LANTERN: Illuminating Hidden Mathematical Knowledge in Language Models

1 Introduction

• The paper targets the problem that mathematicians still choose which questions to investigate, despite language models’ growing ability to prove or generate mathematical results.Source: §1 · p. 1
• The pipeline produced 62 technically verified relations between OEIS entries the database did not connect; content screening retained 13, and a two-axis assessment classified nine as informative or insightful.Source: §1 · p. 2
• LANTERN ranks pairs using pretrained-model hidden representations, then generates candidate relations for executable and analytical checking.On the OEIS, a classifier trained on Qwen3-32B activations scored 50 million pairs among 10,000 frequently referenced entries.Source: §1 · pp. 1–2
• Four of the nine informative or insightful relations were not found in the OEIS or the targeted literature search.Two were classified as informative and two as insightful.Source: §1 · pp. 1–2

2 Related Work

• Prior systems use human-provided objectives, generate conjectures through several strategies, or predict missing links in knowledge graphs.The paper places LANTERN closest architecturally to HyGRAIL, which predicts missing materials-graph edges using a GNN and an LLM.Source: §2 · p. 2
• Generated conjectures are typically new candidates, with novelty judged by absence from a formal library or expert evaluation.Source: §2 · p. 2
• Prior work probes latent model knowledge, while OEIS mining has also found identities through sequence transformations and matching.The paper notes an OEIS coincidence that led to a proof of a bijection among three classes of lattices.Source: §2 · p. 2
• LANTERN combines mining latent representations with predicting and mathematically verifying missing cross-references within the OEIS.The authors describe this combination as, to their knowledge, the first of its kind.Source: §2 · p. 2

3 Setup

• The OEIS supplies defined sequences, recorded cross-references, and stored terms for inexpensive initial checks of exact claims.Its version history also makes it possible to identify when a cross-reference appeared.Source: §3 · p. 3

3.1 The corpus

• The corpus contains 10,000 frequently referenced entries selected from OEIS snapshot S2, forming 50 million pairs.Entries were filtered by term count, definition length, and status keywords before the 10,000 most-mentioned entries were selected.Source: §3.1 · p. 3

3.2 Labeling the pairs

• The authors graded linked pairs as gold, silver, or trivia by the mathematical depth of their relation, since many cross-references are trivial.Pairs whose definitions were near-duplicates and reserved test pairs were excluded before screening; gold verdicts were reread, and 91–94% held.Source: §3.2 · p. 3
• Gold and silver pairs—2,780 pairs, or 9% of linked pairs—served as positives; trivia, bare cross-references, and random unlinked pairs served as negatives.The assumption that random unlinked pairs rarely contain unrecorded relations may introduce mislabeled negatives.Source: §3.2 · pp. 3–4
• For Qwen3-32B, each pair is represented by centred cosine features at every layer and product and absolute-difference features at two selected layers.A logistic regression maps these features to a score.Source: Figure 4 · p. 4
• The classifier was tested on cross-references added after the training cutoff, including 927 future pairs and 89 links added in 2026.A frontier model selected 74 distinct pairs from these sets as substantive relations; 265 of the 927 future pairs shared no definition words.Source: Figure 3 · p. 3, §3.2 · p. 4

4.1 Embedding

• For each sequence definition, the embedding pass retains normalized residual-stream vectors from every layer, which the classifier can weight.The main embedding model is Qwen3-32B; Qwen3-8B and Qwen3.5-27B are also used for size comparisons.Source: §4.1 · p. 4, §4 · p. 4

4.2 Link classifier

• The link classifier combines layerwise centered cosine similarities and selected principal-component features, then trains logistic regression on standardized features and graded labels.For Qwen3-32B, the feature set contains 321 values; training rebalances the rare substantive links with capped positives and ten negatives per positive.Source: §4.2 · pp. 4–5

4.3 Hypothesis generation

• The pipeline ranks unlinked, non-twin pairs, filters candidates, asks a stronger agent for hypotheses, then checks stored terms and analytically verifies each surviving claim.The executable check runs in a sandbox and compares outputs with every stored target term; analytical checking is also required.Source: §4.3 · p. 5

