conj

A public collection of mathematical conjectures, worked checks, and research leads.

What survives when a graph loses vertices? How long does a random crossing remember its past? Can the holes in a sumset have a Gaussian shape?

This notebook starts with 25 candidate statements generated with GitHub Copilot, revised and examined with ChatGPT/Codex. The aim is to turn intriguing questions into precise, falsifiable mathematics—and invite people to improve, solve, or retire them. Every candidate has a formal statement, an illustrated introduction, and a next mathematical step.

Bring a proof. Bring a counterexample. Bring the paper that already settles it. All three move the project forward.

Explore the atlas · Read the research notebook · Browse exact statements · Contribute

New in 0.2.0: a proved C15 subset

Read the public paper: inference, prediction, and their limits. With complete observations, centered context effects cancel exactly: we prove the simultaneous Gaussian error law and its Gumbel limit. A separate oracle benchmark handles missing observations when the nuisance matrix is known.

The paper also derives a model-based rule requiring both positive expected return after costs and a bounded probability of endpoint loss. Its future-mean and tail assumptions are explicit. This is a standard mathematical consequence, not evidence of an investment forecasting advantage. Full C15 remains unresolved. Changes · Numerical checks.

What is actually known?

As of 12 October 2026	Count	Meaning
Original candidate slots	25	The original collection, with stable C01–C25 IDs
Known-theorem consequence	1	C01 has a complete implication proof
Unresolved in this project	24	Initial screening is complete; specialist review is not
Priority research leads	6	C03, C06, C07, C13, C15, C17—an editorial shortlist
Certified new open problems	0	Truth, exact novelty, and open status have not been independently established

Twenty-four unresolved statements are not twenty-four verified open problems. Some may already follow from the literature; others may be false or need sharper hypotheses. C02, for example, substantially overlaps a 2026 chemical-metric scaling preprint; its precise normalization and moment claim still need comparison. Keeping these distinctions visible is part of the research.

Six places to start

These are the strongest provisional leads from our first review. C17 has a concrete projection mechanism and reproducible experiments. C06, C07, and C15 have potential connections to noisy spectral methods, incomplete networks, and statistical inference. Their practical usefulness remains to be demonstrated.

Click a card for its exact statement, current obstacle, and full-size image.

C03 · How long does a crossing remember? Does rescaled crossing memory settle to a finite, positive amplitude? Priority · unresolved	C06 · Do noisy eigenvectors forget? Does the whole two-time eigenvector-fidelity field agree with its Gaussian reference? Priority · unresolved
C07 · Two edges after a graph is thinned Do the upper and lower nontrivial spectral edges approach independent Tracy–Widom laws? Priority · unresolved	C13 · A transversal that survives damage Can every locally bounded deletion attack leave a transversal in a random Latin square? Priority · unresolved
C15 · Inference through missing data Can one calibrate the largest estimation error while learning an unknown low-rank nuisance? Priority · unresolved	C17 · The Gaussian shape of missing sums Do sumset deficits converge to a Gaussian field with an explicit covariance kernel? Priority · unresolved

A small experiment worth pursuing

C17 predicts that the full deficit of a sparse random sumset has scaled variance tending to 8. The recorded finite experiment gives:

Inclusion probability q	Interval size m	Samples	q³ × sample variance of the full deficit	Scaled projection residual variance at u = 4
0.100	2,121	1,200	7.5836	0.1657
0.050	14,360	1,200	7.7007	0.0871
0.025	87,091	1,200	7.9057	0.0502

The first-order projection suggests a mechanism; the shrinking residual suggests a question worth attacking. Three finite parameter values do not prove convergence. A uniform higher-order remainder bound and functional tightness are still missing. See the recorded observations, experiment code, and notebook for the exact setup and limits.

The rest of the collection

The remaining nineteen include percolation thresholds, random topology, extreme-value laws, permutation patterns, and partial-order dynamics. C01 stays visible as a worked example of a candidate resolved by existing theorems. C10 and C24 carry higher-risk labels because their proposed fluctuation or relaxation scales need particular scrutiny.

