Probability Signatures: Theory & Applications

• Probability Signatures are structured probabilistic descriptors that compress high-dimensional randomness into concise signatures for comparison and diagnostic inference.
• They encompass various constructions, including reliability signatures, rough-path expected signatures, and memory witnesses in stochastic processes.
• Applications span engineering reliability, quantum annealing, language model training, and cosmological observations, offering practical insights across disciplines.

Probability signatures are structured probabilistic descriptors used to encode how randomness, dependence, or stochastic dynamics interact with an underlying object. In the cited literature, the term denotes several technically distinct constructions rather than a single universal definition: an $n$-tuple attached to semicoherent systems in reliability theory, a family of subsignatures for component subsets, probability measures and expected signatures on rough-path signature space, single-time witnesses of memory in classical stochastic processes, time-integrated probability-flux patterns in thermal and quantum annealing, and token-level distributions that govern embedding geometry in LLMs (Marichal et al., 2012, Marichal, 2012, Chevyrev et al., 2013, Smirne et al., 2012, Yao et al., 24 Sep 2025, Horiike et al., 20 Nov 2025). This suggests a family resemblance: each construction compresses high-dimensional probabilistic information into a signature that supports comparison, decomposition, inference, or diagnosis.

1. Terminological scope and recurring structure

Across the cited work, the phrase “probability signature” is domain-specific. In semicoherent system theory it is the vector

$p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$

which records which order statistic of the component lifetimes causes system failure. In annealing theory, the “probability-flux signature” is the full set of time-integrated fluxes $\{\Delta\mathcal J(\mathbf s,\mathbf s')\}$ over state-space edges. In representation learning, “probability signatures” are token-level conditional distributions such as label-given-token and co-occurrence-given-token. In non-Markovianity studies, the phrase refers to distributional witnesses such as revivals of the Kolmogorov distance. In CMB polarization, the relevant observables are probability-distribution distortions and topological statistics derived from excursion sets (Marichal et al., 2012, Smirne et al., 2012, Ganesan et al., 2014, Yao et al., 24 Sep 2025, Horiike et al., 20 Nov 2025).

The common structural feature is not a shared formula but a shared role. Each signature is a reduced object that preserves information judged salient for a particular inference problem: system failure ordering, module-level decomposition, uniqueness of a random rough path law, detection of memory, discrimination of thermal versus quantum fluctuations, or alignment between corpus statistics and learned geometry. The mathematical content therefore depends entirely on the ambient theory.

2. Reliability-theoretic probability signatures

In reliability theory, a probability signature is defined for an $n$-component semicoherent system $S=(C,\varphi,F)$, where $C=[n]$, $\varphi:\{0,1\}^n\to\{0,1\}$ is nondecreasing in each coordinate with $\varphi(0,\dots,0)=0$ and $\varphi(1,\dots,1)=1$, and $F$ is the joint c.d.f. of the continuous component lifetimes $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$0. If $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$1 denotes the system lifetime and $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$2 the $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$3-th order statistic of $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$4, then the probability signature is

$p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$5

Equivalently, one introduces the tail signature

$p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$6

with $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$7 (Marichal et al., 2012).

This construction generalizes Samaniego’s structural signature. For i.i.d. continuous lifetimes, the signature depends only on the system structure; in the general semicoherent setting with dependent lifetimes, the same $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$8-tuple becomes a probability signature and may depend on both the structure of the system and the probability distribution of the component lifetimes (Marichal et al., 2012).

A central object is the relative-quality function

$p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$9

The tail signature satisfies

$\{\Delta\mathcal J(\mathbf s,\mathbf s')\}$0

Hence all of $\{\Delta\mathcal J(\mathbf s,\mathbf s')\}$1 depends on the joint lifetime law $\{\Delta\mathcal J(\mathbf s,\mathbf s')\}$2 only through $\{\Delta\mathcal J(\mathbf s,\mathbf s')\}$3. This representation is the basis for both decomposition theorems and nonexchangeable generalizations (Marichal et al., 2012).

The significance of the reliability-theoretic definition lies in its separation of structural and probabilistic content. The structure function $\{\Delta\mathcal J(\mathbf s,\mathbf s')\}$4 identifies which component states keep the system alive, while $\{\Delta\mathcal J(\mathbf s,\mathbf s')\}$5 measures how likely a given subset is to occupy the top order-statistic positions. The signature then converts an arbitrary lifetime model into an ordered failure-attribution profile.

