Trace conclusiveness of contextual graphs and knowledge.

2026-08-26 by Jack Kowalski

Each argument has its own map of routes that changes depending on the order in which and with what you connect it.


Analytical applications in biotech, fintech, rhetorical manipulation, and physical events. A short strict explanation first, then the same idea in plain language.

The sole difference between correlation-based analytical tools and expert systems is the change in the list of factors that influence the conclusion (the logical 0/1 outcome).

In expert systems the conclusion is conditioned on a single variable and returns true/false. In systems based on co-occurrence of events the number of input arguments is usually large, and the obtained result is NotFalse / NotTrue - where NotFalse does not mean True, only the absence of a False conclusion, and vice versa.

People normally expect some probability measure when they see such a result, reasoning “how often is it true / not true.” That expectation is pointless. In the spaces of correlation trajectories there is no sensible global metric description; the metric is variable and local for every argument.

Exploring the composition of arguments into a trajectory - when those arguments carry different local metrics - allows one to locate the metric of highest information density (the highest conclusiveness of assignment to the considered correlations) [generalized Fisher, appendix 1]. The influence of the arguments used (the context) and the order of their application generates a trajectory that affects conclusiveness. This does not mean that the conclusiveness will actually appear; we are exploring correlations that, when projected onto any space imaginable to humans, are sparse (mostly empty, inconclusive - just as most of the universe is simply empty, or even emptier; larger particle aggregates are rare).

The term “trace conclusiveness” used in the text denotes a situation in which a result appears to exist, yet emerges only under additional, usually undeclared conditions - that is, the classic “it depends, and besides, these and those other things would still have to happen…”.

Technically it is a local, conditional density of information - a trace of a trajectory, not a global truth. In HMFoDG terms: a high value of R_w appears only under a specific sequence and context; outside it the metric collapses to emptiness (low conclusiveness).

Here every chemist immediately understands: only successive physical operations performed on strictly defined arguments lead to the very rare result of obtaining a concrete molecular configuration. Those operations cannot be executed in arbitrary order. The execution trajectory is conclusive solely when the strict sequence is preserved - what, how much, when, in what state (temperature, pressure, activation state). Only then does the statistically known portion of the outcome match what we expect. The outcome is a knowledge graph; knowledge itself is the fact that doing something differently mostly leads to no effect at all (emptiness), while a few paths lead to a different conclusion (a different result). Of course no one records the non-conclusive outcomes in the graphs, so they appear non-existent. In systems that take correlations (for example co-occurrence) as input data, that non-conclusiveness is already contained - it is simply not revealed; therefore the sets of results remain sparse relative to the data.

The same is obvious to every biologist: only certain sequences of events lead “with some probability” (because we cannot account for every argument that may appear in a real trajectory, we impose false but socially workable probability metrics for communication) to certain effects. Change the order of just two arguments and the effect disappears completely (emptiness, non-conclusiveness).

Exactly the same holds in fintech or systems of governance. Offering a loan for a horse to someone who lives in a city may not be a trajectory that reaches any conclusion. A sanction that consists of confiscating a chicken may be of negligible importance to someone who does not raise chickens.

We therefore have an entire class of systems in which the obvious answer is “it depends.” That “it depends” is precisely the set of context that generates a trajectory by means of projections carrying different metrics. Different, because the arguments possess different numbers of inputs (“depends on”) and outputs (“leads to”), numbers that are narrowed only by the trajectory itself (the change, visible in the projection, of the number of inputs and outputs that depend on inputs coming from other related arguments - i.e., a Finsler structure). This is quite obvious in practice: the linkage “green cow” produces a conclusion that is widely imaginable, while “creamy sound” does not.

That is why in biotech, fintech, rhetoric and physics the question “is this true?” is poorly formulated - the proper question is: “along which trajectory of context and in what order of arguments can this conclusion appear at all?”

(The variable exponent that steers the local change of metric - and therefore of the resulting algebra - arises isomorphically to the Boltzmann factor; otherwise the construction would be inconsistent. See Feynman Lectures on Physics, Vol. I, §40–4 “The distribution of molecular speeds” for the classical picture of why a variable exponential is forced by the underlying statistics.)

That is why in HMFoDG the exponent [N(v) → R_w → LCP → adaptive λ/w] is adaptive - exactly as the Boltzmann factor must be variable in order to describe local density fluctuations. Interestingly, already on the first page of the function’s derivation the shape of the well-known Boltzmann factor appeared. Later it became clear that it could not have turned out otherwise - the variable exponent is forced by the local statistics of the metric.

If You are intrested to dig deeper or just need someone to solve analitical assumptions in Your project....

The tool is implemented as a prototype tested for several months. If you are interested in the computational side then the entire loop of the following two equations (aggregation + generalized Fisher with free metric):

  1. Projection onto x0 → radial graph + v_i = x_i - x0.
  2. Local N(v), ordering y_i.
  3. Aggregator: R_w(y) = (∑ α_i |y_i|^w )^{1/w} [w adaptive, w=1 neutral].
    • w<1: diffuse/generalize
    • w>1: selective/amplify dominant
  4. Hierarchical addressing: directional/p-adic prefixes → ultrametric d = p^{-LCP} or e^{-λ LCP}.
    • Lemma: NN ≡ max LCP → prefix tree = efficient hierarchy.
  5. Lambda/w from Fisher-like: local entropy H(N(v)), density, or spectral radius embeddability (ρ² < α).
  6. Flow: repeat aggregation → update metric/prefixes → reprojection with new data (redshift horizon).

Yields:

Advantages for ML/embeddings:
- Deterministic hierarchy (sharp taxonomies, noise control).
- Ordering of training instead of post-hoc censorship.
- Strong local metric deformation (tunnels, contrasts).
- Easily implementable (prefixes + LCP << cosine brute force).

This procedure is applied in the tool, from which I can list embeddings.

The computational and proof explanation of this loop is in:

And curiosities are listed here:

Feel free to contact me if You have an analitical assumptions to solve.

-- Jack Kowalski.

Tags: hmfodg

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