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DOCUMENTATION

[01/01]
Documentation · API Reference · Quick Start

From zero to
first computation.

From zero to your first grounded physics computation in under 2 minutes.

Looking for the full SDK documentation?

This page is the quick-start. The complete reference — every equation, every operator, every endpoint, language guides, MCP integration, kernel internals and the full 1,500+-operator catalogue — lives in the Zeq SDK documentation site. If you only read one thing, read that.

1

Get your API key

Go to Pricing and start your free 14-day trial — no credit card required. Enter your email and your zeq_ak_… API key is generated instantly and shown once in your trial portal. Copy it immediately.

The key is shown exactly once. Save it to a password manager or add it to your environment variables immediately. If you lose it, you can regenerate it from the portal.
2

Add it to your environment

In your project, open the Secrets panel (padlock icon in the left sidebar, or press Ctrl+K and search "environment variables"). Add a new secret:

Key ZEQ_API_KEY
Value zeq_ak_…

Restart your Repl after adding a secret so the environment variable is available.

3

Make your first compute request

Call POST /api/zeq/compute with a physics domain and inputs. Working examples in 10 languages — click a tab to switch:

curl -X POST https://www.zeq.dev/api/zeq/compute \ -H "Authorization: Bearer $ZEQ_API_KEY" \ -H "Content-Type: application/json" \ -d '{ "domain": "quantum_mechanics", "operators": ["KO42"], "inputs": { "E": 13.6, "m": 9.11e-31 } }'
const res = await fetch('https://www.zeq.dev/api/zeq/compute', { method: 'POST', headers: { 'Authorization': `Bearer ${process.env.ZEQ_API_KEY}`, 'Content-Type': 'application/json', }, body: JSON.stringify({ domain: 'quantum_mechanics', operators: ['KO42'], inputs: { E: 13.6, m: 9.11e-31 } }) }); const { zeqState, result } = await res.json(); console.log('R(t):', zeqState.masterSum); console.log('Phase:', zeqState.phase); console.log('Precision:', zeqState.precision); // always ≤ 0.001
interface ZeqState { masterSum: number; phase: number; precision: number; zeqond: number; operators: string[]; } interface ComputeResponse { zeqState: ZeqState; result: Record<string, unknown>; } const res = await fetch('https://www.zeq.dev/api/zeq/compute', { method: 'POST', headers: { 'Authorization': `Bearer ${process.env.ZEQ_API_KEY}`, 'Content-Type': 'application/json', }, body: JSON.stringify({ domain: 'quantum_mechanics', operators: ['KO42'], inputs: { E: 13.6, m: 9.11e-31 } }) }); const { zeqState }: ComputeResponse = await res.json(); console.log('R(t):', zeqState.masterSum); console.log('Precision:', zeqState.precision); // always ≤ 0.001
import requests, os resp = requests.post( "https://www.zeq.dev/api/zeq/compute", headers={"Authorization": f"Bearer {os.environ['ZEQ_API_KEY']}"}, json={ "domain": "quantum_mechanics", "operators": ["KO42"], "inputs": {"E": 13.6, "m": 9.11e-31} } ) data = resp.json() zeq = data["zeqState"] print("R(t):", zeq["masterSum"]) print("Phase:", zeq["phase"]) print("Precision:", zeq["precision"]) # always ≤ 0.001 assert "KO42" in zeq["operators"] # always present
package main import ( "bytes" "encoding/json" "fmt" "net/http" "os" ) func main() { body, _ := json.Marshal(map[string]any{ "domain": "quantum_mechanics", "operators": []string{"KO42"}, "inputs": map[string]float64{"E": 13.6, "m": 9.11e-31}, }) req, _ := http.NewRequest("POST", "https://www.zeq.dev/api/zeq/compute", bytes.NewBuffer(body)) req.Header.Set("Authorization", "Bearer "+os.Getenv("ZEQ_API_KEY")) req.Header.Set("Content-Type", "application/json") resp, _ := http.DefaultClient.Do(req) var data map[string]any json.NewDecoder(resp.Body).Decode(&data) zeq := data["zeqState"].(map[string]any) fmt.Println("R(t):", zeq["masterSum"]) fmt.Println("Precision:", zeq["precision"]) // always ≤ 0.001 }
use reqwest::header; use serde_json::{json, Value}; #[tokio::main] async fn main() -> Result<(), Box<dyn std::error::Error>> { let api_key = std::env::var("ZEQ_API_KEY")?; let client = reqwest::Client::new(); let body = json!({ "domain": "quantum_mechanics", "operators": ["KO42"], "inputs": { "E": 13.6, "m": 9.11e-31_f64 } }); let resp: Value = client .post("https://www.zeq.dev/api/zeq/compute") .header(header::AUTHORIZATION, format!("Bearer {}", api_key)) .json(&body) .send() .await? .json() .await?; let zeq = &resp["zeqState"]; println!("R(t): {}", zeq["masterSum"]); println!("Precision: {}", zeq["precision"]); // always ≤ 0.001 Ok(()) }
import java.net.URI; import java.net.http.*; import java.net.http.HttpRequest.BodyPublishers; public class ZeqCompute { public static void main(String[] args) throws Exception { String apiKey = System.getenv("ZEQ_API_KEY"); String body = """ {"domain":"quantum_mechanics", "operators":["KO42"], "inputs":{"E":13.6,"m":9.11e-31}} """; HttpRequest request = HttpRequest.newBuilder() .uri(URI.create("https://www.zeq.dev/api/zeq/compute")) .header("Authorization", "Bearer " + apiKey) .header("Content-Type", "application/json") .POST(BodyPublishers.ofString(body)) .build(); HttpResponse<String> resp = HttpClient.newHttpClient().send(request, HttpResponse.BodyHandlers.ofString()); System.out.println(resp.body()); // includes zeqState.precision ≤ 0.001 } }
using System.Net.Http.Json; using System.Text.Json; var apiKey = Environment.GetEnvironmentVariable("ZEQ_API_KEY"); using var client = new HttpClient(); client.DefaultRequestHeaders.Add("Authorization", $"Bearer {apiKey}"); var payload = new { domain = "quantum_mechanics", operators = new[] { "KO42" }, inputs = new { E = 13.6, m = 9.11e-31 } }; var response = await client.PostAsJsonAsync( "https://www.zeq.dev/api/zeq/compute", payload); var data = await response.Content.ReadFromJsonAsync<JsonElement>(); var zeq = data.GetProperty("zeqState"); Console.WriteLine($"R(t): {zeq.GetProperty("masterSum")}"); Console.WriteLine($"Precision: {zeq.GetProperty("precision")}"); // ≤ 0.001
require 'net/http' require 'json' require 'uri' uri = URI('https://www.zeq.dev/api/zeq/compute') body = { domain: 'quantum_mechanics', operators: ['KO42'], inputs: { E: 13.6, m: 9.11e-31 } }.to_json req = Net::HTTP::Post.new(uri) req['Authorization'] = "Bearer #{ENV['ZEQ_API_KEY']}" req['Content-Type'] = 'application/json' req.body = body resp = Net::HTTP.start(uri.host, uri.port, use_ssl: true) { |h| h.request(req) } data = JSON.parse(resp.body) zeq = data['zeqState'] puts "R(t): #{zeq['masterSum']}" puts "Precision: #{zeq['precision']}" # always ≤ 0.001
<?php $apiKey = getenv('ZEQ_API_KEY'); $payload = json_encode([ 'domain' => 'quantum_mechanics', 'operators' => ['KO42'], 'inputs' => ['E' => 13.6, 'm' => 9.11e-31], ]); $ch = curl_init('https://www.zeq.dev/api/zeq/compute'); curl_setopt_array($ch, [ CURLOPT_RETURNTRANSFER => true, CURLOPT_POST => true, CURLOPT_POSTFIELDS => $payload, CURLOPT_HTTPHEADER => [ "Authorization: Bearer $apiKey", 'Content-Type: application/json', ], ]); $data = json_decode(curl_exec($ch), true); $zeq = $data['zeqState']; echo "R(t): " . $zeq['masterSum'] . "\n"; echo "Precision: " . $zeq['precision'] . "\n"; // always ≤ 0.001
4

