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The 42+ Kinematic Operators

Zeq ships a kinematic operator catalogue organized by domain — each entry is a named, standard physics formula the framework can identify and compute. When you request a domain, a dedicated closed-form solver evaluates the relevant formula (e.g. NM21 → F = G·m₁·m₂/r², computed with CODATA constants). The operator IDs label and document that physics; they are a reference vocabulary, not a runtime plug-in list.

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 C_k(ϕ) mapping is implemented in the runtime. The physics you get from the API comes from the dedicated domain solvers, verified against CODATA — not from a literal 42-term sum. The differential master-equation engine integrates its own well-posed □ϕ = ∇²ϕ − μ²ϕ − λϕ³ core.

Below is the reference catalogue. For the formal background see the framework paper (DOI 10.5281/zenodo.15825138).

Quantum Mechanics (QM1 – QM17)​

  • QM1 — Schrödinger equation: i hbar × d psi / dt = -(hbar^2 / 2m) × d^2 psi / dx^2 + V psi
  • QM2 — Uncertainty: Delta x × Delta p >= hbar / 2
  • QM3 — Superposition: |psi> = Sum c_i |phi_i>
  • QM4 — Entangled singlet: |psi> = (1/sqrt 2) (|up>_A |down>_B - |down>_A |up>_B)
  • QM5 — Energy eigenequation: H hat |psi> = E |psi>
  • QM6 — Fermionic antisymmetry: psi(x_1, x_2) = -psi(x_2, x_1)
  • QM7 — Spin magnitude: S hat^2 |psi> = s(s+1) hbar^2 |psi>
  • QM8 — Tunneling: T ∝ exp(-2 integral sqrt(2m(V-E))/hbar^2 dx)
  • QM9 — de Broglie wavelength: lambda = h / p
  • QM10 — Photon energy: E = h nu
  • QM11 — Canonical commutator: [x hat, p hat] = i hbar
  • QM12 — Dirac equation: (i gamma^mu d_mu - m) psi = 0
  • QM13 — QED Lagrangian: L = psi bar (i D - m) psi
  • QM14 — Bose-Einstein: n_i = 1 / [exp((E_i - mu)/k_B T) - 1]
  • QM15 — Fermi-Dirac: n_i = 1 / [exp((E_i - mu)/k_B T) + 1]
  • QM16 — Heisenberg equation of motion: d A hat / dt = (i / hbar) [H hat, A hat]
  • QM17 — Born rule: P(x) = |psi(x)|^2

Newtonian Mechanics (NM18 – NM30)​

  • NM18 — Inertia: Sum F = 0 ⇒ v = const
  • NM19 — F = ma
  • NM20 — Third law: F_12 = -F_21
  • NM21 — Gravitation: F = G m_1 m_2 / r^2
  • NM22 — Work: W = F · d
  • NM23 — Kinetic energy: KE = (1/2) m v^2
  • NM24 — Potential energy: PE = m g h
  • NM25 — Conservation: KE + PE = const
  • NM26 — Momentum: p = m v
  • NM27 — Momentum conservation: Sum p_init = Sum p_final
  • NM28 — Angular momentum: L = r × p
  • NM29 — Torque: tau = r × F
  • NM30 — SHO: F = -k x, x(t) = A cos(omega t + phi)

General Relativity (GR31 – GR41)​

  • GR31 — Equivalence principle: a_grav = a_inertial
  • GR32 — Einstein tensor: G_{mu nu} = R_{mu nu} - (1/2) R g_{mu nu}
  • GR33 — Field equations: G_{mu nu} + Lambda g_{mu nu} = (8 pi G / c^4) T_{mu nu}
  • GR34 — Geodesic equation: d^2 x^mu / d tau^2 + Gamma^mu_{alpha beta} (dx^alpha / d tau)(dx^beta / d tau) = 0
  • GR35 — Time dilation: Delta t = Delta t_0 × sqrt(1 - 2GM / r c^2 - v^2 / c^2)
  • GR36 — Length contraction: L = L_0 × sqrt(1 - 2GM / r c^2)
  • GR37 — Schwarzschild radius: r_s = 2 G M / c^2
  • GR38 — Gravitational wave: Box h_{mu nu} + kappa × d_t h_{mu nu} = -(16 pi G / c^4) T_{mu nu}
  • GR39 — Cosmological constant: Lambda = 3 H_0^2 Omega_Lambda / c^2
  • GR40 — Friedmann: (a dot / a)^2 = (8 pi G / 3) rho - k c^2 / a^2 + Lambda c^2 / 3
  • GR41 — Redshift: z = (lambda_obs - lambda_emit) / lambda_emit

