# DRCT v19 — Transformation-Generator Invention

## What was removed

v18 assumed named transformation slots such as row, column and output
permutations.

v19 removes those slots.

The learner sees several worlds, each consisting only of measurements attached
to anonymous finite states.

It is not given:
- a grid;
- coordinates;
- rotations;
- reflections;
- row/column/output roles;
- a group label.

It is allowed one remaining meta-assumption: two worlds may be related by an
unknown bijection of their finite state sets.

---

## Step 1 — Experimental transformation tomography

Let a reference world have state set \(S\) and observable signature

\[
\Phi:S\to A^k.
\]

If the joint signature is injective, and a second world is generated by an
unknown bijection \(g:S\to S'\),

\[
\Phi'(g(s))=\Phi(s),
\]

then \(g\) is identified directly by matching signatures:

\[
\boxed{
g(s)
=
(\Phi')^{-1}(\Phi(s)).
}
\]

Thus the transformation itself is recovered as a permutation without any
predefined geometric interpretation.

---

## Step 2 — Generator invention

Suppose recovered transformations are

\[
g_1,\ldots,g_m.
\]

The learner closes them under composition and inverse and searches for the
smallest recovered subset whose closure explains all reachable transformations.

A generator set \(A\) induces

\[
\langle A\rangle
=
\{a_k\cdots a_1:\ a_i\in A\cup A^{-1}\}.
\]

The minimal generator problem here is finite and exhaustive.

In the v19 laboratory, only three worlds are directly observed:
- identity;
- one unknown transformation \(g_1\);
- one unknown transformation \(g_2\).

The two nontrivial recovered permutations generate eight transformations in
total, five of which were never directly observed.

---

## Step 3 — Algebraic group discovery

After the transformations are recovered, v19 measures:
- closure;
- identity;
- inverses;
- element orders;
- commutativity;
- relations among minimal generators.

The recovered group has order eight, element-order histogram

\[
1^1,\ 2^5,\ 4^2,
\]

is nonabelian, and contains generators \(r,s\) satisfying

\[
r^4=e,\qquad s^2=e,\qquad srs=r^{-1}.
\]

Only after this operational reconstruction is the group identified as

\[
\boxed{D_4}.
\]

The label "rotation/reflection" is not used during discovery.

---

## Hidden-coordinate adversary

Internally the benchmark is generated from square symmetries on sixteen states,
but before the learner sees anything the entire action is conjugated by a
random permutation \(h\):

\[
G_{\rm hidden}=hD_4h^{-1}.
\]

Therefore the learned generators are arbitrary-looking permutations on sixteen
anonymous channels. Grid geometry is unavailable to the learner.

---

## Step 4 — Prospective generated transformations

The learned group contains transformations not directly present in the source
worlds.

New worlds generated by these held-out group elements test whether composition
of invented transformations predicts valid unseen coordinate changes.

Out-of-group random permutations are negative controls.

---

## Step 5 — Few-shot transfer

A target value \(Y:S\to B\) is attached to reference states.

In a new world transformed by unknown \(g\), the target is

\[
Y'(g(s))=Y(s).
\]

After observing only a few world-state signatures, the learner retains group
elements consistent with those correspondences and predicts target labels on
unobserved states.

A baseline that knows only "some arbitrary permutation occurred" has no group
structure; after conditioning on observed correspondences, its remaining
unseen assignment is unconstrained.

Thus learned transformations become an executable transfer mechanism rather
than merely a classification label.

---

## Theorem 45 — Generated Transformations Form a Group

For bijections \(g_i:S\to S\), the closure generated under composition and
inverse is a subgroup of the symmetric group \(\mathrm{Sym}(S)\).

This is standard group theory.

The scientific question is not whether such a closure is mathematically a
group; it is whether the transformation generators themselves can be recovered
from cross-world observational correspondences and whether their closure
predicts unseen worlds.

---

## Scope / remaining prior

v19 demonstrates transformation-generator invention in a finite anonymous
state space.

It does NOT remove every prior. The learner is still told:
- worlds have corresponding finite state sets;
- the latent transformation is a bijection;
- joint observational signatures can be compared across worlds.

The next boundary is therefore sharper: infer the *class of admissible map*
itself — bijection, many-to-one quotient, continuous deformation, stochastic
kernel, etc. — rather than assuming bijectivity.