Mathematical framework

Notation

Let \(P\in\mathbb{R}^{N\times N}\) be a non-negative row-stochastic transition matrix from the RNA-velocity graph. Let \(S\) denote the selected root-plus-fate cells, and let \(A\in\{0,1\}^{N\times O}\) identify the anchor states for \(O\) selected and optional competing outcomes.

Discounted Future-Fate Propagation

Anchor rows are made absorbing to form \(P^*\). Before each transition the process stops with probability \(1-\gamma\), where

\[\gamma = \frac{h}{h+1}.\]

The expected number of continued transitions is \(\gamma/(1-\gamma)=h\), so effective_horizon=h is an interpretable graph scale rather than physical time.

For an anchor state, the outcome vector equals its anchor indicator. For a transient state,

\[H_i = \gamma\sum_j P^*_{ij}H_j.\]

Partitioning the absorbing chain into transient-to-transient block \(Q\) and transient-to-anchor block \(R\) gives

\[(I-\gamma Q)H_T = \gamma R.\]

scCS uses sparse direct solution for modest graphs and sparse fixed-point iteration for large graphs. The unresolved probability is

\[U_i = 1-\sum_o H_{io}.\]

Selected-fate metrics

Let \(p_{if}\) denote the DFFP probability for selected fate \(f\). Discounted Fate Reach is

\[R_i = \sum_f p_{if}.\]

Conditional Fate Affinity is defined when \(R_i\) exceeds the configured minimum reach:

\[q_{if} = \frac{p_{if}}{R_i}.\]

Normalized future-fate entropy and Future-Fate Specificity are

\[E_i = -\frac{\sum_f q_{if}\log q_{if}}{\log K}, \qquad S_i = 1-E_i.\]

Resolved Commitment is

\[C_i = R_iS_i.\]

CFA describes identity, DFR describes resolution, FFS describes decisiveness, and RC combines resolution with decisiveness. These values should be reported separately rather than replacing them with one composite.

Signed Ordering Flux

Let \(s_i\) be the supplied ordering. The one-step selected-path coverage is

\[c_i = \sum_{j\in S}P_{ij}.\]

Conditioning transitions on remaining inside the selected furcation gives

\[\widetilde{P}_{ij}=\frac{P_{ij}}{c_i},\qquad j\in S.\]

Signed Ordering Flux is

\[g_i = \sum_{j\in S}\widetilde{P}_{ij}(s_j-s_i).\]

Positive \(g_i\) is forward, negative \(g_i\) is retrograde, and values near zero may represent stationary, mixed, turning, or loop-like motion. The support-weighted flux \(c_ig_i\) is also available when outside-path mass should reduce the contribution.

Endpoint anchors

For each fate, scCS selects late cells within that annotated fate using the ordering quantile and a minimum anchor count. Anchor diagnostics report transition mass to the same fate, root, other selected fates, and outside the selected path. Non-sink-like anchors are warnings for interpretation, not automatic failures.

Instantaneous mode

For selected star coordinates \(y_i\) and retained normalized transition weights \(\widetilde{P}_{ij}\), instantaneous mode computes

\[v_i^{\mathrm{star}}=\sum_{j\in S}\widetilde{P}_{ij}(y_j-y_i).\]

The fate-directed component is compared with ideal regular-simplex directions. For \(K\) fates, these directions satisfy

\[\begin{split}d_f^\top d_g = \begin{cases} 1, & f=g,\\ -1/(K-1), & f\ne g. \end{cases}\end{split}\]

Cosine-softmax affinity is then

\[a_{if}=\frac{\exp(\beta\cos\theta_{if})} {\sum_g\exp(\beta\cos\theta_{ig})}.\]

This mode measures immediate local direction in the supervised geometry. It is not the same as DFFP and should not be interpreted as long-term fate probability.

Condition inference

PairScorer and MultiScorer use one pooled scientific model and aggregate cell metrics within genuine biological replicates. Formal inference permutes replicate labels, not cell labels. Hierarchical bootstrap resamples replicates first and cells within replicate second. This preserves the experimental unit and avoids pseudoreplication.

Parameter interpretation

effective_horizon controls graph depth, anchor_quantile controls how late endpoint anchors are, min_anchor_cells prevents tiny anchor sets, and min_reach controls when CFA is considered defined. A defensible analysis reports sensitivity across plausible horizons and anchor quantiles.