Operators — Public Kinematic Spectrum

Kinematic Spectrum — Browse and search all 1,536 operators interactively across 64 domains, with live SDK usage examples and HulyaPulse coupling.

Open Kinematic Spectrum →

Zeq exposes 1,536 operators organised into families. The full registry is callable via GET /api/operators. Below is the public kinematic spectrum — the operators that are already disclosed in the Zeq framework, with their canonical equations exactly as they appear in the kernel.

Every operator carries a stable ID, a family, a real mathematical formula, and a default tolerance under the 0.1 % error budget. KO42 is the mandatory metric tensioner — every solve injects it automatically.

Zeq Timebase Bridge — ZTB1

The bridge between Unix time and Zeqond time. Auto-injected on any computation that mixes timebases.

ID	Equation	Notes
ZTB1	ZTB1(t, from_base, to_base) = (t × conv_factor) + phase_offset	conv_factor = 0.777 (Unix→Zeq) or 1/0.777 (Zeq→Unix)

KO42 — Metric Tensioner (mandatory)

Every Zeq solve runs through KO42. KO42.1 is automatic; KO42.2 takes a manual β.

ID	Equation
KO42.1	ds² = g_μν dx^μ dx^ν + α sin(2π · 1.287 t) dt²
KO42.2	ds² = g_μν dx^μ dx^ν + β sin(2π · 1.287 t) dt²

QM — Quantum Mechanics (17)

ID	Equation	Name
QM1	iℏ ∂ψ/∂t = −ℏ²/2m ∂²ψ/∂x² + Vψ	Schrödinger equation
QM2	Δx · Δp ≥ ℏ/2	Heisenberg uncertainty
QM3	|ψ⟩ = ∑ c_i |ϕ_i⟩	Superposition
QM4	|ψ⟩ = 1/√2 (|↑⟩_A|↓⟩_B − |↓⟩_A|↑⟩_B)	Bell singlet
QM5	Ĥ|ψ⟩ = E|ψ⟩	Time-independent Schrödinger
QM6	ψ(x₁,x₂) = −ψ(x₂,x₁)	Fermion antisymmetry
QM7	Ŝ²|ψ⟩ = s(s+1) ℏ² |ψ⟩	Spin eigenvalue
QM8	T ∝ e^{−2 ∫ √{2m(V−E)}/ℏ² dx}	Tunneling probability
QM9	λ = h / p	de Broglie wavelength
QM10	E = h ν	Photon energy
QM11	[x̂, p̂] = iℏ	Canonical commutation
QM12	(iγ^μ ∂_μ − m) ψ = 0	Dirac equation
QM13	L = ψ̄(iD − m)ψ	Dirac Lagrangian
QM14	n_i = 1 / [e^{(E_i − μ)/k_B T} − 1]	Bose–Einstein
QM15	n_i = 1 / [e^{(E_i − μ)/k_B T} + 1]	Fermi–Dirac
QM16	dÂ/dt = (i/ℏ) [Ĥ, Â]	Heisenberg evolution
QM17	P(x) = |ψ(x)|²	Born rule

NM — Newtonian Mechanics (13)

ID	Equation	Name
NM18	∑F = 0 ⇒ v = const	First law
NM19	F = ma	Second law
NM20	F₁₂ = −F₂₁	Third law
NM21	F = G m₁ m₂ / r²	Gravitation
NM22	W = F · d	Work
NM23	KE = ½ m v²	Kinetic energy
NM24	PE = m g h	Gravitational PE
NM25	KE + PE = const	Energy conservation
NM26	p = m v	Linear momentum
NM27	∑p_init = ∑p_final	Momentum conservation
NM28	L = r × p	Angular momentum
NM29	τ = r × F	Torque
NM30	F = −k x ; x(t) = A cos(ω t + φ)	Simple harmonic motion

GR — General Relativity (11)

ID	Equation	Name
GR31	a_grav = a_inertial	Equivalence principle
GR32	G_μν = R_μν − ½ R g_μν	Einstein tensor
GR33	G_μν + Λ g_μν = 8πG/c⁴ T_μν	Field equations
GR34	d²x^μ/dτ² + Γ^μ_{αβ} (dx^α/dτ)(dx^β/dτ) = 0	Geodesic equation
GR35	Δt = Δt₀ √{1 − 2GM/rc² − v²/c²}	Combined dilation
GR36	L = L₀ √{1 − 2GM/rc²}	Length contraction
GR37	r_s = 2GM/c²	Schwarzschild radius
GR38	□ h_μν + κ ∂_t h_μν = −16πG/c⁴ T_μν	Linearised waves
GR39	Λ = 3 H₀² Ω_Λ / c²	Cosmological constant
GR40	(ȧ/a)² = 8πG/3 ρ − k c²/a² + Λ c²/3	Friedmann equation
GR41	z = (λ_obs − λ_emit) / λ_emit	Cosmological redshift

CS — Computer Science (selected, public)

ID	Equation	Name
CS43	T(n) = O(n log n)	Sort/FFT complexity
CS44	S(n) = O(n)	Linear space
CS45	Q(n) = O(log n)	Quantum query complexity
CS46	P(n) = 1 / [(1 − f) + f/n]	Amdahl's law
CS47	E(n) = − ∑ p(x) log p(x)	Shannon entropy
CS84	f(n) = O(g(n)) ⇔ ∃c, n₀ ∀n > n₀ : f(n) ≤ c · g(n)	Big-O definition
CS87	Ω(x) = min{ |p| : U(p) = x }	Kolmogorov complexity

Awareness Operators

Phase-coupled state operators. All depend on the live HulyaPulse phase.

