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Operators

An operator is a named, catalogued piece of mathematics — NM19 is F = ma, GR37 is r_s = 2GM/c², QM9 is λ = h/p. The IDs are the machine's vocabulary: every compute names the operators it ran, every contract names the operator each transition fires, and every audit row names the operator it proves. The live registry carries 1,612 operators across 65 categories — browse the full operator reference, where every category states how it computes.

What an operator ID buys you​

When you request an operator, a dedicated closed-form solver evaluates the real formula with CODATA constants — the ID selects the physics, and the pipeline validates, computes, verifies, and signs it. Reproduce these right now:

# NM19 — F = ma
curl -sX POST https://zeqsdk.com/api/zeq/compute -H "Authorization: Bearer $ZEQ_KEY" \
  -H "Content-Type: application/json" \
  -d '{"operators":["KO42","NM19"],"inputs":{"m":10,"a":3}}'          # → 30 N

# GR37 — Schwarzschild radius of the Sun
  -d '{"operators":["KO42","GR37"],"inputs":{"mass":1.98892e30}}'     # → 2954.0077 m

# QM9 — de Broglie wavelength of an electron at 10⁶ m/s
  -d '{"operators":["KO42","QM9"],"inputs":{"m":9.109e-31,"v":1e6}}'  # → 7.274e-10 m

Type safety​

The registry is the type. An operator ID that does not exist is refused with UNKNOWN_OPERATOR — never silently computed as though it were real. A real operator given the wrong inputs names exactly what it needs (the honesty contract).

The 42+ kinematic bootstrap set​

The core families every physics reader knows — reachable without any dispatch logic:

  • QM1–QM17 (quantum): Schrödinger, uncertainty, superposition, entanglement, tunnelling, de Broglie, E = hν, Dirac, Bose–Einstein / Fermi–Dirac, the Born rule.
  • NM18–NM30 (Newtonian): the three laws, gravitation, work, kinetic & potential energy, momentum and its conservation, angular momentum, torque, the harmonic oscillator.
  • GR31–GR41 (relativity): equivalence, the Einstein tensor and field equations, geodesics, time dilation, r_s, gravitational waves, Friedmann, redshift.
  • CS43+ (computation): complexity classes, Amdahl, Shannon entropy, Big-O, Kolmogorov.

Every formula, verbatim from the registry, lives in the reference — the tables there are generated from the live system.

What the engine does — and does not — evaluate​

note
On the Σ_{k=1..42} C_k(ϕ) term

Some framework literature writes the operators as a coupling term Σ_{k=1..42} C_k(ϕ) inside the master equation. Treat that as notational/organisational, not a defined operator: the catalogue contains inequalities (QM2), implications (NM18), and asymptotic notations (CS84, CS87) that are not real-valued functions of a field ϕ, and no literal C_k(ϕ) mapping is implemented. The physics you get from the API comes from the dedicated domain solvers, verified against CODATA. The differential master-equation engine integrates its own well-posed □ϕ = ∇²ϕ − μ²ϕ − λϕ³ core.

Coverage is measured, not assumed — most of the 67 categories are served by a dedicated closed-form solver, and the rest by the universal numerical engine, so every operator returns a real, deterministic value with its method disclosed. A test cross-checks the coverage model against the live dispatch on every build.

The referential (frontier) family​

The self-referential operators (ON0, XI1, LZ1, the ZEQ-PROTECT/TETHER/POCKET forms…) are the framework's frontier research surface — real, exploratory mathematics for information, complexity, and the thermodynamics of computation. Several are established equations (LZ1 is Landauer's principle; XI1 is Shannon entropy). They are not asserted as verified physics of consciousness, and the standard compute path does not evaluate their specific formulas: selecting one runs the transparent master-equation ODE fallback, which returns a real, deterministic value carrying a NUMERICAL FALLBACK disclosure. The operational security of the product is standard, audited cryptography (AES-256-GCM, HMAC-SHA256, PBKDF2 — see security), never these expressions. One (ZEQ000) is ill-posed as written — its exp(2π·2.083·t) term diverges — and is catalogued with that warning.

Composition​

A call carries KO42 plus up to three more operators (max 4). KO42 is the bounded HulyaPulse modulation, applied once per result; the ≤0.1% figure is the modulation amplitude bound |α·sin| ≤ 10⁻³ by construction — not a proof of physical accuracy. The limit keeps every call a small, auditable set. Valid compositions you'll meet in the wild:

  • KO42 + QM9 + NM23 — de Broglie wavelength inside a kinetic-energy calculation.
  • KO42 + GR35 + NM22 — time-dilated work.
  • KO42 + QM1 + QM17 — Schrödinger evolution of a probability distribution.

Need something deeper? Chain CKOs (each independently signed and verified), or deploy a contract — pre-composed operator logic with a full audit trail per fire.