θ as True Anomaly and Intraday Phase

θ as true anomaly in orbital mechanics

In orbital mechanics, the true anomaly ν (or θ) is the angular position of a body along its orbit measured from periapsis. For a circular reference orbit at a known period, the Temporal Spiral's rung-scale θ is directly proportional to true anomaly — no ephemeris library required for quick phase estimates.

Because the API returns sin(θ) and cos(θ) rather than a raw angle, phase stays continuous across midnight, year, and orbital-period boundaries — essential for on-board attitude-control loops that must not encounter angle wrapping. Full θ reference →

// θ from the Temporal Spiral maps directly to true anomaly for circular
// or near-circular reference orbits at known periods.

const { scales } = await fetch(
  "https://api.temporalblock.com/api/v1/spiral",
  { headers: { "X-API-Key": process.env.TBLK_API_KEY } }
).then((r) => r.json());

// For a LEO satellite with a ~90-minute orbital period, use the
// hour-scale rung (index 5 in the default 12-rung set).
// sin(θ) and cos(θ) give the unit-circle phase of that rung —
// proportional to true anomaly for a circular orbit anchored at epoch.
const { sin, cos } = scales[5]; // hour-scale rung

// Recover θ ∈ [−π, π]; multiply by your orbital period to get elapsed time.
const thetaRad = Math.atan2(sin, cos);
const elapsedSec = (thetaRad / (2 * Math.PI)) * 5400; // 90-min period

The Spiral returns 12 nested rungs spanning microseconds to millennia — pick the rung whose period best matches your orbital period. Combine rungs for hierarchical phase representations in multi-body or resonance calculations.

θ as intraday phase in quantitative finance

Intraday seasonality models frequently express the time-of-day as an angular phase — letting signal features be continuous and periodic across session boundaries. The day-scale rung gives sin(θ_day) and cos(θ_day) relative to UTC midnight, making open/close/lunch dips directly expressible as threshold comparisons on θ without ad-hoc minute-of-day buckets.

// Map θ to intraday session phase — where are we in today's trading session?

const { scales } = await fetch(
  "https://api.temporalblock.com/api/v1/spiral",
  { headers: { "X-API-Key": process.env.TBLK_API_KEY } }
).then((r) => r.json());

// The day-scale rung (index 7) gives sin(θ_day) and cos(θ_day):
// θ_day = 0  → midnight UTC
// θ_day = π  → noon UTC
// US equities open at 14:30 UTC → θ ≈ 0.607π
// US equities close at 21:00 UTC → θ ≈ 0.875π
const { sin: sinDay, cos: cosDay } = scales[7];
const thetaDay = Math.atan2(sinDay, cosDay); // ∈ [−π, π]

// Convert to fraction of day (0 = midnight, 1 = next midnight).
const fracDay = ((thetaDay / (2 * Math.PI)) + 1) % 1;
const minuteOfDay = Math.round(fracDay * 1440);

All rung θ values are derived from the same certified server clock — no skew between your signal pipeline nodes, no dependency on the local system clock of each trading machine.

Strict event ordering for tick data and telemetry

High-frequency order events and satellite telemetry frames both arrive faster than wall-clock millisecond resolution. Causal Block issues Hybrid Logical Clock tuples that always strictly increase — even across distributed nodes writing within the same millisecond — giving you a safe primary sort key without a de-duplication pass.

// Causal Block gives every order, fill, or quote event a strictly
// increasing HLC tuple — no duplicate timestamps at sub-millisecond rates.

const event = await fetch(
  "https://api.temporalblock.com/api/v1/causal-block/event",
  {
    method: "POST",
    headers: {
      "Content-Type": "application/json",
      "X-API-Key": process.env.TBLK_API_KEY,
    },
    body: JSON.stringify({
      label: "order-fill",
      payload: { symbol: "ES", price: 5320.25, qty: 2 },
    }),
  }
).then((r) => r.json());

// event.hlc always strictly increases across all nodes — safe as a
// primary sort key in your tick database without a de-duplication pass.
console.log(event.hlc); // { wallMs: 1750000000012, counter: 7, nodeId: "..." }

Related