Mathematical Reference · Patent Pending

θ — Angular Time Coordinate

Theta (θ) · phase angle · polar time encoding

θ (theta) is an angle. It shows where the current moment sits inside a cycle — for example, how far through the day, or how far through the year. The /v1/spiral API returns a (sin θ, cos θ) pair for each of twelve time scales, from microseconds to millennia.

What θ represents

A raw timestamp is a number counted from a fixed start point. It does not tell a model where the moment falls in the day, week, or year. To answer "is it Friday afternoon?" or "how much time has passed?" the model must run date arithmetic every time.

θ removes that work. For each scale i with cycle length T_i, the API computes the angle:

θ_i = 2π × frac(t / T_i)   ∈ [0, 2π)

frac(x) is the fractional part of x — the remainder after removing whole cycles. The API then encodes the angle as a (sin θ_i, cos θ_i) pair. A model receives two smooth numbers it can differentiate and project forward. It does not need to parse strings or run modulo arithmetic.

The twelve θ angles

Each scale produces one independent θ angle. The twelve scales cover twelve orders of magnitude — from sub-millisecond timing to millennial positioning:

Scale	θ wraps once per…	Typical use
θ_us	1 millisecond	µs-precision inference cadence
θ_ms	1 second	token generation rate
θ_s	1 minute	request/response pacing
θ_min	1 hour	session scheduling
θ_hr	1 day	circadian / daily patterns
θ_day	1 ISO week	weekday/weekend detection
θ_wk	1 year (52 wk)	weekly seasonality
θ_mo	1 year (12 mo)	monthly / seasonal
θ_yr	1 year	annual cycle
θ_dec	1 decade	decade positioning
θ_cen	1 century	century positioning
θ_mil	1 millennium	deep-time positioning

A 13th rung adds a radial coordinate r = log(1 + |t − t₀| / T₀). This gives the encoding a "how far along the timeline am I?" value that the twelve cyclic angles alone cannot provide.

θ in the API response

Every GET /v1/spiral response carries theta values in two forms:

Compact form — compact.sin[] and compact.cos[] are parallel arrays of sin(θ_i) and cos(θ_i), ordered from us to mil. Concatenate them directly into a model input vector.
String form — spiralString is a human-readable line of all θ values for logging and debugging.

curl 'https://api.temporalblock.com/api/v1/spiral' \
  -H 'X-API-Key: tblk_live_…'

// Response (abbreviated):
{
  "compact": {
    "scales": ["us", "ms", "s", "min", "hr", "day", "wk", "mo", "yr", "dec", "cen", "mil"],
    "sin":    [0.000, -0.999, 0.951, -0.309, 0.588, 0.454, -0.208, 0.866, 0.743, -0.978, 0.991, 0.171],
    "cos":    [-1.00,  0.009, 0.309,  0.951, 0.809, 0.891, -0.978, 0.500, 0.669,  0.206, 0.134, 0.985]
  },
  "spiralString": "θ_us=3.14 θ_ms=3.13 θ_s=1.26 θ_min=5.96 θ_hr=0.63 …"
}

θ in practice

AI inference — time-of-day routing

An inference pipeline that batches by time-of-day pattern feeds θ_hr and θ_day into its routing model. The angle is continuous — there is no hard step at midnight, no integer hour to parse. The model learns smooth load curves. The whenPhaseRecurs skill trigger fires when θ crosses a threshold, so agents schedule themselves against the spiral without writing any date logic.

Robotics — joint control

A robot joint controller running at 1 kHz queries /v1/spiral once at startup. It uses θ_ms and θ_s as a continuous cadence signal in the control loop — no wall-clock reads, no modulo arithmetic. The (sin θ, cos θ) form feeds directly into the phase-locked loop math the motor driver already uses.

Orbital mechanics — annual phase

In orbital mechanics, the true anomaly is the angle between a reference direction and the body's current position — the same idea as θ, applied to an orbital period. θ_yr gives simulation systems Earth's annual orbital phase without a full propagator. It is accurate enough for rough scheduling and costs one API call.

Financial systems — intraday position

Trading systems use θ_hr and θ_min to place a timestamp inside the trading day. The angle does not depend on exchange-specific session times. A model can interpolate across pre-market, regular, and after-hours periods as a smooth signal rather than a list of named categories.

The polar coordinate view

The twelve θ angles together place the current moment on a 12-dimensional torus T¹² = (S¹)¹². Each scale contributes one circle. As time advances, the point traces a multi-scale spiral: nearby timestamps map to nearby points (smooth for interpolation), and distant timestamps map to clearly different points (distinct for a model to separate).

The spiral's polar form:

r = a · θ            // Archimedean spiral
r = a · e^(b·θ)      // logarithmic spiral (used in the global rung)

The 13th rung uses the logarithmic form. Near the origin, the radius grows linearly with time. Far from the origin, it grows as a logarithm. This keeps the coordinate useful for both recent timestamps and very old ones.

Temporal Spiral — full coordinate referenceNoise classifier, focus limiter, observer-frame correction, simulation clock
Vision Block — spatial-temporal stampingθ-anchored temporal addresses for sensor fusion pipelines
Time as a Phase-Space Trajectory — U(1) symmetry, T¹² torus, Hamiltonian framingDeeper mathematical treatment: Schrödinger parallel, symplectic structure, and the geometric meaning of θ
Robotics — joint control and phase synchronizationθ as a continuous cadence signal for motor controllers and sensor fusion
Orbital mechanics — true anomaly alignmentθ_yr as a zero-overhead proxy for orbital phase in simulation scheduling
Sinusoidal Time Encoding — Fourier feature map framingWhy the (sin θ, cos θ) pairs function as a Fourier feature map for ML models

The Temporal Spiral encoding is patent pending — U.S. Provisional Application No. 64/065,213 (filed 2026-05-14).