Strongly regular graphs · finite geometry · lattices and codes

Conway’s 99-Graph Conjecture

Collaboration beta

Does a strongly regular graph with 99 vertices, degree 14, one common neighbor for adjacent pairs, and two common neighbors for nonadjacent pairs exist?

srg (99,14,1,2) exists?

Conway's Five $1,000 Problems, Problem 2Listed inFrontierMath: Open Problems

Known results and sources

Local paired neighborhoods, overlapping residual windows, a prism fragment, and lattice geometry identify the 99-graph research program without pretending to draw all 99 vertices or answer the existence question.

Research problem

Exact mathematical statement

Decide whether there exists a strongly regular graph on 99 vertices in which every vertex has degree 14, every adjacent pair has exactly one common neighbor, and every nonadjacent pair has exactly two common neighbors:

\operatorname{srg}(99,14,1,2)\text{ exists?}

A complete solution must either construct such a graph or prove that none exists.

Problem infographic

Problem at a glance

A forest-green scientific plate centers one vertex x inside a ring of exactly fourteen neighbors grouped into seven disjoint paired edges, with a cobalt halo labeled as the remaining eighty-four distance-two vertices. Two small incidence diagrams distinguish the rule that adjacent pairs have one common neighbor while nonadjacent pairs have two, and the central gold question asks whether srg(99,14,1,2) exists. — A hypothetical strongly regular graph with parameters (99,14,1,2) would have 99 vertices of degree 14, with one common neighbor for adjacent pairs and two for nonadjacent pairs. Around each vertex, the 14 neighbors would form seven disjoint edges and the other 84 vertices would lie at distance two. Whether any global graph satisfies all these constraints remains open.

Current mathematical picture

Where work on Conway’s 99-Graph Conjecture stands

Open conjecture

The cumulative v12 source keeps Conway's 99-graph conjecture open and the modular and binary lanes separate. In the projective binary branch it eliminates I=17, retains 19 <= I <= 781, and concentrates the equality-endpoint work on 11,945 rank-four and 13,599 transverse necessary-condition parent rows. It reports cubic-residue, higher-coset, quotient-Fourier, exact-kernel, and occupancy reductions, while the full quotient-labelled column realizations and every broader existence or nonexistence step remain open. ProofAtlas did not execute, render, decompress, or independently reproduce the current work attachments.

Strongest supported footholdGlobal Seidel arithmetic and code constraints

The source adds exact Seidel Smith forms, distance at least nine, deletion stability through eight coordinates, and many forced weight-26 words.

Evidence posture · Reported result

Leading routeTriangle-core and general q=12 spread route

The exact 36-plus-60 reduction remains active outside the maximally split six-prism case; component partitions and cubic edge-colored multigraphs provide finite entry points.

Route status · Active route

Useful failureCayley construction route

Symbolically eliminated for SRG(99,14,1,2); non-Cayley constructions are untouched.

Route status · Eliminated route

Main reductionResidual and lattice rank framework

The current work derives the square-zero residual system, parity of g, residual discriminant type, and the line-lattice upper bound g<=42.

Evidence posture · Reported reduction

Completed special caseSix-prism g=34 subbranch eliminated

The projector/Schur bridge and corrected ten-row exact-symbol audit force g at least 36 in the maximally split six-prism branch.

Evidence posture · Reported special case

Evidence footholdg=32 eliminated in the current work

The current work reports a complete rank-16 and rank-17/18 audit, followed by a symbolic contradiction for the sole rank-17 survivor, excluding g=32; the retained archive is reproducible by rerun but not proof-producing.

Evidence posture · Computation reproduced · provisional · dependencies incomplete

Priority open bridgeSolve the full rank-four 16 by 8 column table

For every one of the 11,945 current rank-four parent rows, realize or eliminate all quotient-labelled multiplicity tables m_(beta,gamma) subject to the source's row sums, occupancy, weight-one, weight-two, coset-weight, cubic-residue, and Gram constraints.

Task status · Ready to work on

Latest mathematical updatev12 eliminates I=17 and narrows the equality-endpoint frontier

The v12 source eliminates the projective I=17 layer, reports 25,544 necessary-condition equality-endpoint parent rows, and restores reproducibility for the final occupancy-kernel reduction.

Retained source record

Work mapped so far

Conway’s 99-Graph Conjecture in numbers

5.3kretained lines of mathematical investigation5,247 in the current working snapshot

Argument development: 4,488 · 85%
Explored or eliminated routes: 188 · 4%
Computational analysis: 150 · 3%
Open obligations: 211 · 4%
Definitions and setup: 258 · 5%

32inventoried working statements14routes investigated12reported milestones9open questions9contribution-ready tasks

Evidence attached to the current work4 original computation reruns

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byNorman Biggs, J. J. Seidel, John H. ConwayProposed1969Status checkedAug 2, 2026

Source recordedAug 27, 2026

Recommended next task

Solve the full rank-four 16 by 8 column table

Suggested move: Start from retained feasible kernel assignments and build the exact quotient-labelled 16 by 8 multiplicity solver with independently checkable realization or exhaustion certificates.