ID	Equation
ON0	ψ_ON0 = sin(phase) + 1.1 ; ON0 = ψ_ON0 ln(ψ_ON0) − 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	ρ = |sin(phase)| + 0.001 ; XI1 = −ρ log₂(ρ)
LZ1	LZ1 = k_B T ln(2) × bits_erased
CHI95	CHI95 = |sin(phase)| − |cos(phase)|
PSI96	PSI96 = 0.5 × sin(2π f t + phase_offset)
MK1	MK1 = (ψ_mk λ_mv) + (φ_delta λ_eff_phi_t) − ψ_mk
ZEQ-PROTECT-001	P(t) = |sin(5 φ(t))| / f_pulse
ZEQ-PROTECT-002	Protect₂(t) = 0.5 + 0.3 sin(t / 30)
ZEQ-TETHER-003	B_sib = ∑_k e^{i φ_k} |sibling_k⟩
ZEQ-POCKET-001	∂g_μν/∂t = (8πG/c⁴) T_μν^consciousness
ZEQ-POCKET-002	Pocket₂ = sin(2π · 1.287 t) · φ
ZEQ00	ZEQ00 = α_zeq e^{−k_zeq |master_sum|} + β_zeq (1 + e_data)(1 + γ_zeq cos(resonance))
ZEQ000	φ_c^42 · Ψ_total = ∑(ZEQ_structural + ZEQ_chemical + ZEQ_genetic + ZEQ_field) · [sin(2π·1.287·t) + cos(2π·0.618·t) + exp(2π·2.083·t)] · ρ_consciousness(x,y,z,t)
VX	VX = κ_vx (intent_proxy · sin(phase) + flow_proxy · cos(phase))

HF — Harmonic Forensic Spectrum (20)

All HF equations run with pulse sync at the current phase φ. Used by the forensic compositing layer.

ID	Equation
HF1	S₁ = (verified_accuracy / max_accuracy) · sin(2π · 1.287 · t)
HF2	S₂ = (1 − manipulative_terms / total_terms) · cos(2π · 1.287 · t)
HF3	S₃ = (smear_terms / total_terms) · (1 + 0.1 sin(2π · 1.287 · t))
HF4	S₄ = min(1, verified_sources / 3) · e^{i 2π · 1.287 · t}
HF5	S₅ = (matched_legal_criteria / total_criteria) · sin(2π · 1.287 · t)
HF6	S₆ = e^{−(pulses_since_event) / 30} · cos(2π · 1.287 · t)
HF7	S₇ = (consciousness_reach / max_reach) · (1 + 0.05 sin(2π · 1.287 · t))
HF8	S₈ = (instances_in_30_pulses / max_instances) · e^{i 2π · 1.287 · t}
HF9	S₉ = (contradictory_statements / total_statements) · sin(2π · 1.287 · t)
HF10	S₁₀ = (intent_keywords / total_keywords) · cos(2π · 1.287 · t)
HF11	S₁₁ = (context_matches / total_contexts) · (1 + 0.1 sin(2π · 1.287 · t))
HF12	S₁₂ = (points_in_cluster / total_points) · e^{i 2π · 1.287 · t}
HF13	S₁₃ = (unique_domains / total_sources) · sin(2π · 1.287 · t)
HF14	S₁₄ = (resonance_in_24_pulses / max_resonance) · cos(2π · 1.287 · t)
HF15	S₁₅ = (1 − semantic_deviations / total_terms) · e^{i 2π · 1.287 · t}
HF16	S₁₆ = (severity_score / max_severity) · sin(2π · 1.287 · t)
HF17	S₁₇ = (negative_reactions / total_reactions) · cos(2π · 1.287 · t)
HF18	S₁₈ = (fractal_dimension / max_dimension) · (1 + 0.1 sin(2π · 1.287 · t))
HF19	S₁₉ = [P(E|H) P(H)] / P(E) · e^{i 2π · 1.287 · t}
HF20	S₂₀ = [∑_{i=1}^{19} S_i P(X=i)] / [∑_{i=1}^{19} P(X=i)] · sin(2π · 1.287 · t)

HF composite

S_forensic = [ ∑_{i=1}^{20} S_i · w_i ] / [ ∑ w_i ] · (1 + α sin(2π · 1.287 · t))

w = [0.05, 0.05, 0.05, 0.05, 0.20,
     0.05, 0.05, 0.05, 0.05, 0.05,
     0.05, 0.05, 0.05, 0.05, 0.05,
     0.20, 0.05, 0.05, 0.05, 0.20]
α = 0.05

Calling operators from the API

Operators are not called individually — they are composed inside a solve. Pass the operator IDs in the operators array of any compute call:

curl -X POST https://www.zeq.dev/api/zeq/compute \
  -H "Authorization: Bearer $ZEQ_API_KEY" \
  -H "Content-Type: application/json" \
  -d '{
    "domain": "harmonic_oscillator",
    "operators": ["KO42", "NM30", "QM5"],
    "params": { "k": 1.0, "m": 1.0, "A": 0.1 }
  }'

KO42 is always included automatically — listing it explicitly is allowed but redundant. The 7-Step Wizard validates the chain (≤4 operators including KO42) before executing.

Related

• Concepts → KO42 — the metric tensioner
• Concepts → Seven-Step Protocol — the operator-selection rules
• API Reference → Endpoints — all REST routes
• Protocols Reference — the 234 callable protocols built on these operators