Read the ZeqState response

Every successful compute request returns a zeqState object alongside the domain-specific result. Here is the full ZeqState structure:

Field Type Description
operators string[] Operator IDs applied to this computation. Always includes KO42.
masterSum number R(t) — the grounded result. Approximately 1.000xxx. This is the HulyaPulse-modulated output of the master equation.
phase number φ — current position within the HulyaPulse cycle. Range: 0.0000–1.0000. Two calls at the same Zeqond have the same phase.
zeqond integer τ — Zeqond count at compute time. One Zeqond = 0.777 seconds. Use this to replay any computation reproducibly.
domain string The physics domain that was computed (e.g. "Quantum Mechanics").
precision number Fractional precision of this result. Always ≤ 0.001 (≤ 0.1%). If this exceeds 0.001, the result should be treated as invalid.
pulseHz number The HulyaPulse frequency. Always 1.287. A constant, included for verification.

Tip: validate KO42 presence and precision ≤ 0.001 in every response before using the result.

5

Connect your AI client via MCP (optional)

If you use Claude Desktop, Cursor, or Windsurf, you can connect Zeq.dev as a native AI tool. Add this to your AI client's config file, replace the placeholder key, and restart:

// ~/Library/Application Support/Claude/claude_desktop_config.json (macOS) // %APPDATA%\Claude\claude_desktop_config.json (Windows) { "mcpServers": { "zeq-dev": { "type": "http", "url": "https://www.zeq.dev/api/mcp", "headers": { "Authorization": "Bearer zeq_ak_YOUR_KEY_HERE" } } } }
// ~/.cursor/mcp.json (or Settings > MCP > Add Server) { "mcpServers": { "zeq-dev": { "type": "http", "url": "https://www.zeq.dev/api/mcp", "headers": { "Authorization": "Bearer zeq_ak_YOUR_KEY_HERE" } } } }
// Windsurf: Settings > AI > MCP Servers > Add { "mcpServers": { "zeq-dev": { "type": "http", "url": "https://www.zeq.dev/api/mcp", "headers": { "Authorization": "Bearer zeq_ak_YOUR_KEY_HERE" } } } }

Once connected, your AI gains 50+ native tools. Start with zeq_api_catalog (search the full public API) and zeq_http (call any endpoint), plus zeq_compute, zeq_spin_up, the zeq_contract_* family, zeq_publish_page, and 40+ domain solvers. Call tools/list against this node for the live set.

Auth & Rate Limits

Endpoints & access rules

All authenticated endpoints require an Authorization: Bearer zeq_ak_… header. Public endpoints have no auth requirement but are still rate-limited per IP. Compute is charged per Zeqond against your compute-hours — this cycle’s bond first, then hours you bought. There are no daily budgets; out of hours returns 402 INSUFFICIENT_FUEL.