3. Modular decomposition and subsignatures

A major development is modular decomposition. Suppose the components are partitioned into $\{\Delta\mathcal J(\mathbf s,\mathbf s')\}$6 disjoint modules $\{\Delta\mathcal J(\mathbf s,\mathbf s')\}$7, with $\{\Delta\mathcal J(\mathbf s,\mathbf s')\}$8, module structure functions $\{\Delta\mathcal J(\mathbf s,\mathbf s')\}$9, marginal laws $n$0, and module-level relative-quality functions $n$1. If the modules are connected by an organizing semicoherent structure $n$2, with multilinear extension $n$3, and if $n$4 is $n$5-decomposable in the sense that

$n$6

for $n$7, then the system tail signature obeys the general modular-decomposition formula

$n$8

where

$n$9

This extends earlier two-module i.i.d. results to arbitrary module networks and general dependent lifetimes (Marichal et al., 2012).

Special cases clarify the structure. In the i.i.d. or exchangeable case, $S=(C,\varphi,F)$0 depends only on $S=(C,\varphi,F)$1, and the weights $S=(C,\varphi,F)$2 become multinomial-hypergeometric coefficients. If $S=(C,\varphi,F)$3 is the series-AND of the module statuses, then $S=(C,\varphi,F)$4, and the decomposition reduces to a product of module tail signatures; if $S=(C,\varphi,F)$5 is the parallel-OR, the dual signature formula is used (Marichal et al., 2012).

A related refinement is the subsignature. For a nonempty subset $S=(C,\varphi,F)$6, with $S=(C,\varphi,F)$7 and order statistics $S=(C,\varphi,F)$8, the $S=(C,\varphi,F)$9-signature is

$C=[n]$0

It records the probability that the $C=[n]$1-th failure among the components in $C=[n]$2 causes system failure. This interpolates between the classical system signature and the Barlow–Proschan importance index: if $C=[n]$3, one recovers the global signature; if $C=[n]$4, one obtains $C=[n]$5, the importance of component $C=[n]$6 (Marichal, 2012).

Subsignatures admit linear expressions in terms of the structure function and relative-quality functions $C=[n]$7. In the exchangeable-lifetimes case they become purely structural, and when $C=[n]$8 is a module, the normalized $C=[n]$9-signature can be written as

$\varphi:\{0,1\}^n\to\{0,1\}$0

which interprets the subsignature as the internal module signature times a conditional importance of the module (Marichal, 2012).

4. Probability on signature space in rough-path theory

In rough-path analysis, the key object is not an $\varphi:\{0,1\}^n\to\{0,1\}$1-tuple of failure probabilities but the signature of a path and the probability law carried by signature space. For a continuous path $\varphi:\{0,1\}^n\to\{0,1\}$2 of bounded $\varphi:\{0,1\}^n\to\{0,1\}$3-variation with values in a real Banach space $\varphi:\{0,1\}^n\to\{0,1\}$4, the $\varphi:\{0,1\}^n\to\{0,1\}$5-th level iterated integral is

$\varphi:\{0,1\}^n\to\{0,1\}$6

and the full signature is the formal series

$\varphi:\{0,1\}^n\to\{0,1\}$7

By Chen’s theorem, $\varphi:\{0,1\}^n\to\{0,1\}$8 is group-like (Chevyrev et al., 2013).

For a $\varphi:\{0,1\}^n\to\{0,1\}$9-valued random variable $\varphi(0,\dots,0)=0$0 with law $\varphi(0,\dots,0)=0$1, weak integrability means that every $\varphi(0,\dots,0)=0$2 has finite expectation on $\varphi(0,\dots,0)=0$3. The barycenter $\varphi(0,\dots,0)=0$4 is then defined by

$\varphi(0,\dots,0)=0$5

Writing the projections of $\varphi(0,\dots,0)=0$6 onto each tensor level gives the expected signature

$\varphi(0,\dots,0)=0$7

A basic result is that, for $\varphi(0,\dots,0)=0$8-valued $\varphi(0,\dots,0)=0$9, weak integrability is equivalent to $\varphi(1,\dots,1)=1$0, in which case $\varphi(1,\dots,1)=1$1 (Chevyrev et al., 2013).

Chevyrev and Lyons define a characteristic functional for probability measures on signature space using finite-dimensional unitary representations. If $\varphi(1,\dots,1)=1$2 denotes the continuous algebra maps

$\varphi(1,\dots,1)=1$3

arising from linear maps $\varphi(1,\dots,1)=1$4, then

$\varphi(1,\dots,1)=1$5

serves as the analogue of a characteristic function. Since these representations separate points of $\varphi(1,\dots,1)=1$6, the corresponding $\varphi(1,\dots,1)=1$7 determine the law on $\varphi(1,\dots,1)=1$8 (Chevyrev et al., 2013).

