# DRCT v21 — Morphism Ontology Induction

## What was removed

v20 supplied a menu:

- bijection;
- injection;
- surjection;
- unrestricted function;
- stochastic kernel.

v21 does not expose those names or classes to the learner.

Every world is represented by one universal empirical relation object:

\[
C:X\times Y\to\mathbb N,
\]

a source-target observation-count table.

Its support is

\[
R(x,y)=\mathbf 1[C(x,y)>0].
\]

The learner receives only low-level relation statistics generated from this
universal representation.

---

## Primitive relation language

v21's primitive predicates are intentionally generic degree constraints:

\[
A_0:\quad \deg_R(x)=1\ \forall x
\]

\[
A_1:\quad \deg_R^{-}(y)\le1\ \forall y
\]

\[
A_2:\quad \deg_R^{-}(y)\ge1\ \forall y
\]

\[
A_3:\quad \deg_R(x)=2\ \forall x.
\]

They are not named "deterministic", "injective", "surjective", or
"stochastic" inside the learner.

A candidate reusable concept is a conjunction over these primitive predicates,
with unspecified predicates left as wildcards.

There are

\[
3^4-1=80
\]

nontrivial candidate conjunctions.

---

## Exact combinatorial model code

For finite source/target cardinalities, v21 computes the number of relation
supports satisfying each primitive pattern.

Examples:

The unrestricted nonempty-row relation universe has size

\[
(2^{|Y|}-1)^{|X|}.
\]

The constraint \(A_0\) has

\[
|Y|^{|X|}
\]

relations.

The conjunction \(A_0\land A_1\) has

\[
\frac{|Y|!}{(|Y|-|X|)!}
\]

relations when \(|X|\le |Y|\).

The conjunction \(A_0\land A_2\) has the number of surjections
\(|X|\twoheadrightarrow|Y|\).

For the row-degree-two primitive,

\[
A_3
\]

has

\[
\binom{|Y|}{2}^{|X|}
\]

supports before additional target-coverage constraints.

All sixteen exact truth signatures over
\((A_0,A_1,A_2,A_3)\) are counted by inclusion-exclusion.

Thus the compression score is not an arbitrary clustering distance. It is a
finite code length derived from the number of relations still possible under a
candidate concept.

---

## Concept promotion

Initially every world is encoded from the universal relation model.

For candidate concept \(C_j\), define its net corpus gain

\[
\Delta L_j
=
L_{\rm current}
-
L_{\rm current+C_j}
-
L(C_j).
\]

The learner greedily promotes the concept with largest positive gain and repeats
until no candidate pays for its own definition.

No morphism labels participate in this process.

---

## Anonymous ontology

A world may belong to several promoted concepts.

Its ontology coordinate is the membership vector

\[
\boxed{
\chi(W)
=
(\mathbf1[W\in C_1],\ldots,\mathbf1[W\in C_m]).
}
\]

Distinct recurring membership vectors form anonymous morphism classes.

Logical implication among learned concepts gives an ontology partial order.

For example, if one discovered concept asserts every constraint asserted by
another, the former is a child/more-specific concept.

---

## Post-hoc interpretation

Only after ontology induction do we inspect the learned predicates.

In the v21 laboratory, the promoted concepts correspond post hoc to structures
such as:

- single-output relations;
- single-output relations with no target collision;
- target-covering single-output relations;
- target-covering two-output-per-source relations.

Their intersections recover the conventional deterministic morphism taxonomy.

For example, a relation satisfying both the learned
"single-output/no-collision" and "single-output/target-covering" concepts is
post hoc a bijection.

The important point is that "bijection" is not a candidate label selected by
the learner. It is an interpretation of an induced anonymous ontology cell.

---

## Prospective transfer

After the concept library is frozen, new worlds are encoded by the same
universal count-table representation.

The frozen concepts assign an anonymous membership coordinate and reduce target
description length.

Novel structural signatures absent from the source ontology are not forced into
an existing class; they remain raw/unclassified.

---

## Scope

v21 breaks the fixed morphism-class menu, but it does not create ontology from
literally nothing.

The remaining prior is the primitive relation-description language itself:
row/column degree predicates and finite combinatorial coding.

The next wall is therefore to invent the structural predicates used to describe
relations, rather than supplying degree constraints in advance.