Computer Science (CS43 – CS87)​

  • CS43 — Complexity: T(n) = O(n log n) (e.g. sorting, FFT)
  • CS44 — Space complexity: S(n) = O(n)
  • CS45 — Quantum query: Q(n) = O(log n)
  • CS46 — Amdahl's law: P(n) = 1 / [(1 - f) + f/n]
  • CS47 — Shannon entropy: E(n) = -Sum p(x) log p(x)
  • CS84 — Big-O: f(n) = O(g(n)) iff exists c, n_0 forall n > n_0: f(n) <= c × g(n)
  • CS87 — Kolmogorov complexity: Omega(x) = min{ |p| : U(p) = x }

Referential operators​

The referential family (self-referential operators) is the framework's frontier research surface — boundary-exploration mathematics for information, complexity, neural dynamics, and the thermodynamics of computation. Several are established equations: LZ1 is Landauer's principle (k_B·T·ln2 per bit erased); XI1 is Shannon entropy (−ρ·log₂ρ); the CBCM-style forms are nonlinear neural-oscillator ODEs (the kind that model neural rhythms and plasticity). Treat them as real, exploratory mathematics.

Scope — a research surface, outside the standard physics compute path

These are experimental/frontier expressions, not claims of verified physics (in particular, they are not asserted as an established physics of consciousness). The standard physics compute path does not evaluate them — zeq_compute on one returns unit: "no-match" by design; they're a separate research surface. A few are ill-posed as written (e.g. ZEQ000 below contains exp(2π·2.083·t), which diverges) — noted where they appear.

  • ON0 — psi = sin(phase) + 1.1; ON0 = psi × ln(psi) - phase × f
  • QL1 — density = |sin(phase × 3)| + 0.1; QL1 = 0.1 × density × ln(density / 0.1) + cos(phase) × 0.5
  • TM1 — TM1 = -t + current_utp × period
  • TX — TX = 0.01 × sin(phase × 2) × cos(t / 100)
  • XI1 — rho = |sin(phase)| + 0.001; XI1 = -rho × log_2(rho)
  • LZ1 — LZ1 = k_B T × ln(2) × bits_erased (Landauer limit)
  • CHI95 — CHI95 = |sin(phase)| - |cos(phase)|
  • PSI96 — PSI96 = 0.5 × sin(2 pi f t + phase_offset)
  • MK1 — MK1 = (psi_mk × lambda_mv) + (phi_delta × lambda_eff_phi_t) - psi_mk
  • VX — VX = kappa × (intent_proxy × sin(phase) + flow_proxy × cos(phase))

Tether / Pocket / Protect operators​

These are frontier expressions — the real product security is standard cryptography

This ZEQ-PROTECT / ZEQ-TETHER / ZEQ-POCKET family is part of the experimental research surface. What actually protects Zeq Mail, Message, and the vault is standard, audited cryptography — AES-256-GCM, HMAC-SHA256, PBKDF2 (see ZSP and the security model) — so treat these as exploratory mathematics, not the operational security mechanism. ZEQ000 is ill-posed as written (its exp(2π·2.083·t) term diverges).

  • ZEQ-PROTECT-001 — P(t) = |sin(5 × phi(t))| / f_pulse
  • ZEQ-PROTECT-002 — P_2(t) = 0.5 + 0.3 × sin(t / 30)
  • ZEQ-TETHER-003 — B_sib = Sum_k exp(i phi_k) |sibling_k>
  • ZEQ-POCKET-001 — d g_{mu nu} / dt = (8 pi G / c^4) × T_{mu nu}^{consciousness}
  • ZEQ-POCKET-002 — Pocket_2 = sin(2 pi × 1.287 × t) × phi
  • ZEQ00 — ZEQ00 = alpha × exp(-k × |master_sum|) + beta × (1 + e_data)(1 + gamma × cos(resonance))
  • ZEQ000 — phi_c^42 × Psi_total = Sum(ZEQ_structural + ZEQ_chemical + ZEQ_genetic + ZEQ_field) × [sin(2 pi × 1.287 × t) + cos(2 pi × 0.618 × t) + exp(2 pi × 2.083 × t)] × consciousness_field_density(x, y, z, t)

Composition rules​

The 7-step wizard protocol restricts composition to 1 to 3 operators plus KO42 per call (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 (see The constants), not a proof of physical accuracy. The composition limit keeps a call to a small, auditable set of operators; deeper work chains multiple independently-signed CKOs.

If you need a deeper composition:

  • Chain multiple CKOs together. Each is independently KO42-verified.
  • Use a protocol — protocols are pre-composed, pre-verified operator products with known error bands.

Examples of valid compositions:

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

You'll see these compositions inside every protocol's CKO.