The expected signature also addresses a noncommutative moment problem. If $\varphi(1,\dots,1)=1$9 are $F$0-valued random variables with $F$1 and $F$2 has infinite radius of convergence, then $F$3 and $F$4 have the same law. A method of moments for weak convergence is also established: if $F$5 weakly in $F$6, then along a subsequence $F$7, where $F$8 is the unique $F$9-valued random variable with barycenter $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$00 (Chevyrev et al., 2013).

Applications include Lévy rough paths, Gaussian rough paths, and Markovian rough paths. In each case, exponential or Gaussian tail estimates on suitable decomposition counts imply positive or infinite radius for the expected signature, and hence uniqueness and analyticity properties for the associated characteristic functionals (Chevyrev et al., 2013).

5. Dynamical signatures: memory witnesses and probability flux

For classical finite-state stochastic processes, Smirne, Stabile, and Vacchini use the evolution of single-time probability distributions to define signatures of non-Markovianity. Given two probability vectors $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$01 and $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$02 on a finite set $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$03, the Kolmogorov distance is

$p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$04

If the evolution is described by a family of stochastic matrices $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$05 satisfying the Chapman–Kolmogorov property, then $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$06 is contractive: $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$07 Therefore any interval on which

$p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$08

signals a back-flow of information and serves as a signature of memory. The total amount of such memory is quantified by

$p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$09

where $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$10 are the intervals with $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$11 (Smirne et al., 2012).

A second sufficient witness is violation of $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$12-divisibility. The time-local master equation

$p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$13

is $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$14-divisible exactly when all rates $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$15. Negative rates therefore signal non-Markovianity, but they do not in general coincide with revivals of $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$16. In a two-state semi-Markov process, however, the two signatures become equivalent: $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$17 exactly on the intervals where $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$18 rises (Smirne et al., 2012).

A different dynamical usage appears in annealing. For an $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$19-spin Ising model with Glauber-type dynamics, the thermal probability flux from $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$20 to $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$21 is

$p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$22

with the time-integrated flux

$p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$23

For transverse-field quantum annealing, if

$p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$24

then the quantum probability flux is

$p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$25

with

$p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$26

For either thermal or quantum annealing, the full set $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$27 is a vector in $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$28, where

$p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$29

and this high-dimensional object is called the probability-flux signature (Horiike et al., 20 Nov 2025).

The annealing paper uses these signatures to examine all possible interaction networks of $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$30 Ising spin systems up to seven spins. The main finding is that thermal and quantum annealing are broadly similar, but quantum tunnelling produces qualitative differences for particular interaction networks. In thermal annealing, the sign of every flux is determined by the Boltzmann preference and one always “goes downhill” or thermal-activates over small uphill barriers. In quantum annealing, coherent mixing near degeneracies can produce tunnelling flux that reverses direction once populations accumulate. In the five-spin network labeled 5-219, the thermal flux diagram points steadily toward the true ground state, whereas the quantum flux diagram shows oscillatory back-and-forth behavior on edges adjacent to the true ground state; numerically, $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$31 changes sign whereas $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$32 does not (Horiike et al., 20 Nov 2025).

The paper further proposes dimensional reduction for visualization. Using the cumulative occupancy

$p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$33

one forms a covariance matrix, extracts its top two eigenvectors, embeds the states into $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$34, and draws arrows of width $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$35. The resulting flux diagram is presented as an experimentally verifiable signature in AMO systems, and the authors suggest that classifying $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$36 graphs by flux signatures may help predict whether a given Ising encoding benefits more from thermal activation or quantum tunnelling (Horiike et al., 20 Nov 2025).

6. Probability signatures in language-model representation learning

In language modeling, Yao and Xu introduce “probability signatures” as token-level distributions extracted from the data distribution $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$37 over input-label pairs $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$38. For a token $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$39 and label $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$40, the four signatures are

$p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$41

described respectively as label-given-token, co-occurrence-given-token, co-occurrence-given-token+label, and token-given-label. These quantities are intended to capture token-level relationships intrinsic to the data (Yao et al., 24 Sep 2025).

The central claim is mechanistic rather than merely correlational. For embedding-based models

$p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$42

trained by population cross-entropy, the continuous-time gradient-flow equations for embedding columns and unembedding rows contain these signatures explicitly. In the linear model, the evolution of $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$43 is driven primarily by the label signature $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$44, with a smaller co-occurrence term scaled by $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$45. In the feedforward network, the three signatures $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$46, $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$47, and $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$48 appear with different prefactors, and in the modular-addition regime the $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$49 term is leading. For the linear unembedding, the dynamics are controlled by $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$50 (Yao et al., 24 Sep 2025).