4.4 Search configurations

• The main search processed 500 ranked pairs and yielded 44 verified statements; two 100-pair variants yielded 3 and 15, for 62 statements across configurations.The configurations differed in corpus size and training objective; their outputs are combined rather than treated as one run.Source: §4.4 · p. 5

4.5 Content screen

• A separate content screen retains only exact, pair-specific relations using substantive structure from both objects; significance and documentation novelty are assessed on separate axes.Significance ranges from routine to insightful, while novelty distinguishes documented, previously undocumented, and not-found-in-search connections; N2 does not establish priority.Source: §4.5 · p. 6

5 Results

• 62 verified relations yielded a 62–13–9 funnel: 13 passed the content screen, and nine were informative or insightful (S1 or S2).The 62 statements combine outputs from three search configurations; the main configuration contributed 44.Source: §5 · p. 6

5.1 Significance of the retained relations

• The nine S1/S2 relations comprise five specific routes between objects and four structural interpretations; the two N2/S1 routes and two N2/S2 bridges make up the four reported novel connections.N2 means not found in the OEIS or targeted literature search; it does not establish priority.Source: §5 · p. 6, §5.1 · p. 7
• Two retained cross-domain connections were classified N2/S2: one links one- and two-dimensional cellular automata, and the other connects analytic q-series with 5-core partitions.The authors did not find these connections in the corresponding OEIS records or targeted literature search, but do not claim the underlying identities are new theorems.Source: §5.1 · p. 7, §5.2 · p. 7

5.2.1 Reconstructing a 2D population sequence from a 1D one

• For the cellular-automata bridge, every two-dimensional Fredkin population count can be reconstructed exactly from one-dimensional Rule 150 counts using a difference-of-squares relation and binary-run factorization.At time 13, the reported counts are A(13) = 15 and B(13) = 192.Source: §5.2.1 · p. 7

5.2.2 A continued fraction that counts partitions

• The continued fraction and theta quotient combine to give the classical generating function for 5-core partitions, assigning a concrete counting interpretation to the analytic product.The coefficient of q^5 is two, corresponding to the two qualifying partitions of five.Source: §5.2.2 · pp. 8–9, Figure 6 · p. 9

6.1 Activations against the text of the definitions

• On 793 held-out links, text scoring reached AUC = 0.51 while activation classifiers reached 0.57–0.58; on 42 substantive pairs, the values were 0.61 and 0.67–0.69.Evaluation matched unlinked negatives on the same two topics, testing relatedness within a topical background.Source: §6.1 · p. 10

6.2 Is it memory of the pages?

• On 2,379 entries created after the model cutoff, the classifier reached AUC = 0.88, versus 0.89 on an older-entry split; removing 30 edited-definition links left the figure results unchanged.The authors interpret the new-entry result as consistent with signal transfer from representations of mathematical definitions rather than page retrieval.Source: §6.2 · p. 10

6.3 Is it in the model, or in our labels?

• On 74 substantive future pairs, the label-free question and classifier achieved median percentiles of 88.5 and 88.0, respectively.The direct question used no training labels; the classifier's practical advantage is throughput across all pairs after one embedding pass.Source: §6.3 · pp. 10–11

7 Limitations

• The OEIS has unusually structured, human-written sequence entries and explicit links, and the corpus favors frequently mentioned sequences; generalization to less formal objects or databases without explicit links is unclear.The authors also report evaluating only one family of pretrained models.Source: §7 · p. 11

• 62 technically verified relations between previously unlinked OEIS sequences were produced; a content screen retained 13, of which nine were informative or insightful.Source: §8 · p. 11
• Four of the nine informative or insightful relations were not found in the OEIS or targeted literature search: two offer computational routes and two structural interpretations.Source: §8 · p. 11
• LANTERN ranked millions of OEIS-entry pairs using hidden-state representations and verified promising candidates with an agent.Source: §8 · p. 11
• The results suggest pretrained representations can guide discovery and investigation of unrecorded mathematical relations in large structured knowledge bases.Source: §8 · p. 11