Endpoint Auth Rate limit Consumes tokens
POST /api/zeq/computeAuth60 / min1 per call
GET  /api/zeq/pulsePublic60 / min0 — always free
GET  /api/zeq/pulse/streamAuth60 / min0 — SSE keepalive
POST /api/zeq/verifyAuth60 / min0 — proof check only
POST /api/zeq/latticeAuth60 / minN tokens (1 per node, 2–5)
POST /api/zeq/shiftAuth60 / minsteps tokens (1–64)
GET  /api/zeq/usageAuth60 / min0
GET  /api/operatorsPublic60 / min0
GET  /api/operators/categoriesPublic60 / min0
GET  /api/demo-keyPublic30 / min0 — issues trial key
Free
1 h compute-hour / 30 days
Starter · $29/mo
9.8 h compute-hours / cycle · $2.97/h
Builder · $79/mo
26.6 h compute-hours / cycle · $2.97/h
Advanced · $199/mo
67.0 h compute-hours / cycle · $2.97/h
Architect · $499/mo
168.0 h compute-hours / cycle · $2.97/h
Operators API

Browse 1,500+ catalogued operators

Two public endpoints expose the full operator library — no API key required. Use them to build domain selectors, search UIs, or populate AI tool registries.

GET /api/operators Public 60 req/min · No tokens consumed

Returns the full operator list with optional filtering. All query parameters are optional and combinable. Underscore and space notation are both accepted for multi-word domains.

Query paramTypeDescription
searchstringSubstring match on operator name, ID, domain, or description (case-insensitive)
domainstringFilter by domain — underscores or spaces accepted: quantum_mechanics and Quantum Mechanics both work. Alias of category.
categorystringSame as domain. Underscores or spaces accepted: fluid_dynamics or thermodynamics.
domainGroupstringFilter to a group: Core Physics · Extended Physics · Applied Sciences · Industry · Frontier
limitintegerCap the number of operators returned after all filters. Omit or pass 0 to return all matching results. Response always includes the untruncated filtered count.
# search across all domains curl "https://www.zeq.dev/api/operators?search=entropy&limit=5" # filter by domain (underscores and spaces both accepted) curl "https://www.zeq.dev/api/operators?domain=quantum_mechanics&limit=10" # filter by group curl "https://www.zeq.dev/api/operators?domainGroup=Core+Physics&search=entropy"
{ "operators": [ { "id": "sm-entropy-calc", "name": "Entropy Calc", "domain": "Statistical Mechanics", "domainGroup": "Core Physics", "equation": "S = k_B ln \u03a9", "precision": 0.000129, "description": "Boltzmann entropy for discrete microstates", "sdkUsage": "zeq.compute({ domain: 'statistical_mechanics', operators: ['sm-entropy-calc'] })" } ], "total": 1536, "filtered": 1, "returned": 1, "source": "local" }
GET /api/operators/categories Public 60 req/min · No tokens consumed

Returns all 64 domain definitions sorted by group then name — including prefix codes and per-domain operator counts.

