Delta

Rate of change. The language's soul. Sequential code reasons about state — what IS the value right now? Z+ reasons about change — what is the value DOING right now?

Delta is not a library function. It's a keyword. The OS maintains temporal history on every signal. t, t-1, t-2 exist. Delta is just t - t-1.

Derivatives

.zpderivatives

// first derivative — velocity
delta(temp)                  // how fast is it changing?

// second derivative — acceleration
delta(delta(temp))           // is the change speeding up?

// third derivative — jerk
delta(delta(delta(temp)))    // is this losing control?

A temperature climbing at a steady rate is predictable. One where the rate is accelerating is concerning. One where the acceleration is jerking is a system on the edge of chaotic failure. Three levels of delta(). No calculus library. No numerical differentiation. No buffer management.

Delta vs. State Logic

.zpdelta vs state

// state logic — triggers when temperature crosses threshold
temp -> gate(> 80C) -> alarm

// delta logic — triggers when temperature is rising fast
delta(temp) -> gate(> 2C) -> alarm
// 40C and climbing at 3C/s is more dangerous
// than 79C and stable. delta knows this.

Baseline + Deviation

Learn what's normal over time, then measure how far the current signal deviates from that learned normal.

.zpbaseline

// learn what's normal
signal -> baseline(window: 7d) -> normal

// measure deviation from normal
signal -> deviation(from: normal) -> dev

// gate on standard deviations
dev -> gate(> 2σ, knee: 1σ) -> anomaly

Seasonal Baselines

Monday at 9am is different from Sunday at 3am. Baselines can be segmented by temporal dimensions.

.zpseasonal baseline

// seasonal baseline — accounts for patterns
cpu -> baseline(window: 7d, by: day_of_week, hour) -> seasonal_normal

Sigma

The σoperator measures standard deviations from baseline. Combined with a knee gate, it creates smooth anomaly detection that doesn't trigger on normal variance.

.zpsigma gating

// anomaly: more than 2 standard deviations from normal
// with a 1-sigma knee for smooth transition
dev -> gate(> 2σ, knee: 1σ) -> anomaly

// different severity levels
dev -> gate(> 2σ) -> warning
dev -> gate(> 3σ) -> critical
dev -> gate(> 4σ) -> emergency

Temporal Access

Every signal carries its history. Access any previous value directly.

.zptemporal access

// t is now, t-1 is previous, t-N is N steps back
signal.t        // current value
signal.t-1      // previous value
signal.t-2      // two steps back

// delta is simply t - t-1
delta(signal) == signal.t - signal.t-1

Failure Horizon

Predictive logic built from delta. Not machine learning — just calculus expressed as language primitives on signals with temporal history.

.zpfailure horizon

// predict when a metric will reach a threshold
// using rate and acceleration
goya_3.thermal -> failure_horizon(85C) -> {
    gate(< 30s) -> emergency_migrate,
    gate(< 5m)  -> gentle_migrate,
    gate(< 1h)  -> note_it
}

// "how long until this card overheats?"
// answered continuously, in real time

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