"
},
{
"type": "integer",
"minimum": 0,
"maximum": 99999999999999
}
]
},
"description": "One to 32 exact nonnegative budgets in the SAME source-selected resource unit. This is not monetary spending authorization."
}
}
}
]
},
"example": {
"method": "POST",
"url": "https://market.561.group/v1/mathematics/exact-resource-flow",
"headers": {
"accept": "application/json",
"content-type": "application/json"
},
"body": {
"nodes": [
"lake",
"ridge",
"hamlet"
],
"links": [
{
"id": "a",
"from": "lake",
"to": "ridge",
"capacity": "10",
"unitResourceCost": "1/5"
},
{
"id": "b",
"from": "ridge",
"to": "hamlet",
"capacity": "10",
"unitResourceCost": "3/10"
}
],
"sources": [
{
"node": "lake",
"capacity": "10"
}
],
"demands": [
{
"node": "hamlet",
"amount": "8"
}
],
"budget": "2"
}
}
}Unsigned request example (copy and run)
curl --include --request POST --header 'accept: application/json' --header 'content-type: application/json' --data-raw '{"nodes":["lake","ridge","hamlet"],"links":[{"id":"a","from":"lake","to":"ridge","capacity":"10","unitResourceCost":"1/5"},{"id":"b","from":"ridge","to":"hamlet","capacity":"10","unitResourceCost":"3/10"}],"sources":[{"node":"lake","capacity":"10"}],"demands":[{"node":"hamlet","amount":"8"}],"budget":"2"}' --url 'https://market.561.group/v1/mathematics/exact-resource-flow'This request reads the seller's current HTTP response or 402 terms. It does not authorize or settle a payment. A buyer must independently approve current x402 terms.
Declared output
{
"schema": {
"oneOf": [
{
"type": "object",
"additionalProperties": true,
"required": [
"operation",
"version",
"result",
"verification",
"marketOperation",
"admittedMode",
"evidence",
"baselineVerification"
],
"properties": {
"operation": {
"const": "exact-resource-flow"
},
"version": {
"const": "1"
},
"result": {
"type": "object",
"additionalProperties": true,
"required": [
"model",
"withBudget",
"withoutBudget",
"allDemandFeasible"
],
"properties": {
"model": {
"const": "exact-capacitated-directed-linear-resource-flow/v1"
},
"allDemandFeasible": {
"type": "boolean"
},
"withoutBudget": {
"type": "object",
"required": [
"maximumDelivery",
"minimumResourceForMaximumDelivery"
],
"properties": {
"maximumDelivery": {
"type": "string"
},
"minimumResourceForMaximumDelivery": {
"type": "string"
}
}
},
"withBudget": {
"type": "object",
"required": [
"delivered",
"resourceCost",
"flows",
"served",
"supplied",
"reducedCostPotentials"
],
"properties": {
"delivered": {
"type": "string"
},
"resourceCost": {
"type": "string"
},
"flows": {
"type": "array",
"items": {
"type": "object"
}
},
"served": {
"type": "array",
"items": {
"type": "object"
}
},
"supplied": {
"type": "array",
"items": {
"type": "object"
}
},
"reducedCostPotentials": {
"type": "array",
"items": {
"type": "string"
}
}
}
}
}
},
"verification": {
"type": "object",
"required": [
"verified",
"verifiedDelivered",
"verifiedResourceCost"
],
"properties": {
"verified": {
"const": true
},
"verifiedDelivered": {
"type": "string"
},
"verifiedResourceCost": {
"type": "string"
},
"claim": {
"type": "string"
},
"scope": {
"type": "string"
}
}
},
"marketOperation": {
"const": "original-mathematical-resource-work/v1"
},
"admittedMode": {
"const": "exact-resource-flow"
},
"evidence": {
"type": "object",
"additionalProperties": false,
"required": [
"kind",
"requestId",
"resultId",
"modelId",
"numericalId",
"modelRows",
"numericalRows",
"rows",
"site",
"rules",
"settlementId",
"settledAt",
"claims",
"scope",
"issuerAuthentication"
],
"description": "Compact native generators for source-bound replay: readmitted SignChain rows, SignGraph site presentation, original rules and native settlement identity. Replay binds this evidence to the request and mathematical certificate. Reading it alone does not authenticate an issuer or establish payment.",
"properties": {
"kind": {
"const": "signcraft.exact-resource-evidence/v1"
},
"requestId": {
"type": "string",
"pattern": "^ni:///sha-256;[A-Za-z0-9_-]{43}$"
},
"resultId": {
"type": "string",
"pattern": "^ni:///sha-256;[A-Za-z0-9_-]{43}$"
},
"modelId": {
"type": "string",
"pattern": "^ni:///sha-256;[A-Za-z0-9_-]{43}$"
},
"numericalId": {
"type": "string",
"pattern": "^ni:///sha-256;[A-Za-z0-9_-]{43}$"