curl "https://www.zeq.dev/api/operators/categories"
{ "categories": [ { "domain": "Atmospheric Physics", "domainGroup": "Applied Sciences", "prefix": "AT", "count": 24, "color": "#ffd700" }, { "domain": "Biophysics", "domainGroup": "Applied Sciences", "prefix": "BP", "count": 24, "color": "#ffd700" }, { "domain": "Chemical Physics", "domainGroup": "Applied Sciences", "prefix": "CP", "count": 24, "color": "#ffd700" }, { "domain": "Climate Modeling", "domainGroup": "Applied Sciences", "prefix": "CL", "count": 24, "color": "#ffd700" }, { "domain": "Earth Sciences", "domainGroup": "Applied Sciences", "prefix": "ERS", "count": 24, "color": "#ffd700" }, { "domain": "Environmental Physics", "domainGroup": "Applied Sciences", "prefix": "EN", "count": 24, "color": "#ffd700" }, { "domain": "Geophysics", "domainGroup": "Applied Sciences", "prefix": "GE", "count": 24, "color": "#ffd700" }, { "domain": "Hydrology", "domainGroup": "Applied Sciences", "prefix": "HY", "count": 24, "color": "#ffd700" }, { "domain": "Materials Science", "domainGroup": "Applied Sciences", "prefix": "MS", "count": 24, "color": "#ffd700" }, { "domain": "Oceanography", "domainGroup": "Applied Sciences", "prefix": "OC", "count": 24, "color": "#ffd700" }, { "domain": "Seismology", "domainGroup": "Applied Sciences", "prefix": "SE", "count": 24, "color": "#ffd700" }, { "domain": "Soil Physics", "domainGroup": "Applied Sciences", "prefix": "SOP", "count": 24, "color": "#ffd700" }, { "domain": "Acoustics", "domainGroup": "Core Physics", "prefix": "AC", "count": 24, "color": "#00ff88" }, { "domain": "Classical Mechanics", "domainGroup": "Core Physics", "prefix": "CM", "count": 24, "color": "#00ff88" }, { "domain": "Electromagnetism", "domainGroup": "Core Physics", "prefix": "EM", "count": 24, "color": "#00ff88" }, { "domain": "Fluid Dynamics", "domainGroup": "Core Physics", "prefix": "FD", "count": 24, "color": "#00ff88" }, { "domain": "Gravitational Physics", "domainGroup": "Core Physics", "prefix": "GP", "count": 24, "color": "#00ff88" }, { "domain": "Newtonian Mechanics", "domainGroup": "Core Physics", "prefix": "NM", "count": 24, "color": "#00ff88" }, { "domain": "Nuclear Physics", "domainGroup": "Core Physics", "prefix": "NP", "count": 24, "color": "#00ff88" }, { "domain": "Optics", "domainGroup": "Core Physics", "prefix": "OP", "count": 24, "color": "#00ff88" }, { "domain": "Particle Physics", "domainGroup": "Core Physics", "prefix": "PP", "count": 24, "color": "#00ff88" }, { "domain": "Quantum Mechanics", "domainGroup": "Core Physics", "prefix": "QM", "count": 24, "color": "#00ff88" }, { "domain": "Quantum Resonance", "domainGroup": "Core Physics", "prefix": "QR", "count": 24, "color": "#00ff88" }, { "domain": "Relativistic Physics", "domainGroup": "Core Physics", "prefix": "RP", "count": 24, "color": "#00ff88" }, { "domain": "Statistical Mechanics", "domainGroup": "Core Physics", "prefix": "SM", "count": 24, "color": "#00ff88" }, { "domain": "Thermodynamics", "domainGroup": "Core Physics", "prefix": "TH", "count": 24, "color": "#00ff88" }, { "domain": "Wave Physics", "domainGroup": "Core Physics", "prefix": "WP", "count": 24, "color": "#00ff88" }, { "domain": "Zeq Resonance Theory", "domainGroup": "Core Physics", "prefix": "ZR", "count": 24, "color": "#00ff88" }, { "domain": "Astrophysics", "domainGroup": "Extended Physics", "prefix": "AS", "count": 24, "color": "#00d4ff" }, { "domain": "Condensed Matter", "domainGroup": "Extended Physics", "prefix": "CD", "count": 24, "color": "#00d4ff" }, { "domain": "Cosmology", "domainGroup": "Extended Physics", "prefix": "CO", "count": 24, "color": "#00d4ff" }, { "domain": "Crystallography", "domainGroup": "Extended Physics", "prefix": "CR", "count": 24, "color": "#00d4ff" }, { "domain": "Nanophysics", "domainGroup": "Extended Physics", "prefix": "NA", "count": 24, "color": "#00d4ff" }, { "domain": "Photonics", "domainGroup": "Extended Physics", "prefix": "PT", "count": 24, "color": "#00d4ff" }, { "domain": "Plasma Physics", "domainGroup": "Extended Physics", "prefix": "PL", "count": 24, "color": "#00d4ff" }, { "domain": "Quantum Computing", "domainGroup": "Extended Physics", "prefix": "QC", "count": 24, "color": "#00d4ff" }, { "domain": "Quantum Field Theory", "domainGroup": "Extended Physics", "prefix": "QF", "count": 24, "color": "#00d4ff" }, { "domain": "Spintronics", "domainGroup": "Extended Physics", "prefix": "SP", "count": 24, "color": "#00d4ff" }, { "domain": "Superconductivity", "domainGroup": "Extended Physics", "prefix": "SC", "count": 24, "color": "#00d4ff" }, { "domain": "Analogue Gravity", "domainGroup": "Frontier", "prefix": "AG", "count": 24, "color": "#b44fff" }, { "domain": "Chaotic Systems", "domainGroup": "Frontier", "prefix": "CH", "count": 24, "color": "#b44fff" }, { "domain": "Observer-Effect Models", "domainGroup": "Frontier", "prefix": "CN", "count": 24, "color": "#b44fff" }, { "domain": "Dark Energy Models", "domainGroup": "Frontier", "prefix": "DE", "count": 24, "color": "#b44fff" }, { "domain": "Emergent Complexity", "domainGroup": "Frontier", "prefix": "EC", "count": 24, "color": "#b44fff" }, { "domain": "Extra Dimensions", "domainGroup": "Frontier", "prefix": "XD", "count": 24, "color": "#b44fff" }, { "domain": "Fractal Mechanics", "domainGroup": "Frontier", "prefix": "FM", "count": 24, "color": "#b44fff" }, { "domain": "Neurophysics", "domainGroup": "Frontier", "prefix": "NR", "count": 24, "color": "#b44fff" }, { "domain": "Non-Euclidean Dynamics","domainGroup": "Frontier", "prefix": "NED","count": 24, "color": "#b44fff" }, { "domain": "Quantum Gravity", "domainGroup": "Frontier", "prefix": "QG", "count": 24, "color": "#b44fff" }, { "domain": "Quantum Information", "domainGroup": "Frontier", "prefix": "QI", "count": 24, "color": "#b44fff" }, { "domain": "Time Crystal Physics", "domainGroup": "Frontier", "prefix": "TC", "count": 24, "color": "#b44fff" }, { "domain": "Topological Matter", "domainGroup": "Frontier", "prefix": "TM", "count": 24, "color": "#b44fff" }, { "domain": "Aerospace Engineering", "domainGroup": "Industry", "prefix": "AE", "count": 24, "color": "#ff6b35" }, { "domain": "Automotive Engineering","domainGroup": "Industry", "prefix": "AUTO","count": 24, "color": "#ff6b35" }, { "domain": "Biomedical Engineering","domainGroup": "Industry", "prefix": "BE", "count": 24, "color": "#ff6b35" }, { "domain": "Chemical Engineering", "domainGroup": "Industry", "prefix": "CHE", "count": 24, "color": "#ff6b35" }, { "domain": "Communications", "domainGroup": "Industry", "prefix": "CM2", "count": 24, "color": "#ff6b35" }, { "domain": "Control Systems", "domainGroup": "Industry", "prefix": "CS2", "count": 24, "color": "#ff6b35" }, { "domain": "Nuclear Engineering", "domainGroup": "Industry", "prefix": "NE", "count": 24, "color": "#ff6b35" }, { "domain": "Power Systems", "domainGroup": "Industry", "prefix": "PW", "count": 24, "color": "#ff6b35" }, { "domain": "Robotics", "domainGroup": "Industry", "prefix": "RO", "count": 24, "color": "#ff6b35" }, { "domain": "Semiconductor Design", "domainGroup": "Industry", "prefix": "SD", "count": 24, "color": "#ff6b35" }, { "domain": "Signal Processing", "domainGroup": "Industry", "prefix": "SI", "count": 24, "color": "#ff6b35" }, { "domain": "Structural Engineering","domainGroup": "Industry", "prefix": "SW", "count": 24, "color": "#ff6b35" } ], "groups": ["Core Physics", "Extended Physics", "Applied Sciences", "Industry", "Frontier"], "totalOperators": 1536, "totalDomains": 64, "source": "local" }
GroupDomains
Core Physics16
Extended Physics11
Applied Sciences12
Industry12
Frontier13
Total64
7-Step Protocol

