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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 1,600+ 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 65 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.