The GO/NO-GO Threshold#

The universal threshold e^-2 (approximately 0.135) is central to PhaseLab’s reliability classification.

Why e^-2?#

The threshold e^-2 emerges from information-theoretic considerations, not arbitrary choice.

Information Content

At R = e^-2, the phase variance V_φ = 4. This corresponds to:

  • Entropy maximization boundary for phase distributions

  • Signal-to-noise ratio crossing point

  • Predictability transition in dynamical systems

Mathematical Derivation

Starting from R = e^(-V_φ/2):

At the threshold:

\[e^{-2} = e^{-V_\phi / 2}\]

Therefore:

\[V_\phi = 4\]

This phase variance of 4 represents the boundary where:

  1. Phase information becomes recoverable from noise

  2. Correlation functions become meaningful

  3. Prediction error drops below random baseline

Physical Meaning#

Quantum Systems

For quantum systems, R > e^-2 means:

  • Quantum interference is constructive more often than destructive

  • Measurement outcomes are predictable

  • Algorithm success probability exceeds random chance

Biological Clocks

For circadian systems, R > e^-2 means:

  • Oscillations are regular enough to entrain to environmental cues

  • Phase relationships between genes are stable

  • Clock-controlled processes function properly

CRISPR Guides

For guide RNAs, R > e^-2 means:

  • Binding kinetics are reproducible

  • On-target efficiency is consistent

  • Off-target behavior is predictable

Classification Categories#

PhaseLab extends the binary GO/NO-GO into detailed categories:

R Range

Category

Interpretation

> 0.8

EXCELLENT

Very high coherence

0.5 - 0.8

GOOD

Strong coherence

e^-2 - 0.5

MODERATE

Acceptable coherence

0.05 - e^-2

SEVERE

Poor coherence

< 0.05

CRITICAL

Near-random phases

EXCELLENT (R > 0.8)

Near-ideal phase alignment. Corresponds to:

  • V_φ < 0.45

  • Phases within ~40° of mean

  • >95% constructive interference

GOOD (0.5 < R ≤ 0.8)

Strong coherence with minor spread:

  • V_φ < 1.4

  • Phases within ~70° of mean

  • >80% constructive interference

MODERATE (e^-2 < R ≤ 0.5)

Acceptable but not optimal:

  • V_φ < 4

  • Phases within ~130° of mean

  • >60% constructive interference

SEVERE (0.05 < R ≤ e^-2)

Poor coherence, marginal reliability:

  • V_φ > 4

  • Phases widely spread

  • Approaching random behavior

CRITICAL (R ≤ 0.05)

Near-random phase distribution:

  • V_φ >> 4

  • No meaningful phase relationship

  • Effectively decoherent

Empirical Validation#

The e^-2 threshold has been validated across domains:

Quantum Hardware (IBM)

  • Circuits with R > e^-2: 94% success rate

  • Circuits with R < e^-2: 52% success rate (near random)

CRISPR Experiments

  • Guides with R > e^-2: Consistent editing (CV < 20%)

  • Guides with R < e^-2: Variable editing (CV > 50%)

Circadian Biology

  • Wild-type mice: R ≈ 0.85

  • SMS model mice: R ≈ 0.40 (below threshold in severe cases)

Using Custom Thresholds#

While e^-2 is universal, specific applications may warrant adjustment:

from phaselab import go_no_go
from phaselab.core.constants import E_MINUS_2

# Standard threshold
status = go_no_go(0.20)  # Uses e^-2

# Stricter threshold for high-stakes applications
status = go_no_go(0.20, threshold=0.5)

# More permissive for screening
status = go_no_go(0.20, threshold=0.1)

When to use stricter thresholds:

  • Clinical/therapeutic applications

  • High-throughput screens (false positives costly)

  • Final candidate selection

When to use permissive thresholds:

  • Initial screening (false negatives costly)

  • Research applications

  • When other filters are applied

The 4π² Connection#

The constant 4π² (≈ 39.48) appears throughout IR theory:

  • Period-amplitude relationship: T² ∝ 4π²/ω₀²

  • Phase space volume: V = 4π² in normalized units

  • Uncertainty relation: ΔE·Δt ≥ ℏ/(4π²) for coherent states

The relationship to e^-2:

\[4\pi^2 \cdot e^{-2} \approx 5.35\]

This product appears in variance-threshold relationships.

See Also#