},
"settlementId": {
"type": "string",
"pattern": "^ni:///sha-256;[A-Za-z0-9_-]{43}$"
},
"modelRows": {
"type": "array",
"minItems": 1,
"items": {
"type": "object",
"additionalProperties": false,
"required": [
"subject",
"predicate",
"object",
"side",
"witness"
],
"properties": {
"subject": {},
"predicate": {
"type": "string"
},
"object": {},
"side": {
"enum": [
"for",
"against"
]
},
"witness": {
"type": "object"
}
}
}
},
"numericalRows": {
"type": "array",
"minItems": 1,
"items": {
"type": "object",
"additionalProperties": false,
"required": [
"subject",
"predicate",
"object",
"side",
"witness"
],
"properties": {
"subject": {},
"predicate": {
"type": "string"
},
"object": {},
"side": {
"enum": [
"for",
"against"
]
},
"witness": {
"type": "object"
}
}
}
},
"rows": {
"type": "array",
"minItems": 1,
"items": {
"type": "object",
"additionalProperties": false,
"required": [
"subject",
"predicate",
"object",
"side",
"witness"
],
"properties": {
"subject": {},
"predicate": {
"type": "string"
},
"object": {},
"side": {
"enum": [
"for",
"against"
]
},
"witness": {
"type": "object"
}
}
}
},
"site": {
"type": "object",
"additionalProperties": false,
"required": [
"kind",
"cells",
"specializations"
],
"properties": {
"kind": {
"const": "signgraph.site"
},
"cells": {
"type": "array",
"minItems": 1,
"items": {
"type": "object",
"additionalProperties": false,
"required": [
"subject",
"predicate",
"object"
],
"properties": {
"subject": {},
"predicate": {},
"object": {}
}
}
},
"specializations": {
"type": "array",
"items": {
"type": "object",
"required": [
"kind",
"domain",
"codomain"
],
"properties": {
"kind": {
"const": "signgraph.specialization"
},
"domain": {},
"codomain": {}
}
}
}
}
},
"rules": {
"type": "array",
"minItems": 1,
"items": {
"type": "object",
"additionalProperties": false,
"required": [
"kind",
"name",
"antecedent",
"consequent"
],
"properties": {
"kind": {
"const": "signchain.rule"
},
"name": {
"type": "string",
"minLength": 1
},
"antecedent": {
"type": "array",
"items": {
"type": "object",
"required": [
"cell",
"atLeast"
],
"properties": {
"cell": {
"type": "object",
"additionalProperties": false,
"required": [
"subject",
"predicate",
"object"
],
"properties": {
"subject": {},
"predicate": {},
"object": {}
}
},
"atLeast": {
"enum": [
"absurd",
"potential",
"active",
"stable"
]
}
}
}
},
"consequent": {
"type": "object",
"required": [
"cell",
"reading"
],
"properties": {
"cell": {
"type": "object",
"additionalProperties": false,
"required": [
"subject",
"predicate",
"object"
],
"properties": {
"subject": {},
"predicate": {},
"object": {}
}
},
"reading": {
"enum": [
"absurd",
"potential",
"active",
"stable"
]
}
}
}
}
}
},
"settledAt": {
"type": "integer",
"minimum": 0
},
"claims": {
"type": "object",
"additionalProperties": false,
"required": [
"conditional",
"customerAcceptance",
"payment"
],
"properties": {
"conditional": {
"type": "object",
"additionalProperties": false,
"required": [
"subject",
"predicate",
"object"
],
"properties": {
"subject": {},
"predicate": {},
"object": {}
}
},
"customerAcceptance": {
"type": "object",
"additionalProperties": false,
"required": [
"subject",
"predicate",
"object"
],
"properties": {
"subject": {},
"predicate": {},
"object": {}
}
},
"payment": {
"type": "object",
"additionalProperties": false,
"required": [
"subject",
"predicate",
"object"
],
"properties": {
"subject": {},
"predicate": {},
"object": {}
}
}
}
},
"scope": {
"type": "string",
"minLength": 1
},
"issuerAuthentication": {
"const": "not-established"
}
}
},
"baselineVerification": {
"type": "object",
"required": [
"verified",
"verifiedDelivered",
"verifiedResourceCost"
],
"properties": {
"verified": {
"const": true
},
"verifiedDelivered": {
"type": "string"
},
"verifiedResourceCost": {
"type": "string"
},
"claim": {
"type": "string"
},
"scope": {
"type": "string"
}
}
}
}
},
{
"type": "object",
"additionalProperties": true,
"required": [
"operation",
"version",
"originalOptimum",
"marginalCostFrontier",
"budgetQuotes",
"checkedUnbudgetedOriginal",
"checkedUnbudgetedReference",
"marketOperation",
"admittedMode",
"evidence"
],
"properties": {
"operation": {
"const": "exact-resource-frontier"
},
"version": {
"const": "1"
},
