RRE on Mathematical Reasoning Graphs — making mathematical reasoning visible, reusable, and teachable.
Graph-RRE brings together mathematical reasoning graphs and Recursive Reasoning Exploration to support mathematical research with AI. It records proof dependencies, assumptions, sources, and documented exploration routes, then uses this evolving structure to investigate new problems. Built on the Math Instruments project, it connects reusable mathematical knowledge with new discoveries and potential applications in human research, collaboration, and education.
Mathematical instruments → reasoning graphs → recursive exploration and discovery.
The animation draws every reasoning graph in the integrated research atlas as one structure: 64,309 claims, 75,893 links and 4,217 argument groups. The spiral arms hold 4,200 extraction batches in chronological order. The bright core holds the later autonomous RRE and ETH research runs. Gold arcs connect arguments that build on each other. Colours mark each claim's role: results, reviews, computations, scopes and definitions, open obligations, limits and counterexamples, and sources. The spiral arrangement is a display choice; positions inside each cluster follow the recorded links. Snapshot of 4 November 2026.
Phase I: Mathematical instruments
Ying Xie — What the Machine Invented: Mining New Mathematical Instruments from OpenAI's October 2026 Release of AI-Generated Mathematics. Preprint, 2026.
Identifies reusable mathematical devices and recurring proof mechanisms in the source collection.
Phase II: Math reasoning graph
Ying Xie — From Mathematical Instruments to Reusable Reasoning: A Second Phase of Extraction from AI-Generated Mathematics. Preprint, 2026.
Reconstructs source-linked reasoning graphs with claims, joint-premise inferences, scope conditions and evidence. The six-paper study includes four OpenAI papers and two retrospective application papers by the author. Multiscale visualization connects the collection, individual arguments, inference steps and source passages. Read about the second direction paper.
Phase III: Recursive reasoning exploration
Ying Xie — Recursive Reasoning Exploration: Mathematical Research through Evolving Reasoning Graphs. Case study: An Exact Certificate-Degree Theorem for Two-Term Tensor Decompositions. Preprint, 2026.
Develops Recursive Reasoning Exploration (RRE), in which research targets, arguments, checks and revisions extend the reasoning graph and inform subsequent exploration. The tensor-decomposition case study establishes an exact certificate-degree theorem and records six graph revisions with a replayable exploration history. Read about the third direction paper.
Research direction · Extraction method · How to cite
Find an instrument for your problem: a guide to exact constraints, adaptive choices and operator estimates, with concrete starting points and the hypotheses needed for reuse.
Ying Xie. Preprint, 2026. Quantum information theory.
An RRE investigation extends SR-012 by equating complete, conditional relative-entropy, and conditional mutual-information contraction for finite quantum channels, using the inherited complex-pencil inequality and the prior conditional equivalence. It determines the exact ordinary mutual-information coefficient of a depolarizing qubit: at half-depolarization, the ordinary value is 1/3 while the complete and conditional values are 3/8. For channels with input dimension at least two, an exact Choi-support criterion classifies when near-optimal complete contraction can retain nonvanishing input divergence. Explicit deficits hold uniformly over growing finite references, with product criteria and logarithmic copy bounds. The mutual-information lower family converges pointwise with vanishing input information; the positive-divergence family has vanishing trace distance. Neither construction establishes a fixed-error or capacity statement.
The animation follows two connected tracks: the conditional lift and exact depolarizing mutual-information coefficient, and the positive-divergence construction leading to the general information criterion. It distinguishes jointly required proof premises from research progression, records the disproved fixed-axes route and the earlier restricted classifications, and traces the product and copy bounds. The three views separate the mathematical routes, proof dependencies, and provenance.
Ying Xie. Preprint, 2026. Quantum information theory.
An RRE investigation determines the exact complete equal-reference Umegaki relative-entropy contraction coefficient of every finite-dimensional quantum channel, including coherent inputs, nonunital channels, and singular Choi matrices. A Choi-matrix optimization gives a finite semidefinite representation, a radius formula, an exact product law, and the depolarizing coefficient throughout its physical range. Reference dimension n² suffices for the supremum. The general matching family has vanishing input divergence and a logarithmic rate for a fixed faithful density. A different construction retains positive limiting divergence when the coefficient is one, while a nonlinear binary bound shows that this cannot be done for every channel. The positive construction still has vanishing trace distance and exponentially small regularization; no capacity or fixed-error operational advantage follows.