ZEQOND Computation Protocol

Every query to /api/zeq/compute, /api/zeq/lattice, /api/zeq/shift, and POST /api/mcp (JSON-RPC tools/call) runs through all seven steps in sequence. No query can skip a step. Constants are sourced directly from NIST CODATA 2018. The response always includes a protocol_steps[] array so you can audit what ran at each step.

STEP 1
SELECT

Resolve operator IDs from the 1,500+ verified physics operators. If none are specified, intent-aware selection picks the best match for your domain and inputs.

STEP 2
BIND

Bind NIST CODATA 2018 physical constants (ℏ, c, G, k_B, ε₀, N_A…) to the selected operators. Constants are scoped to the domain group (Core / Extended / Applied / Industry / Frontier).

STEP 3
VALIDATE

Dimensional analysis and domain constraint checks. Mass must be ≥ 0, velocity must be < c, temperature must be ≥ 0 K, and all values must be finite. Violations are clamped and flagged in warnings[].

STEP 4
COMPUTE

Domain-specific physics computation. Core domains use precise analytical solvers (quantum: E_n = n²π²ℏ²/2mL², orbital: v = √(G_mod·M/r), thermal: P = nRT/V, EM: F = q²/4πε₀r², …). Other domains use KO42-modulated formula evaluation.

STEP 5
VERIFY

Cross-check the result against the KO42 precision bound (≤ 0.1%). If the relative uncertainty exceeds 0.001, a precision warning is added to protocol_steps[4].detail. The result is never rejected — flagging is informational.

STEP 6
PULSE

Synchronise to the 1.287 Hz HulyaPulse. Applies the KO42 modulation: R(t) = S(t) × [1 + α·sin(2π·f_H·t)] where α = 1.29×10⁻³ and τ = 0.777 s. Phase and Zeqond index are recorded.

STEP 7
RETURN

Assemble the structured result: value, unit, uncertainty, operator_id, zeqState, result, protocol_steps[], and zeqProof HMAC. API version 2.0. VX mode label applied if the query ran in degraded (quota-exhausted) mode.

[VX MODE]
An account out of compute-hours receives mode: "VX" or a 402 — never a silent or degraded answer.

When an account runs out of compute-hours, Step 4 runs with the VX operator (KO42 ground state only) and the response includes "mode":"VX" where the surface supports it; otherwise the call returns 402 INSUFFICIENT_FUEL. There are no daily allowances — compute is metered per Zeqond against your hours.

{ "mode": "full", "value": 1.3606e-18, "unit": "J", "uncertainty": 3.4015e-22, "operator_id": "QM-001", "protocol_steps": [ { "step": 1, "name": "SELECT", "status": "ok", "detail": { "domain": "quantum-mechanics", "operators_selected": ["QM-001"] }, "durationMs": 1 }, { "step": 2, "name": "BIND", "status": "ok", "detail": { "nist_release": "CODATA 2018", "constants_bound": ["hbar","c","G","k_B","e","epsilon_0","mu_0","m_e","m_p"] }, "durationMs": 0 }, { "step": 3, "name": "VALIDATE","status": "ok", "detail": { "input_count": 2, "warnings": [] }, "durationMs": 0 }, { "step": 4, "name": "COMPUTE", "status": "ok", "detail": { "solver": "QM domain solver", "value": 1.3606e-18, "equation": "E_n = n\u00b2\u03c0\u00b2\u210f\u00b2 / (2mL\u00b2)" }, "durationMs": 0 }, { "step": 5, "name": "VERIFY", "status": "ok", "detail": { "precision_target": "\u22640.1% (0.001)", "precision_actual": 0.000025, "precision_ok": true }, "durationMs": 0 }, { "step": 6, "name": "PULSE", "status": "ok", "detail": { "hulyapulse_hz": 1.287, "R_t": 1.000201, "modulation": "R(t) = S(t) \u00d7 [1 + \u03b1_K \u00d7 sin(2\u03c0 \u00d7 f_H \u00d7 t)]" }, "durationMs": 0 }, { "step": 7, "name": "RETURN", "status": "ok", "detail": { "mode": "full" }, "durationMs": 0 } ], "zeqState": { "operators": ["KO42", "QM-001"], "masterSum": 1.000201, "domain": "quantum-mechanics", "precision": 0.000129, "pulseHz": 1.287, "zeqond": 2156842, "phase": 0.4312 }, "result": { "S_t": 1.0, "R_t": 1.000201, "value": 1.3606e-18, "unit": "J", "uncertainty": 3.4015e-22, "inputs": { "n": 1, "L": 5.29e-9 } }, "meta": { "computedAt": "2026-03-30T00:00:00.000Z", "apiVersion": "2.0", "totalMs": 2, "nistRelease": "CODATA 2018", "sdkVersion": "6.3.0" }, "zeqProof": "a1b2c3...64chars" }
curl -X POST https://www.zeq.dev/api/zeq/compute \ -H "Authorization: Bearer $ZEQ_API_KEY" \ -H "Content-Type: application/json" \ -d '{ "domain": "quantum-mechanics", "inputs": { "n": 1, "L": 5.29e-9 }, "operators": ["QM-001"] }'