• 13 of the 62 statements were retained for useful exact routes or interpretations; nine were rated S1 or S2, while 49 were primarily definitional, generic, or mechanically reproducible.Source: Table 4 · p. 16
• A007325 × A227216 gives T(q)^2R(q)^5 as the generating function of 5-core partitions, a retained structural interpretation.Source: A The 62 statements · pp. 14–16
• A039598 × A000139 forms a retained bridge from planar maps to ternary trees and Catalan numbers.Source: A The 62 statements · pp. 14–16
• A071053 × A160239 links cellular automata whose live-cell counts factor over blocks of ones in n’s binary expansion.The ledger retains this as a structural bridge between one- and two-dimensional cellular automata.Source: A The 62 statements · pp. 15, 17
• A066633 × A185651 provides a retained route from partition counts through divisibility and totients to necklaces.Source: A The 62 statements · pp. 15, 17

Screening standard

• A relation is retained only when it is exact, specific to the pair, and offers a useful mathematical route or interpretation; novelty is assessed separately.A standard method or absence of an OEIS cross-reference is not sufficient on its own to determine whether a relation is substantive or novel.Source: B Content-screening criteria · pp. 18–19

Representation and elementary transformations

• Relations that merely recode or re-represent the same object, or apply elementary index or value changes, are excluded when they add no useful route.Examples include relabelling, standard representations, shifts, scaling, reversals, and extracting subsequences.Source: B Content-screening criteria · p. 18

Definitions and reconstruction

• A relation is excluded when it follows directly from a definition or uses a universal reconstruction procedure without a specific assertion requiring both entries.General transforms or coefficient-recovery procedures are not disqualifying when they yield a specific assertion in which both entries are essential.Source: B Content-screening criteria · p. 18

Mechanical carriers and common families

• A relation is excluded when one entry is only a mechanical intermediate carrier, or when the pair merely names members of a common family.A canonical classical object or shared family remains eligible when the relation supplies a useful exact route, specific bijection, identity, or structural interpretation.Source: B Content-screening criteria · p. 18

Small-value coincidences

• Coincidences among a few constants or initial terms are excluded unless additional structure supports the relation.A shared period alone is not substantive because two periodic sequences can be matched by a finite table.Source: B Content-screening criteria · p. 19

Redundant restatements

• A relation is excluded when it only restates an existing connection or repeats its template with changed parameters or notation.This includes mechanically changing representations in a relation already recorded between another pair.Source: B Content-screening criteria · p. 19

Graded links

• Pairs whose FORMULA or COMMENTS lines name the partner in S0 were graded; training labels also reflect downgraded gold verdicts and exclusions of text twins and Sec. 3.2 test pairs.Bare cross-references appearing only in CROSSREFS were not graded.Source: Table 5 · p. 19

Table 6 comparison

• Table 6 tests scorers trained on the same 321 activation features against 74 substantive future pairs and random or popularity-matched peers; averages favor boosting.The linear classifier places the fewest most-referenced pairs in its top 1,000; two probes share one top-100 pair and have scores correlated at 0.5 over 500,000 random pairs.Source: Table 6 · p. 19

Held-out averages

• On the 48 substantive pairs and 89 links added after S1, all four scorers reached 0.55–0.62 AUC, within one standard error of each other.Model size did not order the classifiers on any of the four sets.Source: D Scorer benchmark · pp. 19–20

Earlier classifier comparison

• For the earlier classifier trained on ungraded S0 cross-references, activation/text AUC was 0.876/0.799 on 927 later links and 0.758/0.498 on 265 pairs with no shared definition word.The gap closed when both scorers were trained on graded labels.Source: D Scorer benchmark · p. 20

Popularity-matched exam

• On the popularity-matched exam, the text scorer reached the 83rd percentile with logistic regression and the 90th with boosting.The 74 pairs represented about 45 distinct facts, with more than half following two templates.Source: D Scorer benchmark · p. 20

Search configurations

• The outputs of the three configurations described in Sec. 4.4 together formed a total of 62 verified statements.Source: E Search configurations · p. 20