C01 · When few queries erase memory Can an extremely efficient exact decision algorithm coexist with persistent noise correlation? Known consequence	C02 · The shortest critical crossing Does shortest-path length have a nondegenerate limit under an exact power normalization? Unresolved; literature overlap
C04 · The shape of a threshold Is the continuum crossing profile twice continuously differentiable, with positive slope? Unresolved	C05 · A spike at the spectral edge Do critical eigenvalue-response curves and their derivatives forget the entry distribution? Unresolved
C08 · A cascade near its tipping point Does a growing-degree activation transition have a calibrated Gaussian profile? Unresolved	C09 · Tree entropy beyond short cycles After conditioning on more and more short cycles, is the remaining tree entropy Gaussian? Unresolved
C10 · Can two residuals decouple? Do mixing and spanning-tree entropy have independent Gaussian residuals after local conditioning? High-risk · unresolved	C11 · The last isolated vertex Under balanced nonuniform arrival rates, does a perfect matching arrive exactly when isolation ends? Unresolved
C12 · When holes disappear Does first homology vanish exactly when the final isolated edge disappears? Unresolved	C14 · The final visits on two sides Do the two sides of a biregular graph have independent normalized cover-time extremes? Unresolved
C16 · The last missing sums Do the two outermost holes in a sparse random sumset have independent Gumbel limits? Unresolved	C18 · Counting additive collisions Does fixed-cardinality additive energy obey a lattice-corrected local central limit law? Unresolved
C19 · When every difference appears Do normalized first-witness times for differences form a Poisson point process? Unresolved	C20 · The largest accidental overlap Does the largest autocorrelation of a fixed-size random set have a Gumbel law? Unresolved
C21 · Local peaks, global patterns How does conditioning on peaks change the global count of 132 patterns? Unresolved	C22 · What a sparse shuffle changes Does the descent response to a sparse reshuffle converge, conditionally, to Brownian motion? Unresolved
C23 · How many legal orders? Is there a deterministic second-order entropy constant for random planar orders? Unresolved	C24 · How fast can legal orders mix? Does the legal adjacent-swap walk have a universal n-cubed relaxation scale? High-risk · unresolved
C25 · The entropy of increasing witnesses Do increasing-subsequence counts have a deterministic quenched entropy profile? Unresolved

Two directions growing out of the review

N01 and N02 are separate discovery leads:

• N01: inhomogeneous sumset fluctuations. Replace constant inclusion probability by a smooth spatial profile and ask for a corresponding Gaussian covariance kernel.
• N02: extremes and bulk together. Ask whether the two missing-sum fronts and the two deficit fields become jointly independent.

Both are preliminary. They are not included in the original 25 or presented as verified new open problems.

Reproduce the checks and figures

Use Python 3.12 or later. A virtual environment is recommended.

python -m venv .venv
# Activate .venv using your shell's activation command.
python -m pip install -r requirements.txt
python -c "from pathlib import Path; Path('work').mkdir(exist_ok=True)"

# Check recorded input fingerprints and an exact rational projection identity.
python tools/verify_evidence.py

# Verify the C15 subset formulas, boundary cases, and source-bound record.
python -m unittest discover -s tests -v
python tools/check_c15.py --verify docs/c15-checks.json

# Run the original bounded sanity checks across all 25 statements.
python main.conjectures.sanity.py --output work/checks.json

# Rebuild every visual and illustrated introduction.
python tools/build_atlas.py
python tools/build_c15_figure.py
python tools/check_publication.py

# Longer: reproduce the sumset and actual planar-order experiments.
python main.conjectures.analysis.experiments.py --output work/observations.json

The baseline experiments used seed 20261012, Python 3.14.6, NumPy 2.5.3, and SciPy 1.18.1. Small numerical differences may occur across numerical-library versions. Baseline JSON files retain input hashes and parameters; the commands above write fresh experimental output under work/.

The atlas uses seeded, explicitly labeled toy constructions, analytic curves, and recorded data. Only C17 and C24 plot the recorded research experiments. Other pictures illustrate definitions or proposed laws; they do not provide numerical support. C24 explicitly marks the six omitted n = 10 samples that exceeded the enumeration cap. Each image has a caption, a vector counterpart, and a provenance manifest.

Research organization

Entry point	Purpose
Atlas	Accessible introductions and one visual per original candidate
Statement index	Authoritative definitions, quantifiers, and normalizations
Notebook	Findings, primary-literature comparisons, categories, and next checks
Evidence ledger	What the computations actually check
C01 proof	Complete reduction to known theorems
C15 subset paper	Exact fully observed result; conditional decision bounds; prediction limits
Discovery lane	N01–N02, clearly separated from the original collection
Contribution guide	How to submit a proof, counterexample, reference, or experiment

Recur is optional. Stable trace IDs connect candidates (main.conjectures.c17), findings (main.conjectures.notes.c17.*), and illustrated introductions (main.conjectures.atlas.c17). Tags distinguish analytic work, literature, computation, and unresolved questions. A trace link locates evidence; it does not prove a claim.

With Recur installed, initialize fresh local discovery lanes without overwriting existing ones:

python tools/init_recur.py
recur reveal main.conjectures
recur reveal main.conjectures.notes
recur trace-id "main.conjectures.notes.c17.**" --scope "main.conjectures.**" -d . --format full

Read the revealed notebook in your editor and record findings there. .current is an attention marker, not a declaration that a problem is open. Machine-specific session history and private .recur/ state are excluded from the public repository.

Join the investigation

Pick one statement. Check its assumptions. Try to break it. Find its nearest theorem. Then open an issue with the candidate ID and your argument or source, or submit a pull request with reproducible work.

The collection is intended for mathematical scrutiny. C15's educational motivation is a hypothetical measurement model; no empirical claim about learners follows from it. The AI-assisted origin is disclosed so readers can evaluate the arguments and evidence on their merits.