A finite setting for invariant rediscovery [ftip-00NS]
AGENTDRAFTED
A first concrete family can use linear chain complexes over the two-element
field \(\mathbb F_2\). This adapts the incidence-matrix concept-discovery
setting of Aggarwal et al. into an
exact finite experiment. The chosen field, transformations and certificate
tasks below are part of this proposal, not a reproduction of that paper.
An object consists of vector spaces of dimensions \(n_0,n_1,n_2\) and
matrices \(D_1:\mathbb F_2^{n_1}\to \mathbb F_2^{n_0}\) and
\(D_2:\mathbb F_2^{n_2}\to \mathbb F_2^{n_1}\) satisfying \(D_1D_2=0\).
All entries, dimensions and allowed operations are public. Arithmetic is
exact. Zero-dimensional spaces and zero maps are included. The experiment
can provide only matrices and arithmetic, or additionally ranks and
nullities; these are different initial endowments and receive separate
results. Neither regime establishes discovery of the entire mathematical
representation from unstructured observations.
The middle cycles and boundaries are \(Z_1=\ker D_1\) and
\(B_1=\operatorname {im}D_2\). The chain identity gives \(B_1\subseteq Z_1\),
so the quotient \(H_1=Z_1/B_1\) is defined. Rank-nullity yields
\[
\beta _1=\dim H_1
=n_1-\operatorname {rank}D_1-\operatorname {rank}D_2.
\notag\]
Indeed, \(\dim Z_1=n_1-\operatorname {rank}D_1\) and
\(\dim B_1=\operatorname {rank}D_2\); taking the quotient subtracts these
dimensions. At the ends, \(\beta _0=n_0-\operatorname {rank}D_1\) and
\(\beta _2=n_2-\operatorname {rank}D_2\). These elementary identities explain
the evaluator's target and remain withheld as explicit answers where
rediscovery is being measured. They are not new mathematical results.
Allowed changes include invertible changes of basis \(P_i\) in each
space. They replace the matrices by
\[
\begin {aligned}
D'_1&=P_0D_1P_1^{-1},\\
D'_2&=P_1D_2P_2^{-1}.
\end {aligned}
\notag\]
The product remains zero and the ranks remain unchanged. Another allowed
change adjoins or removes a direct summand consisting of an identity map
between two adjacent one-dimensional spaces, with zero maps elsewhere.
That summand has zero homology in every degree. Direct sums add dimensions
of homology, so these moves preserve all three \(\beta _i\). A candidate
invariant can be challenged with new valid objects, basis changes and
such elementary additions.
A target asks whether two presented objects are related by a sequence
of these allowed moves. A positive certificate lists legal moves and their
exact matrices, including inverses for basis changes and the displayed
summand for a removal. A negative certificate supplies unequal homology
dimensions with checked rank witnesses. A rank witness can give invertible
row and column transformations, their inverses and a diagonal normal form
with an identity block and zeros elsewhere; the checker verifies these
matrix identities and counts the block size. Agreement of a proposed invariant
on a few examples is insufficient, and equality of the dimensions alone
is not accepted as a positive certificate. The agent must construct the
required transformation or another certificate justified by the fixed
mathematical checker.
A concrete sampler first draws uniformly from the finite set of
nonnegative integer tuples \((h_0,h_1,h_2,r_1,r_2)\) satisfying the declared
dimension cap, where
\[
\begin {aligned}
n_0&=h_0+r_1,\\
n_1&=h_1+r_1+r_2,\\
n_2&=h_2+r_2.
\end {aligned}
\notag\]
It builds a direct sum of zero-differential spaces of dimensions
\(h_i\) in degree \(i\), \(r_1\) identity pairs in degrees one and zero,
and \(r_2\) identity pairs in degrees two and one. Independently sampled
invertible binary basis matrices scramble this presentation; uniform
sampling by rejection from all binary square matrices is one exact choice.
The resulting \(\beta _i=h_i\) are evaluator facts, not additional observations
supplied to either agent. The distribution and sampling algorithm themselves
are public, so reconstructing this decomposition is an admitted strategy.
Positive instances apply a sampled legal move sequence to an object,
retaining that sequence privately. At each step the sampler chooses
uniformly among the declared finite encodings of dimension-bounded legal
moves; the identity move permits padding to the specified length. Negative
instances draw two canonical tuples with different homology vectors and
randomize their presentations independently. Their dimensions are matched
where the chosen profile permits, and results are also stratified by
dimension differences to detect easy size cues. Retained rank witnesses
certify the labels. The mixture, move count, dimension profile and
certificate limits are fixed before final seeds are sampled.
A single middle invariant is not complete even for this family.
Objects concentrated in degree zero can have identical \(\beta _1=0\)
and different \(\beta _0\), and hence cannot be related by the allowed moves.
This provides a concrete counterexample to premature abstraction. A learner
may retain the whole homology vector, refine its proposal or use direct
algebraic reasoning; successful certification, rather than one preferred
formula, determines the outcome.
This initial family has an efficient classical alternative. Gaussian
elimination computes bases for cycles and boundaries and decomposes a
finite complex over a field into homology summands and adjacent identity
summands. It therefore supplies both the invariants and constructive
transformations when appropriate. Its arithmetic and certificate costs
must be measured as an admitted baseline. The setting can reveal how an
agent develops and transfers a method, but cannot support a claim that
all autonomous methods face a prohibitive search barrier merely because
one language model initially fails it.