Table 7

• Table 7 presents the three configurations and their labels; the main configuration adds the Sec. 4.2 caps and is the one tested in Sec. 6.“Bare” denotes pairs named only in CROSSREFS.Source: Table 7 · p. 20

Stage timing

• In the main configuration, labeling took about 40 minutes for popular entries and about two hours for pairs with a lesser-known entry.The filter read 500 queued pairs and 84 calibration pairs in batches of ten with four workers; the agent ran six pairs in parallel, with a median of seven minutes per pair including sandbox verification.Source: Table 8 · p. 20

F Derivations for the 62 statements

• The companion appendix gives one entry per statement, in the same order: first by novelty level, then by content class.Source: F Derivations for the 62 statements · p. 21
• The derivations were produced by a language model and have not been checked by a mathematician.The formulas recorded in the OEIS entries are taken as given rather than re-proved.Source: F Derivations for the 62 statements · p. 21
• Each derivation establishes a relation from the formulas recorded in the two OEIS entries, without re-proving those formulas.The recorded formulas retain their own OEIS provenance.Source: F Derivations for the 62 statements · p. 21
• Proofs use exact chains of formal-power-series identities, factorial algebra, induction, or closure by the Sturm bound.Source: F Derivations for the 62 statements · p. 21

Index of classical results used

• For the A125106 partition λ(n) with k parts, w(n)=k max λ(n)+k(k+1)/2−|λ(n)|; the partitions ordered by sum, length, and reverse lexicographic order form A334302.The encoding is proved bijective onto the nonempty partitions.Source: Index of classical results used · p. 22
• Odd and even power sums T(2p+1,n) and T(2p,n) are expressed using central factorial numbers CF and A001498, with the even case carrying a factor of (2n+1).The odd formula is a weighted sum over CF(p+1,k)B(n,k); the even formula has the corresponding CF(p,k) terms.Source: Index of classical results used · p. 23
• For h=x(1+3h+h^2), the coefficients satisfy [x^n]h^k=A124733(n−1,k−1)+A124733(n−1,k) for k≥1.The derivation obtains the recurrence from the Riordan-array representation of A124733.Source: Index of classical results used · p. 24
• The sequence satisfies y(n)=1+⌊(r(n+4) mod 5)/2⌋, with its proof aligning the residues to a period-5 Lyness cycle.The argument establishes 5-periodicity of the relevant residues and checks the five phases.Source: Index of classical results used · p. 24
• For the Pascal matrix P and exponential Riordan array M, M^-1P^2M=I+2T.The derivation uses the exponential Riordan group law and identifies T with the logarithm of P.Source: Index of classical results used · pp. 24–25
• Both triangles reduce to the generating function e^x sec x; its Euler numbers yield the whole first triangle via Deutsch’s formula.The derivation also relates secant and tangent numbers to A000111.Source: Index of classical results used · p. 25
• For m0 ≡ 3 (mod 8), r3(m0) = 8S(m0), where S(m0) sums class numbers over square divisors.The argument uses the class-number formula for an order of conductor 2 and unit-index bookkeeping, including the case M = 3.Source: Index of classical results used · pp. 27–28
• A001764(n) = (n + 1)A000139(n)/2, relating the map count to ternary trees.The proof uses the Fuss–Catalan substitution C(y) = T3(y/C(y)).Source: Index of classical results used · p. 28
• Deconvolving by {2c + 2d : c even, c ≤ d} recovers unordered triangular-number representations U(m), and t(n) = 2U(m) − [m pronic].The formula also gives t(n) = 0 when the odd part of 3n + 2 is 3 modulo 4.Source: Index of classical results used · p. 28
• The Laguerre triangle becomes the signed Pascal matrix after scaling each entry by k! and each row by its leading entry.Other matrix identities in this part identify a matrix T with the Pascal matrix and its generator: Pascal = exp(A132440). It also gives T(n, k)k!Ck = n! and an additional triangle formula.Source: Index of classical results used · p. 29
• The sequence S satisfies S(n) = c(n − 1) + 1, where c(m) is the binary expansion of m read in base 3.The part also gives the rising-factorial identity k(k + 1) ··· (k + m − 1) = Σ_i s(m − 1, i)k^(i+1).Source: Index of classical results used · pp. 29–30