All API endpoints

MethodEndpointAuthTokensDescription
POST/api/zeq/computeAuth17-step physics computation. Returns value, unit, uncertainty, protocol_steps[].
GET/api/zeq/pulsePublic0Live HulyaPulse snapshot. Zeqond τ, phase φ, R(t), precision.
GET/api/zeq/pulse/streamAuth0SSE stream — live HulyaPulse event every 777 ms.
POST/api/zeq/verifyAuth0Verify a zeqProof HMAC returned by /compute or /lattice.
POST/api/zeq/latticeAuthN nodesZeqLattice — multi-node coherence grid. Each node runs the full 7-step wizard.
POST/api/zeq/shiftAuthN stepsZeqShift — time-series projection across N future Zeqond steps.
GET/api/zeq/usageAuth0Token budget: used, remaining, plan, reset time.
GET/api/operatorsPublic0All 1,500+ catalogued operators. Filter by domain with ?domain=…
GET/api/operators/:idPublic0Single operator by ID (e.g. QM-001, KO42, ZR-003).
GET/api/operators/categoriesPublic0All 64 domain categories with group, prefix, and operator counts.
POST/api/mcpAuth1MCP JSON-RPC endpoint (method: "tools/list" / "tools/call") — 50+ native tools, led by zeq_api_catalog + zeq_http (discovery), plus zeq_compute, zeq_spin_up, zeq_contract_*, zeq_publish_page, and 40+ domain solvers. Call tools/list for the live set.
ZeqField Protocols

Six protocol endpoints

The ZeqField Protocols extend the core compute API with live pulse sync, tamper-evident proofs, multi-node coherence, time-series projection, and usage monitoring.

GET /api/zeq/pulse Public ZeqPulse · 60 req/min · No tokens

Live HulyaPulse snapshot — current Zeqond τ, phase φ, R(t), precision, and time to next tick. No API key required. Ideal as a health-check pulse or for temporal anchoring in distributed agents.

curl https://www.zeq.dev/api/zeq/pulse
{ "protocol": "ZeqPulse", "zeqond": 2156842, "phase": 0.4312, "R_t": 1.000487, "fieldStrength": 1.000487, "timeToNextZeqond": 0.3421, "pulseHz": 1.287, "zeqondSec": 0.777, "alpha": 0.00129, "precision": 0.000129, "timestamp": "2026-03-29T12:00:00.000Z", "modulation": "R(t) = [1 + \u03b1\u00b7sin(2\u03c0\u00b7f\u00b7t)] where f=1.287 Hz, \u03b1=0.00129" }
GET /api/zeq/pulse/stream Auth SSE · 777 ms tick · No tokens

Server-Sent Events stream — a live HulyaPulse event every 777 ms. Requires your API key. Ideal for real-time agent clock synchronisation. The server sends data: lines continuously; close the connection when done.

curl -H "Authorization: Bearer $ZEQ_API_KEY" \ -H "Accept: text/event-stream" \ https://www.zeq.dev/api/zeq/pulse/stream
data: {"zeqond":2156842,"phase":0.4312,"R_t":1.000487,"pulseHz":1.287,"timestamp":"2026-03-29T12:00:00.000Z"} data: {"zeqond":2156843,"phase":0.1124,"R_t":1.000201,"pulseHz":1.287,"timestamp":"2026-03-29T12:00:00.777Z"} data: {"zeqond":2156843,"phase":0.4717,"R_t":0.999874,"pulseHz":1.287,"timestamp":"2026-03-29T12:00:01.554Z"}
GET /api/zeq/usage Auth Usage · 60 req/min · No tokens

Returns your current token budget for the day — calls used, remaining, plan name, and UTC reset time. Use as a preflight check in automated pipelines to avoid unexpected 429 mid-batch.

curl -H "Authorization: Bearer $ZEQ_API_KEY" \ https://www.zeq.dev/api/zeq/usage
{ "used": 42, "limit": 500, "remaining": 458, "plan": "starter", "resetAt": "2026-03-31T00:00:00.000Z" }
POST /api/zeq/verify Auth ZeqProof · HMAC-SHA256 · No tokens

Verify the zeqProof HMAC attached to every compute response. Proves a physics result was generated at a specific Zeqond by the authenticated key — tamper-evident and auditable without storing a secret client-side.

Request body fieldTypeRequiredDescription
zeqProofstringYes64-char HMAC-SHA256 hex returned in the zeqProof field of the compute response
operatorIdsstring[]YesOperator IDs in the exact order returned by the compute response zeqState.operators
R_tnumberYeszeqState.masterSum from the compute response — pass all 6 decimal places
zeqondintegerYeszeqState.zeqond from the compute response
curl -X POST https://www.zeq.dev/api/zeq/verify \ -H "Authorization: Bearer $ZEQ_API_KEY" \ -H "Content-Type: application/json" \ -d '{ "zeqProof": "a3f2b1c8d4e9f012...64-char hex from compute response...", "operatorIds": ["qm-wave-func-solver", "ko42"], "R_t": 1.000487, "zeqond": 2156842 }'
// valid: true { "protocol": "ZeqProof", "valid": true, "zeqond": 2156842, "R_t": 1.000487, "operatorIds": ["qm-wave-func-solver", "ko42"], "keyPrefix": "zeq_ak_abc1", "verifiedAt": "2026-03-29T12:00:01.234Z" } // valid: false also returns: // "hint": "Proof mismatch — check operatorIds ordering, R_t precision (6dp), and zeqond value."
POST /api/zeq/lattice Auth ZeqLattice · 2–5 nodes · N tokens

Multi-domain coherence grid. Pass 2–5 node specs, each with a domain and inputs. All nodes share one Zeqond tick with staggered phase offsets. Returns per-node R_t values, a coherenceScore (0–1), and the lattice equation. Consumes N tokens — one per node.