The animation follows Kraus cancellation and the trace-zero kernel construction into the exact coefficient theorem, then the radius, product law, strict-contraction criterion, and finite-accuracy bounds. Its information-retention branches distinguish the positive construction, the binary obstruction, the stopped perturbation route, and the remaining open classification. The three views separate mathematical progression, proof dependencies, and provenance.
Ying Xie. Preprint, 2026. Quantum information theory.
An RRE investigation determines the exact complete equal-reference relative-entropy contraction coefficient of finite preparation channels after input dephasing, including noncommuting and singular quantum outputs. The optimization over arbitrary finite reference dimensions reduces to a convex problem over the input simplex. The paper proves a Petz–Rényi radius formula and an exact product law, and exhibits faithful binary qubit channels whose complete coefficient tends to one while their ordinary coefficient tends to zero. Reference dimension two suffices for these binary examples. The matching construction converges pointwise with input relative entropy tending to zero; it gives no capacity or fixed-error operational guarantee.
The animation follows the noncommuting upper bound, support reduction, and weighted operator-simplex construction into the exact coefficient theorem, then the radius, product law, and faithful qubit amplification. Its three views distinguish mathematical progression, proof dependencies, and provenance, with open extensions and the slow-convergence limitation kept explicit.
Ying Xie. Preprint, 2026. Quantum information theory.
An RRE investigation determines the exact complete equal-reference relative-entropy contraction coefficient of every finite classical channel: how much relative distinguishability the channel can retain when the two hypotheses share a quantum reference marginal. The optimization over states and arbitrary finite reference dimensions reduces to a convex minimization over the input simplex. The paper proves an exact product law and a sharp comparison with ordinary classical contraction; reference dimension equal to the input alphabet size suffices for the supremum. The matching construction converges pointwise with input relative entropy tending to zero, so it does not itself give a capacity or fixed-error operational guarantee.
The animation follows the residual upper bound and weighted simplex lower construction into the exact coefficient theorem, then the Rényi-radius duality, product law and sharp alphabet comparison. Its three views distinguish the discovery history, proof dependencies and provenance.
Ying Xie. Preprint, 2026.
An open-ended autonomous RRE run generated a counterexample to the 2019 Conca–Welker conjecture about equations describing mutually perpendicular vectors. For nine vectors in eight dimensions, these equations contain hidden algebraic structure that is invisible when we look only at their solution points. The result holds over every field of characteristic different from two, for every nondegenerate symmetric bilinear form, and extends to all dimensions d >= 8 with n >= d + 1 vectors. An explicit certificate uses exact integer arithmetic to verify the central algebraic claims. The discovery changes our understanding of how the geometry of orthogonal vectors relates to the algebra of their equations.
The visualization makes the reasoning easier to follow through three complementary views: All math routes shows the successful route alongside inconclusive approaches; Proof route highlights the steps and dependencies establishing the result; and Provenance identifies inherited knowledge and published inputs. The animation presents the argument in logical order; the paper explains the exact certificate and the recorded research replay.
Ying Xie. Preprint, 2026. Quantum information theory.
An RRE investigation produced an explicit classical probability table with three outcomes per party whose entanglement of purification per copy can be reduced by treating many copies together. The paper certifies a one-copy value of at least 0.07163132826 nats and a regularized value of at most 0.071592778856911368 nats. Rational entropy certificates establish the lower bound, while a direct classical covering construction supplies the upper bound without invoking the Chen–Winter theorem. The construction proves a strict gain for every copy count n >= 10^12; it is not a practical small-copy witness. The separation persists for strictly positive tables, while binary additivity remains open.
The animation follows the two branches needed for the result: a global one-copy lower bound and a cheaper many-copy construction. Its three views distinguish the full reasoning routes, proof dependencies and provenance, including the unresolved binary branches.
The Math Instruments catalogue is the foundation of the project: a growing collection of mathematical instruments extracted from OpenAI’s research papers, with source references, structured descriptions, and studies of subsequent applications.