• A transformed rational triangle is a shifted Pascal triangle, while the sequence a(n) has even- and odd-index formulas in ballot numbers.The rational triangle’s Pascal rule has an exception at cell (2, 1); the ballot-number formulas are a(2m) = T(3m, m)/(m + 1) and a(2m + 1) = 2T(3m + 1, m)/(m + 2).Source: Index of classical results used · p. 30
• The triangle’s row sums equal central binomial coefficients, and two arrays have reversed rows related by a factor depending on n and k.The reversal relation holds for all 0 ≤ k ≤ n; the supplied statement gives the factor as (k + 1)/(n + 1).Source: Index of classical results used · pp. 31–32
• The BB-numbers recover ⌊nφ⌋ and the whole array, with A101864(n) = 3A(n) + 2n.The third column of the array plus 2 gives the BB-numbers; the proof also derives T(n, 3) = A101864(n) − 2.Source: Index of classical results used · p. 31
• An infinite product of values of the partner at x^(m·2^k) gives the first series, and its factorization is unique.No finite monomial in the partner values suffices: their eta-exponent sum is 0, while the first series has weight 3.Source: Index of classical results used · p. 32
• The full Euler product is assembled from partner values at q^k because every positive integer uniquely factors as a 3-smooth number times an integer coprime to 6.The product over integers coprime to 6 uses exactly the residue classes 1 and 5 modulo 6.Source: Index of classical results used · p. 32
• A087808 is computed by folding the reversed binary-run lengths of A227736: each run of ones adds its length, and each run of zeros multiplies by 2^length.The proof establishes the fold formula for all n ≥ 1 by induction.Source: Index of classical results used · pp. 33–34
• Replacing unordered multisets of subtrees with sequences yields a quadratic tree-grammar equation whose solution is the generating function of the large Schröder numbers.The coefficient relation is [x^n]1/(1−S) = A006318(n−1) for n ≥ 1.Source: Index of classical results used · pp. 34–35
• The generating function of A080054 equals A(−x)/A(x) = A(x^2)/A(x)^2, where A is the generating function of A081362.The second equality follows from the recorded identity A(x^2) = A(x)A(−x).Source: Index of classical results used · p. 35
• The generating function of A004018 is that of A227216 multiplied by an Euler product with period-20 exponents c = [−1, 4, −6, 5, −6, 9, −6, 0, −1, 4, −1, 0, −6, 9, −6, 5, −6, 4, −1, 0].The exponents arise by subtracting periodic sequences of periods 4 and 5, giving period lcm(4, 5) = 20.Source: Index of classical results used · p. 36
• The A240009 bivariate generating function equals P(u/q, q^2)P(1/u, q^2), with coefficient extraction yielding the stated identity.The product is well defined as a formal series in Z[u, u^-1][[q]] because each q-coefficient is a finite Laurent polynomial in u.Source: Index of classical results used · p. 37
• A006352 and A121361 yield a coefficient formula that reconstructs the generating function B from A alone.The derivation uses the logarithmic derivative of the eta-product formula recorded in A121361 and compares coefficients in qB′ = B · (qB′/B).Source: Index of classical results used · pp. 37–38
• A122856 determines representation counts r2(k) and gives A000118(n) = r2(3n + 2)/4.The reconstruction uses the 3-adic valuation and residue class of k to recover the relevant A002654 term from A122856.Source: Index of classical results used · p. 38
• A130534 determines every A(n, k) in A144150 through Stirling-transform inversion and triangular recursion.The column relation is the Stirling transform, and the seeds A(0, k) = 1 and A(n, 0) = 1 complete the induction.Source: Index of classical results used · p. 39
• A193842 rows expand in a basis whose coefficients are Fibonacci-polynomial values.The passage identifies its rows with the homogenization of the corresponding Fibonacci-polynomial expansion.Source: Index of classical results used · pp. 39–40
