ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pendingjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Fixed-length universality and unbounded sequence limits

Example

Assume the Axiom of Countable Choice and fix an integer maximum length $N \geq 2$ . For each individual integer length $2 \leq n \leq N$ , the fixed-length theorem concerns a compact-domain approximation problem and permits a Transformer chosen for that particular $n$ and error tolerance. The hard-attention language limit instead fixes one finite-depth model, with one set of parameter maps independent of length, and rules out exact recognition of PARITY and 2DYCK for all sufficiently long strings. For each fixed source parameter $p \in (0, 1)$ , the soft-attention parity result fixes one model satisfying the one-symbol influence hypotheses across lengths and bounds the limit inferior of its expected cross-entropy on the stated parity source from below by the sharp unigram chance entropy, attained by a constant predictor. These statements concern different quantifiers and different notions of approximation.

Example references

Yun et al., Theorems 2–3: fixed-length integrated $L_{p}$ approximation

Hahn, §5, Corollary 2: fixed-depth hard-attention language limits

The Axiom of Countable Choice ( ${A C}_{ω}$ ): the inherited premise for the local integrated-Lebesgue approximation theorem.

Facts & Assumptions

Given: Countable Choice, an integer cutoff $N \geq 2$ , the fixed-length Transformer approximation theorem, and the fixed-model hard-attention PARITY theorem.

[F1]

Under Countable Choice, at each fixed $n \geq 2$ , $d \geq 1$ , and $1 \leq p < \infty$ , continuous compactly supported permutation-equivariant maps admit integrated $L_{p}$ approximants with two heads of size one, width-four ReLU feedforward layers, and no layer normalization; trainable positions give the compact-domain non-equivariant version (Fixed-width Transformer approximation on compact domains).

[F2]

No single fixed finite-depth hard-attention model with parameters fixed independently of input length recognizes PARITY or 2DYCK on all sufficiently long strings (Fixed-depth hard attention cannot recognize PARITY or 2DYCK).

[F3]

For the stated parity next-symbol distribution with fixed source parameter $p \in (0, 1)$ , every fixed soft-attention predictor satisfying the one-symbol influence hypotheses has ${lim inf}_{n \to \infty} E CE \geq H (u)$ , where $u = (p / 2, (2 - p) / 4, (2 - p) / 4)$ in the symbol order $(e o s, 0, 1)$ . The constant predictor $u$ attains $H (u)$ at every length; convergence of the expected loss for an arbitrary fixed predictor is not asserted (Fixed soft attention has no parity advantage over the unigram predictor).

Verification

1.1givenF1construct

Under Countable Choice, apply [F1] separately for each integer $2 \leq n \leq N$ , each fixed $d \geq 1$ , each $1 \leq p < \infty$ , and each chosen positive tolerance, to a target satisfying its corresponding continuity, compactness, and symmetry hypotheses. The theorem's existential network may depend on both $n$ and the tolerance; it gives no shared parameter set or depth bound uniform in $n$ .

2.1step 1.1F2F3algebra∎

[F2] fixes one model across all lengths and asserts failure of exact recognition for sufficiently large PARITY and 2DYCK inputs. [F3] likewise fixes one soft model and gives a liminf lower bound for expected cross-entropy on a specified source. Neither statement contradicts the separate fixed-length approximation claims, whose metric is integrated $L_{p}$ error and whose parameters may vary with length.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

21 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Yun et al., Theorems 2–3: fixed-length approximation (standard reference, not scraped)
Hahn, Theorems 1 and 6: hard and soft sequence limits (standard reference, not scraped)