The aim is to identify new lemmas, constructions, inequalities, objects, structures and proof mechanisms introduced in the source arguments. Extraction comes first; validity, novelty and possible reuse are then subjects for further study. The structured descriptions also give readers a route into the source papers.
| Browse | Contents |
|---|---|
| Mathematical instruments | 799 extracted entries, each with a stable identifier |
| Source papers | 526 manuscript records across the selected families |
| Problem families | 372 family accounts with claims, antecedents and caveats |
| Recurring patterns | 14 approaches connecting the source arguments |
| Tables and review status | Family maps, formalization, dependencies and antecedents |
| Extraction batches | Extraction rounds and source versions |
| Release coverage | 372 of 372 families catalogued; 0 pending |
| Use case studies | Subsequent investigations using extracted instruments |
| Catalogue exports | JSON and CSV for search and further analysis |
How can a stationary process move downward on almost every step, and how much dependence must that behavior retain? CS-002: Stationary descent and its information cost follows two extracted instruments, MI-000098 and MI-000014, into this probability question. The study develops a sharp atomless information bound and an explicit finite-state optimum whose value is invariant under rearranging the state probabilities.
Read Ying Xie's preprint, From Extracted Instruments to Mathematical Insight: A Case Study in Stationary Descent, for the exploration, proofs, physical interpretation and comparison with earlier work. It illustrates how researchers can select a mechanism from the catalogue, adapt the question and investigate the resulting mathematics; historical priority of the finite formula remains unresolved.
Can an instrument from a depth-three circuit proof help at every fixed depth? CS-001: Signed restriction and fixed-depth circuit lower bounds follows MI-000083 from family 112 into this question. The signed restriction estimate remains useful when earlier restrictions produce CNFs with many clauses. Adaptive averaging and quantitative sparsification extend the argument to the same polynomial-time language at every fixed depth, with a logarithmic improvement in the lower-bound exponent.
Ying Xie's preprint, From Signed Restrictions to Lower Bounds at Every Fixed Depth, supplies the theorem, proofs and an explicit sufficient length threshold. The case study explains which ingredients come from the source and what the application adds.
Can errors on smaller quantum signal alphabets certify the error for the full alphabet? CS-003: Quantum decoding and sharp subset bounds follows MI-000329 and MI-000787 from partial differential equations and geometry into quantum state discrimination. A constrained spectral calculation and exact polynomial certificates support two parts of the investigation; additional matrix arguments establish sharp subset bounds for mixed states under explicit assumptions.
Ying Xie's preprint, From Quantum Decoding to Sharp Subset Bounds: A Use Case for Extracted Mathematical Instruments, supplies the proofs, an exact certificate for a restricted four-signal family and a noisy qutrit decoding example. It demonstrates cross-domain reuse while leaving the unrestricted sharp mixed-state subset problem open.
The catalogue continues beyond the first direction paper through numbered extraction batches, covering quantum information, geometric inequalities, dimension reduction and analytic mechanisms. Each new batch records exact source locations, targeted antecedent comparisons and the scope of its checks. The coverage table now accounts for all 372 October-release families, with selected instruments from each family. The catalogue contains 799 instruments and 526 manuscript records; this does not mean all 719 release manuscripts received a complete extraction or proof review.
The completion checkpoint and audit record the final 161 families, 328 added instruments and 40 merged batches. All 84 finite verification programs passed a fresh run; the provenance audit matched 1,618 source files and 1,586 label bindings. The earlier October 16 checkpoint remains available as a historical record.
Batch B001 imports the first direction manuscript’s 38 family entries and all 115 instrument descriptions. The 81 source-paper records include related and companion manuscripts; 81 is a source-coverage count, not a count of separately extracted papers. The initial maps and all four manuscript tables are retained in the fixed import edition.
The source is the public OpenAI mathematics repository, pinned to revision fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb for B001. Its initial release contained 722 manuscripts; the inspected snapshot contains 719 after three withdrawals. This project is maintained independently by Ying Xie.
The descriptions retain the source claims, qualifications and preliminary novelty assessments. Inclusion does not establish novelty or certify a proof. Assessment and evidence explains the separate treatment of mathematical review, formalization and reuse.
Use the record templates and contribution guide to add instruments, source comparisons, corrections or case studies. Keep existing identifiers when updating an entry. A later batch can revisit an existing family or introduce new families.
python -m pip install -r requirements.txt
python scripts/build_indexes.py
python scripts/validate_catalog.pyCite the Phase I direction paper for instrument extraction, the Phase II direction paper for reusable reasoning graphs, and the Phase III direction paper for recursive reasoning exploration and its tensor certificate-degree case study. Identify the repository release or commit when using its curated records, and cite the linked source manuscripts for their mathematical contributions. Source references and hashes are recorded under sources/.
Maintainer: Ying Xie, Kennesaw State University — yxie2@kennesaw.edu.