The empirical formula

The equation of resolution authority

The discipline does not start from a conclusion, but from an equation awaiting verification. Here it is written in mathematical language, with measurable variables and the exact threshold that would falsify it.

The model

Let $G = (V, E)$ be the knowledge graph. Each entity is a node; each directed edge $j \to i$ carries a semantic type $τ_{j i}$ (author, sameAs, member, citation…). The resolution authority $A (i)$ is defined by the recursion:

A (i) = (1 - d) b (i) + d \sum_{j \in In (i)} \frac{w (τ_{j i})}{W (j)} A (j)

(1)

$A (i)$ — resolution authority: How strongly the system "resolves" entity i. A latent quantity; its observable is defined below.
$b (i)$ — intrinsic authority: Signal a node accumulates on its own: domain age, history, backlinks. For a new domain it is ≈ 0.
$w (τ)$ — weight of the relation type: The heart of the generalisation: each type of edge weighs differently. PageRank is the case where all types weigh the same.
$W (j) = \sum_{k \in Out (j)} w (τ_{j k})$ — outgoing normalisation: The sum of the weights of the edges leaving j: the authority j distributes is split among its neighbours.
$d$ — damping: Factor $d \in (0, 1)$ weighing how much authority comes from the network versus the node itself.

Closed form

Setting the typed transition matrix $M_{i j} = w (τ_{j i}) / W (j)$ , equation (1) solves in closed form:

A = (1 - d) {(I - d M)}^{- 1} b

(2)

Each node's authority is thus a linear transformation of the vector of intrinsic authorities $b$ : what is yours, propagated along every path of the graph, weighted by relation type.

Continuity: PageRank as the degenerate case

If a single edge type exists and $w \equiv 1$ , then $w (τ_{j i}) / W (j) = 1 / | Out (j) |$ and (1) returns exactly classic PageRank. The lineage is coherent: the neural network learns the weights $w (τ)$ (the end of uncontrollable weights), PageRank is the propagation structure (the end of unit weights), and Ontopoietica stands in between: propagation over typed edges.

The empirical pivot: the virgin domain

Here the formula meets the experiment. For a new domain $v$ (this domain), intrinsic authority is null:

b (v) \approx 0

(3)

Substituting into (1), the first term vanishes and only propagation remains:

A (v) \approx d P (v), P (v) = \sum_{j \in In (v)} \frac{w (τ_{j v})}{W (j)} A (j)

(4)

The virgin entity's authority, at the start, cannot come from itself: it comes entirely from the propagation term $P (v)$ — the edges toward already-resolved entities (the author Paolo Galbiati; the sameAs to the already-settled term on profpaul.icu). It is the variable isolated by construction: if $A (v)$ becomes positive with $b (v) \approx 0$ , it was propagation, not age.

The observable

$A (v)$ is latent: it is not measured directly. It is measured through the resolution indicator at time $t$ :

R (v, t) \in {0, 1, 2}

(5)

0 — not resolved (no mention)
1 — mentioned and disambiguated as an entity
2 — cited as a source on a neutral query

and through the time to first resolution:

τ_{res} = \inf {t : R (v, t) \geq 1}

(6)

The falsification threshold

The formula is not an opinion: it makes two checkable predictions, decided before the data.

Strong hypothesis (existence). With $b (v) \approx 0$ and the edges in place at time $t_{0}$ , the entity resolves within horizon $T$ :

\exists t \leq t_{0} + T : R (v, t) \geq 1

(7)

Graded hypothesis (monotonicity). Across several virgin entities with different propagation $P (v)$ , the time to resolution decreases as $P (v)$ grows:

\frac{\partial τ_{res}}{\partial P (v)} < 0

(8)

No grounds to proceed

The formula is falsified if, with edges in place and crawled and $b (v) \approx 0$ , we observe $R (v, t) = 0$ for every $t \leq t_{0} + T$ . In that case propagation alone is not enough, intrinsic authority $b$ is necessary, and age cannot be bypassed by edges. It is not a defeat: it is a verdict, and it says exactly which term of (1) needed revising.

What the formula is, and is not

Equation (1) is a model consistent with the observed behaviour of graph-based retrieval systems — not a claim to be the internal equation of a specific engine, whose weights $w (τ)$ are not accessible. It is falsifiable in its predictions, not as a reconstruction of a proprietary implementation. This distinction is part of the method: a model that predicts and lets itself be refuted has value; one that claims to describe what it cannot observe does not.

References

L. Page, S. Brin, R. Motwani, T. Winograd — The PageRank Citation Ranking: Bringing Order to the Web (Stanford InfoLab, 1999).
Google — Introducing the Knowledge Graph: things, not strings (2012).
Schema.org — DefinedTerm: the standard used to structure the entities of this site.

This formalisation renders in notation the argument of the essay The Data-Identity Principle, refined by the discipline's four principles. The numerical check is in the experiment.