"optimizationPasses": {
"const": 1
},
"originalOptimum": {
"type": "object",
"required": [
"maximumDelivery",
"minimumResourceForMaximumDelivery",
"remainingDemand",
"witness"
],
"properties": {
"maximumDelivery": {
"type": "string"
},
"minimumResourceForMaximumDelivery": {
"type": "string"
},
"remainingDemand": {
"type": "string"
},
"witness": {
"type": "object",
"required": [
"delivered",
"resourceCost",
"flows",
"served",
"supplied",
"reducedCostPotentials"
],
"properties": {
"delivered": {
"type": "string"
},
"resourceCost": {
"type": "string"
},
"flows": {
"type": "array",
"items": {
"type": "object"
}
},
"served": {
"type": "array",
"items": {
"type": "object"
}
},
"supplied": {
"type": "array",
"items": {
"type": "object"
}
},
"reducedCostPotentials": {
"type": "array",
"items": {
"type": "string"
}
}
}
}
}
},
"marginalCostFrontier": {
"type": "array",
"items": {
"type": "object",
"required": [
"fromDelivered",
"untilDelivered",
"fromResourceCost",
"untilResourceCost",
"resourceCostPerDeliveredUnit"
],
"properties": {
"fromDelivered": {
"type": "string"
},
"untilDelivered": {
"type": "string"
},
"fromResourceCost": {
"type": "string"
},
"untilResourceCost": {
"type": "string"
},
"resourceCostPerDeliveredUnit": {
"type": "string"
}
}
}
},
"budgetQuotes": {
"type": "array",
"minItems": 1,
"maxItems": 32,
"items": {
"type": "object",
"required": [
"budget",
"maximumDelivery",
"minimumResourceForMaximumDelivery",
"unspentResource"
],
"properties": {
"budget": {
"type": "string"
},
"maximumDelivery": {
"type": "string"
},
"minimumResourceForMaximumDelivery": {
"type": "string"
},
"unspentResource": {
"type": "string"
}
}
}
},
"checkedUnbudgetedOriginal": {
"type": "object",
"required": [
"verified"
],
"properties": {
"verified": {
"const": true
}
}
},
"checkedUnbudgetedReference": {
"type": "object",
"required": [
"verified"
],
"properties": {
"verified": {
"const": true
}
}
},
"marketOperation": {
"const": "original-mathematical-resource-work/v1"
},
"admittedMode": {
"const": "exact-resource-frontier"
},
"evidence": {
"type": "object",
"additionalProperties": false,
"required": [
"kind",
"requestId",
"resultId",
"modelId",
"numericalId",
"modelRows",
"numericalRows",
"rows",
"site",
"rules",
"settlementId",
"settledAt",
"claims",
"scope",
"issuerAuthentication"
],
"description": "Compact native generators for source-bound replay: readmitted SignChain rows, SignGraph site presentation, original rules and native settlement identity. Replay binds this evidence to the request and mathematical certificate. Reading it alone does not authenticate an issuer or establish payment.",
"properties": {
"kind": {
"const": "signcraft.exact-resource-evidence/v1"
},
"requestId": {
"type": "string",
"pattern": "^ni:///sha-256;[A-Za-z0-9_-]{43}$"
},
"resultId": {
"type": "string",
"pattern": "^ni:///sha-256;[A-Za-z0-9_-]{43}$"
},
"modelId": {
"type": "string",
"pattern": "^ni:///sha-256;[A-Za-z0-9_-]{43}$"
},
"numericalId": {
"type": "string",
"pattern": "^ni:///sha-256;[A-Za-z0-9_-]{43}$"
},
"settlementId": {
"type": "string",
"pattern": "^ni:///sha-256;[A-Za-z0-9_-]{43}$"
},
"modelRows": {
"type": "array",
"minItems": 1,
"items": {
"type": "object",
"additionalProperties": false,
"required": [
"subject",
"predicate",
"object",
"side",
"witness"
],
"properties": {
"subject": {},
"predicate": {
"type": "string"
},
"object": {},
"side": {
"enum": [
"for",
"against"
]
},
"witness": {
"type": "object"
}
}
}
},
"numericalRows": {
"type": "array",
"minItems": 1,
"items": {
"type": "object",
"additionalProperties": false,
"required": [
"subject",
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{
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}
}How to buy with x402
- Send the source's unsigned request. An admitted payable request receives HTTP 402 and a
PAYMENT-REQUIREDheader with its current amount, asset, network and payee. - Your x402 client signs the accepted terms and repeats the same request with
PAYMENT-SIGNATURE. - A successful settled response contains the purchased result and its
PAYMENT-RESPONSEreceipt.
Source and machine access
The source SKU, claim cell and operation law above are the identities used by the machine catalogue and this page.