# DRCT v18 — Learning the Equivalence Relation Between Concepts

## Problem

v17 learned a reusable concept only after being told which transformations
should count as superficial coordinate changes.

v18 makes that "sameness rule" latent.

A binary residual concept is a table

\[
T:A\times A\to A.
\]

Suppose a candidate coordinate transformation has three permutation slots:

\[
(\pi_r,\pi_c,\pi_o)
\]

acting on row labels, column labels, and output labels, plus an optional
argument-reversal bit.

v18 is NOT told which permutation slots must represent the same coordinate
gauge.

Instead it searches every set partition of

\[
\{r,c,o\}.
\]

For example:

- \(\{\{r\},\{c\},\{o\}\}\): all three relabelings independent;
- \(\{\{r,c\},\{o\}\}\): row and column share a gauge;
- \(\{\{r,c,o\}\}\): one carrier relabeling acts on both inputs and output.

There are \(B_3=5\) set partitions. A second latent decision is whether argument
reversal belongs to the nuisance transformation family.

Thus the equivalence-language learner compares ten candidate transformation
rules.

---

## Exact mapping tomography

For two observed tables \(T,U\), v18 enumerates row and column permutations.
For each pair it infers whether a bijective output permutation exists that makes

\[
U(\pi_r a,\pi_c b)
=
\pi_o(T(a,b))
\]

or the argument-reversed version hold for every table entry.

This recovers all exact coordinate mappings between the observed relations
without access to their generating transformations.

---

## MDL rule learning

A transformation rule \(\mathcal R\) pays one permutation parameter for each
block in its learned partition.

If the partition has \(k\) blocks, mapping one concept instance costs roughly

\[
k\log_2(|S_{|A|}|)
\]

plus a swap bit when reversal is admitted.

A table that cannot be explained by the rule must be encoded independently.

Therefore

\[
\boxed{
\mathcal R^\star
=
\arg\min_{\mathcal R}
L(\text{concept corpus}\mid\mathcal R).
}
\]

Too-rigid rules fail to merge true coordinate variants and pay for repeated raw
tables.

Too-flexible rules spend unnecessary transformation parameters and can merge
isotopic-but-nonisomorphic traps.

The minimum-description rule identifies which coordinate transformations are
actually shared.

---

## Learned result in the v18 laboratory

The hidden source corpus is generated by one common carrier relabeling acting on
row, column, and output coordinates, with optional argument reversal:

\[
\boxed{
\pi_r=\pi_c=\pi_o.
}
\]

The learner is not told this equality.

The learned partition is

\[
\boxed{
\{\{r,c,o\}\}
}
\]

with argument reversal admitted.

This defines the action

\[
U(\pi a,\pi b)=\pi(T(a,b)),
\]

possibly after transposition.

---

## Theorem 44 — Orbit Sameness Is an Equivalence Relation

Let a group \(G\) act on a concept space \(\mathcal C\). Define

\[
T\sim_G U
\iff
\exists g\in G:\ U=g\cdot T.
\]

Then \(\sim_G\) is an equivalence relation.

### Proof

Reflexive: the identity \(e\in G\) gives \(T=e\cdot T\).

Symmetric: if \(U=g\cdot T\), then
\(T=g^{-1}\cdot U\).

Transitive: if \(U=g\cdot T\) and \(V=h\cdot U\), then
\(V=(hg)\cdot T\).

Therefore learned group orbits provide a mathematically valid notion of
"same concept modulo coordinate convention."

---

## Over-flexibility trap

v18 includes a relation obtained from the latent concept by independent row,
column and output permutations.

It is an **isotope** of the concept, so the fully independent transformation
rule merges it.

But it is not in the same orbit under the learned shared-coordinate rule.

Thus the experiment distinguishes:

\[
\text{same abstract concept under one carrier relabeling}
\]

from

\[
\text{merely transformable by an overly flexible coordinate map}.
\]

---

## Prospective transfer

After the equivalence rule is frozen, new concept tables are generated using
carrier relabelings not present in the source corpus.

The learned orbit is used for few-shot recognition and completion from only a
subset of table entries.

Negative isotopes and unrelated relations test false-positive rejection.

---

## Prior-art boundary

Symmetry discovery is an active field. Recent work studies learning unknown
invariance groups from data, including exact subgroup recovery, unknown
continuous symmetries, and unsupervised symmetry discovery.

Program abstraction also has a long tradition of reasoning modulo user-provided
equational theories.

DRCT does not claim the general notion of discovering symmetries or quotienting
by group actions as new.

The DRCT-specific synthesis under test is:

\[
\boxed{
\text{closure residual concepts}
\to
\text{all exact inter-concept coordinate maps}
\to
\text{learn constraints among transformation slots}
\to
\text{MDL-selected group action}
\to
\text{orbit equivalence}
\to
\text{prospective canonical transfer}
\to
\text{over-flexibility rejection}.
}
\]

The remaining prior is now narrower: v18 still supplies the primitive notion
that candidate coordinate changes are finite label permutations and that
"sameness structure" can be expressed as equality constraints among
transformation slots plus optional argument reversal.