Request body fieldTypeRequiredDescription
nodesobject[]YesArray of 2–5 node objects, each with domain and inputs
nodes[].domainstringYesPhysics domain name (e.g. quantum_mechanics) or prefix code (e.g. QM)
nodes[].inputsobjectYesDomain-specific input parameters (key-value pairs of numeric values)
curl -X POST https://www.zeq.dev/api/zeq/lattice \ -H "Authorization: Bearer $ZEQ_API_KEY" \ -H "Content-Type: application/json" \ -d '{ "nodes": [ { "domain": "quantum_mechanics", "inputs": { "E": 13.6, "m": 9.11e-31 } }, { "domain": "thermodynamics", "inputs": { "temp_K": 500, "pressure_Pa": 101325 } }, { "domain": "fluid_dynamics", "inputs": { "velocity_ms": 340, "altitude_m": 10000 } } ] }'
{ "protocol": "ZeqLattice", "nodeCount": 3, "zeqond": 2156842, "coherenceScore": 0.998421, "latticeEquation": "L(t) = (1/3) \u03a3\u1d62 R\u1d62(t+\u0394\u1d62) \u0394\u1d62=i\u00b7\u03c4/3 coherence=0.9984", "nodes": [ { "node": 1, "domain": "Quantum Mechanics", "zeqond": 2156842, "phase": 0.4312, "R_t": 1.000487, "operator": "qm-wave-func-solver" }, { "node": 2, "domain": "Thermodynamics", "zeqond": 2156842, "phase": 0.6905, "R_t": 1.000201, "operator": "th-first-law" }, { "node": 3, "domain": "Fluid Dynamics", "zeqond": 2156842, "phase": 0.9498, "R_t": 0.999874, "operator": "fd-navier-stokes" } ], "callsConsumed": 3, "pulseHz": 1.287, "zeqondSec": 0.777, "computedAt": "2026-03-29T12:00:00.000Z" }
POST /api/zeq/shift Auth ZeqShift · 1–64 steps · steps tokens

Forward time-series projection. Pass steps (1–64), a domain, and inputs. Returns a per-step array with exact deterministic R_t values 0.777 s apart. Consumes steps tokens.

Request body fieldTypeRequiredDescription
stepsintegerYesNumber of projection steps. Range: 1–64. Each step is 0.777 s (one Zeqond) forward in time.
domainstringYesPhysics domain name (e.g. quantum_mechanics) or prefix code (e.g. QM)
inputsobjectYesDomain-specific input parameters (key-value pairs of numeric values)
curl -X POST https://www.zeq.dev/api/zeq/shift \ -H "Authorization: Bearer $ZEQ_API_KEY" \ -H "Content-Type: application/json" \ -d '{ "steps": 4, "domain": "quantum_mechanics", "inputs": { "E": 13.6, "m": 9.11e-31 } }'
{ "protocol": "ZeqShift", "domain": "Quantum Mechanics", "stepCount": 4, "callsConsumed": 4, "summary": { "minRt": 0.999214, "maxRt": 1.000914, "meanRt": 1.000087 }, "projection": [ { "step": 0, "zeqond": 2156842, "phase": 0.4312, "R_t": 1.000487, "delta": 0, "t": 1743249600.777 }, { "step": 1, "zeqond": 2156843, "phase": 0.1124, "R_t": 1.000201, "delta": -0.000286, "t": 1743249601.554 }, { "step": 2, "zeqond": 2156843, "phase": 0.7717, "R_t": 0.999874, "delta": -0.000327, "t": 1743249602.331 }, { "step": 3, "zeqond": 2156844, "phase": 0.4309, "R_t": 1.000611, "delta": +0.000737, "t": 1743249603.108 } ], "equation": "R_t(step) = S(t) \u00d7 [1 + \u03b1\u00b7sin(2\u03c0\u00b7f\u00b7(t\u2080 + step\u00b7\u03c4))] f=1.287 Hz, \u03b1=0.00129, \u03c4=0.777s", "computedAt": "2026-03-29T12:00:00.000Z" }
Reference

All 65 domains & example inputs

Pass any of these domain names (or their prefix codes) to domain in a compute request. Values shown are realistic representative inputs.