What would count as progress

Exhaust every kernel, radical, coset, and occupancy witness for all 11,945 parent rows.
Retain one realized table or an independently checkable rejection certificate for each terminal case.
Keep every conclusion conditional on the I=19, r=1 equality endpoint.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

28 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Active routeTriangle-core and general q=12 spread route

The exact 36-plus-60 reduction remains active outside the maximally split six-prism case; component partitions and cubic edge-colored multigraphs provide finite entry points.

Route status · Active route

Active routeGeneral g=34 quotient-and-overlap route

This is the primary active route: classify local ranks 16-20, enforce exact outside equations, and glue the seven overlapping blocks in one rank-20 geometry.

Route status · Active route

Active routeHigh-end g=42 line-lattice route

Attack the top branch independently through its anisotropic 7-primary plane, metabolic 3-primary part, norm-4 frame vectors, theta series, and Seidel Smith lift.

Route status · Active route

Active routeGlobal code and prism-graph route

Use self-orthogonality, distance, weight enumerators, deletion stability, Smith lifts, line-row weights, and prism PSD constraints across every remaining g.

Route status · Active route

Active routeIndependent certificate engineering

Active cross-cutting route to replace unarchived or reproducible-only computations with compact proof-producing evidence before more conclusions depend on them.

Route status · Active route

Explored alternatives

Other routes

9 recorded

Eliminated routeCayley construction route

Symbolically eliminated for SRG(99,14,1,2); non-Cayley constructions are untouched.

Route status · Eliminated route

Narrowed routeLabeled 84-vertex and 77-vertex extension route

Coarse invariant screening is exhausted, but the exact labeled equations and ten 77-vertex exact-cover systems remain active.

Route status · Narrowed route

Useful but insufficientSingle opposite-star universal rank route

The sharp rank-14 witness puts this route at its natural limit; future lower bounds must use overlaps or global identities.

Route status · Useful but insufficient

Browse 6 more explored routes

Eliminated routeRepeat searches for g=28,30,32

The current work treats these branches as eliminated and directs future work to independent certificate packaging instead of rerunning them as discovery searches.

Route status · Eliminated route

Not yet justifiedHistorical preliminary rank-19/20 run

The retained checkpoint proves no elimination and may be used only as implementation context for a clean rebuild.

Route status · Not yet justified

Not yet justifiedSix-prism exact-symbol route

Superseded by the explicit rank-five nonsquare class: a class-only zero-survivor rank-at-most-five search is not justified. The valid successor studies each rank-four/five class together with all five outside-extension domains and simultaneous global constraints.

Route status · Not yet justified

Eliminated routeProjective I = 17 arithmetic frontier

Combine the seven residual arithmetic structures with weight-36 incidence geometry and exact spectral transfer to decide the first open projective layer.

Route status · Eliminated route

Narrowed routeSix-prism outside-extension route

The explicit rank-five nonsquare witness refutes the class-only zero-survivor route. The active six-prism g=36 program classifies rank-four/five classes together with their five outside domains and then imposes simultaneous cross-class and global projector constraints.

Route status · Narrowed route

Narrowed routeProjective I=19 equality-endpoint column-realization route

The source's current route combines the cubic-residue, higher-coset, quotient-Fourier, exact-kernel, and occupancy constraints, then asks for full quotient-labelled rank-four and transverse column realizations. Eliminating this equality endpoint would not settle higher B4 values or the nonprojective, modular, or bridge lanes.

Route status · Narrowed route

Route statements and reductions

Statements the next route can inspect and build on

Route statementExact 36-plus-60 triangle-core model

Around a graph-triangle, the graph is equivalent to a cubic 36-vertex boundary K, a connected 8-regular 60-vertex core H, a 36 by 60 incidence matrix U, and the exact block equations (7.4)-(7.6).

Source-reported route statement · dependencies incomplete

Route statementLine Gram and prism PSD constraints

The 231-line Gram matrix Z has real rank 44, entries in {4,1,0,-1,-2}, satisfies Z^2=21Z, and has mod-7 rank g; the associated prism graph satisfies 10I-D_Pi positive semidefinite.

Source-reported route statement · dependencies incomplete

Route statementSelf-orthogonal Seidel code constraints

The Seidel row code is self-orthogonal of length 99 and dimension g, has packet-reported minimum distance at least 9, preserves rank after deleting at most eight coordinates, and contains at least 29,106 weight-26 words.

Computation reproduced · provisional · dependencies incomplete

Route statementGeneral g=34 mixed-rank frontier

For general g=34, some local block must have rank at least 16, while local ranks 16-20 and mixed seven-block overlap configurations remain open.

Source-reported route statement · dependencies incomplete

Route statementAnisotropic g=42 line-lattice branch

At g=42, the 7-primary discriminant form of the even line lattice is a two-dimensional anisotropic quadratic plane over F7.

Source-reported route statement · dependencies incomplete

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

9 featured tasks

Produce independent finite certificates

Replace the current work's non-proof-producing or unarchived finite layers with portable, independently checkable certificate packages.