• Rows of A227543 and A251592 have weighted sums equal to the Catalan numbers, although their relation is between row sums, not flattened terms.The arrays are one-parameter deformations of the same Catalan functional equation and meet at q = 1 and t = 2.Source: Index of classical results used · p. 41
• For both cellular automata, live-cell counts factor over binary blocks of ones, with block factors given by Jacobsthal-number expressions.The strip and plane formulas are recorded in A071053 and A160239, respectively.Source: Index of classical results used · p. 41
• The partition-count array A066633 recovers the divisibility matrix A051731 and, through Gauss’s identity, determines A185651.The same section relates A295862 to Fibonacci numbers: its term a(n) equals L(n + 1).Source: Index of classical results used · pp. 40, 42
• Riesel numbers with a covering set are characterized by a finite residue check over one period using the orders of 2 modulo covering primes.The prime orders are read from A001917, yielding a characterization of membership in A101036 in terms of those orders and the primes.Source: Index of classical results used · pp. 42–43
• A245562 lists the odd-positioned runs in A101211, while A245563 lists those same runs in reverse order.The runs of ones in a binary expansion occupy the odd positions of the full alternating run list.Source: Index of classical results used · p. 43
• A302997’s k=1 column reconstructs θ3 termwise; θ3(x)=E(x2)^5/(E(x)^2E(x4)^2) then recovers E and its powers, yielding A286354.The stated product identity links the theta and Euler products; the reconstructed E powers give the whole A286354 array.Source: Index of classical results used · p. 44
• For odd n, A002129(n)=σ(n), with n+1 identifying odd primes whose squares sieve A056911; A069279 consists exactly of n with Ω(A046523(n))=18.A046523 preserves the prime-exponent multiset, so applying Ω yields the stated characterization of A069279.Source: Index of classical results used · pp. 44, 47, 48
• Bernoulli numerators recover B2m and powers of ζ(2) from ζ(2m), while Bernoulli denominators normalize E4, E6, and the τ generating function.The section uses von Staudt–Clausen and Euler’s zeta formula for the Bernoulli recovery, and gives the tau generating function as (E4^3−E6^2)/(24(b(2)+b(3))).Source: Index of classical results used · pp. 45–46
• A172236(n, k) equals the (k−1)th Fibonacci polynomial evaluated at n, with coefficients given by row k−1 of A168561.The recurrence and initial values establish the polynomial identity for all k≥0 and n≥1.Source: Index of classical results used · pp. 45–46
• A030203’s generating function is the Euler product ∏(1−x^k)^(1+e(k+1)), where e is the Möbius transform of A001817.The exponent identity uses 1+[3|k]=1+e(k+1).Source: Index of classical results used · p. 46
• A002175(n) counts ordered positive pairs coprime to 6 whose squares sum to 24n+2.The result follows by identifying A065330’s values with positive integers coprime to 6 and translating the sum-of-squares condition.Source: Index of classical results used · pp. 46–47
• The map ν sends A341783 bijectively onto the primes, enabling π(p) and A048675 to be recovered from that sequence.A341783 contains primes in residue classes 0, 1, and 4 modulo 5 and squares of primes in classes 2 and 3.Source: Index of classical results used · p. 47
• The denominator of B_k is the product of distinct primes p such that (p−1) divides k, equivalently factors d+1 for d dividing k when d+1 is prime.The prime factors are distinct, so the denominator is squarefree.Source: Index of classical results used · p. 48
• For m ≥ 2, σ(m) = m + 1 exactly when m is prime, so d+1 is prime precisely when A000203(d+1) = d+2.This replaces the primality condition in the divisor-indexed denominator formula with a divisor-sum condition.Source: Index of classical results used · p. 48
• The formula gives denom(B_1) = 2 because 1+1 is prime; for n = 0, B_0 = 1 is an integer.Source: Index of classical results used · p. 48

• Entries 1–9 (T1) include substantive, structural, and mechanical pair classifications.Source: Summary table · p. 48
• Entries 10–43 (T2) include substantive, structural, and mechanical pair classifications.Source: Summary table · pp. 48–49
• Entries 44–62 (T3) include substantive and mechanical pair classifications.Source: Summary table · p. 49