Core Physics · 16 domains
ZRZeq Resonance Theory
{ "f_hz": 1.287, "alpha": 0.00129 }
QMQuantum Mechanics
{ "E": 13.6, "m": 9.11e-31 }
QRQuantum Resonance
{ "omega_0": 1e9, "Q_factor": 1000 }
WPWave Physics
{ "frequency": 440, "wavelength": 0.77 }
EMElectromagnetism
{ "charge": 1.6e-19, "velocity": 3e6 }
THThermodynamics
{ "temp_K": 500, "pressure_Pa": 101325 }
FDFluid Dynamics
{ "velocity_ms": 340, "altitude_m": 10000 }
CMClassical Mechanics
{ "mass": 10, "velocity": 5 }
NMNewtonian Mechanics
{ "force_N": 100, "mass_kg": 5 }
SMStatistical Mechanics
{ "temp_K": 300, "n_states": 1e23 }
RPRelativistic Physics
{ "mass_kg": 1.67e-27, "beta": 0.99 }
NPNuclear Physics
{ "protons": 92, "neutrons": 146 }
PPParticle Physics
{ "mass_GeV": 125.1, "coupling": 0.13 }
OPOptics
{ "focal_m": 0.05, "obj_dist_m": 0.2 }
ACAcoustics
{ "frequency_Hz": 1000, "pressure_Pa": 0.02 }
GPGravitational Physics
{ "mass1_Msun": 30, "mass2_Msun": 25 }
Extended Physics · 11 domains
PLPlasma Physics
{ "density_m3": 1e19, "temp_eV": 10 }
CDCondensed Matter
{ "eff_mass": 0.067, "k_m": 1e8 }
SCSuperconductivity
{ "temp_K": 4.2, "B_field_T": 0.02 }
CRCrystallography
{ "d_spacing_A": 3.5, "theta_deg": 14.3 }
ASAstrophysics
{ "luminosity_Lsun": 1, "radius_Rsun": 1 }
COCosmology
{ "redshift": 1.0, "H0": 67.4 }
QFQuantum Field Theory
{ "mass_GeV": 0.511, "coupling": 0.0073 }
QCQuantum Computing
{ "qubits": 5, "gate_fidelity": 0.999 }
PTPhotonics
{ "wavelength_nm": 1550, "gain_dBm": 15 }
SPSpintronics
{ "spin_current_uA": 10, "B_T": 0.5 }
NANanophysics
{ "length_nm": 10, "voltage_mV": 2.5 }
Applied Sciences · 12 domains
MSMaterials Science
{ "stress_MPa": 250, "strain": 0.002 }
CPChemical Physics
{ "Ea_kJmol": 85, "temp_K": 500 }
BPBiophysics
{ "conc_mM": 1.5, "Km_mM": 0.5 }
GEGeophysics
{ "depth_km": 35, "density_kgm3": 2900 }
ATAtmospheric Physics
{ "altitude_m": 5000, "temp_K": 255 }
OCOceanography
{ "salinity_ppt": 35, "temp_C": 10 }
ENEnvironmental Physics
{ "conc_ugm3": 12.5, "lifetime_d": 7 }
CLClimate Modeling
{ "CO2_ppm": 420, "albedo": 0.3 }
SESeismology
{ "magnitude": 6.5, "depth_km": 10 }
HYHydrology
{ "K_ms": 1e-4, "gradient": 0.01 }
ERSEarth Sciences
{ "depth_km": 100, "viscosity_Pas": 1e21 }
SOPSoil Physics
{ "theta": 0.35, "matric_kPa": -50 }
Industry · 12 domains
SDSemiconductor Design
{ "Vgs_V": 1.2, "Vt_V": 0.4 }
PWPower Systems
{ "voltage_V": 11000, "current_A": 200 }
CM2Communications
{ "bandwidth_MHz": 20, "SNR_dB": 25 }
SISignal Processing
{ "sample_kHz": 44.1, "freq_kHz": 1.0 }
CS2Control Systems
{ "Kp": 2.0, "Ki": 0.5 }
RORobotics
{ "joint_deg": 45, "link_m": 0.5 }
AEAerospace Engineering
{ "velocity_ms": 250, "alt_m": 10000 }
NENuclear Engineering
{ "enrichment": 0.04, "flux": 1e13 }
BEBiomedical Engineering
{ "freq_MHz": 63.8, "B0_T": 1.5 }
SWStructural Engineering
{ "load_kN": 500, "span_m": 12 }
CHEChemical Engineering
{ "temp_K": 600, "pressure_bar": 30 }
AUTOAutomotive Engineering
{ "torque_Nm": 350, "radius_m": 0.33 }
Frontier · 13 domains
DEDark Energy Models
{ "redshift": 0.5, "w_eos": -1.0 }
QGQuantum Gravity
{ "mass_kg": 1e-5, "L_planck": 1e-35 }
TMTopological Matter
{ "chern_n": 1, "filling": 0.5 }
NRNeurophysics
{ "membrane_mV": -65, "cond_mS": 0.3 }
FMFractal Mechanics
{ "iterations": 1000, "scale_ratio": 0.5 }
CHChaotic Systems
{ "sigma": 10, "rho": 28 }
ECEmergent Complexity
{ "n_agents": 1000, "coupling": 0.3 }
TCTime Crystal Physics
{ "drive_Hz": 100, "amplitude": 0.5 }
NEDNon-Euclidean Dynamics
{ "curvature": -1.0, "radius": 2.5 }
XDExtra Dimensions
{ "n_extra": 2, "scale_m": 1e-15 }
CNObserver-Effect Models
{ "phi": 3.2, "n_nodes": 100 }
QIQuantum Information
{ "qubits": 4, "entropy": 1.8 }
AGAnalogue Gravity
{ "flow_ms": 0.5, "sound_ms": 0.35 }

Browse the full operator library across 65 domains in the Operator Explorer ↗.

Authentication

Bearer token

All compute requests require a valid zeq_ak_ API key in the Authorization header:

Authorization: Bearer zeq_ak_…
401 Unauthorized
Missing or invalid key. Check the key value and ensure there is no extra whitespace.
429 Too Many Requests
Burst cap hit — retry after the window resets. Out of compute-hours returns 402 INSUFFICIENT_FUEL, not a 429.