# DRCT v15 — Operator Invention From Closure Residuals

## Starting point

Let \((A,+)\) be an already available algebraic language and let \(M_1,M_2,M_3\)
be primitive measured coordinates.

Assume the entire current grammar contains only affine expressions generated by
addition, subtraction and constants.

If exhaustive search over that grammar leaves

\[
\boxed{
\min_{g\in\mathcal G_{\rm add}}H(F\mid g(M))>0,
}
\]

then closure is impossible in the current operator language.

This is stronger than one failed symbolic-regression search because, for a
finite prime field, all additive/subtractive expressions reduce to affine
forms and can be exhaustively enumerated.

---

# Unknown-operation synthesis

Instead of adding a named operation such as multiplication to the grammar,
introduce an **uninterpreted** binary operation

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

On a finite alphabet of size \(p\), its semantics are an unknown \(p\times p\)
Cayley table.

A candidate relational descriptor may have the meta-form

\[
Q
=
(M_i\odot M_j)+\alpha M_k+\beta,
\]

but the values of \(\odot(a,b)\) are inferred directly from predictive
constraints rather than selected from an operator library.

For each observed state,

\[
\boxed{
M_i\odot M_j
=
F-\alpha M_k-\beta.
}
\]

If repeated observations with the same \((M_i,M_j)\) imply conflicting table
values, that proposed operator placement is falsified.

If every table cell is consistently determined, a new operation has been
operationally synthesized.

---

# Definition — Operator-Language Closure

Let \(\mathcal O\) be the current set of executable operations.

\[
D_{\mathcal O}
=
\min_{g\in{\rm Expr}(\mathcal O)}
H(F\mid g(M)).
\]

An operator extension \(\mathcal O'=\mathcal O\cup\{\odot\}\) is useful iff

\[
\boxed{
D_{\mathcal O'}<D_{\mathcal O}.
}
\]

It reaches predictive closure when

\[
D_{\mathcal O'}=0.
\]

---

# Algebra identification after synthesis

Once a Cayley table is inferred, its structural properties can be tested
without presupposing its name:

- commutativity;
- associativity;
- identity;
- absorbing zero;
- distributivity over an existing addition;
- zero divisors;
- inverses.

If the existing addition is the additive group of a finite field and the new
operation satisfies the corresponding field axioms, then the pair of operations
forms a finite field structure.

Calling the new operation "multiplication" is then an interpretation of its
algebraic role, not an assumption made during its discovery.

---

# Theorem 42 — Additive-Language Impossibility in the v15 Lab

Over \(GF(5)\), every expression generated solely from variables, constants,
addition and subtraction is affine:

\[
q=a_0x_0+a_1x_1+a_2x_2+b.
\]

There are only

\[
5^4=625
\]

such scalar functions.

Exhausting all 625 therefore certifies whether the additive language can
contain a sufficient scalar descriptor for the finite laboratory.

---

# Theorem 43 — Finite Cayley Identification

For a \(p\)-element carrier set, an unrestricted binary operation has \(p^2\)
table entries.

If the experimental data contain at least one consistent identifying
constraint for every input pair \((a,b)\), the entire operation table is
identified.

Prospective states using already identified table inputs test whether the
operator composes correctly with the remaining coordinates.

This is finite operation tomography.

---

# Polynomial compression

Every function

\[
T:GF(p)^2\to GF(p)
\]

has a unique bivariate polynomial representative with degrees at most \(p-1\)
in each variable.

After the operation table is discovered, finite-field interpolation can
compress its  \(p^2\) entries into a symbolic law.

This compression occurs **after** operator invention; it is not used to choose
the operation.

---

# Scope

The experiment demonstrates autonomous invention of the semantics of an
unknown finite binary operation inside a fixed meta-template.

It does not prove that an unconstrained machine can invent arbitrary
mathematics from no prior representational assumptions. The meta-language still
supplies concepts such as "binary operation", composition, predictive closure
and algebraic-axiom testing.

That remaining meta-language is the next boundary.