The empirical study uses three composite addition tasks: simple addition, same-range addition, and modular addition. Each dataset has $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$51 examples, hidden dimension $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$52, small Gaussian initialization $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$53, and $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$54 training epochs. Both Linear and FFN models achieve $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$55 training accuracy on simple addition and same-range addition, whereas only the FFN fits modular addition. The embedding order metric

$p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$56

rapidly approaches $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$57 in simple addition for both models, indicating an ordered embedding line; the same structure emerges more slowly in same-range addition; and in modular addition only the FFN eventually develops the ordered geometry predicted by the theory (Yao et al., 24 Sep 2025).

The paper then scales the analysis to Qwen2.5-12L models trained on five subsets of the Pile, including arXiv, Math, CC, PubMed, and Wikipedia. For each token $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$58, the empirical next-token distribution $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$59 and previous-token distribution $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$60 are computed, and pairwise cosine-similarity matrices are compared with $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$61 and $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$62. The global alignment statistics

$p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$63

fall in the $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$64 range across all five subsets. High-average-similarity tokens also show especially strong alignment between embedding geometry and signature geometry. For the pretrained Qwen2.5-3B-base model with tied embeddings, the combined signature $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$65 recovers the main sub-block structure of $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$66, including the ordered digit block “0–9” (Yao et al., 24 Sep 2025).

The significance of this usage is that the signature is neither an importance index nor a flux observable. It is a collection of corpus-derived distributions that enters the training dynamics directly and is used to explain why embeddings become semantically organized.

7. Probability-based signatures in observational cosmology and related physics

In CMB polarization, probability-based signatures of local-type primordial non-Gaussianity are extracted from the one-point distribution of the total polarization intensity and from geometric-topological observables on excursion sets. If $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$67 and $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$68 are statistically independent zero-mean Gaussian random fields with common variance $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$69, then the total polarization intensity

$p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$70

has the Rayleigh density

$p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$71

For local-type non-Gaussianity parameterized by $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$72, the leading non-Gaussian correction to the PDF of $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$73 appears only at order $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$74, because the linear corrections from the two independent Cartesian components average out (Ganesan et al., 2014).

The same paper studies Minkowski Functionals and Betti numbers of the excursion sets of the $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$75-mode field and of the mean-subtracted intensity $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$76. In two dimensions, the Minkowski Functionals are the area fraction $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$77, the perimeter $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$78, and the genus $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$79; the Betti numbers are $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$80, the number of connected regions, and $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$81, the number of holes. The non-Gaussian deviations

$p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$82

scale linearly in $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$83 for the $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$84-mode field, with shapes, amplitudes, and cosmic-variance error bars very similar to the temperature case. By contrast, the signal in the total polarization intensity is much weaker: its PDF correction is suppressed to order $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$85, its non-Gaussian deviations are an order of magnitude smaller than in $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$86, and the signal-to-noise is correspondingly poor (Ganesan et al., 2014).

A related but terminologically distinct use of signature language appears in black-hole ringdown. In the quasilocal-probability framework, horizon-induced probability flux produces an effective non-Hermitian dynamics

$p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$87

and yields three linked signatures: correlated multi-mode deviations, weak amplitude dependence, and a mismatch between waveform damping and energy accounting. Because these effects arise from a single boundary-flux mechanism, they lie on a low-dimensional manifold in the space of mode observables, in contrast to generic modified-gravity deformations in which mode shifts are typically independent. The paper argues that current LVK observations constrain the relevant leakage scale at the $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$88 level, while multimode spectroscopy, stacking, Cosmic Explorer, Einstein Telescope, and LISA may push sensitivity to $p_k=\Pr[T_S=T_{k:n}],\qquad k=1,\dots,n,$89 or below (Trivedi et al., 22 Apr 2026).

Taken together, these examples show that “probability signatures” is best understood as a family of domain-dependent constructions: reliability signatures of failure order, module-level subsignatures, expected signatures and characteristic functionals on rough-path groups, memory witnesses from single-time distributions, probability-flux signatures of annealing dynamics, token-level distributions steering embeddings, and probability-based observables in cosmology and gravitational physics. The unifying theme is methodological rather than definitional: each signature is a structured proxy for probabilistic information that would otherwise remain distributed across a much larger state, path, or observation space.