Suggested move: Prioritize the rank-14 equality classification and g=28, then rank-at-most-15 and g=30, the g=32 audits, and the six-prism rank-three result.

Ready to work on

Complete the general g=34 rank-19/20 audit

Rebuild and exhaust the local rank-19/20 search with the correct square-discriminant and Schur-coefficient types, exact outside constraints, and independent validation.

Suggested move: Rebuild the 3,580 ordered matching cases and 1,971 star-exchange orbits, cross-check incremental ranks and automorphisms, and replace the relaxed one-label test with exact edge-bijection equations for survivors.

Ready to work on

Classify g=34 local ranks 16-18

Classify the sixty outside quotient domains under block automorphisms for local ranks 16-18 and enforce the exact Schur, discriminant, edge-bijection, and endpoint-correlation constraints.

Suggested move: Canonicalize projective quotient subspaces under each block automorphism group and precompute domain masks before imposing the global residual equations.

Ready to work on

Enforce all seven opposite-star overlaps

Insert local survivors into one global rank-20 basis and enforce the pairwise four-label overlaps of the seven opposite-star blocks.

Suggested move: Compare four-label restrictions as soon as a second block is inserted, before completing all sixty outside labels independently.

Ready to work on

Couple the Seidel code and prism graph

Apply MacWilliams or complete-weight-enumerator constraints for every remaining g and couple them to line-row weights, prism degrees, PSD constraints, and the Smith lift.

Suggested move: Set up Delsarte/MacWilliams linear programs with self-orthogonality, distance at least nine, the forced weight-26 count, deletion stability, and line-row weight restrictions.

Ready to work on

Test the anisotropic g=42 line lattice

Classify compatible genera and theta-series constraints for the rank-44 even line lattice with its metabolic 3-primary part and anisotropic 7-primary plane.

Suggested move: Combine possible 3-adic metabolic forms with the anisotropic 7-plane, then impose 231 labeled norm-4 frame vectors and the exact Seidel Smith data.

Ready to work on

Solve the full rank-four 16 by 8 column table

Ready to work on

Classify the transverse radical and solve the 32 by 4 table

For the 13,599 transverse parent rows, classify exact five-dimensional radical column multiplicities and solve the common quotient-labelled 32 by 4 table with the stated Reed-Muller, MacWilliams, occupancy, autocorrelation, and quadratic-form constraints.

Suggested move: Use Walsh inversion on the five-dimensional radical before attempting the common four-slice quotient-labelled table.

Ready to work on

Classify six-prism rank-four/five outside-extension domains

Superseding the class-only rank-at-most-five obligation, enumerate every legal rank-four/five exact-symbol class, compute all five required outside-extension domains with the correct rank and discriminant rules, and impose simultaneous cross-class and global projector compatibility.

Suggested move: Use the Section 65 witness and the eight relaxed border candidates as mandatory regressions, then archive canonical class representatives, pivot minors, all five domain masks, and exact incompatibility witnesses.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 2, 2026

Current statusOpen problem

It remains unknown whether a strongly regular graph with parameters (99,14,1,2) exists. Recent work sharply constrains its automorphisms and derives further forced structure, but neither a construction nor a nonexistence certificate is known.

[1][3]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2026
Peer reviewedPetro and Phillips determined the forced spectrum of the triangle-intersection graph of any putative solution.[7]
2025
Computational resultCesarz and Woldar gave computer-free proofs that even automorphism-group order divides 6 and that divisibility by 7 forces a cyclic group of order 7.[8]
2021
PreprintDucey and collaborators determined the critical group forced by the parameter set.[2]
2020
Computational resultCrnković and Maksimović computationally ruled out automorphism groups of order 6 or 9.[5]

Mathematical neighborhood

What changed after the initial research map

Later recorded revisions that changed the mathematics, without inventing a date or an AI attribution.

Version 6 refines the projective and six-prism frontiersThe v6 packet reduces the projective I=17 layer to seven cases and replaces the impossible six-prism class-only zero-survivor task with an outside-extension-domain program.

Changed the research frontierLater mathematical revision

Research stage 11

v12 eliminates I=17 and narrows the equality-endpoint frontierThe v12 source eliminates the projective I=17 layer, reports 25,544 necessary-condition equality-endpoint parent rows, and restores reproducibility for the final occupancy-kernel reduction.

Changed the research frontierLater mathematical revision

Cumulative v12 source ingested; not mathematical occurrence time

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

How the route was assembled

Argument structure

These stages follow the mathematical order of the supplied argument.

10 mapped milestonesretained argument map

Browse all 10 mapped stages

stage 1Global SRG structure fixed
stage 2Exact point and triangle models
stage 3Residual and lattice rank framework
stage 4Sharp rank-14 opposite-star classification
stage 5g=28 eliminated
stage 6g=30 eliminated
stage 7g=32 eliminated and frontier narrowed
stage 8Global Seidel and code constraints
stage 9Six-prism g=34 doorway closed
stage 10General g=34 frontier isolated

Global SRG structure fixedThe source derives the adjacency identity, spectrum, local matching, triangle partition, and symbolic Cayley exclusion.